Possibility and ContentMetaphysics without Deep Metaphysics

Agustın Rayo

February 14, 2010

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

I Main Texts 1

1 Identity 3

1.1 Identity Generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Varieties of Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Understanding Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Deciding Between Rival Identity-Statements . . . . . . . . . . . . . . . . 16

1.5 Logical Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.6 What Does the Truth of an Identity-Statement Consist In? . . . . . . . . 29

1.6.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.6.2 Metaphysical Privilege . . . . . . . . . . . . . . . . . . . . . . . . 32

1.6.3 The Upshot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Possibility 37

2.1 Supervenience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2 Possible Worlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 The Supervenience Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.1 The Principle of Maximality . . . . . . . . . . . . . . . . . . . . . 50

2.3.2 The List of Modal Truths . . . . . . . . . . . . . . . . . . . . . . 52

i

ii CONTENTS

2.4 Modal Language and Modal Fact . . . . . . . . . . . . . . . . . . . . . . 53

2.5 Beyond First-Order Languages . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Metaphysics 61

3.1 Tractarianism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Life without Tractarianism . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 Ontological Commitment . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3.1 Quine’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3.2 Is Quine’s Criterion Adequate? . . . . . . . . . . . . . . . . . . . 77

4 Mathematics 81

4.1 Trivialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Benacerraf’s Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Neo-Fregeanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.1 Mixed Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3.2 Abstraction Principles . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4 A Semantics for Trivialists . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4.1 The Trivialist Semantics . . . . . . . . . . . . . . . . . . . . . . . 90

4.4.2 Philosophical Commentary . . . . . . . . . . . . . . . . . . . . . . 93

4.5 Paraphrase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.6 Beyond Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.6.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Content 109

5.1 Folk-Psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Rational Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.3 Belief-Attributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.4 Cognitive Accomplishment in Logic and Mathematics . . . . . . . . . . . 126

CONTENTS iii

5.5 Mary and the Tomato . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

II Detours 139

6 Deep Metaphysics 141

7 A-worlds and the Dot Notation 149

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2 A Kripke-semantics for actualists . . . . . . . . . . . . . . . . . . . . . . 151

7.3 Admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.4 Interlude: The Principle of Representation . . . . . . . . . . . . . . . . . 160

7.5 The dot-notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.5.1 Inference in a language with the dot-notation . . . . . . . . . . . 165

7.5.2 The expressive power of the dot-notation . . . . . . . . . . . . . . 167

7.5.3 Limitations of the proposal . . . . . . . . . . . . . . . . . . . . . . 171

8 Translation 173

9 Introducing Mathematical Vocabulary 181

9.1 Linguistic Stipulation for Anti-Tractarians . . . . . . . . . . . . . . . . . 181

9.2 Success Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

9.3 Internal Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

9.4 Applied Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.5 Compositional Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

III Appendices 199

A A Semantics for a Language with the Dot Notation 201

B If Lewis Can Say It, You Can Too 207

iv CONTENTS

C The Canonical Space of Worlds 213

D Modal Sentences and Modal Facts 219

List of Figures

2.1 Examples of Metaphysical Necessities . . . . . . . . . . . . . . . . . . . . 57

2.2 Examples of Metaphysical Necessities (Continued) . . . . . . . . . . . . . 58

2.3 Examples of Metaphysical Necessities (Mereological Principles) . . . . . . 59

7.1 Examples of A-worlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

v

vi LIST OF FIGURES

Introduction

Here is my best effort to come up with a slogan for the book: I will attempt a post-

Quinean revival of Carnap.

Don’t expect any history of philosophy, though. There will be no meaningful discus-

sion of Quine or Carnap in this book. Here is what the slogan really means. Carnap

believed that true sentences can be usefully divided into those that are true in virtue

of the meaning of their constituent vocabulary, and those whose truth depends not just

on meaning but also on the way the world is. The former—sentences like ‘bachelors

are unmarried’—were thought of as consequences of ‘meaning postulates’. They were

described as analytic and said to be knowable a priori. The latter—sentences like ‘the

Sun’s mass is approximately 1.9891 × 1030 kilograms’—were described as synthetic and

said to be knowable only a posteriori.

Quine objected to Carnap by complaining that our understanding of notions like

meaning postulate or analyticity is not robust enough to do the work that Carnap de-

manded of them. I think Quine was right about this, but I also think it is easy to

overestimate the reach of his criticism. The lesson to take from Quine is that talk of

meaning postulates and analyticity is a bad way of characterizing the sort of distinction

Carnap was after, not that the the project of finding such a distinction is worthless.

What I mean when I say that I will attempt a post-Quinean revival of Carnap is that I

will try to find a better way of characterizing a Carnap-style distinction, and use it to

do philosophical work.

vii

viii LIST OF FIGURES

The distinction I will focus on is a distinction between sentences with trivial truth-

conditions—truth-conditions whose satisfaction requires nothing of the world—and sen-

tences with non-trivial truth-conditions. The notion of triviality won’t give us everything

Carnap hoped to get from the notion of analyticity. Significantly, it won’t deliver the

result that the truth of sentences with trivial truth-conditions is generally knowable a

priori. But what one gets in exchange is a notion that is better understood, and that

can be put to real philosophical use. I will argue, in particular, that it can be used to

deliver each of the following:

1. An account of the limits of metaphysical possibility.

2. An account of ontological commitment, and its relationship to reference.

3. An account of the truth-conditions of mathematical sentences.

4. An account of representational content, and of what one learns when one learns a

necessary truth.

Like Carnap and the Logical Empiricists, I am suspicious of a certain brand of meta-

physics: Deep Metaphysics, as I like to call it. Deep Metaphysics seems to have enjoyed

a resurgence in recent years, and I will be reacting against it throughout the book. Un-

like the Logical Empiricists, however, I will not be offering a quick and easy criterion

of metaphysical respectability. Nor will I be engaging with the offending views. The

only effective way of countering Deep Metaphysics, it seems to me, is by bringing out

the fact that it has no interesting role to play in our theorizing. I hope my discussion

will illustrate that difficult philosophical questions can be fruitfully addressed without

resorting to the dubious concepts of Deep Metaphysics. (It is in this sense that project

of the book may be described as ‘Metaphysics without Deep Metaphysics’.)

LIST OF FIGURES ix

Truth-Conditions

A sentence’s truth-conditions, as they will be understood here, consist of a requirement

on the world—the requirement that the world would have to satisfy in order to be as the

sentence represents it to be. The truth-conditions of ‘snow is white’, for example, consist

of the requirement that snow be white, since that is how the world would have to be in

order to be as ‘snow is white’ represents it to be.

Two sentences might be thought to have the same truth-conditions even if they are

thought to have different meanings in some pre-thoeretic sense of ‘meaning’. Consider

‘the Sun is hot’ and ‘the Sun has high mean kinetic energy’. These two sentences play

very different roles in our linguistic practice. So there is room for thinking that they

mean different things. But one should still think that the two sentences have the same

truth-conditions. For to be hot just is to have high mean kinetic energy. So there is no

difference between what would be required of the world in order to be as ‘the Sun is hot’

represents it to be and what would be required of the world in order to be as ‘the Sun

has high mean molecular motion’ represents it to be.

In this book I will focus on the notion of truth-conditions, to the exclusion of other

aspects of meaning. But this is not because I think that truth-conditions are all there

is to meaning. It is because I think truth-conditions are an especially useful aspect of

meaning, and deserve special attention.

Triviality

For a sentence’s truth-conditions to be trivial is for them to be satisfied provided only

that the world is not incoherent. (And, of course, it is trivially the case that the world

is not incoherent; so—in an interesting sense of ‘nothing’—for a sentence to have trivial

truth conditions is for nothing to be required of the world in order for the truth-conditions

to be satisfied.)

Consider, for example, the sentence ‘white things are white’. To assume that its truth

x LIST OF FIGURES

conditions fail to be satisfied is to assume that there are white things that fail to be white.

This is something I regard as incoherent. So I see the truth-conditions of ‘white things

are white’ as imposing no real demands on how the world is: I take them to be satisfied

trivially.

A sentence can have trivial truth-conditions even if its truth is not knowable a priori.

Consider, for example, ‘things composed of water are composed of H2O’. A scenario in

which this sentence’s truth-conditions fail to be satisfied is a scenario in which something

composed of water fails to be composed of H2O. But to be composed of water just is to

be composed of H2O. So the scenario is one in which something composed of H2O fails

to be composed of H2O. And that is something I regard as incoherent. So I take ‘things

composed of water are composed of H2O’ to have trivial truth-conditions.

I will argue in the book that the notion of triviality draws a distinction that can be

drawn in many different ways, and that our grasp of the distinction can be strengthened

by combining our understanding of the different ways in which it might be drawn.

Here are five different ways of drawing the distinction (all relativized to a subject, so

as to simplify the exposition):

1. Triviality

The distinction between sentences that the subject regards as having trivial truth-

conditions—i.e. truth-conditions whose satisfaction requires nothing of the world—

and the rest.

2. De mundo Intelligibility

The distinction between sets of sentences that describe scenarios that the subject

takes to be intelligible (in a particular sense of ‘intelligibility’, to be elucidated

later), and the rest.

LIST OF FIGURES xi

3. Identity

The distinction between sentences that the subject takes to be logical consequences

of identity-statements she accepts, and the rest.

4. Why-closure

The distinction between sentences that the subject takes to be why-closed—i.e. sen-

tences ψ such that the subject is unable to make sense of the question pI can see

that ψ is the case; what I want to know is why?q—and the rest.

5. Necessity

The distinction between sentences whose truth-conditions that the subject takes to

be satisfied as a matter of metaphysical necessity, and the rest.

I will argue that—once they have been subjected to various qualifications and elucidations—

these five distinctions come to the same thing:

A sentence is taken to have trivial truth-conditions iff its negation is regarded

as de mundo unintelligible iff it is taken to follow logically from identity

statements one accepts iff it is regarded as why-closed iff it is regarded as

metaphysically necessary.

I will not be presupposing that one of these notions is ‘fundamental’ and the rest are

‘derived’. It is no part of the picture that necessity, to pick an arbitrary example, is the

fundamental notion, and that the others—triviality, identity, why-closure and de mundo

intelligibility—should be understood on the basis of a firm and independent grasp of the

distinction between the necessary and the contingent. Instead, I will treat each of the

five notions as contributing to our understanding of the rest, so that even if our grasp

of any one of them is somewhat limited when considered in isolation, they are all better

understood when considered in light of the connections with their peers.

xii LIST OF FIGURES

The family of biconditionals cannot be all there is to our understanding of the five

notions. At least some of the notions must be subject to independent constraints. To a

certain extent, the needed constraints come from our pre-theoretic usage of the relevant

terms. But there will be some regimentation as we go along, because the notions will

have to be subjected to qualifications and precisifications before the biconditionals turn

out to be true. So caution must be applied when appealing to pre-theoretic usage.

A more robust source of constraint on our understanding of the individual notions

comes from their role in our theorizing. Let me give you an example.

Identity and our Scientific Practice

The notions of identity and why-closure play an important role in our scientific practice.

Detailed discussion will have to wait until sections 1.3 and 1.4, but it is easy to convey

a sense of the main idea.

The sentence ‘to be hot just is to have high mean kinetic energy’ is why-closed. It

would be wrong-headed to ask: “I can see as clearly as can be that for something to be

hot just is for it to have high mean kinetic energy; what I want to understand is why?”.

The problem is not just that we wouldn’t know how to answer the question. It isn’t even

clear what sort of information is being requested. One is inclined to reject the question,

and respond by reasserting the identity statement: ‘to be hot just is to have high mean

kinetic energy’. (In contrast, it would be perfectly intelligible to ask: “I can see as clearly

as can be that the Sun is composed primarily of Hydrogen and Helium; what I want to

understand is why”.)

The fact that we treat ‘to be hot just is to have high mean kinetic energy’ as why-

closed is correlated with the fact that we close off certain lines of inquiry (conspicuously,

the project of better understanding why hot substances are made up of rapidly moving

particles). And this, in turn, is correlated with the adoption of a certain kind of scientific

outlook, according to which heat-related phenomena are better accounted for by focusing

LIST OF FIGURES xiii

on molecular motion than by, say, postulating a new kind of substance, such as ‘caloric

fluid’. I will argue below that there is a general connection, via why-closure, between the

identity statements one accepts and the lines of inquiry one takes to be most fruitful in

one’s theorizing about the world.

Conclusion

There are limits to our pre-theoretic understanding of the notions of triviality, de mundo

intelligibility, identity, why-closure and necessity. But some of these notions are con-

strained by their role in our theorizing about the world. And by recognizing connections

between the different notions, our understanding of each of them can contribute to our

understanding of the rest. The result is an improved (though somewhat regimented)

understanding of the relevant notions: an understanding which I hope will prove robust

enough to address a family of puzzles in the metaphysics of modality, the philosophy of

language, the philosophy of mathematics and the philosophy of mind—and do so while

steering clear of Deep Metaphysics.

How to Read this Book

I have divided the book into three parts:

Part I: Main Texts (chapters 1–5)

Part II: Detours (chapters 6–9)

Part III: Appendices

All the main themes of the book are developed in Part I. If you’d like to see the overall

picture while limiting your time-commitment, what I recommend is that you focus on

chapters 1–5.

Part II is for enthusiasts. It discusses issues arising from Part I that are important

for a detailed understanding the project, but may be skipped by casual readers. These

xiv LIST OF FIGURES

detours should not be thought of as extended footnotes, though. Each of them develops a

self-standing idea of independent interest. If you are tempted by the additional material

what I recommend is that you read it in conjunction with Part I. You’ll find plenty of

pointers in the text, but here are some natural combinations:

Read . . . in conjunction with . . .

chapter 6 section 1.6.2 (or chapter 3)

chapter 7 section 2.2

chapter 8 section 4.5

chapter 9 section 4.6.

Part III consists mostly of technical material, and is intended only for the true aficionado.

Part I

Main Texts

1

Chapter 1

Identity

1.1 Identity Generalized

Statements of the form ‘a = b’, for a and b singular terms, are identity-statements. But

they are only a special case. Consider the following sentences:

Sibling

To be a sibling just is to share a parent.

[In symbols: ‘Sibling(x) ≡x ∃y∃z(Parent(z, x) ∧ Parent(z, y) ∧ x 6= y)’]

Marriage

For a marriage to take place just is for someone to get married.

[In symbols: ‘a marriage takes place ≡ someone gets married’]

In these three sentences the expression ‘just is’ (or its formalization ‘≡x’) is functioning

as an identity-predicate of sorts.

To accept Sibling it is not enough to believe that all and only the siblings share a

parent. You must also believe that there is no difference between being a sibling and

sharing a parent; you must believe that if someone is a sibling it is thereby the case that

3

4 CHAPTER 1. IDENTITY

she shares a parent. (More colorfully: when God created the world, and made it the case

that some of Her creatures shared a parent, there was nothing extra She had to do, or

refrain from doing, in order to ensure that there were sisters. She was already done.) If

you accept Sibling, you believe that the feature of reality that is fully and accurately

described by saying ‘Susan is a sibling’ can also be fully and accurately described by

saying ‘Susan shares a parent’. What is required of the world in order for the first of

these sentences to be true is precisely what is required of the world in order to make the

second of these sentences true.

Similarly, if you accept Marriage, you believe that there is no difference between

a marriage taking place and someone’s getting married. When someone gets married

it is thereby the case that a marriage takes place. The feature of reality that is fully

and accurately described by saying ‘A marriage took place’ is also fully and accurately

described by saying ‘Someone got married’.

It is useful to compare Marriage and Sibling with a regular first-order identity

such as ‘Hesperus is Phosphorus’. If you accept ‘Hesperus is Phosphorus’, you believe

that there is no difference between traveling to Hesperus and traveling to Phosphorus.

Someone who travels to Hesperus has thereby traveled to Phosphorus. The feature of

reality that is fully and accurately described by saying ‘A Russian spaceship traveled to

Hesperus’ is also fully and accurately described by saying ‘A Russian spaceship traveled

to Phosphorus’.

The English expression ‘just is’ is sometimes used asymmetrically in ordinary con-

versation. It is, for instance, more natural to say ‘to be a sibling just is to share a

parent’ than to say ‘to share a parent just is to be a sibling’. (Perhaps what explains

the difference is the presence of some sort of convention to the effect that statements of

the form ‘to be φ just is to be ψ’ should be used to clarify what it takes to satisfy φ by

using ψ, rather than the other way around.) I will be disregarding any such asymmetries

here. I will be thinking of ‘≡’ as an identity predicate (and therefore as an equivalence

1.2. VARIETIES OF IDENTITY 5

relation), so I will be treating ‘φ ≡ ψ’ and ‘ψ ≡ φ’ as interchangeable. If you find that

you are unable to rid yourself of the asymmetrical reading of ‘just is’, please treat ‘≡’ as

primitive, or find some other paraphrase for ‘φ ≡ ψ’ (for instance: pthere is no difference

between its being the case that φ and its being the case that ψq).

1.2 Varieties of Identity

Second-order Identity

As I did in the case of Sibling, I shall sometimes index ‘≡’ with variables. (I might say,

for instance, ‘≡x’, or ‘≡z,x’.) This is to indicate that occurrences of the relevant variables

in the formulas that ‘≡’ takes as arguments are to be regarded as bound by ‘≡’. Thus,

whereas

x is a sibling ≡x x shares a parent

is read ‘to be a sibling just is to share a parent’ and expresses a complete thought, its

index-free correlate

x is sibling ≡ x shares a parent

is read ‘for it to be a sibling just is for it to share a parent’, and expresses an incomplete

thought.

Whenever ‘≡’ is indexed with first-order variables, I shall refer to the relevant identity-

statements as second-order identities. Here are some additional examples of second-order

identities:

Composed-of-Water(x) ≡x Composed-of-H2O(x)

(Read: To be composed of water just is to be composed of H2O.)

Hot(x) ≡x High-Mean-Kinetic-Energy(x)

(Read: To be hot just is to have high mean kinetic energy.)

6 CHAPTER 1. IDENTITY

Sisters(x, y) ≡x,y Female(x) ∧ Female(y) ∧ ∃z(Parent(z, x) ∧ Parent(z, y))

(Read: For x and y to be sisters just is for x and y to be female and for them

to share a parent.)

It is natural to think of second-order identities as expressing identities amongst properties

(e.g. ‘the property of being a sibling = the property of sharing a parent’.) I have no qualms

with this description, as long as property-talk is understood in a suitably deflationary

way. (To have the property of Fness just is to be F.) But it is important to be aware that

property-talk is potentially misleading. It might be taken to suggest that one should only

assert Sibling if one is prepared to countenance a naıve realism about properties—the

view that even though it is intelligible that there be no properties, we are lucky enough

to have them. The truth of Sibling, as I understand it, is totally independent of such

a view.

Semi-Identity

Sometimes one is in a position to endorse something in the vicinity of an identity-

statement even though one has only partial information. Suppose you know that the

chemical composition of water includes oxygen but don’t know what else is involved.

You can still say:

Part of what it is to be composed of water is to contain oxygen.

[In symbols: ‘Composed-of-water(x)�x Contains-Oxygen(x).’]

I shall call this as a semi-identity statement. Think of it as a more idiomatic a way of

saying:

To be composed of water just is (to contain oxygen and to be composed of

water).

1.2. VARIETIES OF IDENTITY 7

In general, I shall treat the semi-identity statement ‘φ(x)�x ψ(x)’ as a syntactic abbre-

viation of the identity-statement ‘φ(x) ≡x (ψ(x) ∧ φ(x))’.

Please keep in mind that ‘F (x)�x G(x)’ should not be understood as entailing that

being G is, in some sense, ‘metaphysical prior’ to being F , or as entailing that something

is G ‘in virtue’ of being F.

As in the case of second-order identity-statements, it is natural to think of semi-

identity-statements in terms of properties (e.g. ‘the property of being water has the

property of containing oxygen as a part’.) Again, I have no objection to this sort of

description, as long as property-talk is taken in a suitably deflationary spirit.

We have seen that ‘�’ can be defined in terms of ‘≡’. The converse is also true.

‘To be composed of water just is to be composed of H2O’, for example, is equivalent

to the conjunction of ‘part of what it is to be water is H2O’ and ‘part of what it is to

be H2O is to be water’. In general, ‘F (x) ≡x G(x)’ is equivalent to the conjunction of

‘F (x)�x G(x)’ and ‘G(x)�x F (x)’.

First-Order Identity-Statements

There are two different readings of the first-order identity predicate ‘=’. On the stronger

reading, ‘Hesperus = Phosphorus’ is false at worlds in which Venus fails to exist; on

the weaker (Kripkean) reading ‘Hesperus = Phosphorus’ is true at all worlds. On the

weaker reading, ‘Hesperus = Phosphorus’ can be paraphrased as ‘x = Hesperus ≡x x =

Phosphorus’ (regardless of whether ‘=’ takes the weaker reading in the paraphrase).

On the stronger reading, ‘Hesperus = Phosphorus’ is equivalent to the conjunction of

‘x = Hesperus ≡x x = Phosphorus’ and ‘∃x(Hesperus = x)’.

One can think of the weaker reading as claiming only that what it takes to satisfy the

condition of being identical to Hesperus is precisely what it takes to satisfy the condition

of being identical to Phosphorus. If such a claim is true, it will be true even at worlds in

which Venus fails to exist (though, of course, the existence of Venus in the actual world

8 CHAPTER 1. IDENTITY

is a precondition for our being able to talk about the relevant conditions).

When ‘=’ takes the weaker reading and a and b are proper names, pa = bq can always

be paraphrased as pa = x ≡x b = xq. But this need not hold when we have definite

descriptions. ‘Obama = the 44th president of the United States’, for example, cannot be

paraphrased as ‘to be Obama just is to be the 44th president of the United States’. For

even though ‘Obama’ and ‘the 44th president’ refer to the same individual, the predicates

‘x = Obama’ and ‘x = the 44th president’ have different satisfaction-conditions: what it

takes to satisfy the condition of being identical to Obama is different from what it takes

to satisfy the condition of being the 44th president.

When I speak of identities (or identity-statements) below, what I will have in mind

is always statements of the form ‘φ(x) ≡x ψ(x)’. This means that both semi-identity

statements and weak first-order identity-statements featuring proper names will count as

‘identities’, but contingently true first-order identity-statements will not.

Constitutive Properties and Conditional Semi-identity Statements

P is a constitutive property if the assumption that z has P is enough to license the

conclusion that part of what it is to be z is to be P . Being human, for example, is a

constitutive property. For the assumption that Socrates is human is enough to license:

x = Socrates�x Human(x)

(Read: part of what it is to be Socrates is to be human.)

Being snub-nosed, on the other hand, is not a constitutive property. For the assumption

that Socrates is snub-nosed does not license:

x = Socrates�x Snub-Nosed(x)

1.3. UNDERSTANDING IDENTITY 9

(Read: part of what it is to be Socrates is to be snub-nosed.)

The claim that P is a constitutive property can be formulated as a conditional semi-

identity statement:

P (z)

x = z �x P (x)

(Read: assume z is P ; then part of what it is to be z is to be P .)

Having w as a biological parent is a parameterized constitutive property. For the as-

sumption that z has w as a biological parent is enough to warrant the conclusion that

part of what it is to be z is to have w as a parent. Parameterized constitutive properties

can also be captured by conditional semi-identity statements. One can say, for example:

B(z, w)

x = z �x B(x,w)

(Read: assume z has w as a biological parent; then part of what it is to be z is to have

w as a biological parent.)

Although conditional identity-statements can take other forms, here I will restrict my

attention to the reflexive case: the case in which the antecedent is a first-order formula

φ(z, ~w) and the consequent is px = z �x φ(x, ~w)q.

Some notation: a predicate will be said to be constitutive just in case it expresses a

constitutive property. Also, I shall use ‘identity-statement’ to cover both conditional and

unconditional identity-statements.

1.3 Understanding Identity

Identity-statements pervade our pre-theoretic, scientific and philosophical discourse. Yet

they have been given surprisingly little attention in the literature, and are in much need

10 CHAPTER 1. IDENTITY

of elucidation. I think it would be hopeless to attempt an explicit definition of ‘≡’.

This is not because true and illuminating equivalences couldn’t be found—I have already

suggested some in the Introduction to Part I—but because any potential definiens can

be expected to contain expressions that are in at least as much need of elucidation

as ‘≡’. The right methodology, it seems to me, is to explain how our acceptance of

identities interacts with the rest of our theorizing, and use these interconnections to

inform our understanding of ‘≡’. (I make no claims about conceptual priority: the

various interconnections I will discuss are as well-placed to inform our understanding of

‘≡’ on the basis of other notions as they are to inform our understanding of the other

notions on the basis of ‘≡’.)

In what follows I will show that the notion of identity is closely related to three other

notions: truth-conditionality, de mundo intelligibility and why-closure. In chapter 2 we

will discuss a connection with necessity, and later in the book we will discuss ways in

which the family of notions interacts with the other regions of our philosophical and

non-philosophical theorizing.

Truth-conditions

A sentence’s truth-conditions, as they will be understood here, consist of a requirement

on the world—the requirement that the world would have to satisfy in order to be as the

sentence represents it to be. The truth-conditions of ‘snow is white’, for example, consist

of the requirement that snow be white, since that is how the world would have to be in

order to be as ‘snow is white’ represents it to be.

There is a close relationship between the identities one accepts and one’s views about

which sentences have the same truth-conditions. Suppose, for example, that you think

that for something to be composed of water just is for it to be composed of H2O. Then

you should think that ‘A is composed of water’ and ‘A is composed of H2O’ have the

same truth-conditions. For what the former requires of the world is that A be composed

1.3. UNDERSTANDING IDENTITY 11

of water. But to be composed of water just is to be composed of H2O, which is what the

latter requires of the world.

Conversely: suppose you think ‘A is composed of water’ and ‘A is composed of H2O’

have the same truth-conditions. Then you think there is no difference between satisfying

the requirement that A be composed of water and satisfying the requirement that A be

composed of H2O. And this can only be true if what it is to be composed of water is to

be composed of H2O.

If you accept ‘to be composed of water just is to be composed of H2O’, you should

also think that the sentence ‘things composed of water are composed of H2O’ has trivial

truth-conditions. For you will take the condition that things composed of water be

composed of H2O to place the same demands on the world as the condition that things

composed of H2O be composed of H2O—and nothing is required of the world in order for

the condition that things composed of H2O be composed of H2O to be satisfied. (“Wait!

Doesn’t the world have to satisfy the condition of being such that things composed of

H2O are composed of H2O?” Yes, but—unless you are a dialetheist—you will think that

this is a condition that the world could only fail to satisfy by being incoherent, and it

is trivially the case that the world is not incoherent. It is in this sense that you will

think that nothing is required of the world. There is no particular way the world needs

to be—nothing God needs to have done or refrained from doing—in order for things

composed of H2O to be composed of H2O. )

De mundo intelligibility

Let a story be a set of sentences in some language we understand. I shall assume that

stories are read de re: that every name used by the story is used to say of the name’s

actual bearer how it is according to the story, and that every predicate used by the story

is used to attribute the property actually expressed by the predicate to characters in the

story. Accordingly, in order for a story that says ‘Hesperus is covered with water’ to be

12 CHAPTER 1. IDENTITY

true it must be the case that Venus itself is covered with H2O. (I shall ignore names that

are actually empty, such as ‘Sherlock Holmes’, and predicates that are actually empty,

such as ‘. . . is composed of phlogiston’ or ‘. . . is a unicorn’.)

Sometimes we describe a story as unintelligible on the grounds that it is too com-

plicated for us to understand. That is not the notion of unintelligibility I will have in

mind here. I will say that a story is de mundo unintelligible for a subject if her best

effort to make sense of a scenario verifying the story would yield something she regards as

incoherent. (I use ‘incoherent’ as interchangeable with ‘absurd’; you should also think of

it as interchangeable with ‘inconsistent’ and ‘trivially false’, unless you are a dialetheist

or a paracompletist. For more on dialetheism and paracompletism, see Priest (2006),

Beall (2009) and Field (2008).) De mundo intelligibility is the complement of de mundo

unintelligibility.

Here is an example. Consider a story that says ‘Hesperus is not Phosphorus’. My best

effort to make sense of a scenario verifying this story ends in incoherence. For a scenario

in which the story is true would have to be a scenario in which Hesperus itself (i.e. Venus)

fails to be identical with Phosphorus itself (i.e. Venus), and the nonselfidentity of Venus

is something I regard as incoherent. (Of course, it would be easy enough to make sense

of a scenario in which language is used in such a way that the expression ‘Hesperus is

not Phosphorus’ is true. But that wont help with the question of whether the original

story is de mundo intelligible.)

Another example: consider a story that says ‘there is a hot substance with low mean

kinetic energy’. For something to be hot just is for it to have high mean kinetic energy. So

a scenario verifying ‘there is a hot substance with low mean kinetic energy’ would have to

be a scenario in which someone hot—i.e. something with high mean kinetic energy—fails

to have high mean kinetic energy, which is something I regard as incoherent.

There is an intimate connection between identity and de mundo intelligibility: a

subject regards a story as de mundo unintelligible just in case she takes the story to

1.3. UNDERSTANDING IDENTITY 13

entail something she regards as incoherent in light of the identities she accepts.

Three clarifications: (1) The notion of de mundo intelligibility is deeply non-a priori.

For whether or not one takes a story to be de mundo intelligible will depend on whether

one believes that, say, Hesperus is Phosphorus—and this is not the sort of thing that

one could come to know a priori. (2) Perhaps you think there is a sense of intelligibility

(distinct from de mundo intelligibility) with respect to which it would be right to say that

‘Hesperus is not Phosphorus’ is an intelligible story. You could claim, for example, that

‘Hesperus is not Phosphorus’ is intelligible in the following sense: it would be verified (in

a special sense of ‘verify’) by a scenario in which the first heavenly body to be visible

in the evenings is not the last heavenly body to be visible in the mornings. I myself am

skeptical that the needed notion of verification—which is, in effect, an implementation of

the idea that our sentences have ‘primary intensions’—is in good order. But no interesting

claims in this book depend on this skepticism. If you think you can understand a notion

of intelligibility distinct from de mundo intelligibility, that’s fine. Just keep in mind

that it’s not the notion under discussion here. (3) So far we have talked about the

intelligibility of stories, but not the intelligibility of the scenarios that the stories depict.

When a scenario is picked out by a story, we shall say that the scenario is intelligible just

in case the story used to pick it out is de mundo intelligible.

Why-Closure

Suppose someone says: “I can see as clearly as can be that Hesperus is Phosphorus;

what I want to understand is why.” It is not just that one wouldn’t know how to comply

with such a request—one is unable to make sense of it. The natural reaction is to either

find a charitable reinterpretation of the question (“why does one planet play both the

morning-star and the evening-star roles?”) or reject it altogether (“What do you mean

why? Hesperus just is Phosphorus”.)

Contrast this with a case in which someone says: “I can see that the window is broken;

14 CHAPTER 1. IDENTITY

what I want to understand is why,” or “I can see that the radioactive isotope decayed at

time t rather than a second later; what I want to understand is why”, or even “I can see

that there is something rather than nothing; what I want to understand is why”. In all

three of these cases one can make sense of the request, taken at face-value. In the first

case, one may even have a satisfying reply (e.g. “The window is broken because it was

hit by soccer ball”). In the second case, it is harder to think of a good reply, but one can

at least think of a bad one (e.g. “because God willed it so”). Potential replies are even

scarcer in the third case (even bad ones), but one can at least state that there is no good

answer to be given without refusing to make sense the question (e.g. “Well, that’s just

the way things turned out.”). Contrast this with the initial Hesperus/Phosphorus case,

where it isn’t even appropriate to say “Well, that’s just the way things turned out”.

Say that a sentence φ is treated as why-closed just in case one is unable to make sense

of the question pWhy is it the case that φ?q when it is understood as follows:

I can see exactly what it would take to satisfy φ’s truth-conditions, but I wish

to better understand why the world is such as to satisfy them.

There are various other ways of reading the question pWhy is it the case that φ?q, and

it is important to be clear that they are not relevant to why-closure. In particular:

1. I wish to to understand why φ’s ‘primary intension’ holds.

[For instance: ‘Why is it the case that Hesperus is Phosphorus?’, meaning ‘Why is

there a single planet playing both the morning-star and the evening-star roles.’]

2. I wish to better understand what it would take to satisfy φ’s truth-conditions.

[For instance: ‘Why is it the case that the mean score is different from the median

score?’, meaning ‘I don’t understand ‘mean’ and ‘median’ well enough to know

what it would take for ‘the mean score is different from the median score’ to be

true.’]

1.3. UNDERSTANDING IDENTITY 15

3. I wish to better understand why φ has the truth-conditions that it in fact has.

[For instance: ‘Why is it the case that (p ⊃ q) ∨ (q ⊃ p)?’, meaning ‘Help me

understand how it comes about that the truth-functional operations corresponding

of ‘⊃’ and ‘∨ conspire to make it the case that every row in a truth-table for

‘(p ⊃ q) ∨ (q ⊃ p)’ turns out to be be true.”]

4. Convince me that φ is true.

[For instance: ‘Why is it the case that Hesperus is Phosphorus?’, meaning ‘Give

me grounds for thinking that Hesperus is Phosphorus.’]

When pWhy is it the case that φ?q is understood in accordance with any of these alternate

readings, I shall say that it is read as a grounding question. It is often tempting to

interpret why-questions as grounding questions—and the temptation is especially great

when the intended reading is unavailable (as it will be cases of why-closure). But, as

I noted above, grounding questions are not relevant for the purposes of assessing why-

closure.

There is an intimate connection between identity and why-closure: the sentences

one treats as why-closed are precisely those one regards as logical consequences of the

identity-statements one accepts.1 Suppose, for example, that grounding readings have

been excluded by context and someone says: “Why are things composed of water com-

posed of H2O?” The natural reaction is to reject the question altogether, and reiterate

the identity-statement: “What do you mean why? To be composed of water just is to

be composed of H2O!”.

1Dialetheists: please add “and whose negations one takes to be incompatible with the identity-statements one accepts”.

16 CHAPTER 1. IDENTITY

1.4 Deciding Between Rival Identity-Statements

It will be useful to say something about the considerations that go into determining

which identity-statements to accept. (My discussion is very much indebted to Block and

Stalnaker (1999).)

Part of the story is straightforward. One will only accept pF (x) ≡x G(x)q if one

also accepts p∀x(F (x)↔ G(x))q. But this cannot be the end of the story, since one

might reject pF (x) ≡x G(x)q even if one accepts p∀x(F (x)↔ G(x))q. Our problem is

to determine what else is required.

In rough outline, the story is this. In an effort to satisfy our goals, we develop

strategies for interacting with the world. Fruitful strategies allow us to control what

the world is like and predict how it will evolve under specified circumstances. They also

allow us to direct our research in ways that lead to the development of further fruitful

strategies. In order to articulate the strategies we adopt, we do three things at once:

firstly, we develop a language within which to formulate theoretical questions; secondly

we set forth theoretical claims addressing some of these questions; finally, we endorse a

family of identity-statements. The third task is connected to the other two because the

identity-statements we endorse will help determine which theoretical questions are treated

as why-closed, and therefore which theoretical questions are treated as not requiring

answers.

It is useful to consider some examples.

Example 1: The chemistry crank

A chemistry crank believes that the chemical composition of various substances varies

with temperature. Methanol, she thinks, is normally composed of hydrogen, oxygen and

carbon; but at a temperature of precisely√

2 degrees celsius, its chemical composition

changes to hydrogen and platinum. Similarly, our crank expects the chemical composition

1.4. DECIDING BETWEEN RIVAL IDENTITY-STATEMENTS 17

of water to vary with temperature. Her model predicts that at a temperature of precisely

π degrees celsius, water is composed of oxygen and gold. Our crank sets out to test

her hypothesis by carrying out water-electrolysis at a range of temperatures. What she

finds, of course, is that hydrogen and oxygen bubble up, regardless of how closely the

temperature approximates π degrees celsius. She concludes that every portion of water

on Earth is composed of hydrogen and oxygen. She takes this to be a remarkable fact,

in need of explanation. “Perhaps Earth’s gravitational field is getting in the way”—she

thinks—“perhaps under low-gravity conditions water is composed of oxygen and gold at

π degrees celsius.”

What identity-statements will our crank accept? The decision will be based on her

explanatory needs. She wishes to make sense of ‘why is this portion of water composed of

oxygen and hydrogen?’. So she had better not to accept ‘part of what it is to be water is

to be composed of oxygen and hydrogen’. Perhaps she will instead be inclined to accept

‘part of what it is to be water is to be a colorless, odorless, liquid’. This would lead

her to think that there is sense to be made of ‘why is this portion of water composed of

hydrogen and oxygen’, but not of, e.g. ‘why is this portion of water a liquid?’.

Now consider a real chemist. She does not think it at all remarkable that every

portion of water on Earth be composed of hydrogen and oxygen, and does not think it

is in need of explanation. In fact, she believes that there is no sense to be made of ‘why

is every portion of water on Earth composed of hydrogen and oxygen?’. Accordingly,

she is inclined to accept ‘part of what it is to be water is to be composed of hydrogen

and oxygen’. She also believes that ‘why is this portion of water a liquid?’ is a sensible

question to ask. (Remember that we are ignoring grounding questions!) So she had

better not accept ‘part of what it is to be water is to be a liquid’.

The crank and the chemist use different strategies for investigating chemical phenom-

ena. Because of these differences, they articulate their methods of inquiry in different

ways: they both accept the theoretical claim ‘every portion of water on Earth is com-

18 CHAPTER 1. IDENTITY

posed of hydrogen and oxygen’, but only the chemist treats the claim as why-closed

because only the chemist accepts ‘part of what it is to be water is to be composed of

hydrogen and oxygen’. It is important to be clear, however, that nothing in their meth-

ods of inquiry mandates a particular selection of identity-statements. The crank could,

if she really wanted, accept ‘part of what it is to be water is to be composed of oxygen

and hydrogen’. By doing so she would be committed to thinking that there is no sense

to be made of ‘why is every portion of water on Earth composed of oxygen and hydro-

gen?’. But this needn’t interfere with her ability to articulate her methods of inquiry.

For rather than asking ‘why is every portion of water on Earth composed of hydrogen

and oxygen?’, she could ask, e.g. ‘why is every portion of colorless, odorless, liquid on

Earth with such-and-such additional properties composed of hydrogen and oxygen?’.

One’s reason for preferring a particular set of identity-statements over its rivals is

not that it is somehow enforced by one’s explanatory methods. One’s reason is that it

yields an articulation of one’s explanatory methods that one finds especially congenial.

Accepting ‘part of what it is to be water is to be composed of oxygen and hydrogen’

is uncongenial for the crank because it forces her to reformulate some of her chemical

claims. Rather than saying, e.g. ‘under ideal conditions, water is composed of oxygen and

gold at π degrees celsius’, she will now have to say, e.g. ‘under ideal conditions, colorless,

odorless liquids with such-and-such additional properties are composed of oxygen and

gold at π degrees celsius’. Such reformulations are undesirable because of the extra

work, but also because it may not be obvious to the crank how a given claim is best

reformulated. There would be no problem if—before the change—the crank had an

exhaustive characterization of what it is to be water. For then she would be in a position

to reformulate her chemical claims by replacing every occurrence of the world ‘water’

with the relevant characterization. But suppose that—before the change—all the crank

has is a partial characterization of what it is to be water: she believes that part of what

it is to be water is to be a colorless, odorless liquid, but leaves open that there might be

1.4. DECIDING BETWEEN RIVAL IDENTITY-STATEMENTS 19

more to being water than that. Then she won’t be sure how to cash out the ‘such-and-

such’ in a claim like ‘under ideal conditions, colorless, odorless liquids with such-and-such

additional properties are composed of oxygen and gold at π degrees celsius.’

If our crank comes to accept ‘part of what it is to be water is to contain hydrogen’ she

will change her views about the satisfaction-conditions of ‘. . . is a portion of water’. She

might start out believing that part of what it takes to satisfy ‘. . . is a porition of water’

is to be a colorless, odorless liquid. But after accepting ‘part of what it is to be water is

to contain oxygen and hydrogen’, she will come to believe that part of what it takes to

satisfy ‘. . . is a portion of water’ is to be composed of oxygen and hydrogen. This is not

to say that ‘water’, as used by the crank, undergoes a change in satisfaction-conditions.

(It rigidly designates H2O throughout.) The point is that if the crank accepts deviant

identity-statements, she will also have deviant beliefs about the satisfaction-conditions

of the expressions of her language. (A similar point can be made with respect to the

crank’s concepts : if she accepts deviant identity-statements, she will also have deviant

beliefs about the satisfaction-conditions of her concepts.)

Example 2: The zoology crank

A zoology crank believes that snails placed in a ‘lobster environment’ will evolve into

lobsters after a few generations, and that lobsters placed in an ‘elephant environment’

will evolve into elephants after a few generations. He also believes that ‘original creatures’

spontaneously come into existence from time to time, and their descendants evolve into

members of various species as the generations go by. As a result, our crank believes that

it is rare for all the members of a species to share a common ancestry. They are typically

composed of the descendants of several different ‘original creatures’. All the same, our

crank decides to sample the DNA of extant elephants in order to determine whether they

are, in fact, related. When the results come in, he concludes that elephants do, as a

matter of fact, share a common ancestry. This conclusion strikes him as remarkable, and

20 CHAPTER 1. IDENTITY

in need of explanation. “Perhaps it was an incredible coincidence”—he thinks—“perhaps

it is only the descendants of a particular original creature that happened to end up in an

elephant environment.”

What identity-statements will our crank accept? As before, the decision will be based

on his explanatory needs. He wishes to makes sense of ‘why do elephants share a common

ancestry?’. So he had better not to accept ‘part of what it is to be an elephant is to have

a certain lineage’. Perhaps he will instead be inclined to accept, e.g. ‘part of what it is

to be an elephant is to have such-and-such a phenotype’. This would lead him to think

that there is sense to be made of ‘why do elephants share a common ancestry’, but not

of, e.g. ‘why do elephants have trunks?’.

Unfruitful research might lead our zoology crank to change his approach. He may

alter his zoological beliefs and explanations. He might come to believe, for example, that

changes in phenotype between an individual and its offspring are much slighter (and far

more random) than he previously thought, and that the link between surviving under

environmental conditions of a given type and having a particular phenotype is much less

robust than he previously thought. Accordingly, he might come to think that the sharing

of elephant-phenotypes—and the ability of elephants to interbreed—is best explained by

elephants’ common ancestry.

What identity-statements will our crank now accept? After the change of approach,

the notion of having such-and-such a lineage can be expected to play a more central

role in his zoological thinking than the notion of having such-and-such a phenotype. So

by accepting ‘part of what it is to be an elephant is to have a certain lineage’ rather

than ‘part of what it is to be an elephant is to have such-and-such a phenotype’ he can

expect to articulate his methods of zoological inquiry in a relatively congenial way. He

will be committed to thinking that there is no sense to be made of ‘why do elephants

have a common ancestry?’. But he can still hold that there are intelligible questions

in the vicinity (e.g. ‘why are elephants the only extant creatures with an elephantine

1.4. DECIDING BETWEEN RIVAL IDENTITY-STATEMENTS 21

phenotype?’).

One respect in which the zoology example is different from the chemistry example is

that the considerations that recommend accepting ‘part of what it is to be an elephant

is to have a certain lineage’ over some of its rivals are not particularly decisive. There is

room for arguing, for instance, that the notion of being a member of a group of organisms

that are able to interbreed and produce fertile offspring will play a more central role in

the zoologist’s thinking than the notion of having such-and-such a lineage, and therefore

that his methods of zoological inquiry would be best articulated by accepting ‘part of

what it is to be an elephant is to be a member of a certain group of organisms that

are able to interbreed and produce fertile offspring’ (Two groups, actually: African and

Indian elephants can’t interbreed).

In general, the question of what identities to accept may turn on the purposes at

hand. The same theorist might find it useful to accept ‘what it is to be an elephant is to

be a member of a certain lineage’ for the purposes thinking about evolution by natural

selection, and find it useful to accept ‘what it is to be an elephant is to be a member of a

certain group of organisms that are able to interbreed and produce fertile offspring’ for

the purposes of studying ancient patterns of elephant-migration. Moreover, one would

expect that situations in which one’s methods of inquiry can be articulated with similar

success by way of rival identity-statements would be the rule rather than the exception.

Example 3: The vixen-conspiracy crank

A conspiracy crank believes that vixens are super-intelligent creatures with magical pow-

ers who are out to dominate the Earth. She believes they are extraterrestrials—no

Earthly creature has ever been vixen-like. (Male foxes, on the other hand, are not alive

at all; they are robots created by the vixens as a cover for their operations.) Vixens

have the ability to assume any form they like, and have chosen to assume the form of

mammalian females. This is a fact in need of explanation.

22 CHAPTER 1. IDENTITY

What identity-statements will our crank accept? She wishes to make sense of ‘why

are vixens female?’. So she had better reject ‘part of what it is to be a vixen is to be

female’. Perhaps she will claim instead that what it is to be a vixen is to be a member of

a certain extra-terrestrial lineage. She can then be expected to have unorthodox views

about the satisfaction-conditions of ‘vixen’. In particular, she might deny that part of

what it takes to satisfy ‘vixen’ is to be female—she might claim instead that all it takes

to satisfy ‘vixen’ is to be a member of the right extra-terrestrial lineage, whether or not

one happens to assume a female form. This is not to say, however, that ‘vixen’, as used

by the crank, undergoes a change in satisfaction-conditions. Even in her mouth, what it

takes to satisfy ‘vixen’ is to be a female fox.

As in previous examples, nothing in the crank’s methods of inquiry mandates a partic-

ular choice of identity-statements. (She could, if she really wanted, accept ‘part of what

it is to be a vixen is to be female’. By doing so she would be committed to thinking that

there is no sense to be made of ‘why are vixens female?’. But this needn’t interfere with

her ability to articulate her methods of inquiry. For rather than asking ‘why are vixens

female?’ she might ask ‘why have the extraterrestrials chosen to assume female form?’.)

But the present example can be used to illustrate an important point: even if nothing

in the crank’s methods of inquiry mandates a particular choice of identity-statements,

there is room for thinking that certain choices are excluded by facts about the crank’s

language, together with her methods of inquiry. Suppose, for example, that one believes

that ‘vixens are female’ is analytic. One might then think that—on pain of betraying the

meanings of her words—the crank is barred from accepting, e.g. ‘to be a vixen just is to

be a member of a certain extraterrestrial lineage’.

More generally, suppose, one believes that our words have primary intensions. (See

Chalmers (1996) and Jackson (1998).) One believes, in particular, that the meaning of

‘vixen’ determines more than just an intension for ‘vixen’ (i.e. a function assigning to each

possible world the set of objects ‘vixen’ applies to at that world); it also determines, for

1.4. DECIDING BETWEEN RIVAL IDENTITY-STATEMENTS 23

each possible world, what the intension of ‘vixen’ should be taken to be on the assumption

that that world is actual. One might then think that—on pain of betraying the meanings

of her words—the crank is not free to choose between rival identity-statements. Suppose

the meaning of ‘vixen’ determines that, on the assumption that the crank’s beliefs are

true, one should take the intension of ‘vixen’ to assign each world the set of female

members of a certain terrestrail lineage. Then—on pain of betraying the meanings of her

words—the crank is barred from accepting ‘to be a vixen just is to be a member of a

certain extra-terrestrial lineage’.

I am myself skeptical of the notion of analyticity as traditionally understood, and I

can see no good reason for believing that the meaning of ‘vixen’ determines what the

intension of ‘vixen’ should be taken to be on the assumption that that the crank’s beliefs

are true. But I shall remain neutral about such questions throughout the book.

Example 4: First-order Identity

The examples we have considered so far have all concerned second-order identity-statements,

and they have all had the same form. We discussed scenarios in which a statement of

the form p∀x(F (x) → G(x))q is taken for granted, and asked about the sorts of con-

siderations that would lead one to take the additional step of accepting or rejecting

pF (x)�x G(x)q. The first-order analogue of these examples would be a case in which,

for each of a suitable stock of predicates φ(x), pφ(a)↔ φ(b)q is taken for granted, and

one asks about the sorts of considerations that would lead one to take the additional step

of accepting or rejecting pa = bq. (On pain of making the exercise uninteresting, one had

better not allow, e.g. ‘x = a’ to be one of the ‘suitable predicates’.)

Familiar examples of this kind include cases of personal identity. (“This human animal

and this person are physically indistinguishable, but are they the same individual?”)

They also include the puzzle of the statue and the clay. (“This statue and this lump of clay

share are physically indistinguishable, but are they the same individual?”) Unfortunately,

24 CHAPTER 1. IDENTITY

both of these examples are tied up with philosophical controversies which are best avoided

in the present context. I shall therefore use an example from science-fiction.

By the dawn of the twenty-seventh century, the old theories of quantum mechanics had

been superseded by superquantum-theory. This new fundamental physics countenances

four different kinds of fields (numbered 1 through 4). Type-1 and type-2 fields are

generated, respectively, by particles of type-1 and type-2. These particles usually have

different locations, but on occasion they undergo a ‘superquantum merger’, in which they

come to occupy the same location. (‘Location’ in superquantum-theory is a messy affair:

particles occupy ‘clouds’ in space rather than exact regions of space, and particles ‘merge’

when they come to share a ‘cloud’.)

There are also fields of type-3 and type-4. Everyone agrees that they are different.

(Type-3 fields can be blocked by a lead barrier, for example; but type-4 fields cannot.)

Everyone agrees, moreover, that type-3 fields and type-4 fields are both generated by

particles, and there is a stipulation in place to the effect that the source of a type-3 fields

is to be referred to as a ‘type-3’ particle and the source of type-4 fields is to be referred

to as a ‘type-4’ particle. There is, however, an important disagreement. As far as anyone

has been able to tell, type-3 fields and type-4 fields are always generated from the same

location. Monists suggest that this is because type-3 particles and type-4 particles are

one and the same. Dualists suggest that type-3 particles and type-4 particles are distinct,

but that they are always ‘merged’ with one another, in the same sort of way that type-1

particles and type-2 particles are sometimes ‘merged’.

What sorts of considerations would lead one to embrace one of these positions over

its rival? Here the crucial observation is that monists and dualists face different ex-

planatory burdens. From the dualist’s perspective, the colocation of type-3 particles and

type-4 particles is a fact that calls for explanation. (Perhaps she has an explanation:

perhaps the theoretical model which she uses to explain the occasional merging of type-1

and type-2 particles predicts that the particles generating type-3 and type-4 fields will

1.5. LOGICAL TRUTH 25

be systematically merged, or that they will be merged in all but extraordinary circum-

stances.) From the perspective of the monist, on the other hand, there is no need to

explain the colocation of type-3 and type-4 particles. For there is no sense to be made of

‘I can see that this type-3 particle is identical to this type-4 particle, but why are they

colocated?’.

If she really wanted to, the dualist could try to articulate her methods of inquiry in

monistic terms. Whether or not she would find a monistic articulation congenial depends

on the details of her methods of inquiry. The change may prove to be a welcome one

if her theorizing is unable to supply a satisfying answer to the question ‘why are type-

3 particles and type-4 particles systematically colocated?’. For in that case the shift to

monism would relieve her of an unwelcome explanatory burden. But suppose instead that

she has a satisfying answer to the question—perhaps her answer involves a bit of theory

which is ripe with interesting predictions, some of which have been confirmed. Then

whether the move is congenial will depend on whether she is able to find an attractive

way of articulating the insight captured by the extra bit of theory in a monistic setting.

What this story suggests is that the sorts of considerations that would lead one to go

from accepting suitable instances of pφ(a)↔ φ(b)q (e.g. ‘this type-3 particle is located

here↔ this type-4 particle is located here’) to accepting pa = bq (e.g. ‘this type-3 particle

is this type-4 particle’) are similar to the sorts of considerations that would lead one to

go from accepting p∀x(F (x)→ G(x))q to accepting pF (x)�x G(x)q. In both cases, one

will be motivated to take the additional step to the extent that the resulting explanatory

landscape offers good prospects for a congenial articulation of one’s methods of inquiry.

1.5 Logical Truth

When one treats a sentence as logically true, one does more than simply treat it as true.

One takes it to have trivial truth-conditions, and takes it therefore to be why-closed: one

26 CHAPTER 1. IDENTITY

thinks there is no intelligible question as to why the world is such as to satisfy its truth-

conditions. If, for example, you are a classical logician, you will think it wrong-headed to

say “I can see as clearly as can be that if there are elephants, then there are elephants;

what I want to understand is why.”

That sentences regarded as logically true are generally treated as why-closed is a point

that is easily obscured. The problem is that when φ is taken to be a logical truth, pWhy

is it the case that φ?q is most naturally read as a grounding question. And although

the relevant grounding questions do indeed make sense, they are irrelevant to assessing

why-closure. (See section 1.3.)

Suppose φ is a relatively complex logical truth, such as ‘∃x(φ(x) ⊃ ψ) ≡ (∀x(φ(x)) ⊃

ψ)’ or ‘DP ⊃ there is no largest prime’ (where ‘DP ’ abbreviates the conjunction of an

interesting subset of the first-order Dedekind-Peano Axioms). When someone asks pWhy

is it the case that φ?q, it can be natural to read the question as follows:

I do not fully understand how the meanings of φ’s lexical items conspire to

deliver the truth-conditions that they in fact deliver.

Thus read, the question certainly makes sense. And it can be adequately addressed by

offering a sufficiently illuminating proof of φ, and making sure that it is fully understood

by one’s interlocutor. But all one has done so far is address a grounding question. It

would be hard to make sense of one’s interlocutor if she went on to ask the question

that is relevant for assessing why-closure: “I fully understand why it is that φ has the

truth-conditions that it has, and I can fully understand what it would take to satisfy such

truth-condtions. What I want to know is why the world is such as to satisfy them.” For

one should think that nothing is required of the world in order for the truth-conditions

of a sentence one regards as logically true to be satisfied. (For further discussion, see

chapter 5.)

The claim that sentences regarded as logically true are taken to have trivial truth-

1.5. LOGICAL TRUTH 27

conditions is open to two potential sources of misunderstanding. First, one might worry

that it leads to the conclusion that logical truths all have the same meaning. But there

is no such entailment. To say that two sentences have the same truth-conditions is not

to say that they have the same meaning. It is only to say that there is no difference

between what would be required of the world to satisfy the constraints determined by

one of the meanings and what would be required of the world to satisfy the constraints

determined by the other.

Second, one might worry that it leads to the conclusion that coming to know of a

logical truth that it is true should be a trivial matter. But the conclusion only follows on

the assumption that determining whether a sentence has trivial truth-conditions is itself

a trivial affair. And this is not generally true: it can be a highly non-trivial matter to

work out whether the truth-conditions of a logical truth are indeed trivial.

The connection between logical truth and triviality can be restated as a connection

between logical truth and identity. When one regards a sentence as a logical truth, one

is, in effect, accepting an identity-statement. Part of what it is to treat the sentence

pφ↔ ψq as a logical truth is to commit oneself to the identity-statement pφ ≡ ψq. More

generally, part of what it is to treat the sentence φ as a logical truth is to commit oneself

to the identity-statement pφ ≡ >q (where > is a sentence one takes to have trivial truth-

conditions and ‘≡’ is the material biconditional).

As a result, when one treats a sentence like p¬¬φ↔ φq as a logical truth, it is not

just that one takes oneself to be justified in accepting φ provided one feels justified in

acepting p¬¬φq (and vice-versa). One will think that an understanding of why p¬¬φq’s

truth-conditions are satisfied is already an understanding of why φ’s truth-conditions are

satisfied (and vice-versa). There is no need to add an explanation of why the transition

from p¬¬φq to φ is valid: that φ’s truth-conditions be satisfied is what it is for p¬¬φq’s

truth-conditions to be satisfied.

28 CHAPTER 1. IDENTITY

Deciding between rival logics

In the preceding section we considered examples in which the decision to accept an

identity-statement can be closely tied to empirical considerations. But when it comes to

accepting the family of identities corresponding to the adoption of a logical system, the

decision is more organizational in nature.

Generally speaking, there is a delicate balance to be struck in deciding which identities

to accept. If one accepts too many, one will be committed to treating as unintelligible

scenarios that might have been useful in theorizing about the world. If one accepts too

few, one opens the door to a larger range of intelligible scenarios, all of them candidates

for truth. In discriminating amongst these scenarios one will have to explain why one

favors the ones one favors. And although the relevant explanations could lead to fruitful

theorizing, they could also prove burdensome.

The adoption of a suitable logic plays an important role in finding the right balance

between these competing considerations. Consider, for example, the question whether to

treat p¬¬φ↔ φq as a logical truth (and therefore accept every identity-statement of the

form ‘for ¬p to fail to be the case just is for p to be the case’). A friend of intuitionistic

logic, who denies the logicality of p¬¬φ↔ φq, thinks it might be worthwhile to ask why

it is the case that p even if you fully understand why it is not the case that ¬p. In the

best case scenario, making room for an answer will lead to fruitful theorizing. But things

may not go that well. One might come to see the newfound conceptual space between

a sentence and its double negation as a pointless distraction, demanding explanations in

places where there is nothing fruitful to be said.

Even if none of the decisions one makes in adopting a family of identity-statements is

wholly independent of empirical considerations, some decisions are more closely tied to

empirical considerations than others. And when it comes to identity-statements corre-

sponding to logical truths, one would expect the focus to be less on particular empirical

1.6. WHAT DOES THE TRUTH OF AN IDENTITY-STATEMENT CONSIST IN?29

matters and more on the question of how to best organize one’s methods of inquiry.

So there is room for a picture whereby an epistemically responsible subject can believe

identity-statements on the basis of considerations that aren’t grounded very directly in

any sort of empirical investigation.

1.6 What Does the Truth of an Identity-Statement

Consist In?

Throughout this chapter I have talked about about triviality, de mundo intelligibility

and why-closure according to a subject. But one can also talk about triviality, de mundo

intelligibility and why-closure simpliciter. A sentence has trivial truth-conditions just in

case it is a logical consequence of true identity-statements; a story is de mundo intelligible

just in case it is logically consistent with the set of true identity-statements; a sentence

is why-closed just in case it it is a logical consequence of true identity-statements.

The issue of what the truth or falsity of an identity-statement consists in is a delicate

one. The aim of this section is to help clarify what is going on.

1.6.1 Preliminary Remarks

True identity-statements have trivial truth-conditions; false identity-statements have im-

possible truth-conditions. Consider, for example, ‘to be hot is to have high mean kinetic

energy’. What is required of the world in order for the truth-conditions of this sentence

to be satisfied is that there be no difference between having high mean kinetic energy

(i.e. being hot) and being hot. Equivalently: that there be no difference between being

hot and being hot.

The result is that controversies surrounding the truth of an identity-statement will

never concern the question of whether the world is such as to satisfy the truth-conditions

30 CHAPTER 1. IDENTITY

that are actually expressed by the identity-statement: everyone agrees that the trivial

truth-conditions are satisfied, and that the impossible truth-conditions are not. This

is not to say, however, that controversies surrounding the truth identity-statement are

divorced from disagreements about how the world is. They typically concern two kinds

of non-linguistic issues: (1) empirical questions not expressed by the identity-statement,

and (2) differences in the lines of inquiry that are regarded as fruitful in theorizing about

the world. Disagreements about the truth of ‘to be hot is to have high mean kinetic

energy’, for example, concerned straightforwardly factual questions (e.g the existence of

caloric), and differences in the lines of research that were regarded as fruitful (e.g. whether

it would be fruitful to engage in the project of accounting for heat-related phenomena

by postulating a new substance).

There is also disagreement about language—specifically, about the satisfaction-con-

idiitions of the expressions involved in the identity-statement. The caloric theorist will

think that what it takes to satisfy ‘is hot’ is to contain sufficient quantities of caloric;

the contemporary scientist will think that what it takes is to have high mean kinetic

energy. (And, of course, the disagreement about the satisfaction-conditions of ‘is hot’ is

also a disagreement about the satisfaction-conditions of ‘to be hot is to have high mean

kinetic energy’: if ‘is hot’ and ‘has high mean kinetic energy’ are taken to have the same

satisfaction-conditions, the identity-statement’s truth-conditions will be regarded as triv-

ial; otherwise, the identity-statement’s truth-conditions will be regarded as impossible.)

Debates about the truth-conditions of an identity-statement are a subtle affair. In

simple examples, such as the one considered above, or a disagreement about the truth

of ‘Hesperus is Phosphorus’, it is relatively straightforward to see how one’s views about

whether or not the identity-statement’s has trivial truth-conditions turn on non-linguistic

matters. In the case of ‘Hesperus is Phosphorus’ it can be agreed on all sides that the

truth-conditions are trivial if a single planet plays the morning-star and evening-star

roles, and impossible if there is a different planet playing each role.

1.6. WHAT DOES THE TRUTH OF AN IDENTITY-STATEMENT CONSIST IN?31

Unfortunately, matters are not always that straightforward. Consider the examples

we discussed in section 1.4. The chemist and the crank can agree about the most ob-

viously relevant non-linguistic fact—that every extant portion of water is composed of

H2O—and still disagree about whether the truth-conditions of ‘water is H2O’ are trivial

or impossible. Likewise, the zoologist and the crank can agree about the most obviously

relevant non-linguistic fact—that all elephants have a common ancestry—and still dis-

agree about whether the truth-conditions of ‘part of what it is be an elephant is to have

a certain lineage’ are trivial or impossible. In both cases, the disagreement comes from

a difference in the lines of research that the subjects consider worthwhile. Earlier I mo-

tivated these differences by ascribing the cranks false beliefs: that gold can be extracted

from water at the right temperature, and that sea-slugs, say, lack a common ancestry.

But in the general case there is no guarantee that differences in the lines of research that

the subjects take to be fruitful will be traceable to false beliefs of this kind. They may

ultimately rest on differences in the sorts of methodologies that the various subjects find

useful in theorizing about the world.

Perhaps one zoologist thinks it fruitful to place considerations of lineage at the center

of his theorizing whereas another finds it more useful to emphasize the ability of individu-

als to interbreed and produce fertile offspring. Neither of them has any false beliefs about

how organisms are related to each other, or about who is able to interbreed with whom.

The different methodologies might be more or less fruitful relative to different purposes,

and it might turn out that neither of them is clearly dominant in zoology as a whole. If

so, there will be no clear empirical pressure to adopt one candidate identity-statement

instead of one of its rivals. It may well come down to a matter of deciding which way of

using language would be most convenient for the purposes at hand.

32 CHAPTER 1. IDENTITY

1.6.2 Metaphysical Privilege

Perhaps you believe that there is such a thing as ‘metaphysically privileged’ properties—

‘natural kinds’, in the metaphysician’s sense—and that one of the property of sharing a

common lineage and the property of having genetic material that could be combined with

that of such-and-such individuals to produce fertile offspring enjoys greater ‘metaphysical

privilege’ than the other. If so, you might think that a debate about whether to accept

‘to be an elephant just is to be a member of thus-and-such a lineage’ or ‘to be an

elephant just is to have the sort of genetic material that could be combined with that of

such-and-such individuals to produce fertile offspring’ turns on more than just linguistic

preference. It is, in part, a debate about where metaphysical privilege lies. You might

think, for example, that if the property of being a member of thus-and-such a lineage is

endowed with greater metaphysical privilege, it is thereby a more eligible candidate for

the referent of ‘elephant’.

I myself am utterly unable to make sense of the requisite notion of metaphysical

privilege. As far as I can tell, it is a piece of Deep Metaphysics. (‘Deep Metaphysics’

is my label for the sort of metaphysics that makes non-metaphysicians cringe: the kind

that outsiders see as relying on distinctions without a difference, and that the Logical

Empiricists reacted against in the first third of the Twentieth Century. For discussion,

see chapter 6.)

Calling metaphysically privileged properties ‘natural kinds’ masks just how obscure

the notion of metaphysical privilege really is. For the use of a label from the natural sci-

ences might be taken to suggest that metaphysically privileged properties are scientifically

distinguished: that they lead to more fruitful theorizing than their less metaphysically

exalted counterparts. But this is precisely not the case in the situation at hand. For

we wheeled in the notion of metaphysical privilege in an effort to buttress the idea that

one of the candidate referents for ‘elephant’ could be deemed superior to its peers even

1.6. WHAT DOES THE TRUTH OF AN IDENTITY-STATEMENT CONSIST IN?33

though it did not lead to better scientific theorizing.

The problem with metaphysical privilege, as I see it, is that it is not subject to

independent constraints. It is bit like saying: “the reason there are facts about which

outfits are objectively fashionable—not just fashionable relative to the tastes of some

community or other—is that certain outfits are metaphysically privileged: they carve the

world at the joints.” Be a fashion objectivist if you must, but don’t pretend that talk of

metaphysical privilege makes your view any less obscure.

I am under no illusions about the ability of remarks like these to convince the uncon-

vinced. Talk of metaphysical privilege is currently in vogue, and many metaphysicians

would report having no trouble whatsoever understanding what is going on. Because I

am pessimistic about the prospects of making any meaningful progress with the initiated,

I will limit my discussion to a brief remark, aimed at people who have yet to take sides.

Conversation with my metaphysician friends has sometimes resulted in dialogue such

as the following:

What is it to carve the world at the joints? It is to describe the world in

metaphysically fundamental terms. But what is it for a term to be metaphys-

ically fundamental? It is a term that God would use when describing the

world. Yes, but what is so special about God’s description of the world? She

would only use predicates expressing perfectly natural properties. What do

you mean, ‘perfectly natural’? Oh, perfectly natural properties are properties

that carve the world at the joints.

Perhaps there is good sense to be made of this family of interrelated terms. But it is

important to be clear that it would be no thanks to the proposed circle of definitions. As

far as the definitions go, ‘carves at the joints’, ‘metaphysically fundamental’, ‘expression

in God’s language’ and ‘perfectly natural’ are nothing more than labels. No light has

been shed on the underlying notions because pieces of metaphysical jargon are defined

34 CHAPTER 1. IDENTITY

in terms of other pieces of metaphysical jargon: none of the terms can be used by the

uninitiated to break into the circle.

Before feeling satisfied about understanding a notion of metaphysical privilege, I urge

you to make sure you are told what work the notion is meant to be doing. And not

any job-description will do. If you are having trouble understanding objectivism about

fashion, for example, it won’t be very helpful to be told that objectively fashionable

outfits carve at the joints and that joint-carving is to be understood by way of its job in

explicating objectivism about fashion.

Some metaphysicians have made a serious effort to elucidate the job that a notion

of metaphysical privilege might be use to perform. (See, for instance, Lewis (1983a)

and Sider (typescript).) While I applaud their methodology, I must report that I am

totally unconvinced. Of the many applications that have been suggested for notions of

metaphysical privilege, I have never been able to find one that strikes me as plausible

and outside the realm of Deep Metaphysics.2 But I may be prejudiced because I started

out as a skeptic. Perhaps you can do better. (If you are able to do better, please keep

in mind it that nothing in this book presupposes that your views about metaphysical

privilege are mistaken.)

1.6.3 The Upshot

We have identified several different issues that might be at stake when there is a discussion

about whether a given identity-statement is true:

1. Empirical questions not expressed by the identity-statement.

[Is there one planet or two? Is there is such a thing as caloric fluid?]

2Steve Yablo’s recent work on aboutness speaks of some facts being true ‘in virtue’ of others. Itseems to me that Yablo has identified enough interesting work for the notion of ‘in virtue of’ to do thathe has succeeded in elucidating what it means. I doubt, however, that Yablo’s notion is a notion ofmetaphysical privilege of the kind that more metaphysically minded philosophers have been interestedin defending.

1.6. WHAT DOES THE TRUTH OF AN IDENTITY-STATEMENT CONSIST IN?35

2. Differences in the lines of research that are regarded as fruitful.

[Would it be fruitful to engage in the project of accounting for heat-related phe-

nomena by postulating a new substance?]

3. Disagreements about which way of using language is most convenient for the pur-

poses at hand.

[Is it easier to express interesting zoological claims by using ‘elephant’ to mean

‘member of thus-and-such a lineage’ or ‘has the sort of genetic material that could

be combined with that of such-and-such individuals to produce fertile offspring’?]

4. Disagreement, amongst Deep Metaphysicians, about which properties are endowed

with ‘metaphysical privilege’.

[Does the property of having thus-and-such a lineage enjoy greater metaphysical

privilege—and therefore greater eligibility as a referent for the word ‘elephant’—

than other candidate properties?]

It is important to be clear, however, that none of these issues are actually expressed by

the identity-statements themselves. As we have seen, the truth-conditions of an identity-

statement are always either trivial or impossible, and there is never any disagreement

about whether the world satisfies either of these conditions. What goes on is, rather,

that one’s views about whether a particular identity-statement should be thought of as

expressing the trivial truth-conditions or the impossible truth-conditions can be tied up

with one’s views about the sorts of issues listed above.

In certain cases, it is possible to isolate the substantial questions that are actually

fueling the debate, and leave identity-statements out of the picture, to be revisited only

once substantial issues have been settled. But there is no reason to expect this to be true

in general. One might accept an identity-statement because it is part of one’s overall

scientific outlook without being in a position to articulate the particular empirical claims,

36 CHAPTER 1. IDENTITY

methodological principles and linguistic preferences that one’s outlook presupposes. In

such cases one would be unable to leave identity-statements out of the debate.

Another context in which it can be a good idea to place identity-statements at the

center of the debate is when one wishes to defend a view about the sorts of distinctions

that one is able to make using one’s language. Suppose, for example, that you wish to

defend the view that ‘is hot’ and ‘has high mean kinetic energy’ make the same distinction.

You could perspicuously put the point by saying ‘there is no difference between being

hot and having high mean kinetic energy’, or by saying ‘to be hot just is to have high

mean kinetic energy’.

Chapter 2

Possibility

The focus of this chapter will be on the notion that is referred to in the literature as

metaphysical possibility. The use of ‘metaphysical’ is potentially misleading here be-

cause lends itself to two very different readings. On the first—and, in my opinion, less

interesting—reading, ‘metaphysical’ is to be understood as restricting the range of pos-

sibilities that are to be considered. Just like that the notion of physical possibility might

be understood as restricted to possibilities that are compatible with the physical laws,

one might think that there is a notion of possibility broader than metaphysical possibility

(conceptual possibility?), and thinking of metaphysical possibility as restricted to those

of the possibilities in the broad sense that satisfy the ‘metaphysical laws’. One might

claim, for example, that it is possible in the broad sense that there be zombies, or that

God fail to exist, but that such scenarios are to be counted as metaphysically impossible,

on the grounds that they violates the ‘metaphysical laws’. (For an example of a restric-

tive approach to metaphysical possibility, see Kment (2006); for illuminating discussion,

see Rosen (2006). I believe the term ‘metaphysical law’ is due to Hartry Field.)

The second way of reading ‘metaphysical’ in ‘metaphysical possibility’—the one I

prefer—is as supplying a contrast class, rather than a restriction. It is meant to dis-

tinguish a notion of possibility that applies to ways for the world to be, which is what

37

38 CHAPTER 2. POSSIBILITY

we want, from a notion of possibility that applies to representations (and is sensitive

not just to the representations’ truth-conditions but also to how the truth-conditions are

represented), which is not what we want. Ideally, one would use a label like ‘possibility

de mundo’ to refer to the former, and a label like ‘possibility de representatione’ to refer

to the latter. (The notion of logical consistency, for instance, is best understood as a

form of possibility de representatione, since ‘Hesperus 6= Phosphorus’ and ‘Hesperus 6=

Hesperus’ differ in terms of logical consistency, even though satisfaction of their truth-

conditions imposes the same requirement on the world.) On the view we are considering,

the role of ‘metaphysical’ in ‘metaphysical possibility’ is to be understood as demanding

that metaphysical possibility be thought of as a form of possibility de mundo.

It seems to me that the first way of thinking about metaphysical possibility—where

‘metaphysics’ is treated as a restrictor, rather than as supplying a contrast class—is

doubly objectionable. Firstly, I think one should feel uncomfortable about admitting

a notion of possibility de mundo broader than metaphysical possibility. The proposal,

presumably, is that ‘φ is possible-in-the-broader-sense’ is to be cashed out in terms of

something along the lines of ‘¬φ is not analytic’ or ‘¬φ has a non-empty primary inten-

sion’. This sounds perilously like possibility de representatione, since ‘is not analytic’ and

‘has a non-empty primary intension’ are properties of sentences, not properties of ways

for the world to be. Notice, moreover, that analyticity-talk and primary-intension-talk

both rely on the assumption that one can factor the requirement that a sentence’s truth-

conditions impose on the world into two parts: a component that is knowable a priori,

merely in virtue of one’s mastery of the language, and a component that is knowable

only a posteriori. And I see no good reason for thinking that this is true in general, even

if it seems plausible when one focuses on toy examples such as ‘Hesperus is Phosphorus’.

Secondly, and more importantly, the notion of a ‘metaphysical law’ strikes me as

both obscure and unhelpful. It strikes me as obscure because I have never been able

to find a use for it outside Deep Metaphysics, in the sense of chapter 6. It strikes me

39

as unhelpful because it drives a wedge between the notion of possibility and one of its

most useful applications: the representation of content. For in stating a ‘metaphysical

law’ one would presumably be saying something non-trivial about the way the world

is. But if every possible scenario is a scenario which verifies the metaphysical laws, one

cannot hope to represent the non-trivial requirement that the world must meet in order

for a metaphysical law to hold by dividing up the possibilities into those that meet the

requirement and those that do not.

On the alternative way of thinking about metaphysical possibility—when ‘metaphys-

ical’ is thought of as supplying a contrast class, rather than a restriction—metaphysical

possibility is the most inclusive form of possibility de mundo there is. Going beyond

metaphysical possibility is not a matter of violating the ‘metaphysical laws’: it is a

matter of lapsing into incoherence.

The material developed in chapter 1 allows us to articulate this idea further. The

stories one treats as de mundo unintelligible are the stories one regards as incoherent.

So one should count a story as describing a metaphysically possible scenario just in

case one regards it as de mundo intelligible. In light of the connection between de mundo

intelligibility and identity, this allows us to establish a connection between possibility and

identity: a story describes a metaphysically possible scenario just in case it is logically

consistent with the set of true identity-statements.

Thinking of metaphysical possibility in this way has several important advantages.

First, it dispels some of the mystery surrounding modal knowledge. For if a story is

metaphysically possible just in case it is logically consistent with the set of true identities,

then our knowledge of which scenarios are possible should be no more mysterious than

our knowledge of which identities are true. And we saw in chapter 1 that there is a

substantial story to be told about our knowledge of identities.

A second advantage of this way of thinking about metaphysical possibility is that one

is able to steer clear of the analytic/synthetic distinction and its siblings. One works

40 CHAPTER 2. POSSIBILITY

with a notion which is non-a priori to begin with, rather than construing necessity as

a marriage between a priori truths and non-a priori Kripkean necessities. The result is

that one gets a distinction between necessity and contingency without having to forego

the Quinean insight that conceptual questions cannot be cleanly separated from questions

of fact.

A third advantage of this way of thinking about metaphysical possibility is that one

retains the connection between possibility and content. To see this, think of a sentence’s

truth-conditions as a requirement imposed on the world: a requirement that the world

be a certain way. (See introduction and section 1.3.) Knowing whether a scenario we

regard as de mundo intelligible must fail to obtain in order for the requirement to be met

is valuable because it gives us an understanding of how the world would need to be in

order for the requirement to be met. But knowing whether a scenario we don’t regard

as de mundo intelligible must fail to obtain in order for the requirement to be met is not

very helpful. For when one is unable to describe a scenario in a way one finds de mundo

intelligible, the claim that the scenario must fail to obtain gives one no understanding of

how the world would need to be in order for the requirement to be satisfied. The result is

that one can model a sentence’s truth-conditions as a subset of the space of scenarios one

regards as de mundo intelligible. So the notion of possibility that is useful for modeling

truth-conditions is precisely the notion that is correlated with de mundo intelligibility.

(For further discussion, see section 5.2.)

In what follows I shall use ‘possibility’ to refer metaphysical possibility in the proposed

sense: as the most inclusive kind of de mundo possibility there is.

2.1 Supervenience

In suggesting that a story describes a possible scenario just in case it is logically consistent

with the set of true identity-statements, I am thinking of a ‘story’ as a set of (interpreted)

2.1. SUPERVENIENCE 41

first-order sentences, with no boxes or diamonds. So the claim so far is that the set of

true identity statements fixes the truth-value of every de dicto modal sentence. (More

specifically: when φ contains no boxes or diamonds, p♦φq is true just in case φ is logically

consistent with the set of true identity statements, and p�φq is true just in case p¬φq is

not logically consistent with the set of true identity statements.)

The central claim of this chapter is that the connection between identity and possibil-

ity need not be limited to de dicto modal sentences. One can use the set of true identity

statements to fix the truth value of every sentence in a first-order modal language. More

precisely, let L be an arbitrary first-order language and L♦ be the modal language that

results from adding ‘♦’ to L. Then one can prove the following:

Supervenience Theorem

By fixing the truth-value of every sentence in L and every identity statement

built up from vocabulary in L, one fixes the truth-value of every sentence in L♦.

The Supervenience Theorem turns out to have an interesting corollary. One gets the

result that the truth-conditions of arbitrary sentences in L♦ can be specified using only

vocabulary in L.

To see what I have in mind, it is useful to begin with an example. Ask yourself: what

is required of the world in order for the truth-conditions of the following sentence to be

satisfied?

Modal

∃x(Mammal(x) ∧ ♦(Human(x)))

(Read: something is a mammal and might have been a human.)

It is clear that something or other is required of the world, since whether or not there

are any mammals who might have been human depends on what the world is like.

A perfectly accurate way of specifying truth-conditions for Modal is by stating that

what is required of the world in order of Modal’s truth-conditions to be satisfied is that

42 CHAPTER 2. POSSIBILITY

there be something that is a mammal and might have been a human. But suppose one is

aiming for more than mere accuracy. One wants one’s specification to take the following

form:

What is required of the world in order for Modal’s truth-conditions to be

satisfied is that it be such that p.

where ‘p’ is replaced by a sentence containing no modal operators. Then one might

reason as follows:

Being a non-human is constitutive property: if you’re a non-human, part of

what it is to be you is to be a non-human. So the requirement that the world

be such that there is a mammal that might have been a human boils down

to the requirement that the world be such that there is a mammal that is

also a human. But part of what it is to be a human is to be a mammal. So

the requirement that the world be such that there is a mammal that is also a

human boils down to the requirement that the world be such that there are

humans.

Accordingly, all that it is required in order for Modal’s truth-conditions to

be satisfied is for the world to be such that there are humans—and this gives

us what we want, since ‘there are humans’ contains no modal operators.

The proof of the Supervenience Theorem can be used to show that an analogous result

holds for arbitrary modal sentences. More specifically, for φ an arbitrary sentence of L♦,

the truth-conditions of φ are correctly specified by some clause of the form:

What is required of the world in order for φ’s truth-conditions to be satisfied

is that it be such that p.

2.1. SUPERVENIENCE 43

where ‘p’ is replaced by a (possibly infinite) sentence built out of the vocabulary of L.1

It is worth emphasizing that not every sentence of L♦ will turn out to have truth-

conditions as interesting as Modal. Consider, for example,

�(∀x(x is composed of water → x is composed of H2O)).

If you accept ‘To be composed of water just is to be composed of H2O’, you should believe

that nothing is required of the world in order for the truth-conditions of this sentence to

be satisfied. For consider an arbitrary scenario σ, and assume that, at σ, some object

x is composed of water. Since there is no difference between being composed of water

and being composed of H2O, x is thereby composed of H2O. There is no such thing as a

transition from x’s being composed of water to x’s being composed of H2O, and therefore

no requirement that σ would have to satisfy for the transition to be valid. (When God

made sure that x was made of water, there is nothing extra she had to do, or refrain

from doing, to get the result that x was composed of H2O.) So the truth-conditions of

‘∀x(x is composed of water → x is composed of H2O)’ are trivially satisfied at σ. But σ

was arbitrarily chosen. So the truth-conditions of ‘�(∀x(x is composed of water → x is

composed of H2O))’ are trivially satisfied.

One might be tempted to describe the Supervenience Theorem as a reduction of the

modal to the non-modal. But I think that would be a tendentious way of putting the

point. It ignores the fact that a predicate such as ‘is human’ might be thought to have

modal content. (You might think this, for example, if you think that part of what it is

to be human is to be essentially human.) The safe way of putting the point is by saying

that the truth-conditions of sentences in L♦ can be specified using only vocabulary in L.

1Here is a proof for those familiar with the material in section 2.3. Let a complete L-theory be amaximally consistent set of sentences of L. Let S be the set of true identity statements built fromvocabulary in L. Then a sentence guaranteed to do the job is the (possibly infinite) disjunction of(infinite) sentences, each of which consists of the (infinite) conjunction of sentences in a complete L-theory T such that that φ is true according to the Kripke-model based on the canonical space of worldsgenerated by S, and an ‘actual’ world in accordance with T .

44 CHAPTER 2. POSSIBILITY

2.2 Possible Worlds

Philosophers call on possible worlds to perform different kinds of jobs. One of these jobs

is foundational : the job of explaining what it takes for modal truths to be true. “For it

to be possible that p”, a possible-worlds-foundationalist would say, “just is for there to

be a possible world at which p.” Possible worlds are also used as semantic machinery.

The semanticist needs entities for the quantifiers of her metalanguage to range over, and

possible worlds—or, more generally, possibilia—can be used to construct them.

In this section I will argue that neither of these tasks requires a specialized modal

ontology. Some philosophers might be interested in this sort of claim because of ontolog-

ical scruples. Not me. I take up the issue here because it will help set things up for our

discussion of the Supervenience Theorem.

The foundational job

In the case of the foundational job—the job of explaining what it takes for modal truths

to be true—we have already seen why a specialized modal ontology is unnecessary. Recall

Modal. A possible-worlds-foundationalist would use possible worlds to explain what it

takes for Modal to be true: that there be a possible world at which some actual mammal

is a human. But we have seen that by making use of certain identity-statements—‘if

you’re human, part of what it is to be you is to be human’ and ‘part of what it is to be a

human is to be a mammal’—one can reach the conclusion that all it takes for Modal to

be true is for there to be a human. And the Supervenience Theorem shows that Modal

is not a special case. By using suitable identity statements, one can explain what it takes

for an arbitrary first-order modal sentence to be true without indulging in possible-worlds

talk.

There is a different way of making what is essentially same point. Kripke taught

us how to define a formal semantics for first-order modal languages. One starts with

2.2. POSSIBLE WORLDS 45

a space of mathematical points (often called ‘worlds’), and uses it to give a recursive

characterization of the notion of satisfaction-at-a-point. (p♦φq, for instance, is said to

be satisfied at point p just in case φ is satisfied at some point p′ accessible from p—all

of this relative to a variable assignment.) A sentence is then said to be true at a point p

just in case it is satisfied at p relative to every variable assignment, and the sentence is

said to be true simpliciter just in case it is true at a point labeled as ‘actual’.

Whether or not this semantics delivers the right assignment of truth-values to modal

sentences—the assignment that captures the fact that ‘♦’ is intended to express meta-

physical possibility—will depend on the space of points one chooses to work with, since

different spaces of points yield different results. How is the right space to be selected?

A possible-worlds-foundationalist might address the problem by appeal to metaphysics.

She might claim that the world is endowed with a specialized modal ontology—the pos-

sible worlds—and go on to suggest that the possible worlds should be used as points.

What the Supervenience Theorem shows is that there is an alternative: one can use the

set of true identity statements to fix a suitable space of points.2

The Semantic Job

Possible worlds are also used as semantic machinery. Kripke-semantics is the obvious

example, but the semantic applications of possible worlds extend far beyond that: the

most natural way of producing an intensional semantics—a semantics that assigns truth-

2Beware: Whether or not you think this succeeds in addressing the foundational problem—the prob-lem of explaining what it takes for modal truths to be true—will depend on your views on metaphysicalpossibility. As noted earlier, if you think of metaphysical possibility as the result of restricting a more in-clusive notion of possibility with the metaphysical laws, you will find no reason to think that a sentence’struth conditions can be modeled by the set of metaphysical possibilities at which the sentence is true.But suppose you think that the metaphysical possible scenarios are the de mundo intelligible scenarios,and that a sentence’s truth-conditions can be modeled by the set of de mundo intelligible scenarios atwhich the sentence is true. Then you will think that one can describe a sentence’s truth-condtions byspecifying the metaphysical possibilities at which a sentence is true. Accordingly, you will think thatwhen one uses the Kripkean semantic clauses to ascertain which points a modal sentence is true at, anduses the set of true identity statements to fix the relevant space of points, one succeeds in specifyingwhat it takes for the modal sentence to be true.

46 CHAPTER 2. POSSIBILITY

conditions, rather than simply truth-values, to sentences in the language—is by using

possible worlds (and, more generally, possibilia) in the range of one’s metalinguistic

quantifiers. (For instance, one might take a proposition to be a set of possible worlds,

or some complication thereof, and one might take the reference of a predicate to be a

function that assigns each world a set of objects in that world, or some complication

thereof. See Lewis (1970) and Heim and Kratzer (1998).)

Unfortunately, possibilia are philosophically controversial, and by using possibilia as

semantic machinery one runs the risk of dragging one’s semantic work into the contro-

versy. A modal actualist, for example, would say that there is no such thing as a merely

possible object. But when one uses possibilia as semantic machinery in a Kripke seman-

tics, merely possible objects figure in the domains of non-actual worlds; and when one

uses possibilia as semantic machinery in an intensional semantics, merely possible objects

figure in the intensions of predicates.

The good news is that the needs of the semanticist can be satisfied without using

possibilia. One can use representations of possibilia instead. The variety of represen-

tationalism that I prefer is model-theoretic: the role of possible worlds is played by

a-worlds—set-thereotic constructs similar to models in the logician’s sense—and the role

of possibilia is played by the entities in the domains of a-worlds. A thorough discus-

sion of a-worlds is supplied in chapter 7, but the main thing to know about a-worlds is

that you don’t need to worry about them. You can pretend that you are working with

the possible worlds they represent and rest assured that—as far as the semantic job is

concerned—you will always get the right results. (I mean this in a very precise sense: I

prove in section 7.5.2 that anything that can be expressed by quantifying over Lewisian

possibilia can also be expressed by quantifying over objects in the domains a-worlds.)

You can also rest assured that a-worlds won’t offend your philosophical sensitivities.

They are designed to be acceptable to a modal actualist (‘a-world’ is short for ‘actualist-

world’), and presuppose very little by way of ontology: they use a bit of set-theory, but

2.2. POSSIBLE WORLDS 47

do not assume a specialized modal ontology, or an ontology of properties, or anything

else that might be controversial. And they do not presuppose potentially controversial

expressive resources such as infinitary languages or non-standard modal operators. Just

about everyone should be happy to use a-worlds as semantic machinery.

You may, however, be one of those philosophers who are generally squeamish about

representationalism. If so, your squeamishness may be traceable to Lewis’s On the Plu-

rality of Worlds. It is surprisingly common for philosophers to have a distant but firm

recollection that Lewis showed representationalism to be untenable. “I can’t remember

the details” one often hears “but it’s all there in the chapter on ersatzism.”

This is a misreading of Lewis. The first thing to note is that Lewis himself was a

representationalist: Lewisian worlds represent possibilities, they are not the possibilities

themselves. Here is a passage from Plurality :

How does a world, [Lewisian] or ersatz, represent, concerning Humphrey, that

he exists?. . . A [Lewisian] world might do it by having Humphrey himself as

a part. That is how our own world represents, concerning Humphrey, that

he exists. But for other worlds to represent in the same way that Humphrey

exists, Humphrey would have to be a common part of many overlapping

worlds. . . I reject such overlap. . . There is a better way for a [Lewisian] world

to represent, concerning Humphrey, that he exists. . . it can have a Humphrey

of its own, a flesh-and-blood counterpart of our Humphrey, a man very much

like Humphrey in his origins, in his intrinsic character, or in his historical role.

By having such a part, a world represents de re, concerning Humphrey—

that is, the Humphrey of our world, whom we as his worldmates may call

simply Humphrey—that he exists and does thus-and-so. (p. 194; where I

write ‘[Lewisian]’ Lewis writes ‘genuine’.)

If you are still in any doubt that Lewis was a representationalist, note that there is

48 CHAPTER 2. POSSIBILITY

not a one-one correlation between Lewisian worlds and possibilities: the same Lewisian

world will correspond to different possibilities depending on one’s choice of counterpart

relation. When the mustached man in a Lewisian world is taken to be my counterpart, it

represents the possibility that I have a mustache; when it is taken to be my twin brother’s

counterpart, it represents the possibility that he have a mustache. (Here I am indebted

to Damien Rochford.)

Part of the reason it is easy to lose track of Lewis’s representationalism is that Lewis

also subscribed to a striking foundationalist claim: he believed that what it is for it to

be possible that p is for there to be a Lewisian world at which (i.e. representing that) p.

In light of this claim, it is natural to fudge the difference between Lewisian worlds and

possibilities. But there is a difference nonetheless.

The second thing to note is that Lewis was clear about the fact that for the purposes

of doing Kripke-semantics there is no need for possible worlds:

For that job [the job of giving a metalogical ‘semantical analysis of modal

logic’], we need no possible worlds. We need sets of entities which, for heuristic

guidance, ‘may be regarded as’ possible worlds, but which in truth may be

anything you please. We are doing mathematics, not metaphysics. (Plurality

p. 17).

When Lewis criticizes linguistic representationalism (or linguistic ersatzism, as he calls

it), his complaint is not that it wouldn’t be adequate for the job of doing semantics.

His worry is that it would be inadequate in two other respects. The first is to do with

the foundational job. He worried that ‘modality must be taken as primitive’ in deciding

which linguistic representations represent genuine possibilities and which ones do not.

We can agree with Lewis that linguistic representationalism does not, by itself, solve

the foundational problem: it does not, by itself, explain what the truth of modal truths

consists in. But that is not something we were hoping representationalism would do for

2.3. THE SUPERVENIENCE THEOREM 49

us in the present context. As noted in the preceding section, the foundational problem

is to be addressed by appeal to identity statements.

The second respect in which Lewis finds linguistic representationalism wanting is the

fact that it conflates possibilities that cannot be expressed by the language doing the

representing. This will be a problem for certain metaphysical applications of possibilia,

such as Lewis’s analysis of properties. But it need not be an obstacle when it comes to

doing semantics: “When I complain, as I shall, that there are various ways for different

possibilities to get conflated in their linguistic descriptions, that may be harmless when

we want to use ersatz possibilia to characterize the content of thought for a subject

who has no way to distinguish the conflated possibilities in his perception and conduct.

(Plurality p. 144, footnote.)” Similarly, as long as the language doing the representing is

able to express every distinction that can be expressed in the object language, potential

conflations won’t be a problem for the purposes of doing semantics.

2.3 The Supervenience Theorem

The Supervenience Theorem states that, by fixing the truth-value of every sentence in

a first-order language L and every identity statement built up from vocabulary in L,

one fixes the truth-value of every sentence in L♦. Until more has been said about the

principles that govern the ‘fixing’ of truth-values, however, the theorem is of limited

interest. It would be no fun, for example, to ‘fix’ the truth-values of modal statements

by assuming that p♦φq is generally equivalent to φ.)

What gives content to the statement of the theorem is the assumption that the fixing

of truth-values is to be governed by the following two principles:

Principle of Identity

Every sentence of L♦ corresponding to a true identity statements should turn

out to be true.

50 CHAPTER 2. POSSIBILITY

[In other words, p�(∀~x(φ(~x)↔ ψ(~x)))q should be true whenever pφ(~x) ≡~x ψ(~x)q is

true, and p�(∀z∀~w(φ(z, ~w)→ �(∃y(y = z)→ φ(z, ~w))))q should be true whenever

φ(z, ~w) is constitutive.]

Principle of Maximality (Informal Version)

There are no limits on possibility beyond the Principle of Identity.

I have stated the Principle of Maximality informally so as to make its intuitive content

clear. But, as we shall see, it is in need of further elucidation.

2.3.1 The Principle of Maximality

There is a procedure for making the Principle of Maximality precise that immediately

suggests itself. Start with a space of logically consistent ‘worlds’. (I find it useful to

think of a worlds as a-worlds, as explained in section 2.2, but you may think of worlds

differently if you like.) Go on to eliminate just enough of them to get the result that

the Principle of Identity is satisfied. (More precisely: take the maximal set of worlds

with the property that every sentence of L♦ corresponding to a true identity statement

is counted as true by a Kripke-semantics based on that set.) The worlds you’re left with

are the worlds that depict genuine possibilities.

When one restricts one’s attention to the special case of unconditional identity state-

ments, this procedure is well-defined. For given a space of worlds W and an arbitrary

set of unconditional identity statements S, there is a unique maximal subset of W that

satisfies the Principle of Identity. It is the result of eliminating from W all and only

worlds that fail to verify p∀~x(F (~x)↔ G(~x))q for some pF (~x) ≡~x G(~x)q in S.

Unfortunately, things get messier when conditional identities are brought into the

picture. To see this, consider a conditional identity statement to the effect that F-ness

is constitutive, and suppose that the corresponding sentence of L♦ (i.e. ‘�(∀z(F(x)) →

�(∃y(y = x) → F(x)))’) would be counted as false by a Kripke-semantics based on a

2.3. THE SUPERVENIENCE THEOREM 51

space of worlds W . Then W must contain a counterexample to the categoricity of F-

ness: there must be worlds w and w′ in W such that according to w there is something

that is an F, and according to w′ that very individual is not an F. The problem is that

there is more than one way to get rid of the counterexample. One could do so either by

eliminating w or by eliminating w′. So it is not immediately clear how one should go

about eliminating ‘just enough worlds to get the result that the Principle of Identity is

satisfied’.

Happily, there turns out to be a natural way of generating a ‘canonical’ space of worlds

from the set of true identity-statements. (See Appendix C for details.) The Principle of

Maximality can then be made precise as follows:

Principle of Maximality (Formal Version)

Let I be the set of true identity-statements built from vocabulary in L. A

sentence of L♦ is true just in case it is counted as true by the Kripke-semantics

based on the canonical space of worlds generated from I.

So defined, the Principle of Maximality entails the Supervenience Theorem as an imme-

diate corollary.

Does this formal version of the Principle of Maximality succeed in doing justice to

the informal version that I mentioned earlier? I think it does. But it is important to

keep in mind that the principle that possibility is to be limited only by identity is pretty

rough. (It is far from obvious, for example, that we have a clear sense of which modal

statements should be counted as expressing ‘possibilities’.) We will see in section 2.3.2

that the notion of canonicity I develop in the appendix delivers sensible results. But I

would be hesitant to declare that a better notion of canonicity couldn’t be found.

In any case, it would be a mistake to put too much weight on any particular way of

spelling out the notion of canonicity. It is useful to have an existence proof: it is useful to

know that there is at least one reasonable way of using the set of true sentences in L and

52 CHAPTER 2. POSSIBILITY

the set of true identity statements to fix the truth-values of sentences in L♦. And there

may well be value in the project of comparing the virtues of different such proposals.

But one shouldn’t be too bent on finding the one true notion of canonicity when the

discussion turns on issues that go beyond the region with respect to which the notion of

possibility is robustly understood.

2.3.2 The List of Modal Truths

Say we adopt the proposed formalization of the Principle of Maximality. Which modal

sentences turn out to be true?

1. The ‘actual’ world of the Kripke-semantics is chosen so as to ensure that every true

sentence of L turns out to be true.

2. The use of a Kripke-semantics guarantees that one gets a normal modal system,

and therefore that every sentence of L♦ that is theorem of classical logic will count

as true.

Also, the canonical space of worlds is based on a trivial accessibility relation. So

one every theorem of S5 will count as true.

3. One gets a sentence of L♦ corresponding to every true identity statement. In

other words, one gets p�(∀~x(φ(~x) ↔ ψ(~x)))q whenever pφ(~x) ≡~x ψ(~x)q is true,

and one gets p�(∀z∀~w(φ(z, ~w) → �(∃y(y = z) → φ(z, ~w))))q whenever φ(z, ~w) is

constitutive.

For instance, by assuming that ‘Elephant(x) �x Mammal(x)’ is true one gets

the result that ‘�(∀x(Elephant(x)→ Mammal(x)))’ is true, and by assuming that

‘Human(x)’ is constitutive one gets the result that (an L♦ rendering of) ‘necessarily,

humans are essentially human’ is true.

2.4. MODAL LANGUAGE AND MODAL FACT 53

As far as I can tell, any first-order modal sentence that constitutes a relatively

uncontroversial example of a metaphysical necessity can be recovered from suitable

identity statements in this sort of way. Fgures 2.1–2.3 list some examples.

4. Every possibility statement of the form

∃ ~x1(φ1( ~x1) ∧ ♦(∃ ~x2(φ2( ~x1, ~x2) ∧ ∃ ~x3(φ3( ~x1, ~x2, ~x3) ∧ . . .)))

will will turn out to be true, provided it satisfies a some of reasonable constraints.

(Specifically: (1) ‘φ1( ~x1)’ is satisfied by the actual world, (2) each of the ‘φi( ~x1, . . . , ~xi)

is consistent with the set of true identity-statements, and (3) there are no clashes

amongst the φi about the constitutive properties demanded of the referent of a

given variable, or about which pairs of variables must be correferential. Proofs and

details are supplied in Appendix C.)

Assuming reasonable identity-statements, this ensures that one gets L♦-renderings

of sentences like ‘I might have had a sister’ and ‘I might have had a sister who was

a cellist but might have been a philosopher’.

There is no official catalogue of recognized modal truths. But I hope this section lends

some plausibility to the claim that the formal version of the Principle of Maximality

delivers a list of modal truths that is in line with the standard literature on metaphysical

possibility.

2.4 Modal Language and Modal Fact

The Supervenience Theorem is a result about language. It shows that the set of identity

statements can be used to fix the truth-values of every sentence in L♦. But one might

worry that not everything there is to be said about possibility can be expressed in L♦, and

54 CHAPTER 2. POSSIBILITY

therefore that there is a version of the foundational problem that has been left untouched

by the present proposal.

My own view is that a non-language-based version of the proposal would be of limited

interest. I offer a detailed discussion in Appendix D. But the main idea is this. When one

focuses on the space of properties, and abstracts away from the predicates that might be

used to express these properties, possibility is a pretty boring concept: a possible assign-

ment of properties to objects is simply one that steers clear of incoherence. One might

think that there is an interesting project of ascertaining which property-assignments are

incoherent: whether, for example, it would be incoherent for an object to instantiate both

the property of being composed of water and the property of not being composed of H2O.

This could be construed as an interesting debate if it was thought of as a debate about

whether to accept the identity-statement ‘to be composed of water just is to be composed

of H2O’. For, as noted in section 1.6, one’s views about whether an identity-statement

has trivial or impossible truth-conditions might be tied up with a number of interest-

ing questions. But none of these questions has anything particular to do with identity:

all there is to be said about property-identity itself is that every property is identical

to itself and nothing else. When one abstracts away from the question of whether the

identity-statement should be accepted, and focuses on identity itself, there are nothing

interesting to be discussed.

However that may be, it is important to acknowledge a limitation of the language-

based approach I have set forth in this chapter. Namely: whether the right sentences

of L♦ get counted as true depends essentially on whether L has a suitably rich stock of

non-logical predicates. Say you think that there might have been an essentially lonely

object:

Lonely

♦(∃x ∧�(∃y(y = x)→ (∀y(y = x))))

2.4. MODAL LANGUAGE AND MODAL FACT 55

Since Lonely contains no non-logical predicates, it will be statable in L♦ regardless of

which non-logical predicates are in L. But—on reasonable assumptions—one won’t be

able to express the identity-statements necessary to ensure that Lonely gets counted as

true unless some non-logical predicate of L is available, and one is able to say something

along the following lines:

P (z)

z = x�x P (x)P (x)�x ∀y(y = x)

(Read: ‘being P is constitutive of its bearers’, and ‘part of what it is to be a

P is to be lonely’)

Similarly, one won’t be able to state the identity-statements necessary to make the fol-

lowing sentence of L♦ true:

Nemeses

♦(∃x♦(∃y�(x 6= x ∨ y 6= y)))

(Read: There might have been incompatible objects: objects each of which

might have existed but such that they couldn’t have existed together.)

unless L contains non-logical predicates, and one is able to say something along the

following lines:

A(z)

z = x�x A(x)

B(z)

z = x�x B(x)A(x)�x ¬∃yB(y)

(Read: ‘being A is constitutive of its bearers’, ‘being B is constitutive of its

bearers’ and ‘part of what it is to be an A is for there to be no Bs’)

56 CHAPTER 2. POSSIBILITY

2.5 Beyond First-Order Languages

An interesting project, which I do not develop here, is that of proving a version of the

Supervenience Theorem for languages with higher-order variables. Identity statements of

the form ‘φ(X) ≡X ψ(X)’, in which the identity predicate binds second-order variables,

are particularly interesting. For instance, the higher-order predicate ‘≡X ’ can be used to

capture the difference between Hume’s Principle,

∀F∀G(#x(F (x)) = #x(G(x))↔ F (x) ≈x G(x))

[Read: the number of the Fs equals the number of the Gs just in case the Fs

are in one-one correspondence with the Gs.]

and the identity-statement corresponding to Hume’s Principle,

#x(F (x)) = #x(G(x)) ≡F,G F (x) ≈x G(x)

[Read: for the number of the Fs to equal the number of the Gs just is for the

Fs to be in one-one correspondence with the Gs.]

(For more on Neo-Fregeanism see Wright (1983) and Hale and Wright (2001).)

The higher-order predicate ‘≡X ’ can also be used to define an analogue of Kit Fine’s

‘�FA’ (‘A is true in virtue of the nature of the objects which F ’). Namely: ‘∀z(Xz ↔

Fz)�X A[X/F ]’, where second-order quantification is cashed out in plural terms. (See

Fine (1995a) and Fine (2000); see also Fine (1994) and Fine (1995b).)

Unfortunately, the technique I use in Appendix C to generate a canonical space of

worlds won’t automatically carry over to a higher-order setting. So one would need

a somewhat different set of tools to prove a higher-order version of the Supervenience

Theorem.

2.5. BEYOND FIRST-ORDER LANGUAGES 57

In order to get the result that . . . isa true sentence of L♦

it is enough to count . . . as a trueidentity-statement.

Analyticity

�(∀x(V(x)→ F(x))) V(x)�x F(x)

(necessarily, every vixen is female) (part of what it is to be a vixen is to befemale)

Analyticity

�(∀x∀y(S(x, y)→ ∃z(P(z, x) ∧ P(z, y)))) S(x, y)�x,y ∃z(P(z, x) ∧ P(z, y))

(necessarily, sisters share a parent) (part of what it is for objects to be sistersis for them to share a parent)

Determinates and determinables

�(∀x(E(x)→ M(x))) E(x)�x M(x)

(necessarily, every elephant is a mammal) (part of what it is to be an elephant is tobe a mammal)

Cross-category prohibitions

�(∀x(E(x)→ ¬O(x))) E(x)�x ¬O(x)

(necessarily, every elephant is not an oc-topus)

(part of what it is to be an elephant is tonot be an octopus)

Supervenience

�(∀x(Φ(x)→ Ψ(x)) Φ(x)�x Ψ(x)

(necessarily, if you have physical propertyΦ, you have psychological property Ψ)

(part of what it is to have physical prop-erty Φ is to have psychological propertyΨ)

Supervenience

�(Φ→M) Φ�M

(necessarily, if physical fact Φ obtains,moral fact M obtains)

(part of what it is for physical fact Φ toobtain is for moral fact M to obtain)

Identity

�(∃y(y = h)→ h = p) h = p

(necessarily, if Hesperus exists, it is iden-tical to Phosphorus)

(Hesperus is Phosphorus)

Kind identity

�(∀x(Water(x)↔ H2O(x)) Water(x) ≡x H2O(x)

(necessarily, water is H2O) (what it is to be water is to be H2O)

Figure 2.1: Examples of Metaphysical Necessities

58 CHAPTER 2. POSSIBILITY

In order to get the result that . . . isa true sentence of L♦

it is enough to count . . . as a trueidentity-statement.

Essentiality of kind

�(∀z(H(z)→ �(∃y(y = z)→ H(z)))H(z)

x = z �x H(x)

(necessarily, if you’re human, you couldn’thave failed to be human)

(assume z is human; then part of what itis to be z is to be human)

Essentiality of kind

�(∀z(M(z)→ �(∃y(y = z)→ H(z))) M(x)�x H(x);H(z)

x = z �x H(x)

(necessarily, if you’re a man, you couldn’thave failed to be human)

(part of what it is to be a man is to be hu-man; moreover: assume z is human; thenpart of what it is to be z is to be human)

Essentiality of origin

�(∀z(B(c, z)→ �(∃y(y = z)→ B(c, z)))B(c, z)

x = z �x B(c, x)

(necessarily, if you have Charles as a bio-logical parent, you couldn’t have failed tohave Charles as a biological parent)

(assume z has Charles as a biological par-ent; then part of what it is to be z is tohave Charles as a biological parent)

Essentiality of origin

�(∀x∀y(B(w, z) → �(∃y(y = z) →B(w, z)))

B(w, z)x = z �x B(w, x)

(necessarily, if z has w as a biological par-ent, then z couldn’t have failed to have was biological parent)

(assume z has w as a biological parent;then part of what it is to be z is to havew as a biological parent)

Essentiality of constitution

�(∀z(W(z)→ �(¬I(z)))W(z)

x = z �x W(x); W(x)�x ¬I(x)

(necessarily, if you’re made of wood, youcouldn’t have been made of ice)

(assume z is made of wood; then part ofwhat it is to be z is to be made of wood;moreover, part of what it is to be made ofwood is to not be made of ice)

Essentiality of constitution

�(∀x∀y((C(z) ∧ C(w) ∧ P(w, z)) →�(∃y(y = z)→ P(w, z)))

C(z) ∧ C(w) ∧ P(w, z)x = z �x C(x) ∧ C(w) ∧ P(w, x)

(necessarily, if z and w are portions of clayand w is part of z, then z couldn’t havefailed to have w as a part)

(assume z and w are portions of clay andw is part of z; then part of what it is tobe z is to be made of clay, to have w as apart and for w to be made of clay)

Figure 2.2: Examples of Metaphysical Necessities (Continued)

2.5. BEYOND FIRST-ORDER LANGUAGES 59

In order to get the result that . . . isa true sentence of L♦

it is enough to count . . . as a trueidentity-statement.

Reflexivity

�(∀x(x < x)) x = x�x x < x

(necessarily, anything is part of itself) (part of what it is to be self-identical is tobe be a part of oneself)

Antisymmetry

�(∀x∀y((x < y ∧ y < x)→ x = y)) (x < y ∧ y < x)�x,y x = y

(necessarily, if x is part of y and y is partof x, then x is identical to y)

(part of what it is for x and y to be suchthat x is part of y and y is part of x is forx and y to be identical)

Transitivity

�(∀x∀y∀z((x < y ∧ y < z)→ x < z)) (x < y ∧ y < z)�x,y,z x < z

(necessarily, if x is part of y and y is partof z, then x is part of z)

(part of what it is for x, y and z to be suchthat x is part of y and y is part of z is forx to be part of z)

Strong Supplementation

�(∀x∀y(¬(y < x) → ∃z(z < y ∧¬O(z, x)))

¬(y < x)�x,y ∃z(z < y ∧ ¬O(z, x))

(necessarily, if y is not a part of x, then yhas a part that does not overlap with z)

(part of what it is for x and y to be suchthat y is not a part of x is for y to have apart that does not overlap with x)

Unrestricted Fusions

�(∃x(φ(x))→ ∃z∀x(O(x, z)↔ ∃y(φ(y)∧O(x, y))))

∃x(φ(x)) � ∃z∀x(O(x, z) ↔ ∃y(φ(y) ∧O(x, y))))

(necessarily, if there are any φs, then thereis a sum of the φs: a z such that the thingsthat overlap with z are precisely the thingsthat overlap with some φ)

(part of what it is for there to be a φ is forthere to be a sum of the φs: a z such thatthe things that overlap with z are preciselythe things that overlap with some φ)

Figure 2.3: Examples of Metaphysical Necessities (Mereological Principles)

60 CHAPTER 2. POSSIBILITY

Chapter 3

Metaphysics

3.1 Tractarianism

Suppose you introduce the verb ‘to tableize’ into your language, and accept ‘for it to

tableize just is for there to be a table’ (where the ‘it’ in ‘it tableizes’ is assumed to play

the same dummy role as the ‘it’ in ‘it is raining’). Then you will think that ‘it tableizes’

and ‘there is a table’ have the same truth-conditions. In each case, what is required in

order for the truth-conditions to be satisfied is that there be a table (equivalently: that

it tableize). So you will think that—for the purposes of stating that there is a table—

object-talk is optional. One can state that there is a table by employing a quantifier

that binds singular term positions—as in ‘there is a table’—but also by employing an

essentially different syntactic structure—as in ‘it tableizes’.

If object-talk is optional, what is the point of giving it a place in our language?

According to compositionalists the answer is “compositionality”. A language involving

object-talk—that is, a language including singular terms and quantifiers binding singular

term positions—is attractive because it enables one to give a recursive specification of

truth-conditions for a class of sentences rich in expressive power. But there is not much

more to be said on its behalf. If one could construct a language that never indulged

61

62 CHAPTER 3. METAPHYSICS

in object-talk, and was able to do so without sacrificing compositionality or expressive

power, there would be no immediate reason to think it inferior to our own. Whether or

not we choose to adopt it should turn entirely on matters of convenience. (For an example

of such a language, and illuminating discussion, see Burgess (2005); for an articulation

of compositionalism, see Lewis (1980).)

Compositionalists believe that it takes very little for a singular term to be in good

order. All it takes is a compositional specification of truth-conditions for whichever

sentences involving the term one wishes to make available for use. The reason there

is nothing more that needs to be done is that there was nothing special about using

singular terms to begin with. In setting forth a language, all we wanted was the ability

to express a suitably rich range of truth-conditions. If we happened to carry out this aim

by bringing in singular terms, it was because they supplied a convenient way of specifying

the right range of truth-conditions, not because they had some further virtue.

Tractarians, in contrast, believe that object-talk is subject to a further constraint:

there needs to be a certain kind of correspondence between the semantic structure of

our sentences and the ‘metaphysical structure of reality’. In particular, they presuppose

the following: (1) there is a particular carving of reality into objects which is more apt,

in some metaphysical sense, than any potential rival—the one and only carving that

is in accord with reality’s true metaphysical structure; (2) to each legitimate singular

term there must correspond an object carved out by this metaphysical structure; and

(3) satisfaction of the truth-conditions of an atomic sentence of the form pP (t1, . . . , tn)q

requires that the objects paired with t1, . . . , tn bear to each other the property expressed

by P . (For discussion of Tractarian conceptions of language, see Heil (2003).)

A consequence of Tractarianism is that one cannot accept an identity statement like

‘for Susan to instantiate the property of running just is for Susan to run’. The reason is

that ‘Susan instantiates the property of running’ and ‘Susan runs’ are atomic sentences

with different semantic structures. So there can’t be a single feature of reality they

3.1. TRACTARIANISM 63

are both accurate descriptions of when it is presupposed that correspondence between

semantic and metaphysical structure is a precondition for accuracy.

On natural assumptions about the connection between the truth-conditions of a com-

plex sentence and the satisfaction-conditions of its constituent subformulas, the Tractar-

ian is also barred from accepting identities such as ‘for it to tablezie just is for there to

be a table’, or ‘for some things to be arranged tablewise just is for there to be a table’,

or ‘for a marriage to take place just is for someone to get married’, or ‘for the number of

dinosaurs to be Zero just is for there to be no dinosaurs’. This means that Tractarians

take themselves to be in a position to make distinctions that a proponent of the com-

positional view would be unable to understand. They might take themselves to be in

a position to make sense of a scenario in which someone gets married but no marriages

take place, or a scenario in which there are some things arranged tablewise but no table.

I am ignoring a complication. I am assuming that the semantic structure of a sentence

can be read off more or less straightforwardly from the sentence’s surface grammatical

structure. This is a non-trivial assumption. Say you believe that proper logical analysis

of ‘there is a table’ reveals it to have the same semantic structure as ‘some things are

arranged tablewise’. Then you should think that the Tractarian could accept ‘for there

to be a table just is for some things to be arranged tablewise’ after all. The problem, of

course, is that the needed claims about semantic structure would be highly implausible.

Generally speaking, the project of uncovering exotic semantic structure by logical analysis

has fallen on hard times. There was certainly a period of initial enthusiasm, fueled by

Russell’s analysis of definite descriptions. But it seems to have petered out. Although

contemporary linguistics does suggest that there are certain cases in which there is a real

mismatch between surface structure and semantic structure, it is not to sort of mismatch

that would offer much comfort to the Tractarian. So the assumption that semantic

structure can be read off more or less straightforwardly from grammatical structure is a

harmless simplification in the present context.

64 CHAPTER 3. METAPHYSICS

The difference between compositionalism and Tractarianism is significant. The first

thing to note is that Tractarianism relies on the notion of metaphysical structure, and

compositionalism does not. To my mind, at least, the notion of metaphysical structure

is hopelessly obscure: it is a piece of Deep Metaphysics in the sense of chapter 6: the

sort of metaphysics that the Logical Empiricists reacted against in the first third of the

Twentieth Century.

In addition to being obscure, Tractarianism is bad philosophy of language. As its

name suggests, it is a close cousin of the ‘picture theory’ that Wittgenstein defended in

the Tractatus.1 And it ought to be rejected for just the reason Wittgenstein rejected the

picture theory in his later writings. Namely: if one looks at the way language is actually

used, one sees that usage is not beholden to the constraint that an atomic sentence can

only be true if its semantic structure is in suitable correspondence with the metaphysical

structure of the world.

It is simply not the case that ordinary speakers are interested in conveying informa-

tion about metaphysical structure. The sentences ‘a marriage took place’ and ‘someone

got married’, for example, are used more or less interchangeably in non-philosophical con-

texts. An ordinary speaker might choose to assert one rather than the other on the basis

stylistic considerations, or in order to achieve the right emphasis. But it would be ten-

dentious to suggest that her choice turns on her views about metaphysical structure. It

is not as if an ordinary speakers would only be prepared to assert ‘a marriage took place’

instead of ‘someone got married’ if she has a certain metaphysical view about events :

that they are amongst the entities carved out by the world’s metaphysical structure.

Think about how inappropriate it would be to respond to an assertion of ‘a marriage

took place’ in a non-philosophical context by saying “It am certainly prepared to grant

that someone got married, but I just don’t think that the world contains events amongst

1Here I have in mind a traditionalist interpretation of the Tractatus, as in Hacker (1986) andPears (1987). See, however, Goldfarb (1997).

3.1. TRACTARIANISM 65

its ultimate furniture.” One’s interlocutor would think that one has missed the point of

her assertion, and gone off to a different topic.

If ordinary assertions of ‘a marriage took place’ are not intended to limn the meta-

physical structure of the world, what could be the motivation for thinking that the

truth-conditions of the sentence asserted play this role? As far as I can tell, it is nothing

over and above the idea that semantic structure ought to correspond to metaphysical

structure. Remove this idea and there is no motivation left. To buy into Tractarianism

is to start out with a preconception of the way language ought to work, and impose it on

our linguistic theorizing from the outside—from beyond what is motivated by the project

of making sense of our linguistic practice.

There is a moderate form of Tractarianism according to which the constraint that

there be a correspondence between semantic structure and metaphysical structure ap-

plies only to assertions made by philosophers in the ‘ontology room’. When I use the term

‘Tractarianism’ in this book, the view I have in mind is always non-moderate Tractarian-

ism. My arguments for the claim that Tractarianism is bad philosophy of language do not

apply to moderate Tractarianism. For all I know, there is a special convention governing

discourse in the ‘ontology room’, which demands correspondence between semantic and

metaphysical structure.

Special conventions or not, I think one should stay away from moderate Tractarianism.

For it still trades on the questionable notion of metaphysical structure. Of course, the

fact that a notion fails to be understood by the uninitiated doesn’t entail that the notion

fails to make sense. If you are sympathetic towards moderate Tractarianism, that’s fine.

Nothing in this book presupposes that your view is mistaken. Just make sure you don’t

read covert claims about metaphysical structure into anything I say. I wouldn’t be caught

dead in the ontology room.

A moderate Tractarian is free to accept an identity-statement such as ‘for a marriage

to take place just is for someone to get married’. All she needs to do is insist that at most

66 CHAPTER 3. METAPHYSICS

one side of the identity-statement is taken in an ontology-room spirit. To avoid confusion,

moderate Tractarians might consider introducing a syntactic marker for ontology-room

discourse, as in Fine (2001). They could then say

What it really is for a marriage to take place is for someone to get married

or

What it is, in fundamental terms, for a marriage to take place is for someone

to get married

to indicate that the feature of reality described by ‘a marriage takes place’ gets carved by

the world’s metaphysical structure in a way that corresponds to the semantic structure

of ‘someone got married’. Just to be clear: this is not what I intend when I use identity

statements in this book.

Moral: If you want to indulge in talk of metaphysical structure, that’s up to you.

Just make sure you keep it away from your semantics.

3.2 Life without Tractarianism

Realism

I have given several examples of eminently sensible identity-statements that a full-blooded

Tractarian would be barred from accepting:

1. For Susan to instantiate the property of running just is for Susan to run.

2. For there to be a marriage just is for someone to get married.

3. For there to be a table just is for there to be some things arranged tablewise.

4. For the number of the dinosaurs to be Zero just is for there to be no dinosaurs.

3.2. LIFE WITHOUT TRACTARIANISM 67

Let an anti-Tractarian be a compositionalist who believes that identity statements such

as these are true. A Tractarian might be tempted to complain that if anti-Tractarianism

were correct, there would fail to be a definite fact of the matter about how the world is.

I have often heard arguments such as the following:

Say you believe that for the number of the dinosaurs to be Zero just is for

there to be no dinosaurs. You believe, in other words, that a single fact can

be described fully and accurately by asserting ‘the number of the dinosaurs

is Zero’ and by asserting ‘there are no dinosaurs’. This presupposes that a

single fact can get carved up into objects and properties in different ways.

When the fact is described by asserting ‘the number of the dinosaurs is Zero’,

it gets carved up into an individual (the number Zero), a first-order property

(the property of being a dinosaur), and a second-order function (the function

taking first-order properties to their numbers); when it is described as ‘there

are no numbers’, it gets carved out into a first-order property (the property

of being a dinosaur) and a second-order property (non-existence).

But if this is so, there can’t be an objective, language-independent fact of the

matter about whether there are numbers. It all depends on how we choose

to describe the world.

I am happy to grant everything in the first paragraph of this argument—at least on

the assumption that the metaphor of fact-carving is spelled out properly (see below),

and that one takes a suitably deflationary attitude towards property- and fact-talk. The

argument’s second paragraph, on the other hand, strikes me as deeply misguided.

The anti-Tractarian is certainly committed to the view that a single feature of reality

can be fully and accurately described in different ways. But this does not entail that

there is no fact of the matter about how the world is. On the contrary: it is strictly

and literally true that the number of the dinosaurs is Zero, and therefore that there

68 CHAPTER 3. METAPHYSICS

are numbers. And this is so independently of which sentences are used to describe the

world—or, indeed, of whether there is anyone around to describe it. The point is simply

that the relevant feature of the world could also be fully and accurately described in

another way: by asserting ‘there are no dinosaurs’.

Moral: Don’t confuse realism with Tractarianism. (More specifically: if realism is the

view that there is a definite, subject-independent fact of the matter about how the world

is, then anti-Tractarianism is no less of a realist position than Tractarianism.)

The world as a structureless blob

“Wait a minute!”—you might be tempted to reply—“Isn’t the anti-Tractarian still com-

mitted to the view that the world is a structureless blob?”

Absolutely not. The anti-Tractarian believes that it is strictly and literally true that

there are tables, that a marriage took place, that the number of the dinosaurs is Zero, and

so forth. So if the strict and literal existence of tables, marriages and numbers is enough

for the world not to be a structureless blob, then it is no part of anti-Tractarianism that

the world is a structureless blob.

Of course, there is always Deep Metaphysics. Perhaps what you mean when you say

that the world might be a structureless blob is that the world might fail to be endowed

with metaphysical structure. In that case, you should think that anti-Tractarianism is

neutral with respect to the question of whether the world is a structureless blob. Anti-

Tractarianism does not entail that there is no such thing as metaphysical structure: that

would be to treat the notion of metaphysical structure as intelligible, and use it to take

a stand on matters of Deep Metaphysics. Here we have done no such thing.

A brief aside: I don’t really understand what Putnam has in mind when he talks

about Internal Realism. But perhaps one could interpret some of what he says as an

endorsement of an anti-Tractarian form of realism. (See, for instance, Putnam (1987)

pp. 18–19.)

3.2. LIFE WITHOUT TRACTARIANISM 69

Comprehensivism

Comprehensivism is that view that it is in principle possible to give a comprehensive

description of the world—a description such that: (1) there is precisely one way for the

world to be that would satisfy the description, and (2) the world, as it actually is, satisfies

the description.

A critic might be tempted to think that anti-Tractarianism is incompatible with com-

prehensivism. “According to anti-Tractarianism”—the critic might argue—“the same

fact can be described in many different ways. One can say that there is a table, or that

some things are arranged tablewise, or that the world tabelizes, or that tablehood is

instantiated, or that two half-tables are put together in the right sort of way, and so

forth, with no natural end. But one hasn’t given a exhaustive description of the world

until one has described it in all these ways. So the anti-Tractarian could never give a

comprehensive description of the world.”

To see where the critic goes wrong, it is useful to consider an example. Suppose I

ask you to go to the room next door, and bring me a comprehensive description of its

contents. You come back and say: “There is a table, and nothing else.” It would be

inappropriate for me to respond by complaining that your answer is incomplete on the

grounds that failed to mention at least two objects: the right-half of the table and left-

half of the table. Such a response would be guilty of double-counting. Part of what it is

for there to be a table in the room (a normal table, at least) is for there to be half-tables,

put together in the right sort of way. So when you mentioned that there was a table,

the presence of the half-tables was already included in the information you gave me. It is

true that you never mentioned half-tables explicitly. But that was not required for your

description to be comprehensive. All that comprehensiveness requires is that there be

precisely one way for the room to be that would satisfy your description.

In this case your description of the room was not fully comprehensive. It didn’t specify

70 CHAPTER 3. METAPHYSICS

where in the room the table is located, or what it is made of, or how its constituent atoms

are configured. But these omissions have nothing to do with anti-Tractarianism. As far

as anti-Tractarianism is concerned, it might well be possible to specify all the missing

details.

Moral: Anti-Tractarianism does not entail that comprehensivism is false. What it en-

tails is that there could be more than one way of giving a fully comprehensive description

of the world.

Absolute Generality

Is anti-Tractarianism compatible with the view that there is an all-inclusive domain?

This question can be understood in different ways, depending on how one reads the

claim that there is an all-inclusive domain:

• First Reading [Realism + Comprehensivism]

There is a definite fact of the matter about how the world is, and it is in principle

possible to give a fully comprehensive description of its contents.

• Second Reading [Metaphysical Absolutism]

There is a ‘fundamental domain’—a domain consisting of the entities that are

carved out by the world’s metaphysical structure.

• Third Reading [Recarving-Absolutism]

There is a ‘maxi-domain’—a domain consisting of the entities that result from every

possible way of carving of the world into objects.

We have seen that anti-Tractarianism is compatible with both Realism and Compre-

nensivism. So, on the first of the three readings, there is no tension between anti-

Tractarianism and the existence of an all-inclusive domain.

What about the second reading? Anti-Tractarianism is neutral with respect to the

existence of a ‘fundamental domain’. To address the issue of a fundamental domain

3.2. LIFE WITHOUT TRACTARIANISM 71

would be to treat the notion of metaphysical structure as intelligible, and use it to take

a stand on matters of Deep Metaphysics. Anti-Tractarianism does no such thing.

Let us therefore turn our attention to the third reading. The anti-Tractarian believes

that there are tables. So a ‘maxi-domain’ would have to include tables. But according

to anti-Tractarians, the fact that there are tables could also be described by saying

that there are half-tables put together in the right sort of way, or that the property of

tablehood is instantiated, or that some mereological simples are arranged tablewise, or

that the set of tables is non-empty, or that the number of tables is greater than Zero.

So the maxi-domain would also have to include half-tables and instantiated properties of

tablehood and mereological simples arranged tablewise and non-empty sets and numbers

greater than Zero and Zero itself, and so forth.

Could such a list ever be completed? It seems to me that anti-Tractarians should be

skeptical about the claim that it could. It is not that an anti-Tractarian should think

that the world is somehow incomplete. It is no part of the view that the world is open-

ended—at least in any sense of ‘open-endedness’ that is incompatible with there being a

definite and final fact of the matter about how the world is. The problem is that there

is no reason to think that our concept of ‘carving the world into objects’ is determinate

enough to allow for a final answer to the question of how it might be possible to carve

the world into objects.

As I understand it, a ‘carving’ of the world is nothing more than a compositional

system of representation for describing the world. In the most familiar case, a carving

is a compositional system of linguistic representation: a language in which the truth-

conditions of sentences are generated recursively from the semantic values of a restricted

set of basic lexical items. To say that a subject carves the world into objects is simply

to say that she represents the world using a language that contains singular terms, or

variables that take singular term positions. Similarly, to say that a subject carves the

world into properties is simply to say that she represents the world using a language

72 CHAPTER 3. METAPHYSICS

that contains predicates, or variables that take predicate positions. (A carving can also

be a compositional system of representation in thought—a system of mental representa-

tions with recursively generated truth-conditions. But I shall focus my attention to the

linguistic case here.)

Carving up the world is not like cutting cake. For the purposes of spelling out the

carving-metaphor, one is not to think of the world as a big object—the mereological

fusion of everything there is—and of a carving as a way of subdividing the world into

smaller parts. The world, for these purposes, is to be thought of as ‘the totality of facts,

not of things’, and a carving is to be thought of as a compositional system for describing

these facts.

When the carving-metaphor is spelled out in this way, the existence of a maxi-domain

would require a final answer to the question of what counts as a possible system of

compositional representation. And I see no prima facie reason to think that our notion of

representation (and our notion of linguistic representation, in particular) are constrained

enough for this question to have a definite answer. From the perspective of Tractarianism,

the range of admissible compositional languages is restricted by Deep Metaphysics, since

only languages whose semantic structure is in correspondence with the metaphysical

structure of the world are potential vehicles for truth. From the perspective of anti-

Tractarianism, on the other hand, the only constraint on semantic structure is that it

deliver an assignment of truth-conditions to sentences from the semantic values of basic

lexical items. So it is hard to say in advance what would count as a possible compositional

language. Whenever we dream up a new mechanism for representing reality, the potential

for a new compositional language—and hence for a new way of carving up the world—will

be in place.

You may be worried that my way of cashing out the carving-metaphor is too light-

weight. “If the only relevant difference between asserting ‘there are tables’ and asserting

‘some things are arranged tablewise’ is to do with the system of compositional represen-

3.2. LIFE WITHOUT TRACTARIANISM 73

tation one chooses to employ”—you might be tempted to complain—“then someone who

asserts ‘there are tables’ hasn’t really committed herself to the existence of tables. For

what she says could be true even if there are really no tables.” As far as I’m concerned,

all it takes for there to really [table thump!] be tables is for an English sentence like

‘there are tables’ to be strictly and literally true. And all it takes for ‘there are tables’

to be strictly and literally true is that there be some things arranged tablewise (equiva-

lently: that the property of being a table be instantiated; equivalently: that there be two

half-tables put together in the right sort of way; equivalently: that there be tables). But

presumably you mean something different by ‘really’. Perhaps what you have in mind is

that in order for something to really exist, it must figure in a ‘fundamental’ description of

the world. It must, in other words, be carved out by the world’s metaphysical structure.

Real existence, in this sense, is a creature of Deep Metaphysics. It is a notion I am unable

to understand, and therefore not a notion we will be concerned with here.

“Wait a minute!”—you might be tempted to reply—“Are you setting forth a view

according to which the existence of objects is somehow constituted by language?” Ab-

solutely not. What is ‘constituted by language’ is the use of singular terms. If we had

no singular terms (or variables taking singular term positions) we wouldn’t be able to

describe the world in a way that made the existence of objects explicit. But there would

be objects just the same. Speakers of a language with no singular terms can say things

like ‘Lo, tableization here!’. But for it to tableize just is for there to be a table. So even

without singular terms, they would be in a position to convey information about tables.

For the anti-Tractarian, the existence of tables depends entirely on how the non-

linguistic world is. If there are things arranged tablewise (equivalently: if it tableizes;

equivalently: if there are tables), then there are indeed tables. If no things are arranged

tablewise (equivalently: if it fails to tableize; equivalently, if there are no tables), then it

is not the case that there are tables. The Tractarian’s mistake is to conflate form and

content. Tractarians think there is a difference in content (i.e. truth-conditions) between

74 CHAPTER 3. METAPHYSICS

‘there are tables’ and ‘some things are arranged tablewise’, when in fact there is only a

difference in form (i.e. semantic structure).

3.3 Ontological Commitment

For a sentence to carry commitment to Fs is for satisfaction of the sentence’s truth-

conditions to require the existence of Fs. The sentence ‘Susan runs’, for example, carries

commitment to runners, since part of what would be required for its truth-conditions to

be satisfied is that there be runners.

A sentence’s ontological commitments are an aspect of its truth-conditions: to describe

a sentence’s ontological commitments is to describe those aspects of its truth-conditions

that pertain to ontology. There are other aspects of a sentence’s truth-conditions that

one could in principle be interested in. One could, for example, be concerned with

the ‘megethological’ commitments carried by a sentence, and ask about requirements

imposed by the sentence’s truth-conditions that pertain to the size of the world. The

sentence ‘some people run and some don’t’, for example, carries ontological commitments

to runners and non-runners, and megethological commitments to at least two objects.

Some philosophers might find a sentence’s ontological commitments more interesting

than other aspects of its truth-conditions. But it is important to keep in mind that the

project of ascertaining a sentence’s ontological commitments is not separate from the

project of ascertaining the sentence’s truth-conditions.

3.3.1 Quine’s Criterion

Quine famously suggested a criterion of ontological commitment for first-order sentences.2

2See Quine (1948) p. 32, Quine (1951a) p. 67, Quine (1951b) p. 11 and Quine (1953b) p. 103.For discussion, see Cartwright (1954), Alston (1957), Jackson (1980), Parsons (1982), Routley (1982),Hodes (1990), Lewis (1990), Melia (1995), Azzouni (1998), Yablo (1998) and Priest (2005).

The formulation of Quine’s Criterion I use here is to be thought of as a schema whose instances arethere result of substituting a count-noun for ‘F’. By ‘the variables’, I mean the variables of the language

3.3. ONTOLOGICAL COMMITMENT 75

Quine’s Criterion

A first-order sentence carries commitment to Fs just in case Fs must be

counted amongst the values of the variables in order for the sentence to be

true.

This criterion should not be thought of as a competitor to the claim that ontological com-

mitment is an aspect of truth-conditions. It should be thought of as playing a different

role. Whereas the former is meant to supply an elucidation of what ontological commit-

ment consists in, Quine’s Criterion embodies a substantial claim about the ontological

commitments of first-order sentences.

Consider a disagreement about the ontological commitments of ‘∃x Elephant(x)’.

On one view, ‘∃x Elephant(x)’ is committed to elephants and nothing else. On a

rival view, ‘∃x Elephant(x)’ is committed also to the property of elephanthood. The

claim that ontological commitment is an aspect of truth-conditions won’t decide the

issue. All it tells us is that the matter depends entirely on the truth-conditions of

‘∃x Elephant(x)’. But wheel in Quine’s Criterion and the matter is decided in favor of

the first view. For properties needn’t be counted amongst the values of the variables in

order for ‘∃x Elephant(x)’ to be true. So ‘∃x Elephant(x)’ carries no commitment

to properties.

Quine’s Criterion makes substantial claims about ontological commitment by estab-

lishing a correlation between the ontological commitments of a first-order sentence and

the semantic machinery that must be deployed by a semantic theory if it is to count

the sentence as true. It is important to be clear that such a correlation is in no way

constitutive of the notion of ontological commitment. One should generally distinguish

between the ontological commitments carried by a sentence and the semantic machinery

employed by a semantic theory assigning truth-conditions to that sentence. Note, for

rather than the variables of the sentence, so as to ensure that, e.g. ‘Runs(charles)’ carries commitmentto runners (and to Charles), even though it contains no variables.

76 CHAPTER 3. METAPHYSICS

example, that on standard semantic theories one assigns to each first-order predicate of

the language a set as its semantic value. From this it follows that one’s semantic theory

for ‘∃x Elephant(x)’ carries commitment to sets. But it would be a mistake to conclude

on those grounds alone that ‘∃x Elephant(x)’ itself carries commitment to sets. Just

because a semantic theory uses sets in specifying truth conditions for ‘∃x Elephant(x)’,

it doesn’t follow that the truth-conditions thereby specified demand of the world that it

contain sets. Similarly, just because a semantic theory uses elephants in specifying truth-

conditions for ‘∃x Elephant(x)’ it doesn’t immediately follow that the truth-conditions

thereby specified demand of the world that it contain elephants.

Insofar as one agrees with Quine’s Criterion, one will think that it is nonetheless

a feature of first-order languages that there is an exact correspondence between the

ontological commitments carried by a sentence and the objects that must be counted

amongst the values of the variables in order for the sentence to be true. But this in itself is

not a reason for thinking that such a feature will generalize beyond first-order languages.

On the standard (Kripkean) semantics for modal languages, for example, possibilia (or

objects representing possibilia) must be counted amongst the values of the variables in

order for ‘♦(∃x(Elephant(x) ∧ Purple(x)))’ to be true. But it would be a mistake

to conclude on those grounds alone that ‘♦(∃x(Elephant(x) ∧ Purple(x)))’ itself is

committed to possibilia (or objects representing possibilia). To insist: just because a

semantic theory uses possibilia (or objects representing possibilia) in specifying truth

conditions for ‘♦(∃x(Elephant(x) ∧ Purple(x)))’, it doesn’t follow that the truth-

conditions thereby specified demand of the world that it contain possibilia (or objects

representing possibilia). In the absence of further argumentation, all one gets is the

conclusion that one’s semantic theory is committed to possibilia (or objects representing

possibilia).

3.3. ONTOLOGICAL COMMITMENT 77

3.3.2 Is Quine’s Criterion Adequate?

Quine’s Criterion can be expected to undergenerate when the language contains atomic

predicates expressing non-intrinsic properties. Here is an example. Part of what it

is to be a daughter is to have a parent. So satisfaction of the truth-conditions of

‘∃x(Daughter(x))’ requires that there be parents. But parents needn’t be counted

amongst the values of the variables in order for ‘∃x(Daughter(x))’ to be true. So

Quine’s Criterion delivers the mistaken result that ‘∃x(Daughter(x))’ carries no com-

mitment to parents. (As Gabriel Uzquiano pointed out to me, not all atomic predicates

expressing non-intrinsic properties lead to trouble: ‘Lonely(. . . )’ is an example of one

that does not.)

One could try to avoid the problem of non-intrinsicness by limiting the application of

Quine’s Criterion to cases in which the offending predicates are avoided. But so many of

our predicates express non-fully-intrinsic properties that one would run the risk of ending

up with a criterion with a very limited range of application. (Part of what it is to be a

human is to belong to a certain lineage; part of what it is to be a moon is to orbit around

a planet; part of what it is to be a table is to be used, or designed, as a table.) One

could, of course, embark in a project of philosophical analysis, and attempt to supply

paraphrases for one’s non-intrinsic predicates in terms of intrinsic predicates. But it is

hard to feel optimistic about the prospects of such an enterprise.

Tractarians and anti-Tractarians should agree about everything so far. But their

ultimate assessments of Quine’s Criterion will be very different. A Tractarian might think

that the criterion only undergenerates in the presence of non-intrinsic atomic predicates.

She might think that when such predicates are set aside, the criterion is a sensible one—

and certainly more sensible than a criterion whereby ‘Susan runs’ carries commitment

to sets, or to properties. So even if Quine’s Criterion isn’t everything one might have

hoped for, it is adequate as a rough guide to the ontological commitments of first-order

78 CHAPTER 3. METAPHYSICS

sentences.

Anti-Tractarians will disagree. Suppose one thinks that for there to be a table just

is for there to be two half-tables put together in the right sort of way. Then one should

think that ‘∃x(Table(x))’ carries commitment to half-tables. To see this, recall that for

a sentence to carry commitment to Fs is for satisfaction of its truth-conditions to require

the existence of Fs. What is required in order for the truth-conditions of ‘∃x(Table(x))’

to be satisfied is that there be a table. But part of what it is for there to be a table is

for there to be half-tables. So part of what is required in order for the truth-conditions

of ‘∃x(Table(x))’ to be satisfied is that there be half-tables. So ‘∃x(Table(x))’ carries

commitment to half-tables.

For similar reasons, if one thinks that for the set of tables to be non-empty just is for

there to be a table, or that for the property of tablehood to be instantiated just is for

there to be a table, then one should think that ‘∃x(Table(x))’ carries commitment to

non-empty sets, or to instantiated properties.

The anti-Tractarian can therefore be expected to think that Quine’s Criterion un-

dergenerates very badly indeed. The criterion goes wrong by assuming too much of a

connection between ontological commitment and semantic structure. If Tractarianism

were correct—and if every sentence of the language could be rewritten as a sentence in

which every atomic predicate is fully intrinsic—then one might hope for such a connec-

tion. For one might hope to be able to read off from a sentence’s semantic structure

which of the objects carved out by the world’s metaphysical structure would have to

exist in order for the sentence’s truth-conditions to be satisfied. But, as we have seen,

Tractarianism is bad philosophy of language: it is the result of doing one’s linguistic

theorizing from the metaphysician’s armchair.

Anti-tractarians should think that Quine’s Criterion is not a very good guide to

ontological commitment. But they should also think that the notion of ontological com-

mitment is of more limited interest than is usually supposed. For they will think that

3.3. ONTOLOGICAL COMMITMENT 79

discussion of ontological commitment is often based on a mistaken assumption about

the connection between ontological commitment and truth-conditions: that a sentence’s

ontological commitments generally make the sentence’s truth-conditions more difficult to

satisfy.

It is true that some ontological commitments make a sentence’s truth-conditions

more difficult to satisfy. A sentence that carries commitment to tables, for example,

would fail to be satisfied in a world without tables (equivalently: in a world in which

it doesn’t tableize; equivalently: in a world in which the property of tablehood fails to

be instantiated). So its truth-conditions are more difficult to satisfy than they would

be if the commitment to tables was removed—they impose a stronger demand on the

world. The problem is that not all ontological commitments are burdensome in this way.

Suppose, for example, that for the number of Fs to be Zero just is for there to be no

numbers. Then a sentence like ‘the number of non-self-identical things is Zero’ can both

have trivial truth-conditions—truth-conditions whose satisfaction requires nothing of the

world—and carry commitment to numbers. This is because commitment to numbers is

no commitment at all. It is trivially the case that there are no non-self-identical things.

But we are assuming that for the number of non-self-identical things to be Zero just is

for there to be no self-identical things. So it is trivially the case that the number of non-

self-identical things is Zero. So it is trivially the case that Zero exists. So it is trivially

the case that there are numbers.

I don’t mean to suggest that the notion of ontological commitment is somehow il-

legitimate. A sentence’s ontological commitments are, after all, an aspect of its truth-

conditions. The point is just that ontological commitment is potentially misleading as a

method for gauging the demandingness of a sentence’s truth-conditions.

Moral: If you are a Tractarian you might think a sentence’s semantic structure is a

good guide to its ontological commitments—at least to commitments concerning objects

carved out by the world’s metaphysical structure. But anti-Tractarians have no reason to

80 CHAPTER 3. METAPHYSICS

think that a sentence’s ontological commitments can be read off from its semantic struc-

ture. For ontological commitment is an aspect of truth-conditions, and anti-Tractarians

believe that the same truth-conditions can be expressed by sentences with very different

semantic structures.

Chapter 4

Mathematics

4.1 Trivialism

Mathematical Platonism is the view that mathematical objects exist. Mathematical Triv-

ialism is the view that the truths of pure mathematics have trivial truth-conditions, and

the falsities of pure mathematics have trivial falsity-conditions.

Platonism is compatible with both trivialism and non-trivialism. The easiest way of

getting a handle on non-trivialist Platonism is by imagining a creation myth. On the

first day God created light; by the sixth day, She had created a large and complex world,

including black holes, planets and sea-slugs. But there was still work to be done. On the

seventh day She created mathematical objects. Only then did She rest.

On this view, it is easy to make sense of a world with no mathematical objects: it

is just like the world we are considering, except that God rested on the seventh day. In

the world we are considering, God had to do something extra in order to bring about

the existence of mathematical objects—something that wasn’t already in place when she

created black holes, planets and sea-slugs. The existence of numbers is, in this sense, a

non-trivial affair.

According to the version of Trivialist Platonism we will be considering here, in con-

81

82 CHAPTER 4. MATHEMATICS

trast, every instance of the following schema is true:

For the number of the Fs to be n just is for there to be n Fs.

[In symbols: #x(F(x)) = n ≡ ∃!nx(F(x)).]

So when God created the planets, and made sure that there were exactly eight of them,

She thereby made it the case that the number of the planets was Eight. There was

nothing extra that needed to be done to bring about the existence of numbers.

Proponents of trivialist Platonism will think that the existence of numbers is a trivial

affair. One way to see this is by rehearsing an argument from section 3.3.2. Since it is

a truth of logic that everything is self-identical, it is trivially the case that there are no

non-self-identical things. But we are assuming that for the number of non-self-identical

things to be Zero just is for there to be no self-identical things. So it is trivially the case

that the number of non-self-identical things is Zero. So it is trivially the case that Zero

exists. So it is trivially the case that there are numbers.

Relatedly, it is a consequence of trivialist Platonism that there is no sense to be made

of a world without numbers. Suppose, for reductio, that there are no numbers. We

know that for the number of numbers to be Zero just is for there to be no numbers.

So the number Zero must exist after all, contradicting our assumption. It is de mundo

unintelligible that numbers fail to exist.

Trivialist Platonism is, of course, a form of anti-Tractarianism. It entails that a single

feature of reality—the fact that there are no dinosaurs, for example—can be fully and

accurately described by using mathematical vocabulary (‘the number of the dinosaurs is

Zero’), and also fully and accurately described without using mathematical vocabulary

(‘there are no dinosaurs’).

4.2. BENACERRAF’S DILEMMA 83

4.2 Benacerraf’s Dilemma

A sizable portion of the debate in contemporary philosophy of mathematics has been

shaped by Paul Benacerraf’s ‘Mathematical Truth’. According to Benacerraf, we face an

unhappy choice. We must either give a non-standard semantics for mathematical dis-

course, according to which mathematical statements are not really committed to mathe-

matical objects, or stick to a straightforward semantics and explain how we could come

to have knowledge about the realm of abstract objects, which is causally inert.

As directed towards a non-trivialist, Benacerraf’s Dilemma has force (at least on a

suitably cleaned up version, such as Field (2005)). For if the existence of numbers is

non-trivial—if God would have had to do something extra to make sure they were in

place—then there is room for asking how one could ever check whether the world does

indeed contain numbers—whether God did indeed work on the seventh day. So unless

semantic analysis were somehow to reveal that mathematical sentences are not really

committed to numbers, it is not entirely clear what an epistemology for mathematics

would look like.

As directed towards a trivialist, on the other hand, the dilemma has little force.

For, according to the trivialist, nothing is required of the world in order for the truth-

conditions of a truth of pure mathematics to be satisfied. There is no intelligible possi-

bility that the world would need to steer clear of in order to cooperate with the demands

of mathematical truth. This means, in particular, that there is no need to go to the

world to check whether any requirements have been met in order to determine whether

the truth-conditions of a truth of pure mathematics are satisfied. Once one gets clear

about the sentence’s truth conditions—clear enough to know that they are trivial—one

has done all that needs to be done to establish the sentence’s truth. (It is important

to keep in mind that getting clear about the truth-conditions of a given mathematical

sentence can be highly non-trivial. So determining whether the sentence is true is not,

84 CHAPTER 4. MATHEMATICS

in general, a trivial affair.)

I suspect that the prevalence of Benacerraf’s Dilemma in the literature is partly

to do with a misunderstanding. Philosophers of mathematics tend to be divided into

two groups. Members of the first group are noncommittalists (i.e. they think that a

typical mathematical statement carries no commitment to mathematical objects) and

also nominalists (i.e. they think that there are no mathematical objects); members of the

second group are committalists and Platonists. The problem, I suspect, is that members

of the two groups are sometimes talking about different positions when they use the

term ‘Platonism’. It is agreed on all sides that Mathematical Platonism is the view

that there are mathematical objects. But members of the first group tend to assume

that the only available form of Platonism is non-trivialist Platonism, and therefore see

Benaccerraf’s Dilemma as strong evidence for nominalism. (Since nominalism without

noncommittalism entails the falsity of standard mathematical axioms, they also see the

dilemma as evidence for noncommittalism.) Members of the second group, in contrast,

tend to have some form of trivialist Platonism in mind, and therefore fail to feel the force

Benacerraf’s Dilemma. As far as I can tell, few actual Platonists would endorse the form

of Platonism that is targeted by the dilemma.

It is hard to mention particular people without failing to do justice to the subteties of

their specific views. But it seems to me that Frege (1884), Parsons (1983), Wright (1983)

and Stalnaker (1996) can all be interpreted as defending versions of trivialist Platonism.

4.3 Neo-Fregeanism

Hume’s Principle is the following sentence:

∀F∀G(#x(F (x)) = #x(G(x))↔ F (x) ≈x G(x))

[Read: the number of the Fs equals the number of the Gs just in case the Fs

are in one-one correspondence with the Gs.]

4.3. NEO-FREGEANISM 85

Neo-Fregeanism is the view that when Hume’s Principle is set forth as an implicit defini-

tion of ‘#x(F (x))’, one gets the following two results: (1) the truth of Hume’s Principle

is knowable a priori, and (2) the referents of number-terms constitute a realm of mind-

independent objects. (Neo-Fregeanism was first proposed in Wright (1983), and has since

been championed by Bob Hale, Crispin Wright and others. For a collection of relevant

essays, see Hale and Wright (2001).)

Just like one can distinguish between two different varieties of mathematical Platon-

ism, one can distinguish between two different varieties of neo-Fregeanism: trivialist and

non-trivialist. The two positions agree that numbers—the referents of numerical-terms—

constitute a realm of mind-independent objects. But they disagree about whether the

existence of this realm of objects is a trivial affair.

Trivialist neo-Fregeans go beyond mere acceptance of Hume’s Principle; they accept

an identity-statement corresponding to Hume’s Principle:

#x(F (x)) = #x(G(x)) ≡F,G F (x) ≈x G(x)

[Read: for the number of the Fs to equal the number of the Gs just is for the

Fs to be in one-one correspondence with the Gs.]

A consequence of this identity-statement is that it is unintelligible that there be no

numbers. For it is trivially true that, e.g. the planets are in one-one correspondence with

themselves. But for the planets to be in one-one correspondence with themselves just is

for the number of the planets to be self-identitcal. So numbers exist after all. So it is de

mundo unintelligible that there be no numbers.

Non-trivialist neo-Fregeans, in contrast, accept Hume’s Principle, but shy away from

accepting the corresponding identity-statement. Accordingly, they take themselves to be

able to make sense of a world with no numbers.

It seems to me that there is some confusion in the literature about which of the two

version of the neo-Freganism is being discussed. Critics of neo-Fregeanism have some-

86 CHAPTER 4. MATHEMATICS

times interpreted the program as a version of traditional neo-Fregeanism—the author

of Rayo (2003) and Rayo (2005), for example. (For a survey of neo-Fregean literature,

see MacBride (2003).) But it is not clear that this is what actual proponents of neo-

Fregeanism have had in mind. There are strong indications that trivialist neo-Fregeanism

is closer to the mark. One such indication is the use of ‘neo-Fregeanism’ as a name for

the program. As I hinted above, there is good reason to think that Frege himself was a

proponent of trivialist Platonism.

When Frege claims, for example, that the sentence ‘there is at least one square root of

4’ expresses the same thought as ‘the concept square root of 4 is realized’, and adds that

“a thought can be split up in many ways, so that now one thing, now another, appears

as subject or predicate” (Frege (1892) p. 199), it is natural to interpret him as embracing

the identity-statement:

For the concept square root of 4 to be realized just is for there to be at least

one square root of 4.

And when he claims, in Grundlagen §64, that in treating the judgement ‘line a is parallel

to line b’ as an identity, so as to obtain ‘the direction of line a is identical to the direction

of line b’, we “carve up the content in a way different from the original way”, it is natural

to interpret him as embracing the identity-statement:

For the direction of line a to equal the direction of line b just is for a and b

to be parallel.

In both instances, Frege puts the point in terms of content-recarving, rather than as an

identity-statement. But, as emphasized in section 1.3, one’s views about truth-conditions

are tightly correlated with the identities one accepts.

Neo-Fregeans have been sympathetic towards Frege’s views on content-recarving.

(See, for instance, Wright (1997).) And even though talk of content-recarving has be-

come less prevalent in recent years, with more of the emphasis on implicit definitions, a

4.3. NEO-FREGEANISM 87

version of neo-Fregeanism rooted in trivialist Platonism is clearly on the cards. It seems

to me that such an interpretation of the program would be decidedly advantageous.

4.3.1 Mixed Identities

As noted above, trivialist Platonism is a form of anti-Tractarianism. (For discussion of

Tractarianism and anti-Tractarianism, see chapter 3.)

An important feature of anti-Tractarianism is that it leaves room for meaninglessness

where Tractarianism does not. Suppose it is agreed on all sides that the singular terms

t1 and t2 are both in good order, each of them figuring meaningfully in sentences with

well-defined truth-conditions. A Tractarian is, on the face of it, committed to the claim

that it must be possible to meaningfully ask whether pt1 = t2q is true. For she believes

that each of t1 and t2 is paired with one of the objects carved out by the metaphysical

structure of the world. So the question whether pt1 = t2q is true can be cashed out as

the question whether t1 and t2 are paired with the same such object.

For an anti-Tractarian, in contrast, there is no tension between thinking that t1 and t2

figure meaningfully in sentences with well-defined truth conditions and denying that one

has asked a meaningful question when one asks whether pt1 = t2q is true. For according

to the anti-Tractarian, all it takes for a singular term to be in good order is for there

to be a compositional specification of truth-conditions for whichever sentences involving

the term one wishes to make available for use (see section 3.1). And there is no reason

one couldn’t have a compositional specification of truth-conditions for a large range of

sentences involving t1 and t2 without thereby specifying truth-conditions for pt1 = t2q.

Arithmetic is a case in point. As we will see in section 4.4, trivialist Platonists have a

straightforward way of specifying a compositional assignment of truth-conditions to arith-

metical sentences. On this assignment, every arithmetical sentence a non-philosopher

would care about gets well-definied truth-conditions, as does every sentence in the non-

arithmetical fragment of the language. But no truth-conditions are supplied for mixed

88 CHAPTER 4. MATHEMATICS

identity-statements, such as ‘Julius Caesar = 7’. In fact, there is no natural way of

extending the relevant semantic clauses to cover these cases.

Tractarians will claim that something important has been left out. For in the absence

of well-defined truth-conditions for ‘Julius Caesar = 7’, it is unclear which of the objects

carved out by the metaphysical structure of reality has been paired with ‘the number of

the planets’. But anti-Tractarians will disagree: it is simply a mistake to think that such

pairings are necessary to render a singular term meaningful. In fact, one should expect

there to be mixed identity statements that lack well-defined truth-conditions. For when

a sentence has no clear role to play in communication—as is the case of ‘Julius Caesar

= 7’ in non-philosophical contexts—our linguistic practice generates no pressure for it

to come to be associated with truth-conditions, even when its constituent terms figure

meaningfully in other sentences.

A Tractarian might reply that ‘Julius Caesar = 7’ doesn’t need a role to play in

communication in order to have well-defined truth-conditions. Its truth-conditions will

be determined by whether ‘Julius Caesar’ and ‘7’ are paired onto a single one of the

objects that are carved out by the world’s metaphysical structure—an issue which must

have a determinate answer if the terms are to figure meaningfully in other sentences. But

the anti-Tractarian would disagree. ‘Julius Caesar’ and ‘7’ are paired onto objects with

respect to different ways of carving up the world: they are components of different systems

of representation for describing the world. So the fact that they occur in sentences with

well-defined truth-conditions offers no guarantee that their occurrence in ‘Julius Caesar

= 7’ will result in well-defined truth-conditions. (For further discussion of this issue, see

chapter 9.)

Moral: from the point of view of an anti-Tractarian, there is no pressure for thinking

that a mixed identity-statement such as ‘Julius Caesar = 7’ should have well-defined

truth-conditions. So there is no reason to think—in spite of what Frege suggests in §66

of the Grundlagen and what proponents of neo-Fregeanism have tended to presuppose—

4.3. NEO-FREGEANISM 89

that a characterization of the concept of number will be unacceptable unless it settles the

truth-value of mixed identity-statements. If you are a trivialist neo-Fregean, you should

be resolute about your anti-Tractarianism, and stop worrying about mixed identities.

4.3.2 Abstraction Principles

I have never been able to understand why a non-trivialist neo-Fregean would think it

important to use Hume’s Princple to characterize the meaning of arithmetical vocabulary,

instead of using, e.g. the (second-order) Dedekind Axioms. In the case of trivialist neo-

Fregeans, on the other hand, I can see a motivation. Hume’s Principle, and abstraction

principles more generally, might be thought to be important because they are seen as

capturing the difference between setting forth a quantified biconditional as an implicit

definition of mathematical terms:

∀α∀β(f(α) = f(β)↔ R(α, β))

and setting forth the corresponding identity-statement:

f(α) = f(β) ≡α,β R(α, β).

And this is clearly an important difference. Only the latter delivers a trivialist form of

Platonism, and only the latter promises to deliver an account for the special epistemic

status of mathematical truths.

If the motivation for appealing to abstraction principles is simply to secure a triv-

ialist form of Platonism, it seems to me that there are better ways of doing the job.

Neo-Fregeans have found it difficult to identify abstraction principles that can do for set-

theory what Hume’s Principle does for arithmetic. (For a selection of relevant literature,

see Cook (2007).) But if one waives the requirement that meaning-fixation work be done

90 CHAPTER 4. MATHEMATICS

by abstraction principles, the difficulties vanish. As we shall see below, proponents of

trivialist Platonism have a straightforward way of specifying compositional semantic the-

ories for a large range of mathematical languages that deliver just the right results. (For

the case of arithmetic, see the next section; for the case of set-theory, see section 4.6.1.)

4.4 A Semantics for Trivialists

In this section we will see that the trivialist can give a compositional specification of

truth-conditions for arithmetical sentences which yields the result that every true sentence

of pure arithmetic is assigned trivial truth-conditions and every false sentence of pure

arithmetic is assigned trivial falsity-conditions.

The trivialist’s semantics is also defined for every sentence of applied arithmetic that

a non-philosopher would care about, and whenever it is defined it delivers the intu-

itively correct results. It does not, however, assign truth-conditions to mixed identity-

statements, such as ‘the number of the planets = Julius Caesar’. As we saw in sec-

tion 4.3.1, the anti-Tractarian should see no tension between thinking that the singular

terms t1 and t2 figure meaningfully in sentences with well-defined truth conditions and

denying that one has asked a meaningful question when one asks whether pt1 = t2q is

true.

4.4.1 The Trivialist Semantics

I will now spell out the semantics. Readers uninterested in the details are welcome to

skip ahead to section 4.4.2. But there is really nothing to fear—the material is totally

straightforward.

We work with a two-sorted first-order language with identity, L. Besides the identity-

symbol ‘=’, L contains arithmetical variables (‘n1’, ‘n2’, . . .), individual-constants (‘0’)

and function-letters (‘S’, ‘+’ and ‘×’), and non-arithmetical variables (‘x1’, ‘x2’, . . .),

4.4. A SEMANTICS FOR TRIVIALISTS 91

constants (‘Caesar’ and ‘Earth’) and predicate-letters (‘Planet(. . . )’). In addition, L

has been enriched with the function-letter ‘#v(. . .)’ which takes a first-order predicate in

its single argument-place to form a first-order arithmetical term (as in ‘#x1(Planet(x1))’,

which is read ‘the number of the planets’).

If σ is a variable assignment and w is a world, truth and denotation in L relative to

σ and w can be characterized as follows:

Denotation of arithmetical terms:

1. δσ,w(pniq) = σ(pniq)

2. δσ,w(‘0’) = the number Zero

3. δσ,w(pS(t)q) = δσ,w(t) + 1

4. δσ,w(p(t1 + t2)q) = δσ,w(t1) + δσ,w(t2)

5. δσ,w(p(t1 × t2)q) = δσ,w(t1)× δσ,w(t2)

6. δσ,w(p#xi(φ(xi))q) = the number of zs such that Sat(pφ(xi)q, σz/pxiq, w)

7. δσ,w(p#ni(φ(ni))q) = the number of ms such that Sat(pφ(ni)q, σm/pniq, w)

Denotation of non-arithmetical terms:

1. δσ,w(pxiq) = σ(pxiq)

2. δσ,w(‘Caesar’) = Gaius Julius Caesar

3. δσ,w(‘Earth’) = the planet Earth

Satisfaction:

Where p[φ]wq is read pit is true at w that φq,

92 CHAPTER 4. MATHEMATICS

1. Sat(p∃ni φq, σ, w)↔ there is a number m such that Sat(φ, σm/pniq, w)

2. Sat(p∃xi φq, σ, w)↔ there is a z such that ([∃y(y = z)]w ∧ Sat(φ, σz/pxiq, w))

3. Sat(pt1 = t2q, σ, w)↔ δσ,w(t1) = δσ,w(t2)

4. Sat(pPlanet(t)q, σ, w)↔ [δσ,w(t) is a planet]w (for t a non-arithmetical term)

5. Sat(pφ ∧ ψq, σ, w)↔ Sat(φ, σ, w) ∧ Sat(ψ, σ, w)

6. Sat(p¬φq, σ, w)↔ ¬Sat(φ, σ, w)

A few remarks about the semantics:

Infinite numbers

I assume throughout that ‘number’ includes infinite numbers. This is done in order to

ensure that there are no empty terms in the language. If one wanted to restrict one’s

attention to the natural numbers one could do so by working in a free logic.

Truth-Conditions and Possible Worlds

What I had promised is an assignment of truth-conditions to sentences; what the trivialist

semantics actually delivers an assignment of sets of worlds to sentences. To bridge the

gap we need the assumption that a sentence’s truth-conditions are adequately modeled

by a set of worlds. This is a substantial assumption. But chapters 1 and 2 of this book

can be read as an extended argument for the claim that the assumption is satisfied when

the space of worlds is taken to be the to be the space of metaphysically possible worlds.

(As emphasized in section 1.5, it is important to distinguish between the claim that two

sentences have the same truth-conditions and the claim that they have the same meaning.

To say that they have the same truth-conditions is only to say that there is no difference

between what would be required of the world to satisfy the constraints determined by

4.4. A SEMANTICS FOR TRIVIALISTS 93

one of the meanings and what would be required of the world to satisfy the constraints

determined by the other. For further discussion of these topics, see Part ??.)

Actualism

To simplify the exposition, I have made the tacit assumption that the range of the

metalinguistic quantifiers includes merely possible objects. But it is worth keeping in

mind that non-actualist quantification could be avoided entirely. As advertised earlier,

we will see in chapter 7 that when it comes to doing semantics there is a very precise

sense in which everything that could be achieved by quantifying over possibilia can also

be achieved by quantifying over representatives for possibilia, while working from within a

purely actualist perspective. The trivialist semantics is no exception. (For a development

of the trivialist semantics in explicitly actualist terms, see Rayo (2008).)

4.4.2 Philosophical Commentary

Outscoping

What is distinctive about the trivialist semantics is that mathematical vocabulary always

occurs outside the scope of ‘[. . .]w’. Denis Bonnay once suggested a nice name for the

procedure that gives rise to semantic clauses of this kind: outscoping.

Consider, for example, the object-language sentence ‘#x(Planet(x)) = 0’. Straight-

forward application of the trivialist semantic clauses yields the result that this sentence

is true at a world w just in case w satisfies the following metalinguistic formula:

the number of zs such that [z is a planet]w = 0

We are assuming arithmetic in the metatheory. So all that is required of w in order for

the metalinguistic formula to be satisfied is that it contain no planets.

94 CHAPTER 4. MATHEMATICS

Compare this with the result of using a homophonic semantics to specify truth-

conditions for ‘#x(Planet(x)) = 0’. What follows from the usual semantic clauses is that

the sentence is true at w just in case w satisfies the following metalinguistic formula:

[the number of zs such that z is a planet = 0]w

In this case, arithmetical vocabulary occurs within the scope of ‘[. . .]w’. So—regardless of

whether we help ourselves to arithmetic in the metatheory—we get the result that what

is required of w in order for the metalinguistic formula to be satisfied is that it contain

the number Zero, and that, at w, Zero number the planets.

Of course, if it is true that for the number of the planets to be Zero just is for there to

be no planets, then the two requirements on w utlimately come to the same thing. So it

will be true, both according to the trivialist semantics and according to the homophonic

semantics, that all that is required of w to verify ‘#x(Planet(x)) = 0’ is that it contain

no planets. But there is still an important difference: only the trivialist semantics entails

the result with no appeal to the metatheoretic assumption that for the number of the

planets to be Zero just is for there to be no planets.

Because of this difference, only the trivialist semantics can be used to give a non-trivial

proof of the object-language identity statement ‘#x(Planet(x)) = 0 ≡ ¬∃xPlanet(x)’.

For—unlike the homophonic semantics—the trivialist semantics can be used to show that

‘#x(Planet(x)) = 0’ and ‘¬∃xPlanet(x)’ are true at precisely the same worlds without

assuming the relevant identity statement in the metatheory. One does need to use arith-

metical reasoning in the metatheory to prove the result, but one doesn’t need to assume

that arithmetical truths are trivially true.

In general, the upshot of outscoping is that even though mathematics is used in the

metatheory, all one needs to know about a world w in order to determine whether a given

arithmetical sentence would be true at w is which non-mathematical predicates apply to

which objects.

4.4. A SEMANTICS FOR TRIVIALISTS 95

In the case of ‘#x(Planet(x)) = 0’, we saw that what it takes for the sentence be true

at w is for w to satisfy the following metalinguistic formula:

the number of zs such that [z is a planet]w = 0

And in order to tell whether w satisfies this formula all we need to know about w is which

objects satisfy the non-mathematical predicate ‘is a planet’ at w. Once we have this

information we can ask ourselves—without collecting additional information about w—

whether the number of these objects is Zero, and go on to ascertain whether the sentence

is true at w. An analogous point can be made for every sentence for which the trivialist

semantics is defined. There is therefore a precise sense in which the trivialist semantics

establishes a connection between mathematical and non-mathematical descriptions of the

world.

Attention so far has been focused on applied arithmetic. But it is useful to see how

outscoping plays out in the pure case. Consider the object-language sentence ‘1 + 1 =

2’ (in primitive notation: ‘S(0) + S(0) = S(S(0))’). Since there is no non-mathemtical

vocabulary to remain within the scope of ‘[. . .]w’, application of the semantic clauses yields

the result that the sentence is true at a world w just in case w satisfies a metalinguistic

formula in which all the vocabulary has been outscoped:

1 + 1 = 2

What one gets, in other words, is a formula in which ‘[. . .]w’ does not occur, and therefore

a formula with no free variables. In general, a formula with no free variables is satisfied

by all objects if it is true, and no objects if it is false. Since the metalinguistic formula

‘1 + 1 = 2’ is, in fact, true, it will be satisfied by all objects—and in particular by w

for arbitrary w. It is, in other words, satisfied by w independently of what w is like.

The trivialist semantics therefore delivers the conclusion that nothing is required of w

in order for the object-language sentence ‘1+1 = 2’ to be true at w. And, of course,

96 CHAPTER 4. MATHEMATICS

the point generalizes. The trivialist semantics (plus arithmetic) yields the result that

an arbitrary truth of pure arithmetic is true at w independently of what w is like, and

that an arbitrary falsehood of pure arithmetic is false at w independently of what w is

like. So—on the assumption that truth-conditions are adequately modeled by sets of

worlds—the trivialist semantics entails trivialism.

Using Arithmetic in the Metalanguage

The trivialist semantics makes free use of arithmetic in the metalanguage. This means

that it cannot be used to explain the truth-conditions of arithmetical sentences to some-

one who doesn’t already understand arithmetical vocabulary. One would be able to

supply such an explanation if one had method for paraphrasing arbitrary arithmetical

sentences as sentences containing no mathematical vocabulary. We will see in section 4.5

that there are important limits to what can be achieved when it comes to the project of

characterizing paraphrase-methods of this kind. The thing to keep in mind for present

purposes is that supplying non-mathematical paraphrases for arithmetical sentences is

not part of what the trivialist semantics is meant to achieve.

The aim is, rather, to give a precise statement of trivialism by saying exactly what

truth-conditions a trivialist would associate with each arithmetical sentence—and doing

so in such a way that the resulting assignment of truth-conditions can be recognized as

delivering trivialism regardless of whether one happens to be a trivialist.

A homophonic semantics does not succeed in doing this. The trivialist and the non-

trivialist can both agree that the homophonic semantics is correct. They can both agree,

for example, that what it takes for ‘1 + 1 = 2’ to be true is for it to be the case that One

plus One is Two. (Though, of course, trivialist would go on to add, and the non-trivialist

would not, that nothing is required for it to be the case that One plus One is Two.) In

contrast, when one sets forth a trivialist semantics, one supplies an assignment of truth-

conditions that is unequivocally trivialist. For one gets the conclusion that ‘1 + 1 = 2’ is

4.5. PARAPHRASE 97

trivially true as a logical consequence of one’s semantics (plus arithmetic).

4.5 Paraphrase

Philosophers of mathematics have often been concerned with the question of whether

mathematical statements can be paraphrased as sentences containing no mathemati-

cal vocabulary. A non-trivialist who thinks that typical mathematical assertions carry

no commitment to mathematical objects might wish to claim that what is communi-

cated by a mathematical assertion is the same as what is literally expressed by the

non-mathematical paraphrase of the asserted sentence. But a trivialist might also be

interested in the issue, since she might be hope to be in a position to claim that every

mathematical statement has the same truth-conditions as some sentence containing no

mathematical vocabulary.

For trivialists and non-trivialists alike, it seems to me that putting too much emphasis

on paraphrase is a bad idea. For the existence of suitable paraphrase-function turns

crucially on the expressive resources of one’s non-mathematical vocabulary. And it would

be a mistake to burden one’s philosophy of mathematics with claims about the legitimacy

of a given set of expressive resources. Notice, moreover, that a paraphrase-function is

unnecessary to characterize the desired specification of truth-conditions for mathematical

sentences, since one can instead use a compositional semantics of the sort described above.

An additional danger of focussing too much attention on paraphrase is that specifying

a non-mathematical paraphrase for a mathematical statement is not always the best way

of shedding light on the statement’s truth-conditions. So in setting forth a parahrase-

function one runs the risk of thinking that one has succeeded in clarifying the truth-

conditions of mathematical statements, when in fact one has not.

Chapter 8 is devoted to a detailed discussion of these issues.

98 CHAPTER 4. MATHEMATICS

4.6 Beyond Arithmetic

Our discussion of mathematical trivialism has been focused so far on the special case of

arithmetic. How does trivialism plays out when it comes to other branches of mathemat-

ics?

One issue to be addressed is the question of what it takes for a trivialist understanding

of a given branch of mathematics to be available. Could one give a trivialist account of

set-theory, for example? Could one specify truth-conditions for set-theoretic sentences

in such a way that the standard set-theoretic axioms turn out to be trivially true?

I offer a detailed discussion of the matter in Chapter 9, but the upshot can be stated

very succinctly. From the point of view of an anti-Tractarian, it doesn’t take much for a

trivialist understanding of an axiomatic mathematical theory to be available. In the case

of pure mathematics, all it takes is for the theory to be internally coherent ; in the case

of applied mathematics, all it takes is for the theory to be conservative over sentences of

the original language. The reason this is so is that one can introduce new mathematical

vocabulary by stipulation: one can stipulate that it is to be understood in such a way that

a suitable set of axioms turn out to be trivially true. And the anti-Tractarian can show

that—as long as it gives rise to the right sort of linguistic practice—such a stipulation is

guaranteed to succeed on the assumption the axioms are internally coherent (in the case

of pure mathematics) or conservative (in the case of applied mathematics).

The anti-Tractarian will therefore think that a trivialist interpretation is available for

any minimally well-behaved mathematical theory. It does not follow, however, that anti-

Tracatrains will always be in a position to characterize a compositional semantics that

makes the triviality of a given axiom system explicit. A homophonic semantics will always

be available, of course, and will always be recognized as delivering an accurate assignment

of truth-conditions. The problem is that only the trivialist will take it to deliver a trivialist

specification of truth-conditions—the non-trivialist will take it to deliver a non-trivialist

4.6. BEYOND ARITHMETIC 99

specification of truth-conditions. A homophonic semantics is therefore not a good way of

making the triviality of the relevant axiom system explicit—or, indeed, of shedding any

real light on the truth-conditions of sentences of the language under consideration.

A trivialist semantics of the kind described in section 4.4.1 can be significantly more

illuminating. But such a semantics is only guaranteed to be available in the case of pure

mathematics. One reason problems can arise in the case of applied mathematics is that

there is no general recipe for constructing trivialist semantic clauses for atomic formulas

that mix mathematical and non-mathematical vocabulary.

Things work out for the language of applied arithmetic thanks to the following equiv-

alence:

[#x(F(x)) = n]w ↔ #x([F(x)]w) = n.

Read: at w (the number of the Fs = n) just in case (the number of the xs

such that at w (x is an F)) = n.

(Or an actualist version thereof—see section 4.4.1.) It is because of this equivalence that

we are able to ‘outscope’, and go from the homophonic semantic clause for the mixed

atomic formula ‘#x(F(x)) = n’,

‘#x(F(x)) = n’ is true at w ↔ [#x(F(x)) = n]w,

to a trivialist semantic clause,

‘#x(F(x)) = n’ is true at w ↔ #x([F(x)]w) = n.

But such equivalences are not guaranteed to be available in general. For instance, I know

of no general way of defining a trivialist semantics for a plural language that has been

enriched with non-logical atomic plural predicates. (For discussion of such languages, see

Rayo (2002b).)

100 CHAPTER 4. MATHEMATICS

4.6.1 Set Theory

Happily, one can give a trivialist semantics for the language of set-theory with urelements.

It is an easy modification of the trivialist semantics for the language of arithmetic that

was described in section 4.4.1.

Our object-language will be the language of two-sorted set-theory with urelements:

Roman variables (x, y, . . . ) range over urelements, Greek variables (α, β, . . . ) range

over sets. Let σ be a variable assignment, let w be a metalinguistic variable ranging over

worlds and let p[φ]wq be read pit is true at w that φq. Then satisfaction can be defined

as follows:

1. Sat(p∃xi φq, σ, w)↔ there is a z such that ([∃y(y = z)]w ∧ Sat(φ, σz/pxiq, w))

2. Sat(p∃αi φq, σ, w) ↔ there is a set β such that: (i) for any urelement z in the

transitive closure of β, [∃y(y = z)]w, and (ii) Sat(φ, σβ/pαiq, w)

3. Sat(pF(x)q, σ, w)↔ [σ(x) is an F]w

4. Sat(pα ∈ βq, σ, w)↔ σ(α) ∈ σ(β)

5. Sat(px ∈ βq, σ, w)↔ σ(x) ∈ σ(β)

6. Sat(px = yq, σ, w)↔ σ(x) = σ(y)

7. Sat(pφ ∧ ψq, σ, w)↔ Sat(φ, σ, w) ∧ Sat(ψ, σ, w)

8. Sat(p¬φq, σ, w)↔ ¬Sat(φ, σ, w)

As in the case of arithmetic, I simplify the exposition by making the tacit assumption

that the range of the metalinguistic quantifiers includes merely possible objects. But

it is worth keeping in mind that non-actualist quantification can be avoided altogether

by appeal to the machinery developed in chapter 7. (For a rendering of the trivialist

semantics in explicitly actualist terms, see Rayo (2008).)

4.6. BEYOND ARITHMETIC 101

Our semantics delivers just the assignment of truth-conditions that a trivialist would

hope for. One gets the result that every truth of the pure fragment of set-theory is trivially

true and every falsity of the pure fragment of set-theory is trivially false. And one gets

the desired truth-conditions for sentences involving urelements. For instance, one gets

the result that all that is required of the world in order for ‘∃α∃x(x ∈ α ∧ Elephant(x))’

to be true is for there to be an elephant.

As in the case of arithmetic, one usually needs to prove a mathematical result in the

metatheory in order to show that a given sentence is trivially true, or to show that it is

trivially false. For instance, in order to show that an object-language sentence stating

that there is an inaccessible has trivial truth-conditions one has to show in the metatheory

that there is an inaccessible. The result is that the semantics entails very little on its own

about what an intended model for the object-language should look like. It all depends

on what one assumes about set theory in the metatheory.

The Iterative Conception of Set

A trivialist semantics for the language of set theory is a good way of making explicit the

truth-conditions of set-theoretic sentences, as understood by the trivialist. But it fails to

deliver a conception of set. As mentioned above, it doesn’t offer much guidance about

what the hierarchy of sets should look like, since any substantial information about the

hierarchy is imported from the background metatheory.

There is, however, an illuminating conception of set that is available to the trivi-

alist. Early discussions include Zermelo (1930) (which builds on Zermelo (1908)) and

Godel (1944) (which was partially anticipated in Godel (1933)). More recent discus-

sions include Boolos (1971), Parsons (1974) and Potter (2004). The discussion in Lin-

nebo (forthcoming) and its technical companion Linnebo (typescript) are especially ger-

mane to the present discussion.

On the version we will consider here, the basic idea is that set-talk is to be introduced

102 CHAPTER 4. MATHEMATICS

in stages. At Stage Zero, the only quantification we have available is quantification over

urelements. At Stage One, we introduce the membership-predicate ‘∈’, and the set-

theoretic term-forming operator ‘{z : . . . z . . .}’. We then set forth the following identity

statement, where ‘x’ and ‘z’ range over urelements and ‘F ’ is a second-order variable

taking first-order urelement-variables as arguments:

Stage One Identity Statement

x ∈ {z : F (z)} ≡x,F F (x)

Read: for x to be a member of the set of F s just is for x to be an F .

The result is that we have a new way of carving up the world. Because of the introduction

of additional linguistic resources, the feature of reality that was fully and accurately

described by means of ‘a is F ’ can now also be fully and accurately described by means

of ‘a is a member of the set of F s’. (Does this mean that novel objects have been

brought into existence? Absolutely not. For an extended discussion of these issues, see

section 3.2.)

The Stage One Identity Statement is, of course, a close cousin of Frege’s Basic Law V,

which leads to inconsistency. We avoid inconsistency here because the range of ‘x’ does

not include the referents of the newly introduced set-theoretic terms. A Tractarian will

grant that inconsistency has been avoided, be she will think that it is the result of an ad

hoc restriction. “Why restrict the range of ‘x’ to urelements?” she would ask. “Is there

some reason not to treat it as ranging over absolutely everything, beyond the desire to

avoid paradox?”

But the anti-Tractarian has a reply. It is true that from a Tractarian point of view

it makes good sense to speak of a maximal domain: it is simply the domain of objects

that gets carved out by the world’s metaphysical structure. But from an anti-Tractarian

point of view, different uses of quantification presuppose different ways of carving up the

world, and there is no obvious reason to think that any one such carving will deliver a

4.6. BEYOND ARITHMETIC 103

‘maximal’ domain.

“I grant that the anti-Tractarian needn’t think of every domain as a subdomain of the

‘metaphysically privileged’ domain”—a skeptic might reply—“but she must still think of

every domain as a subdomain of the ‘maxi-domain’: the domain that results from every

possible way of carving the world into objects. So when the trivialist takes ‘x’ to range

over the domain of urelements, she is still placing ad hoc restrictions on the range of the

variables.” As I pointed out in section 3.2, however, the anti-Tractarian has no reason to

think that there is such a thing as a ‘maxi-domain’. For a carving of the world is just a

compositional system of representation for describing the world, and there is no obvious

reason to think that there is a final answer to the question of what counts as a possible

system of representation.

The proper response to the skeptic is therefore this. When we introduce the Stage

One Identity Statement, the situation is not one in which a domain that includes sets

is available to us, and we choose to restrict it to urelements in order to avoid paradox.

Instead, the label ‘urelement’ is applied to everything we are able to quantify over at that

point. It is only after the identity statement is introduced, and we have a way of fixing

truth-conditions for set-theoretic sentences, that our representational resources are rich

enough to quantify over sets.

Once Stage One is in place, one can begin to iterate the process. At each stage µ one

starts by treating every object one is in a position to quantify over as a ‘µ-urelement’. One

then sets forth the following identity statement, where ‘xµ’ and ‘zµ’ take µ-urelements

as values, and ‘F ’ is a second-order variable taking first-order µ-urelement-variables as

arguments:

Stage µ Identity Statement

xµ ∈ {zµ : F (zµ)} ≡xµ,F F (xµ)

Read: for xµ to be a member of the set of F s just is for xµ to be an F .

104 CHAPTER 4. MATHEMATICS

[Beware: it is potentially misleading to think of µ-urelements as constituting

a single domain, since nothing has been done to fix the truth-conditions of

mixed identity statements relating set-theoretic and non-set-theoretic terms.

It is better to think of set-theoretic variables and non-set-theoretic variables as

falling under different sorts—and therefore ranging over separate domains—

and to treat mixed identity statements as ill-formed. I avoid doing so here to

improve the exposition, but further discussion of these matters can be found

in section 9.1.]

How far could this procedure be iterated? The first thing to note is that no limits are

imposed by the way the world is. Pace Tractarianism, we are not operating against the

background of a fixed domain, which might happen to include enough objects for the

procedure to be iterated until a particular point and no further. At each stage of the

process, we introduce a new family of set-theoretic terms by way of an axiom system

that is conservative over the second-order language of µ-urelements,1 and, according to

1The conservativeness claim is easily verified in ZFC. It is enough to show that any model of thesecond-order language of µ-urelements can be extended to a model of the expanded language thatsatisfies the Stage µ Identity Statement. If one wishes to prove syntactic conservativeness, one shouldwork with Henkin-models; if one wishes to prove semantic conservativeness one should work with fullmodels. Either way, one proceeds by letting first-order set-theoretic variables take the same values assecond-order µ-urelement-variables, and letting second-order set-theoertic variables take as values setsof values of first-order set-theoretic variables.

Note, however, that the second-order theory of µ-level sets with µ-urelements can be used to definea truth-predicate for the second-order language of µ-urelements. And by using such a predicate togenerate new instances of the axiom-schemas governing second-order µ-urelement quantification onemight be able to prove sentences containing no µ-set-theoretic vocabulary that one couldn’t prove before(such as the Godel-sentence for the second-order language of µ-urelements). So although the Stage µIdentity Statement is itself (syntactically and semantically) conservative, it is couched in a languagewhich can be used to formulate additional second-order axioms, and thereby increase the deductivepower of one’s second-order theory of µ-urelements.

The new axioms will certainly be semantically conservative over the second-order language of µ-urelements, since they are all true in every full second-order model. So if one thinks that semanticconservativeness is enough to legitimate the introduction of new axioms, one will have no reason to worryin the case at hand. If, on the other hand, one thinks that syntactic conservativeness is a prerequisitefor legitimacy, then there might be reasons for concern, since, as we have seen, the new axioms need notbe syntactically conservative over the second-order language of µ-urelements.

Notice, however, that whether or not the introduction of µ-level set-theoretic vocabulary generatespressure to accept the new axioms depends on how one is thinking about µ-urelement second-orderquantification. One possibility is to think of such quantification open-endedly, and insist that it is part

4.6. BEYOND ARITHMETIC 105

the anti-Tractarian, conservativeness is enough to guarantee that the introduction will

be successful. (See section 9.2.) So the anti-Tractarian will think that the process is

limited only by our ability to introduce further and further identity statements.

Notice, however, that the anti-Tractarian has no obvious reason to think that it would

be helpful to insist that the process is to be iterated ‘all the way up’ or ‘as far as it could

possibly go’. In order for such pronouncements to have definite content there would have

to be a final answer to the question of what counts as a possible system of representation.

And, as emphasized above, the anti-Tractarian has no obvious reason to think that there

is such an answer. When one insists that the process be iterated ‘all the way up’, one

will only succeed in saying something with definite content to the extent that one has

managed to articulate a definite well-ordering, so that the pronouncement can understood

as signaling that the process is to be iterated so as to form a hierarchy that is isomorphic

with that well-ordering. Accordingly, someone who has succeeded in characterizing a

well-ordering isomorphic to the natural numbers will be able to iterate the process far

enough to get the hereditarily finite sets. Someone who has succeeded in characterizing

a well-ordering isomorphic to the smallest strongly inaccessible cardinal will be able to

iterate the process far enough to get enough sets for second-order ZF to have an intended

of one’s original intent that one be committed not only to instances of the second-order axiom-schemasthat can be formulated in one’s current language, but also to instances formulable in arbitrary extensionsof the language. (See McGee (2000), Williamson (2003) and Lavine (2006).) If so, one should think thattruth-conditions are semantically determined (in the sense of section 9.2) and hence that one’s theoryof µ-urelements should be semantically conservative over the base non-mathematical theory. As it turnsout, semantic conservativeness of this kind is enough to guarantee the weak syntactic conservativenessof the new axioms. (Proof: Suppose the new axioms fail to by weakly syntactically conservative over thesecond-order theory of µ-urelements. Then there is a world w such that one can prove a contradictionfrom the result of adding the new axioms to the set Tw of second-order µ-urelement sentences withwell-defined truth-conditions are true at w, even though Tw is itself consistent. Since the theory ofµ-urelements is semantically conservative over the base non-mantematical theory, Tw must have a fullsecond-order model. But each of the new axioms holds in every full second-order model, contradictingthe assumption that the result of adding the axioms to Tw is inconsistent.) This is significant because,as emphasized in section 9.2 the anti-Tractarian should think that weak conservativeness is all it takesto guarantee successful introduction of the new axioms.

Alternatively, µ-urelement second-order quantification may not be thought of open-endedly. If so, ourargument for weak syntactic conservativeness breaks down. But so does the rationale for thinking thatthe introduction of µ-level set-theoretic vocabulary generates pressure to accept the new axioms.

106 CHAPTER 4. MATHEMATICS

interpretation. And so forth.

A consequence of this picture is that the project of developing an iterative conception

of set is inextricably linked to the project of doing set-theory. For the canonical way of

identifying a large well-ordering is by describing an ordinal in set-theoretic terms, and

motivating the idea that its existence wouldn’t lead to inconsistency. The result is that

it would be a mistake to think of the iterative construction described in this section as

a substitute for a theory like ZFC. It is more illuminating than ZFC in some respects,

and less illuminating in others. It is more illuminating as a way of shedding light on the

truth-conditions of a sentence like ‘Caesar is a member of the set of Romans’, or as a

method for understanding the most general constraints on our reasoning about sets. But

it does far worse than ZFC as a method for shedding light on more specific questions

about the set-theoretic landscape. The right attitude, it seems to me, is to use both

kinds of theories in tandem, as complementary descriptions of a single subject-matter.

Paraphrase

Is it possible to characterize a trivialist paraphrase function for the language of set-theory?

Is there, in other words, an algorithmic procedure for mapping each set-theoretic sentence

φ to a sentence whose truth-conditions are uncontroversially the truth-conditions that

the trivialist semantics associates with φ?

If the paraphrase-language includes variables of sufficiently high transfinite order—

and if a trivialist understanding of higher-order quantification is assumed to be uncon-

troversial—then the answer is ‘yes’.

Let L α∈ be a version of the language of set-theory with urelements in which each

occurrence of a quantifier is restricted by some Vβ (β < α). As mentioned above, Lin-

nebo and Rayo (typescript) shows that, for arbitrary α, there is a truth-value-preserving

paraphrase-function φ 7→ φ(α) that maps every sentence of L α∈ onto a sentence in a

language of order α + 2 (or order α, if α is a limit ordinal). Moreover, by assuming a

4.6. BEYOND ARITHMETIC 107

suitable reflection principle, one can show that there is a cardinal ξ such that Vξ satisfies

the same L∈-sentences as the universe. (See Shapiro (1987), pp. 323–4.) So one can

characterize a trivialist paraphrase-function for the language of set-theory with urele-

ments by using the following procedure. First, transform every sentence of the language

of set-theory into a sentence of L ξ+1∈ by restricting the quantifiers to Vξ. Then apply

the paraphrase-function φ 7→ φ(ξ+1). The result is a function that paraphrases every

set-theoretic sentence as a sentence of order (ξ + 3). (Would such a paraphrase-function

count as algorithmic? Yes, assuming one can help oneself to ξ. But, of course, ξ is far

beyond the recursively specifiable ordinals.)

Whether or not a paraphrase-function of this kind succeeds in elucidating the truth-

conditions of set-theoretic sentences will, of course, depend on one’s understanding of

languages of very high order. If one’s grasp of such languages were suitably independent

of set-theory, then the elucidation could be very significant indeed. But it is hard to see

how one could acquire a clear grasp of transfinite type-theory without making substantial

use of set-theory. Conspicuously, one’s ability to characterize a language of order α pre-

supposes that one is able to characterize a well-ordering with an αth member. And, when

α is sufficiently large, it is hard to see how much progress would be made independently

of set-theoretic reasoning.

My own view is that set-theory and transfinite type-theory are best thought of as

different presentations of the same theory. Neither of them should be thought of as being

in some sense ‘prior’ to the other, or as supplying a ‘foundation’. But one can benefit

from having both because they each shed light on different aspects of the theoretical

landscape. For further discussion of these issues, see Linnebo and Rayo (typescript).

108 CHAPTER 4. MATHEMATICS

Chapter 5

Content

5.1 Folk-Psychology

There are a number of different strategies for predicting a subject’s behavior. Each of

them has its own range of advantages and disadvantages.

Folk-psychology is at one end of the spectrum. It has the advantage of being well-

suited for marketplace prediction: prediction of the kind that can be easily carried out

by normal human beings, on the basis of evidence that can be easily acquired on the

basis of informal observation. But it has the disadvantage of being very limited in

its predictive power. And there is room for worrying that—regardless of whether its

behavioral predictions are indeed accurate—it might not be an accurate description of

how the mind works. Neuroscience is at the other end of the spectrum. A fully developed

version would be the ultimate description of how the brain works. It would also deliver

maximally fine-grained predictions of a subject’s behavior, with near-perfect accuracy.

But it would be ill-suited for the marketplace, since its predictions would rely on data

that is very difficult to come by and require computations that a normal human would

find very difficult to carry out.

As long as one thinks of folk-psychology as a systematic description of the subjects’s

109

110 CHAPTER 5. CONTENT

behavioral dispositions, and not as an account of how the mind works, it needn’t be in

competition with cognitive science. Thus understood, folk-psychology could only run

into trouble by making the wrong behavioral predictions, and it doesn’t take much for

there to be an instance of folk psychology that delivers accurate predications. To a first

approximation, it is enough for there to be some assignment of beliefs and desires, some

conception of rationality and some conception of action such that, whenever the subject

is counted as ‘rational’, she ‘acts’ in ways that would tend to bring about her ‘desires’ in

a world compatible with her ‘beliefs’. Maybe our behavioral dispositions are too complex

to be systematized in this sort of way. But there is so much room for adjusting one’s

attributions of beliefs and desires, one’s conceptions of rationality and action, and the

details of one’s folk-theory, that it’s hard not to feel optimistic.

Of course, if one thinks of folk-psychology as more than just a systematic description

of a subject’s behavioral dispositions—if one thinks of it as supplying an account of how

the mind works—then the risk of conflict with cognitive science is very substantial indeed,

since there is no obvious reason to think that a fully developed cognitive science would

look anything like folk psychology. As you read this chapter, please keep in mind that I

will not be thinking of folk-psychology as an account of how the mind works. I will be

thinking of it as a systematic description of the subject’s behavioral dispositions.

I will not, however, be thinking of folk-psychology instrumentally. The view is not, in

other words, that folk-psychology is to be thought of as a (potentially false) theory about

how the mind works which one should be only partially committed to: only insofar as

behavioral predictions are concerned. That would lead to the awkward conclusion that

whether or not people really have beliefs and desires—as opposed to just being such that

they are usefully described in such terms for the purposes of behavioral prediction—

depends on the risky matter of whether folk-psychology turns out to be adequate as a

description of how the mind works. The suggestion, instead, is that for a subject to

have a full set of beliefs and desires just is for the subject’s behavioral dispositions to

5.1. FOLK-PSYCHOLOGY 111

be such as to be describable folk-psychologically, on the basis of those beliefs and those

desires. Thus, a person can be accurately described as believing and desiring irrespective

of whether tomorrow’s cognitive science will look anything like folk-psychology.

What if there is more than one full assignment of beliefs and desires that accurately

systematizes a subject’s behavioral dispositions? Then the successful theories will all

impose the same requirement on the world: that the subject have a certain range of

behavioral dispositions. So the subject can be accurately described in terms of the

beliefs and desires ascribed by any of the theories, provided one is careful not to mix

beliefs and desires from different theories in a single description of the subject.

When folk-psychology is thought of in this way, there is a lot to be said on its behalf.

The first thing to note is that there is a certain respect in which it is the only game in

town. When it comes to the project of making market-place predications—predictions

of the kind that can be easily carried out by normal human beings, on the basis of

evidence that can be easily acquired on the basis of informal observation—it is head and

shoulders above its peers. And there is no escaping marketplace predictions. They are

indispensable, in practice, for the successful navigation of our lives. So folk-psychology

is indispensable, in practice, for the successful navigation of our lives.

The project of better understanding how we manage to navigate our lives is an in-

teresting one, but it is not of special concern to philosophers. There is, however, a

distinctly philosophical reason for being interested in folk-psychology. Intentional no-

tions are a crucial building-block of the philosophical landscape, and they all have their

origins in folk-psychology. One could think that in spite of their origin, the best strategy

for shedding light on intentional notions sidesteps folk-psychology. But it seems to me

that that would be a mistake. At the very least, getting clear on a suitably regimented

version of folk-psychology can shed light on our intentional notions. And it may well

prove advisable to go further, and characterize intentional notions wholly in terms of

their functional role in a well-regimented folk-psychology. (For a nice example of what a

112 CHAPTER 5. CONTENT

regimented folk-psychology might look like, see Lewis (1974).)

Discussion of folk-psychology in what follows should be understood in this spirit. I will

develop a regimented version of our folk-psychological account of mental representation

with the aim of better understanding the family of intentional notions that figure in

the theory. I will argue, in particular, that the notion of fragmentation has a role to

play in our best folk-theoretic account of mental representation. I will then highlight the

notion’s importance by showing that it can be used to address puzzles in the philosophies

of mathematics and mind.

The material in this chapter borrows heavily from Bob Stalnaker. The fundamen-

tal ideas can all be found in Stalnaker (1984) (especially chapters 1 and 5) and Stal-

naker (1999) (especially chapters 13 and 14). (See also Perry (2001) and Parikh (2009).)

My own thinking about these matters has developed in large part as a result of a joint

project with Adam Elga. He has contributed to the project at least as much as I have,

but shouldn’t thereby be burdened with commitment to the idiosyncrasies of the present

discussion.

5.2 Rational Action

In preceding chapters I have tried to articulate the notions of possibility, de mundo

intelligibility and truth-conditions. On the resulting picture, there is no gap between

possibility and de mundo intelligibility. This means, in particular, that one should only

regard as scenario as metaphysically impossible if one takes it to be incoherent in light of

the identity-statements one accepts. It is also a consequence of the picture I have been

defending that a representation’s truth-condtions—i.e. the requirement that the world

would have to satisfy in order to be as the representation represents it to be—can be

modeled as a set of possible worlds: the set of worlds whereby the requirement is satisfied.

Accordingly, to regard a representation as having trivial truth-conditions—to think,

5.2. RATIONAL ACTION 113

in other words, that its content should be modeled by the set of all possible worlds—is

to think that its truth-conditions will be satisfied provided only that the world is not

incoherent. And to see two representations as having the same truth-conditions—to

think, in other words, that their contents should be modeled by the same set of possible

worlds—is to think that a scenario satisfying one set of truth-conditions but not the other

would be incoherent.

In this section I will articulate a regimentation of our folk-psychological account of

behavior, and argue that the contents that are required by the theory can be fruitfully

modeled as sets of possible worlds.

The Fundamental Principle

This is the fundamental principle of folk-psychological accounts of behavior:

Fundamental Principle

The intentional behavior of a rational subject is precisely the behavior that

would constitute the most sensible way of bringing about satisfaction of the

subject’s desires in a world satisfying the subject’s beliefs.

The reason the Fundamental Principle is so useful is that it allows one to make predictions

about the subject’s behavior without knowing anything about the physical mechanisms

that are actually responsible for the behavior in question. It transforms a problem

about the output of a largely intractable neural system into a problem about the sorts

of behaviors that would count as sensible for the attainment of a given aim in a given

situation. And this is an eminently tractable problem for an ordinary human. (An

imperfect but effective strategy is for the theorist to ask herself what she would do to

satisfy the relevant aim in the relevant situation.)

Notice, moreover, that the Fundamental Principle is not just a mechanism for pre-

diction, given an assignment of beliefs and desires. It also supplies the basic method for

114 CHAPTER 5. CONTENT

forming hypotheses about a subject’s beliefs and desires. The theorist can proceed by

determining which combination of beliefs and desires would make it the case that the sub-

ject’s past behavior is, by and large, the most sensible way of bringing about satisfaction

of those desires in a world satisfying those beliefs, and go on to use (sensibly updated

versions of) those beliefs and those desires to make predictions, via the Fundamental

Principle, about the subject’s future behavior.

Coarse-Grained Content

As far as the Fundamental Principle is concerned, the most natural way of modeling the

contents of beliefs and desires is by using sets of possible worlds. To see this, reflect on

the work that the Fundamental Principle demands of such contents. Their one and only

job is to fill in the blanks in the following counterfactual question:

If it were the case that . . . , what would be the most sensible way of bringing

it about that . . . ?

And the only useful way of filling either of these blanks is by writing in a specification of

a way for the world to be. Since sets of metaphysically possible worlds are well-suited to

model ways for the world to be, they are also well-suited to supply such a specification.

(If W is a set of worlds, one can fill in the blanks with pone of the possibilities in W was

actualizedq.)

Notice, moreover, that it wouldn’t be helpful to consider ‘worlds’ that the theorist

takes to be metaphysically impossible. For a scenario that the theorist regards as meta-

physically impossible is a scenario she regards as incoherent, and it is hard to know how

to assess counterfactual questions involving scenarios one regards as incoherent. Nor

would it be helpful to add structure to the contents of beliefs and desires. For even if

the additional structure were to somehow encode valuable information about how the

subject represents the world, it is not information that the Fundamental Principle is able

5.2. RATIONAL ACTION 115

to use.

Beyond the Fundamental Principle

The Fundamental Principle can be supplemented in various ways to produce a more

powerful folk-psychological account of behavior. Here are some representative examples:

1. Rationality and Action

One could try to place independent constraints on the range of application of the

Fundamental Principle by saying something substantial about the circumstances

under which the subject should be counted as rational, and about which of her

behaviors should be counted as intentional. (One could say, for example, that

the subject gets angry under such-and-such circumstances, and that she should be

counted as irrational whenever she is angry. And one could say that the subject

is asleep under such-and-such circumstances and that whenever she is asleep her

behavior fails to be intentional.)

2. Observation

One could add an account of how the subject’s observations impact her beliefs. In

the simplest case, the account is just this:

By and large, a rational subject comes to believe that p whenever she

observes that p.

(This principle can be used to help one decide what beliefs to ascribe to the subject,

given information about her observational situation. But it can also help one decide

what to say about the content of a subject’s observations on the basis of independent

information about what she came to believe after being in the relevant observational

situation.)

116 CHAPTER 5. CONTENT

3. Language

One could add an account of the connection between the subject’s linguistic inter-

actions and the subject’s beliefs. In the simplest case, the account is just this:

Truthfulness

By and large, a rational subject believes that p whenever she makes an

assertion which communicates that p.

Trust

By and large, a rational subject comes to believe that p whenever someone

she trusts makes an assertion which communicates that p.

(As usual, these principles can be used to help one decide what beliefs to ascribe

to the subject, given independent information about what was communicated by

various assertions. But they can also help one decide what correctness-conditions

to associate with assertions—and, indirectly, to decide what meanings to associate

with sentences—given independent information about the subject’s beliefs. For

details, see Lewis (1973) and Lewis (1974).)

4. Belief Kinematics

One could add an account of how the subject would update her beliefs upon learning

that p. In the simplest case, the account is just this:

By and large, a rational subject updates her beliefs in the most sensible

way possible.

(An imperfect but effective way of implementing this principle is for the theorist

to ask herself how she would update her beliefs if she had the same beliefs as the

subject and learned that p.)

5.2. RATIONAL ACTION 117

5. Probabilities

Instead of formulating the Fundamental Principle in terms of belief simpliciter and

desire simpliciter, one could formulated it by using a probability function to play

the role of belief and a value-function to play the role of desire:

Fundamental Principle (probabilistic version)

The intentional behavior of a rational subject is precisely the behavior

that would maximize expected utility (where expected utility is defined

on the basis of the subject’s credences and the subject’s value-function).

(If one goes probabilistic, the principles in 2–4 above will have to be updated

accordingly. In the case of Belief Kinematics, it would be natural to do so by

stating that, by and large, the subject updates by conditionalization.)

Coarse-Grained Content Revisited

Refinements such as the ones described above are all compatible with modeling contents

as sets of metaphysically possible worlds. But once one moves to the probabilistic version

of the Fundamental Principle there is also room for working with a more fine-grained

notion of content, since one need not think of the subject’s credences and value function

as defined over the space of metaphysically possible worlds. One could, for instance, use

a space of ‘worlds’ each of which consists of a set of sentences from the subject’s (public

or mental) language. (A set of sentences might be counted as a ‘world’ just in case the

subject is not able to not rule out a priori that every sentence in the set is true, and if

no proper superset of the set has that property.)

Switching to a finer-grained notion of content would come at a cost. The first thing

to note is that one would be be committed to elucidating the new notion. It is true

that notion of metaphysical possibility is somewhat rough around the edges. But I hope

to have shown in chapters 1 and 2 that it is constrained by its connection with the

118 CHAPTER 5. CONTENT

notions of identity and why-closure, and therefore by its role in our scientific practice. So

understood, it seems to me that the notion of metaphysical possibility is robust enough

to do the job that would be demanded of it by a coarse-grained development of our

folk-psychology. But when it comes to notions of fine-grained content, we may well be

on shakier ground.

To see what I have in mind, consider the proposal I mentioned above: a ‘world’, in the

fine-grained sense, is a set of sentences in the subject’s language which is such that the

subject is not able to rule out a priori that every sentence in the set is true. Any sentence

can be made true by altering its meaning, so in order for this proposal to be interesting

one has to assume that the meanings of the sentences in question remain fixed. And

not any notion of meaning will do. If, for example, one were to take the meaning of a

name to be its referent, fixing the meaning of ‘Hesperus is Phosphorus’ would be enough

to guarantee its truth. So by fixing meanings one would fix more than is knowable a

priori. What one is needs in order to get the right results is a notion of meaning such

that mastery of a language is enough to know the meanings of its sentences. What one

needs, in other words, is a notion like Fregean sense or primary intension. I myself am

pessimistic about the prospects of articulating a characterization of such notions in a way

that would be robust enough for the needs of a well-regimented folk-psychology. But you

don’t have to share my pessimism to agree that by moving towards fine-grained contents

one acquires an explanatory burden, and that it is not obvious that addressing such a

burden would be straightforward.

It is also worth noting that we are stuck with coarse-grained content regardless of

whether we also bring in a finer-grained notion of content. As it is understood here, the

aim of folk-psychology is to predict behavior. And we want the theory to issue predictions

of the following form:

Under such-and-such circumstances, the subject will behave in ways that

cause it to come about that thus-and-such.

5.2. RATIONAL ACTION 119

[Example: After a snow-storm, the subject will behave in ways that cause it

to come about that her sidewalk gets cleared.]

In order for predictions of this kind to be useful to the theorist, it had better be the

case that the blanks are filled with statements that the theorist regards as de mundo

intelligible. Consider what would happen if one’s folk-psychology delivered the following:

While eating a big meal, the subject will behave in ways that cause it to come

about that she is in the proximity of a glass containing water but no H2O.

To be filled with water just is to be filled with H2O. So the best the theorist could do

with such a prediction—if it is to be taken at face value—is conclude that the subject

won’t be eating a big meal.

The alternative is to work on the assumption that the prediction should not be taken

at face value. Perhaps what it really means is something like:

While eating a big meal, the subject will behave in ways that cause it to come

about that she is in the proximity of a glass containing a watery substance

but no H2O.

This is certainly a useful prediction, but notice that the point at which it became useful

was precisely the point at which we were able substitute a de mundo intelligible statement

for the original de mundo unintelligible statement: that there be a glass containing a

watery substance but no H2O.

More generally, the situation is this. Whether or not one’s folk-psychology uses fine-

grained contents to come up with predictions, the predictions themselves must be stated

in terms of counterfactuals that the theorist is able to assess. And since the scenarios

that are counted by the theorist as de mundo intelligible are precisely the scenarios she

regards as metaphysically possible, this means that the theory must issue predictions in

coarse-grained terms. It must, in effect, specify a coarse-grained content (e.g. the set of

120 CHAPTER 5. CONTENT

worlds whereby the subject is in the proximity of a glass containing a watery substance

but no H2O) and claim that under particular circumstances—also specified in coarse-

grained terms—the subject will behave in ways that cause it to come about that that

content is satisfied.

There is, in other words, no escaping coarse-grained content. If one goes in for

fine-grained contents one is acquiring an additional commitment, not substituting one

commitment for another.

5.3 Belief-Attributions

An Obvious Problem?

Someone might believe that water is H2O, and fail to believe that√

81 = 9. And yet

‘water is H2O’ and ‘√

81 = 9’ have the same coarse-grained content, since they are both

necessarily true.

It is tempting to conclude from this that it would be obviously wrong to model

the content of a subject’s belief state in coarse-grained terms. But that would be a

mistake. The belief-ascriptions in question only pose an obvious problem for coarse-

grained accounts of content in the presence of further assumptions—assumptions that a

friend of coarse-grainedness has independent reasons to reject.

Here is an example of an assumption that would lead to trouble:

The Fregean Assumption

What it takes for a belief-attribution of the form pS believes that φq to be true

is for for the subject to have a belief whose content is the content expressed

by φ.

When conjoined with a coarse-grained account of belief, the Fregean Assumption allows

one to go from the uncontroversial observation that ‘water is H2O’ and ‘√

81 = 9’ have

5.3. BELIEF-ATTRIBUTIONS 121

the same coarse-grained content, to the disastrous conclusion that ‘S believes that water

is H2O’ is equivalent to ‘S believes that√

81 = 9’.

Fortunately, there are independent reasons for rejecting the Fregean Assumption.

Consider the question of what one learns about a subject when one is told that she

believes that water is H2O. What one learns is not that in order for the world to be as

the subject’s belief-state represents it to be, it must be the case that things composed

of water (i.e. things composed of H2O) are composed of H2O. One knew that already.

(Trivially, any way for the world to be is such that things composed of H2O are composed

of H2O.) In a typical context, what one learns is something along the the lines of “in

order for the world to be as the subject’s belief-state represents it to be, it must be

the case that certain watery things are composed of H2O” (though one should expect

the details to be highly sensitive to the particularities of the context.) So—contra the

Fregean Assumption—one should refrain from identifying the content of the sentence

embedded in a belief-attribution and what the belief-attribution teaches us about the

subject’s belief-state.

The Plan

Say that a belief-attributoin of the form pS believes that φq is pleaonastic if φ is non-

contingent. I have argued against the claim that pleonastic belief-attributions pose an

immediate problem for coarse-grained accounts of belief. But that doesn’t mean that

pleonastic belief-attributions won’t lead to trouble. A pleonastic belief-attribution such

as ‘S believes that H2O’, or ‘S believes that√

81 = 9’, can be used to report significant

cognitive accomplishments on the part of the subject. So a coarse-grained theory of belief

had better have a story to tell about how such accomplishments are to be modeled. The

chief burden of the present chapter is to outline such a story.

It is important to be clear that explaining how to model the cognitive accomplishments

that get reported by marketplace belief-ascriptions is not the same as giving a semantics

122 CHAPTER 5. CONTENT

for marketplace belief-ascriptions. The project of understanding how expressions of the

form ‘S believes that φ’ work in natural language is an interesting one. But it is a project

in philosophical linguistics, and our present focus is on developing a folk-psychological

account of behavior. Marketplace belief-attributions have played a role in our discussion.

But only because they supply information about a subject’s cognitive state that can

be put to use by folk-psychology, and we want to make sure that our regimented folk-

psychology is able to accommodate the relevant information. There is no reason to

think that anything like marketplace belief-reports will figure in a well-regimented folk-

psychology. (Notice, in particular, that they are not required for successful application

of the Fundamental Principle. Application of the principle requires nothing beyond the

ascription of a content to the subject’s belief-state and the ascription of a content to

the subject’s desire-state. In the original version of the Fundamental Principle, each of

these contents can be modeled by a set of possible-worlds; in the probabilistic version

the content of the subject’s belief-state can be modeled by a probability assignment, and

the content of the subject’s desire-sate can be modeled by a value-function.)

Easy Cases and Hard Cases

I noted earlier that when one is told that a subject believes that water is H2O what

one learns is not that the subject represents the world as satisfying a trivial condition

(i.e. the condition that things composed of water be composed of water). What one learns

is that the subject represents the world as satisfying a certain non-trivial condition (as

it might be: the condition that certain watery things be composed of H2O). The result

is that, even though ‘S believes that water is H2O’ is a pleonastic belief-attribution,

a coarse-grained account of belief would have no trouble accommodating the cognitive

accomplishment that is being reported. When coarse-grained contents are taken to be

sets of possible worlds, this can be done by ensuring that every possible world in the

content of the subject’s belief-state is a world in which, as it might be, certain watery

5.3. BELIEF-ATTRIBUTIONS 123

things are composed of H2O.

This example is representative of a large range of cases in which what one learns

about the subject’s belief-state on the basis of a pleonastic belief-attribution is simply

that the subject represents the world as satisfying a certain non-trivial condition which

is not expressed by the sentence embedded in the belief-attribution. It may not always

be easy to determine exactly which non-trivial condition is to be associated with a given

belief-attribution. But that is not something we need to worry about for present purposes

because marketplace belief-attributions are not part of our regimented folk-psychology.

What matters for present purposes is whether the sorts of cognitive accomplishments

that get reported by marketplace belief-attributions can be modeled in coarse grained

terms. And, as the example of a subject’s believing that water and H2O illustrates,

coarse-grained contents are eminently well-suited for the job when the cognitive accom-

plishment in question consists simply of coming to represent the world as satisfying a

certain non-trivial condition. Unfortunately, not every pleonastic belief-attribution is as

straightforward as that. Consider, for example, ‘S believes that√

81 = 9’. What sort of

cognitive accomplishment might such a belief-attribution be used to report?

To make things interesting, I shall assume that one is a mathematical trivialist, in

the sense of chapter 4. Accordingly, any way for the world to be is, trivially, such that√

81 = 9. So one wouldn’t learn anything interesting about the subject if one learned

that in order for the world to be as the subject represents it to be, it must satisfy the

condition of being such that√

81 = 9. (If you are not a mathematical trivialist, please

use a logical truth in place of ‘√

81 = 9’.)

What makes this case difficult is that—unlike ‘S believes that water is H2O’—it is not

clear than one could find a non-trivial condition which, while not expressed by ‘√

81 = 9’,

could be used to capture the sort of cognitive accomplishment the gets reported by ‘S

believes that√

81 = 9’.

The sentence ‘ ‘√

81 = 9’ expresses a truth’ expresses a contingent truth, since ‘√

81 =

124 CHAPTER 5. CONTENT

9’ would not have expressed a truth had its constituent terms had different meanings.

So one might be tempted to think that the cognitive accomplishment that gets reported

by ‘S believes that√

81 = 9’ consists, at lest in part, of coming to represent the world

as satisfying the (non-trivial) condition of being such that ‘√

81 = 9’ expresses a truth.

But it turns out that not even semantic ascent will do in a case like this. As stressed

in Field (1986), one will run into trouble whenever the subject is assumed to know

that the relevant mathematical axioms are true. For any world in which a suitable

axiomatization of arithmetic is true is also a world ‘√

81 = 9’ expresses a truth. So one

would be left with the unpalatable conclusion that anyone who knows that the axioms

are true should thereby enjoy the cognitive accomplishment that gets reported by ‘S

believes that√

81 = 9’.

It is hard cases like these that pose a real challenge to coarse-grained accounts of

belief.

Fine-Grained Content to the Rescue?

It is tempting to think that the problems would disappear if only we brought in fine-

grained contents. That would be a mistake. Bringing in fine-grained contents would only

postpone the problem. Worse: it might give the illusion of progress where there is none.

The aim, recall, is to model cognitive accomplishments of the sort that get reported by

belief-attributions like ‘S believes that√

81 = 9’. And not any model will do. We want a

model that will allow us to issue predictions about the subject’s behavior. How would one

proceed if one availed oneself of finer-grained contents? Suppose, for example, that one

were to claim that what one learns from the belief-attribution is that a Mentalese analogue

of the English sentence ‘√

81 = 9’ is stored in the subject’s ‘belief box’. (Equivalently:

one learns that every fine-grained ‘world’ compatible with the object’s beliefs contains

the Mentalese analogue of ‘√

81 = 9’.) So far so good. But now what? How is the

theorist to use this information in issuing predictions about the subject’s behavior?

5.3. BELIEF-ATTRIBUTIONS 125

One idea is to emulate the Fundamental Principle. The theory could issue predictions

by claiming that the subject’s actions will constitute the most sensible way of bringing

it about that the sentences in the subject’s ‘desire box’ are verified in a world in which

the sentence’s in the subject’s ‘belief box’ are verified. But then we are back to where

we started. Any way for the world to be is, trivially, such as to verify ‘√

81 = 9’ (or its

Mentalese analogue). So learning that the Mentalese analogue of ‘√

81 = 9’ is stored in

the subject’s ‘belief box’ has absolutely no effect on the predictive power of the theory.

We have not yet managed to come up with a good model of the cognitive accomplishments

that get reported by belief-attributions like ‘S believes that√

81 = 9’.

Another idea is to link information about the contents of the subject’s ‘belief box’

to predictions about the subject’s behavior by doing cognitive science rather than folk-

psychology. I have never seen a proposal of this kind developed in enough detail to allow

for adequate assessment. But it is important to be clear that by appealing to cognitive

science, proponents of fine-grained content would be changing the subject. It can be

agreed on all sides that a fully developed cognitive science would supply a much better

model of cognitive accomplishment—and much better behavioral predictions—than our

best folk-psychology. But the point of the present exercise is not to develop the best

possible theory of cognitive accomplishment, or the best possible account of behavior.

It is to develop a well-regimented folk-psychology, and use it to shed light on some of

the intentional notions that are of interest to philosophers. What matters for present

purposes is whether fine-grained contents have a role to play in folk-psychology, not

whether they have a role to play in cognitive science.

There may well be a way developing a bridge-theory that links information about

the fine-grained contents of a subject’s mental states to predictions about the subject’s

behavior, and does so in a way that is broadly in keeping with folk-psychology. Perhaps

one could start with the claim that the subject will be disposed to assent to φ whenever

a Mentalese analogue of φ is in her belief-box. But then what?

126 CHAPTER 5. CONTENT

In the absence of a minimally developed theory, it would be hasty to suggest that

wheeling in fine-grained content delivers an account of cognitive accomplishment in math-

ematics of the kind we are after. Even if an appeal to fine-grained contents does ultimately

set the stage for a theory of cognitive accomplishment, the real work won’t get done until

the theory itself is developed.

In what follows I will argue that the most perspicuous way of developing the missing

theory makes no use of fine-grained contents.

5.4 Cognitive Accomplishment in Logic and Mathe-

matics

The Desiderata

Consider a subject who knows that√

81 = 9. How might this knowledge be manifested

in behavior? Here are some examples:

1. Assent

The subject is able to give a correct answer to the question ‘Is it the case that√

81 = 9?’ (or to some analogue of this question in a language she understands).

2. Deduction

The subject is able to perform certain kinds of deductions. For instance, she might

be able to derive ‘√

81 is divisible by 3’ from ‘9 is divisible by 3’.

3. Application

The subject is able to use information about the natural world that was acquired

under one set of circumstances for the purposes of tasks that take place under

very different sets of circumstances. For instance, upon discovering that it takes

81 one-square-meter tiles to cover a plot of land which is known to be perfectly

5.4. COGNITIVE ACCOMPLISHMENT IN LOGIC AND MATHEMATICS 127

square-shaped, the subject might acquire the ability to predict how many meters

of fencing would be needed to build a perimiter.

Each of abilities can be usefully modeled as an information-transfer ability. Let us begin

with Assent. Suppose that the subject has known all along that the Dedekind Axioms

are true, and that she comes to see that ‘√

81 = 9’ is true by deriving it from the axioms.

It is then true to begin with that the subject possesses a certain piece of information—the

information that arithmetical vocabulary is used in such a way that the axioms turn out

to be true—and is in a position to deploy this information for the purposes of a certain

range of tasks; for instance: answering the question ‘Are the axioms true?’. But, on

reasonable assumptions, any world in which arithmetical vocabulary is used in such a

way that the axioms are true is also a world in which arithmetical vocabulary is used in

such a way that ‘√

81 = 9’ is true. So the information that the subject possess is to begin

with includes the information that ‘√

81 = 9’ is true. And yet she is unable to deploy this

information for the purposes of answering the question ‘Is it the case that√

81 = 9?’.

What happens after she performs the relevant deduction is that she acquires the ability

to deploy this information—information she already possessed—in the service of new

tasks; in particular: the task of answering the question ‘Is it the case that√

81 = 9?’. So

her cognitive accomplishment can be construed, at least in part, as the acquisition of an

information-transfer ability: she has broadened the range of tasks with respect to which

she is able to deploy the information that arithmetical vocabulary is used in such a way

that the axioms turn out to be true.

Next consider Deduction. How might one model the fact that a subject who learns

that√

81 = 9 acquires the ability to derive ‘√

81 is divisible by 3’ from ‘9 is divisible by

3’? Consider a subject who knows that the axioms are true, and is able to deploy this

information not only for the purposes of answering the question ‘are the axioms true?’,

but also for the purposes of answering the question ‘Is 9 divisible by 3?’. Part of what

happens when she learns that√

81 = 9 is that she is able to extend the range of tasks

128 CHAPTER 5. CONTENT

with respect to which she is able to deploy this information even further. She is now able

to deploy it for the purposes of answering the question ‘Is√

81 divisible by 3’.

Finally, consider Application. So far we have focused on the deployment of linguistic

information—information to the effect that arithmetical vocabulary is used in such a way

that the axioms turn out to be true—in the service of an essentially linguistic task: the

task of answering a linguistically-posed question. But when we describe a subject as

knowing that√

81 = 9 we sometimes expect this knowledge to be manifested in her

non-linguistic behavior as well. Recall our farmer and her square piece of land. She

knows that it takes 81 one-square-meter tiles to cover her land. She therefore possesses

the information that the land is 81m2 in area, and is able to deploy it for the purposes

of tiling. Assume, for simplicity, that we may ignore worlds in which the land is not

flat, or in which the geometry is not Euclidean. Then for a square piece of land to be

81m2 in area just is for each of its sides to be 9m in length. So there is no difference

between possessing the information that the land is 81m2 in area and possessing the

information that each of its sides is 9m in length. But the subject might nonetheless

lack the ability to deploy such information for the purposes of, say, buying just the right

amount of fence to build a perimeter. By doing mathematics, however, she can acquire

an information-transfer ability: the ability to deploy information—information that was

previously available only for the purposes of tiling—in the service of new tasks, such as

perimeter-building. This is a cognitive accomplishment that might be reported by saying

of the subject that she knows that√

81 = 9.

Fragmentation

A subject’s cognitive system is modeled as fragmented when the theorist’s attributions

of content to the subject’s mental states are relativized to tasks that the subject might

be engaged in.

Suppose, for example, that our farmer is confused about the size of her land. When it

5.4. COGNITIVE ACCOMPLISHMENT IN LOGIC AND MATHEMATICS 129

comes to the project of going to the store to buy tiles for covering her land, she behaves

as if she believed that her land has an area of 81m2 (i.e. a side-length of 9m). As it might

be: she loads her truck with 81 one-square-meter tiles. But when it comes to the project

of going to the store to buy fencing to build a perimeter around her land, she behaves as

if she believed that that her land has a side-length of 10m (i.e. an area of 100m2). As it

might be: she loads her truck with 40m of fencing.

Such a subject can be usefully described as fragmented. We can say that relative to

the task of buying tiles the subject’s-belief state represents the land as being 81m2 in

area, and relative to the task of buying fencing the subject’s belief-state represents the

land as being 100m2 in area. Predictions about the subject’s behavior can then be made

on the basis of a suitably modified version of the Fundamental Principle:

Fundamental Principle (relativized version)

When a rational subject is engaged in task τ , her intentional behavior is

precisely the behavior that would constitute the most sensible way of bringing

about satisfaction of the subject’s desires-relative-to-τ in a world satisfying

the subject’s beliefs-relative-to-τ .

An advantage of describing a subject as fragmented is that it gives us an attractive way

of modeling her information-transfer abilities. One can model an information-transfer

ability as the instantiation of a relation of accessibility amongst different fragments within

the subject’s cognitive state.

Suppose, for example, that our farmer learns that√

81 = 9. Earlier I suggested

that a cognitive accomplishment of this kind might involve a family of information-

transfer abilities, and in particular the ability to deploy information that was previously

available only for the purposes of buying tiles, in the service of new tasks, such as buying

fencing. On a fragmentation model, this can be captured by saying that the fragment

corresponding to the task of buying tiles and the fragment corresponding to the task of

130 CHAPTER 5. CONTENT

buying fencing have become accessible to each other.

When different fragments are accessible to each other, there is pressure for them to

become synchronized. In the case of the farmer who is confused about the size of her

land, synchronizing the different fragments will require resolving an internal tension. By

learning that√

81 = 9 she gains access to the fact that she has been proceeding on

different assumptions relative to different purposes. The tension will be resolved if she

updates the beliefs corresponding to one of the fragments so as to make them consistent

with the beliefs corresponding to the other. But she may or may not be able to resolve

it, since she may not be sure which fragment to revise.

In other cases, however, synchronization need not consist in the resolution of an

internal tension. Consider a farmer who is able to deploy the information that her piece

of land is 81m2 in area for the purposes of buying tiles, but not for the purposes of buying

fencing: if she were to be faced with the project of building a perimeter around her land,

she would simply have no idea how much fencing to buy. When the farmer learns that√

81 = 9, the fragment corresponding to the task of tile-purchase and the fragment

corresponding to the task of fencing-purchase become accessible to each other, and there

is therefore pressure for them to become synchronized. But in this case synchronization

is easy: it is a matter of having the better-informed fragment update its less-informed

counterpart. In the case at hand, the fragment corresponding to fencing purchases is

updated in accordance with the content of the fragment corresponding to tile purchases.

So the model will predict—via the updated version of the Fundamental Principle—that

as a result of learning that√

81 = 9, the farmer acquires the ability to deploy the

information that her piece of land is 81m2 in area for the purposes of fencing-purchase.

Let me explain in more general terms how a fragmented cognitive system might be

modeled. (Here I am especially indebted to Elga.) A fragmented belief-state is modeled

as an ordered-triple 〈T, f, α〉. T is a domain of ‘tasks’ that the subject might be engaged

in; f is a function that maps each task in T to a content (in the simplest case the content

5.4. COGNITIVE ACCOMPLISHMENT IN LOGIC AND MATHEMATICS 131

is a set of possible worlds, but a probabilistic version of the proposal might take a belief

state to be a probability distribution over the space of possible worlds); and α is an

reflective and symmetric accessibility relation defined over elements of T . (A fragmented

desire-state is to be modeled analogously.)

The assignment of a particular triple 〈T, f, α〉 is used to issue predictions about a

subject via the relativized version of the Fundamental Principle, and is constrained by

(suitably relativized versions of) folk-psychological principles such as those described in

section 5.2. But there is an important addition to the list:

Synchronization

By and large, the contents assigned to mutually accessible members of T will

tend to become synchronized over time, and become synchronized in the most

sensible way possible. (If, however, there is no salient way of resolving a con-

flict between mutually accessible members of T , the corresponding contents

will tend to remain unsynchronized.)

This principle can be used to make predictions about the subject’s future behavior, via

the Fundamental Principle. But it also supplies the basic method for forming hypothe-

ses about which of the subject’s fragments are mutually accessible, since the theorist

can select present accessibility relations by determining which assignments of mutual

accessibility are good ways of making sense of the subject’s past behavior.

T should also be chosen on the basis of considerations of theoretical fruitfulness. In

particular, it should chosen so as to supply a happy medium between recognizing too

many ‘tasks’ to allow for systematic theorizing and recognizing too few ‘tasks’ to do

justice to the subject’s behavior.

132 CHAPTER 5. CONTENT

Modes of Presentation

The proposal I have been defending is certainly not incompatible with a fine-grained

account of content. In fact, it could be redescribed in terms of ‘modes of presentation’.

Rather than saying that the subject believes a certain coarse-grained content for the

purposes of a given task, one could say that the subject believes that content under a

‘mode of presentation’ that corresponds to the task in question. So one could emulate

the present proposal in a fine-grained setting by thinking of fine-grained contents as pairs

consisting of a coarse-grained content and a ‘mode of presentation’.

The problem with talk of modes of presentation is that one runs the risk of thinking

that progress has been made when in fact there is none. In order for the notion of a

mode of presentation to do any real work one needs to know how the various modes of

presentation are supposed to differ, and how differences in mode of presentation are meant

to result in different predictions about the subject’s behavior. On the proposal I have been

defending, these two issues are explicitly addressed. (Different ‘modes of presentation’

correspond to different tasks in the service of which the subject might deploy a certain

piece of information, and the predictive upshot of a ‘mode of presentation’ is given by

the relativized version of the Fundamental Principle.) In the absence of a theory of this

kind, however, the notion of a mode of presentation is nothing more than a label. One

has supplied a place-holder for a theory of cognitive accomplishment, but no meaningful

progress has been made when it comes to developing the theory itself.

Back to Logic and Mathematics

I have been arguing that folk-psychology can accommodate cognitive accomplishment in

logic and mathematics by: (1) taking the subject to have a fragmented belief-state, and

(2) modeling logical and mathematical feats as instantiations of the accessibility relation

amongst previously unrelated fragments of her cognitive system.

It is nonetheless tempting to think that there must be more to knowing that√

81 = 9

5.4. COGNITIVE ACCOMPLISHMENT IN LOGIC AND MATHEMATICS 133

than having a given range of information-transfer abilities. “I have come to understand a

mathematical fact!”, one sometimes hears it said. Perhaps there is an important insight

behind such intuitions—something that will be captured by tomorrow’s cognitive science.

But that is not something that needs to be decided for present purposes. Our aim is to

develop a well-regimented folk-psychology. So all that is required for present purposes

is that the fragmentation model be enough to deliver the sorts of behavioral predictions

one expects from folk-psychology. And, as far as I can tell, ascribing the subject a

suitable family of information-processing abilities is enough to deliver the right behavioral

predictions.

It is no part of the present proposal that every assertion of ‘S believes that√

81 =

9’ reports a cognitive achievement that correspond to the same range of information-

transfer abilities. Belief-ascriptions of this kind should be thought of as reporting different

cognitive achievements in different contexts. When one describes a linguistically-deprived

farmer by saying ‘S believes that√

81 = 9’ one may be ascribing her the ability to use

the information that it takes 81 one-square-meter tiles to cover a square piece of land

to buy the right amount of fencing, and not the ability to use the information that the

Dedekind Axioms are true to answer the question ‘Is ‘√

81 = 9’ true?’. But the very

same belief-report might be used to describe the cognitive accomplishments of a student

of arithmetic who has learned to derive ‘√

81 = 9’ from the axioms but knows nothing

about applied arithmetic.

If ψ is a truth of pure mathematics devoid of application, the cognitive accomplish-

ment reported by belief ascriptions of the form pS believes that ψq’ will usually get

modeled as a feat of information-transfer amongst fragments corresponding to linguistic

tasks. This is appropriate because knowledge of application-free mathematics is normally

only manifested in the subject’s linguistic behavior. So the only predictions one should

expect one’s folk-psychology to deliver on the basis of the relevant cognitive accomplish-

ments concern the subject’s linguistic behavior.

134 CHAPTER 5. CONTENT

In the simplest case, the cognitive accomplishment reported by pS believes that ψq

can be modeled in two steps. First, we capture the fact that the subject takes ψ to be

true by updating the fragment corresponding to the task of answering the question pIs

ψ true?q so that the world is represented as being such that ψ is true. When the subject

comes to believe that ψ is true as a result of deriving it from the axioms, this can be

done by letting the fragment corresponding to the task of answering the question pIs ψ

true?q become accessible to the the fragment corresponding to the task of answering the

question pAre the axioms true?q.

Second, we capture the fact that the subject is in a position to make inferences on

the basis of ψ. This can be done by:

A. Identifying a family of pairs of sentences 〈θ, ζ〉 such that: (i) ζ is an ‘easy’ conse-

quence of θ and ψ, and (ii) the subject takes θ to be true; and

B. letting the fragment that corresponds to the task of answering the question pIs ζ

true?q become accessible to the fragments that correspond to the tasks of answering

the question pIs θ true?q and answering the question pIs ψ true?q.

When the theorist’s cognitive system is not too different from the subject’s, the theorist

can count an inference as ‘easy’ for the subject just in case she would find it easy herself.

(A more sophisticated version of the theory would take a stand on how smart the subject

is, and model talented mathematicians as having more distant deductive horizons than

their less talented counterparts.) In the general case, however, the problem of deciding

when to count an inference as ‘easy’ is highly non-trivial. If the theorist’s cognitive

system is sufficiently different from the subject’s, then the fact that the theorist regards

an inference as easy supplies no real grounds for thinking that the subject should be

modeled as treating the inference as easy.

When we use folk psychology in ordinary life, our subjects are usually fellow humans,

and they are often members of our own community. So the difficult cases are less likely

5.4. COGNITIVE ACCOMPLISHMENT IN LOGIC AND MATHEMATICS 135

to come up. This obscures the fact that the problem of determining which inferences to

count as easy is non-trivial. But when it comes to subjects with alien cognitive systems,

we may well be in uncharted waters. Ordinary folk-psychology may not supply much

guidance about how to solve the problem in such cases.

Independent Motivation

The postulation of fragmented belief-states is not an ad hoc maneuver. It can be moti-

vated independently of the project of giving a coarse-grained account of belief.

It should be agreed on all sides that a subject might possess information that she is

able to deploy in the service of some tasks but not others. Consider, for example, the

expert gymnast, who is able to perform a perfect back salto but is unable to explain how

she does it. The gymnast possesses a piece of information—information to the effect that

such-and-such bodily movements are required to deliver a back salto—that she is able to

deploy for certain practical purposes (i.e. the task of performing a back salto) but not

for theoretical purposes (e.g. the task of explaining how to perform a back salto). Such

differential access to information is naturally accounted for on a fragmented model.

It should also be agreed on all sides that a subject’s beliefs can be incoherent. I once

saw my friend Pedro eating an enormous breakfast while reporting that he never eats

big breakfasts. He suddenly realized the inconsistency, and we both laughed. Incoherent

belief states of this kind are naturally accounted for on a fragmented model. Before the

crucial realization, Pedro can be described by saying that for the purposes of articulating

a general description of his eating habits he represents the world as being such that his

breakfasts are always light, and for the purposes of reporting how much he is currently

eating he represents the world as being such that he is eating a very substantial break-

fast. Pedro’s realization can then be modeled by saying that the two fragments become

accessible to each other, and that the former is updated so as to represent the world

as being such that he has a light breakfast almost always. (For additional discussion,

136 CHAPTER 5. CONTENT

see Lewis (1982).)

Moral: By modeling a cognitive system as fragmented one can give a unified treatment

of three phenomena that might have initially seemed unrelated—imperfectly accessible

information, incoherence and cognitive accomplishment in logic and mathematics.

5.5 Mary and the Tomato

In fact, there is a fourth kind of phenomenon that can be accounted for by appeal to

fragmentation. Here is Frank Jackson’s Knowledge Argument :

Mary is confined to a black-and-white room, is educated through black-and-

white books and through lectures relayed on black-and-white television. In

this way she learns everything there is to know about the physical nature of

the world. She knows all the physical facts about us and our environment,

in a wide sense of ‘physical’ that includes everything in completed physics,

chemistry, and neurophysiology, and all there is to know about the causal

and relational facts consequent upon all this, including of course functional

roles. If physicalism is true, she knows all there is to know. For to suppose

otherwise is to suppose that there is more to know than every physical fact,

and that is just what physicalism denies. . . It seems, however, that Mary does

not know all there is to know. For when she is let out of the black-and-white

room or given a color television, she will learn what it is like to see something

red, say. This is rightly described as learning—she will not say “ho, hum.”

Hence, physicalism is false. (Jackson (1986), p. 29.)

What the argument brings out is that physicalists face a challenge. They must somehow

accommodate the fact that it seems like Mary acquires information about the world—

information she did not already have—when she first experiences the sensation of seeing

5.5. MARY AND THE TOMATO 137

red, even though physicalism appears to entail that she does not.

Suppose that, before her release, Mary is informed that she will be presented with

a red tomato at noon. Suppose, moreover, that physicalism is true, and, in particular,

that for a normal human to experience the sensation of seeing red just is for her to be

in brain-state R. Mary knows this. So any world compatible with what Mary knows is a

world in which her brain enters state R at noon. Accordingly, Marry possesses a certain

piece of information—the information that her brain will be in state R at noon—and is

able to deploy it for the purposes of certain tasks (for instance, the task of answering

the question ‘Will your brain be in state R at noon?’). She does not, however, have the

ability to deploy this information in the service of certain other tasks.

Suppose, for example, that before seeing the tomato Mary is shown a red ball, but is

not told the color of the ball. Any world in which Mary has a sensation with the same

kind of phenomenal character as the sensation she experiences when she sees the ball is

a world in which she is in brain-state R. So every world in which Mary is in brain-state

R at noon and has an experience with the relevant phenomenal character before noon is

a world in which Mary has two experiences with the same kind of phenomenal character.

It follows that two pieces of information that are already in Mary’s possession—the

information that she will be in brain-state R at noon, and the information that, while

seeing the ball, she experiences a sensation with this phenomenal character—are enough

to determine that the experience of seeing the tomato at noon will have the same kind

of phenomenal character as the experience of seeing the ball. And yet Mary is unable to

deploy the information in her possession for the purposes of answering the question ‘Is

this what it will be like to see the tomato?’ when she is still looking at the ball.

There is an attractive way of modeling Mary’s predicament in a fragmented system.

Whereas she is able to deploy the information that her brain will enter state R at noon

for the purposes of answering the question ‘Will your brain be in state R at noon?’

(or the question ‘Will you see something red at noon’), she is not able to deploy this

138 CHAPTER 5. CONTENT

information for the purposes of answering the question ‘Is this what it will be like to

see the tomato?’ while looking at the ball. What she lacks, in other words, is a certain

information-transfer ability.

When Mary is finally shown the tomato, she accomplishes in a cognitive feat. For she

is now able to to deploy the information that her brain enters state R at noon for the

purposes of answering the question ‘Is this what it will be like to see the tomato?’ while

looking at the ball. This shouldn’t be modeled as a feat of information acquisition, since

Mary had all the relevant information to begin with. It should be modeled by treating

some of the fragments in Mary’s cognitive system as accessible to one another.

Part II

Detours

139

Chapter 6

Deep Metaphysics

‘Deep Metaphysics’ is my label for the sort of metaphysics that makes non-metaphysicians

cringe—the kind that outsiders see as relying on distinctions without a difference, and

that the Logical Empiricists reacted against in the first third of the Twentieth Century.

The aim of this chapter is to further articulate the notion of Deep Metaphysics, and say

something about how I think metaphysical debate ought to be constrained.

A number of classical debates in metaphysics can be thought of as revolving around

identity statements:

1. Mereology

Is it true, in general, that for the fusion of the Fs to exist just is for the Fs to exist?

2. Facts and Properties

Is it true, in general, that for the fact that φ to obtain just is for it to be the case

that φ.

Is it true, in general, that for an object to instantiate the property of running just

is for the object to run?

3. Philosophy of Mathematics

141

142 CHAPTER 6. DEEP METAPHYSICS

Is it true, in general, that for the number of the Fs to be n just is for there to be

n Fs?

4. Physicalism

Is it true, in general, that for such-and-such a mental state to obtain just is for a

certain brain-state to obtain?

5. Modality

Is it true, in general, that for there to be a possible world at which φ just is for it

to be the case that possibly φ?

Is it true, in general, that for it to be the case that possibly φ just is for it to be

the case that it is de mundo intelligible that φ?

6. Time

Is it true, in general, that for a time to be present just is for it to have a certain

relational property?

7. Causation

Is it true, in general, that for E to cause C just is for there to be a certain kind of

counterfactual relationship between E and C?

8. Personal Identity

Is it true, in general, that for x and y to be the same person just is for there to be

continuity of such-and-such a kind between x and y?

When one accepts an identity statement one closes a theoretical gap. So the more

identity statements one accepts when doing metaphysics, the less one will be burdened

by awkward metaphysical questions. Suppose, for example, that you answer ‘yes’ to

question 3 on the list: you think it is it true, in general that for the number of the Fs to

143

be n just is for there to be n Fs. Then you should regard queries such as the following as

misguided: “I can see that there are no dinosaurs. What I want to know is whether it is

also true that the number of the dinosaurs is Zero. I would like to understand, moreover,

how one could ever be justified in taking a stand on this issue, given that we have no

causal access to the purported realm of abstract objects.” You should think that such

questions rest on a false presupposition. They fail to take on board the fact that for the

number of the dinosaurs to be Zero just is for there to be no dinosaurs. (For an account

of mathematics along these lines, see chapter 4.)

Conversely, when one rejects an identity statement one opens a theoretical gap. Sup-

pose, for example, that one answers ‘no’ to question 6: one rejects the idea that for a

time to be present just is for it to have a certain relational property. One is thereby left

with a theoretical gap that would have otherwise been avoided: that of explaining what

it takes for a time to be present simpliciter (as opposed to present relative to some time

or other). One might address the gap by saying something like ‘to be present simpliciter

is to be at the edge of objective becoming’ (a piece of Deep Metaphysics, if you ask me).

By thinking of metaphysics as concerned, in part, with identity statements one can

shed light on the question of how metaphysical debate ought to be constrained—how one

should go about deciding which of two metaphysical positions is correct.

As noted in section 1.6, the truth-conditions of an identity statement are always either

trivial or impossible. And, of course, there is never any disagreement about whether

the world satisfies either of these conditions. What goes on when there is controversy

surrounding an identity-statement is, rather, that one’s views about whether the identity-

statement should be taken to express the trivial truth-conditions or the impossible truth-

conditions can be tied up with issues such as the following:

1. Empirical questions not expressed by the identity statement.

[Does a single planet play both the morning-star and evening star roles? Is there

144 CHAPTER 6. DEEP METAPHYSICS

is such a thing as caloric fluid?]

2. Differences in the lines of research that are regarded as fruitful.

[Would it be fruitful to engage in the project of accounting for heat-related phe-

nomena by postulating a new substance?]

3. Disagreements about which way of using language is most convenient for the pur-

poses at hand.

[Is it easier to express interesting zoological claims by using ‘elephant’ to mean

‘member of thus-and-such a lineage’ or ‘has the sort of genetic material that could

be combined with that of such-and-such individuals to produce fertile offspring’?]

4. Disagreement about which properties are endowed with ‘metaphysical privilege’.

[Does the property of having thus-and-such a lineage enjoy greater ‘metaphysical

privilege’—and therefore greater eligibility as a referent for the word ‘elephant’—

than other candidate properties?]

In the special case of identity-statements pertaining to metaphysics, the first of these

four sources of disagreement—disagreement about empirical matters not expressed by

the identity statement—can be expected to play a fairly limited role. (It isn’t wholly

absent, though: Einstein’s Theory of Special Relativity, for instance, could be relevant

to assessing item 6 on our list.)

Attention to the fourth source of disagreement—disagreement about which properties

are endowed with ‘metaphysical privilege’—has, on the other hand, played a substantial

role in contemporary metaphysical debate. It seems to me that this is an unfortunate

turn of events. By wheeling in metaphysical privilege, one certainly makes room for

the view that debates that might have appeared to be merely terminological turn out

to concern substantial matters. By claiming, for instance, that a debate about what a

species consists in is ultimately a debate about which of a range of candidate-properties

145

enjoys greater metaphysical privilege, one certainly succeeds in identifying a subject-

matter for one’s debate. But the underlying issue remains hopelessly obscure. It is bit

like saying: “the reason the debate about which outfits are objectively fashionable—

not just fashionable relative to the tastes of some community or other—is not merely

terminological is that certain outfits are metaphysically privileged: they carve the world

at the joints.” Be a fashion objectivist if you must, but don’t pretend that talk of

metaphysical privilege makes your view any less obscure. As far as I can tell, the notion

of metaphysical privilege is nothing but Deep Metaphysics. (For further discussion of the

notion of metaphysical privilege, see section 1.6.2.)

When it comes to identity-statements pertaining to metaphysics, it seems to me that

the proper way of tackling the debate is by focusing on the second of the four sources

of disagreement mentioned above: disagreement about the lines of research that are

regarded as fruitful. Consider item 4 from our list as an example. Suppose we are

considering whether to accept the identity statement ‘to experience the sensation of

seeing red just is to be in a certain brain state’. What sorts of considerations might be

used to advance the issue in an interesting way?

The Knowledge Argument immediately suggests itself. (See Jackson (1982) and Jack-

son (1986); for a review of more recent literature, see Byrne (2006).)

Mary is confined to a black-and-white room, is educated through black-and-

white books and through lectures relayed on black-and-white television. In

this way she learns everything there is to know about the physical nature of

the world. She knows all the physical facts about us and our environment,

in a wide sense of ‘physical’ that includes everything in completed physics,

chemistry, and neurophysiology, and all there is to know about the causal

and relational facts consequent upon all this, including of course functional

roles. If physicalism is true, she knows all there is to know. For to suppose

otherwise is to suppose that there is more to know than every physical fact,

146 CHAPTER 6. DEEP METAPHYSICS

and that is just what physicalism denies. . . It seems, however, that Mary does

not know all there is to know. For when she is let out of the black-and-white

room or given a color television, she will learn what it is like to see something

red, say. This is rightly described as learning—she will not say “ho, hum.”

Hence, physicalism is false. (Jackson (1986), p. 29.)

What Jackson’s argument brings out is that physicalists face a challenge. They must

somehow accommodate the fact that it seems like Mary acquires information about the

world—information she did not already have—when she first experiences the sensation

of seeing red, even though physicalism appears to entail that she does not. My own

view is that the challenge can be met. (See chapter 5.5.) But someone who thinks that

the challenge cannot be met might see the argument as motivating the introduction of

possibilities that a physicalist would regard as unintelligible. According to the physicalist,

to experience the sensation of seeing red just is to be in a certain brain state. So it

makes no sense to consider a scenario in which someone is in the brain state but lacks the

sensation. If, however, one were to give up physicalism and countenance the intelligibility

of such a scenario, one might be able to relieve some of the pressure generated by Jackson’s

argument. For one could claim that, even though Mary knew all along that she would be

in the relevant brain state when she was first shown a ripe tomato, she did not yet know

if she would also experience the relevant sensation. It is only after she is actually shown

the tomato, and experiences the relevant sensation, that she is in a position to rule out a

scenario in which she is in the brain state without having the sensation. And this ruling

out of scenarios substantiates the claim that Mary does indeed acquire information about

the world when she is first shown the tomato.

I think there are good reasons for resisting this way of addressing the puzzle. (See,

for instance, Lewis (1988).) But suppose one takes it to work. Suppose one thinks

that by creating a gap between being in the relevant brain state and experiencing the

relevant sensation—and thereby making room for the possibility of being in the brain

147

state without having the sensation—one can adequately account for a case like Mary’s.

Then one will be motivated to give up the identity statement that keeps the gap closed

(‘to experience the sensation of seeing red just is to be in a certain brain state’). Doing

so comes at a cost because it opens up space for awkward questions. For instance: “I

can see that Mary is in the relevant brain state. What I want to know is whether she

is also experiencing the relevant sensation. I would like to understand, moreover, how

one could ever be justified in taking a stand on this issue, given that we would find

Mary completely indistinguishable from her zombie counterpart, or from someone with

‘inverted’ sensations.” But one may well think that the account of Jackson’s puzzle is

attractive enough to make up for the need to address such questions.

More generally, the situation is as follows. Rejecting an identity statement comes at a

cost, since it increases the number of scenarios that are treated as intelligible, and there-

fore the number of questions that are regarded as demanding answers. But having extra

scenarios to work with can also prove advantageous, since it makes room for additional

theoretical positions, some of which could deliver fruitful theorizing. Disagreement about

whether to accept an identity statement often involves disagreement about whether the

additional positions really would be fruitful enough to justify paying the price of having

to answer a new range of potentially problematic questions.

To describe a debate as Deep Metaphysics is to see it as based on a multiplication

of possibilities that is only motivated from within. Whenever one rejects an identity

statement one increases the range of scenarios that one treats as intelligible. What is

distinctive about Deep Metaphysics is that the extra theoretical space does no indepen-

dent work: the only questions it can be used to address are questions in other regions of

Deep Metaphysics, or questions that it itself generates. It consists entirely of issues that

only the initiated would understand, and only the initiated would care about.

148 CHAPTER 6. DEEP METAPHYSICS

Chapter 7

A-worlds and the Dot Notation

7.1 Introduction

Philosophers call on possible worlds to perform different kinds of jobs. One of these jobs

is foundational : the job of explaining what it takes for modal truths to be true. “For it

to be possible that p”, a possible-worlds-foundationalist would say, “just is for there to

be a possible world at which p.” Possible worlds are also used as semantic machinery.

The semanticist needs entities for the quantifiers of her metalanguage to range over, and

possible worlds—or, more generally, possibilia—can be used to construct them.

I argued in chapter 2 that the foundational project can be carried out without making

use of a specialized modal ontology: one can appeal to identity statements instead. The

purpose of this chapter is to defend the claim that the needs of the semanticist can be

satisfied without appealing to a specialized modal ontology.

My proposal is an instance of what David Lewis called ‘ersatzism’. I argue that the

needs of the semanticist can be satisfied by using representatives for possibilia in place

of possibilia. Although there are other ersatzist proposals in the literature,1 I hope that

1See, for instance, Plantinga (1976) and chapter 3 of Lewis (1986). A recent ersatzist proposal isdiscussed in Fine (2002b) and Sider (2002) (but embraced only by Sider). The sort of proposal that Fineand Sider discuss has a more ambitious objective than the proposal developed here, since it is intended tocapture finer-grained distinctions amongst possibilities. It also relies on more substantial expressive and

149

150 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

the machinery developed here will earn its keep by delivering an attractive combination

of frugality and strength.

The proposal is frugal in two different respects. First, it is metaphysically frugal: it

is designed to be acceptable to a modal actualist, and presupposes very little by way of

ontology. (I help myself to set-theory, but do not assume a specialized modal ontology,

or an ontology of properties.) Second, the proposal is ideologically frugal: it does not

presuppose potentially controversial expressive resources such as infinitary languages or

non-standard modal operators. The point of developing machinery that presupposes

so little is that one can put it to work without having to take a stance on difficult

philosophical issues.

As far as strength is concerned, one gets a qualified version of the following claim:

anything that can be said by quantifying over Lewisian possibilia can also be said by

using the machinery developed here. The result is that the proposal can be used quite

freely in the context of semantic theorizing, without having to worry too much about

running into expressive limitations. (An especially useful feature of the proposal is that

it allows one to enjoy the benefits of quantification over sets of possibilia, which are often

appealed to in the course of semantic theorizing.)

Possible worlds theorists sometimes claim that the same individual exists according

to distinct possible worlds. (There is a world according to which I have a sister who

is a philosopher, and a world at which that very individual is a cellist rather than a

philosopher.) Ersatzist representatives for such worlds might be said to be linked. Much

of the chapter will be devoted to the phenomenon of linking.

ontological resources. (It relies, in particular, on an infinitary language and an ontology of properties.)The proposal developed here is similar in spirit to those in Roy (1995) and Melia (2001).

7.2. A KRIPKE-SEMANTICS FOR ACTUALISTS 151

7.2 A Kripke-semantics for actualists

Sadly, I don’t have a sister. But I might have had a sister. In fact, I might have had a

sister who was a philosopher. And, of course, had I had sister who was a philosopher,

she wouldn’t have been a philosopher essentially: she might have been a cellist instead.

The following is therefore true. (See McMichael (1983).)

Sister

♦(∃x(Sister(x,ar) ∧ Phil(x) ∧ ♦(Cellist(x) ∧ ¬Phil(x))))

(Read : I might have had a sister who was a philosopher and might have been

a cellist rather than a philosopher.)

On the most straightforward version of a Kripke-semantics for first-order modal lan-

guages, Sister will only be counted as true if there are worlds w1 and w2 with the

following properties: according to w1, there is an individual who is my sister and a

philosopher; according to w2, that very individual—as one is inclined to put it—is a

cellist rather than a philosopher. It is therefore tempting to say the following:

∃x([Sister(x,ar) ∧ Phil(x)]w1 ∧ [Cellist(x) ∧ ¬Phil(x)]w2)

(Read: There is an x such that: (i) according to w1, x is my sister and a philosopher,

and (ii) according to w2, x is a cellist rather than a philosopher.)

But is there anything to make this existential quantification true? If you believe in

merely possible sisters, you might think that one of my possible sisters can do the job.

But if, like me, you are a modal actualist, then you believe there are no merely possible

sisters.

In spite of this difficulty, there is a certain sense in which it is straightforward to

give an actualistically acceptable Kripke-semantics for modal sentences. The trick is to

152 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

have one’s semantics quantify over representations of possibilities, rather than over the

possibilities themselves. In this this section I will describe such a semantics.

Let L be a first-order language, and let L♦ be the result of enriching L with the

sentential operator ‘♦’. An a-world (short for ‘actualist-world’) for L♦ is an ordered pair

〈D, I〉 such that:

• The domain D is a set of ordered pairs of the form 〈x, ‘actual’〉 (for x an individual

in the domain of L), or of the form 〈x, ‘nonactual’〉 (for x an arbitrary individual).

In both cases x is assumed to be an actually existing individual.

• The interpretation function I assigns a subset of Dn to each n-place predicate-letter

of L, and a function from Dn to D to each n-place function-letter of L.

• If c is an individual constant of L and x is its intended interpretation, I assigns

the pair 〈x, ‘actual’〉 to c.

(The notions of truth and satisfaction at an a-world are characterized along standard

lines, with the proviso that ‘x = x’ is only satisfied at an a-world by objects in the

domain of a-world, with the result that ‘x = x’ can be used as an existence predicate.

See Appendix A for details.)

The easiest way of understanding how a-worlds are supposed to work is by com-

paring them to Lewisian worlds. Like a-worlds, Lewisian worlds can be thought of as

representing possibilities. Here is Lewis:

How does a world, [Lewisian] or ersatz, represent, concerning Humphrey, that

he exists?. . . A [Lewisian] world might do it by having Humphrey himself as

a part. That is how our own world represents, concerning Humphrey, that

he exists. But for other worlds to represent in the same way that Humphrey

exists, Humphrey would have to be a common part of many overlapping

worlds. . . I reject such overlap. . . There is a better way for a [Lewisian] world

7.2. A KRIPKE-SEMANTICS FOR ACTUALISTS 153

to represent, concerning Humphrey, that he exists. . . it can have a Humphrey

of its own, a flesh-and-blood counterpart of our Humphrey, a man very much

like Humphrey in his origins, in his intrinsic character, or in his historical role.

By having such a part, a world represents de re, concerning Humphrey—that

is, the Humphrey of our world, whom we as his worldmates may call simply

Humphrey—that he exists and does thus-and-so. (Lewis (1986), p. 194. Lewis

writes ‘genuine’ where I have substituted ‘Lewisian’.)

(It is easy to lose track of Lewis’s representationalism. As I mentioned in chapter 2,

part of the reason is that Lewis also subscribed to a striking foundationalist claim: he

believed that what it is for it to be possible that p is for there to be a Lewisian world at

which (i.e. representing that) p. In light of this claim, it is natural to fudge the difference

between Lewisian worlds and possibilities. But there is a difference nonetheless.)

Whereas Lewisian worlds represent by analogy, a-worlds represent by satisfaction. A

Lewisian world represents the possibility that I have a sister by containing a person who

is similar to me in certain respects, and has a sister. An a-world, on the other hand,

represents the possibility that I have a sister by satisfying the formula ‘∃x(Sister(x,ar))’,

where ‘Sister’ is a predicate that expresses sisterhood on its intended interpretation and

‘ar’ is a name that refers to me on its intended interpretation. For instance, a-world

〈D1, I1〉 from figure 7.1 represents a possibility whereby I have a sister who is a philoso-

pher.

From the perspective of the Lewisian, an individual with a counterpart in the actual

world represents its actual-word counterpart, and an individual with no counterpart in

the actual world represents a merely possible object. From the present perspective,

a pair of the form ‘〈x, ‘actual’〉’ represents its first component, and a pair of the form

‘〈x, ‘nonactual’〉’ represents a merely possible object (even though the pair itself, and both

of its components, are actually existing objects). Thus, 〈D1, I1〉 represents a possibility

whereby an actual object (i.e. me) has a sister who doesn’t actually exist.

154 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

〈D1, I1〉

D1 = {〈Agustın, ‘actual’〉 , 〈Socrates, ‘nonactual’〉}I1(‘Philosopher’) = {〈Socrates, ‘nonactual’〉}I1(‘Cellist’) = {}I1(‘Sister’) = {〈〈Socrates, ‘nonactual’〉 , 〈Agustın, ‘actual’〉〉}I1(‘ar’) = 〈Agustın, ‘actual’〉

〈D2, I2〉

D2 = {〈Socrates, ‘nonactual’〉}I2(‘Philosopher’) = {}I2(‘Cellist’) = {〈Socrates, ‘nonactual’〉}I2(‘Sister’) = {}I2(‘ar’) = 〈Agustın, ‘actual’〉

〈D3, I3〉

D3 = {〈Plato, ‘nonactual’〉}I3(‘Philosopher’) = {}I3(‘Cellist’) = {〈Plato, ‘nonactual’〉}I3(‘Sister’) = {}I3(‘ar’) = 〈Agustın, ‘actual’〉

〈D4, I4〉

D4 = {〈Agustın, ‘actual’〉 , 〈Plato, ‘nonactual’〉}I4(‘Philosopher’) = {〈Plato, ‘nonactual’〉}I4(‘Cellist’) = {}I4(‘Sister’) = {〈〈Plato, ‘nonactual’〉 , 〈Agustın, ‘actual’〉〉}I4(‘ar’) = 〈Agustın, ‘actual’〉

These examples assume that the only non-logical expressions in L are ‘Philosopher’,‘Cellist’, ‘Sister’ and ‘ar’, and that the domain of L is {Agustın}.

Figure 7.1: Examples of a-worlds.

7.2. A KRIPKE-SEMANTICS FOR ACTUALISTS 155

Say that two representations are linked if—as one is inclined to put it—they concern

the same individual, even if the individual in question doesn’t exist. In order for a Kripke

semantics based on a-worlds to verify Sister, there must be linking amongst a-worlds.

In particular, some a-world must represent a possibility whereby I have a sister who is a

philosopher and another must represent a possibility whereby—as one is inclined to put

it—that very same individual is a cellist rather than a philosopher.

Let us first see how linking gets addressed from a Lewisian perspeective. l1 and l2 are

Lewisian worlds: l1 contains an individual a1 who bears the right sort of similarity to me

and an individual s1 who is a1’s sister and a philosopher; l2 contains an individual s2 who

is a cellist. Accordingly, l1 represents a possibility whereby my sister is a philosopher,

and l2 represents a possibility whereby someone is a cellist. But nothing so far guarantees

linking. Nothing so far guarantees that—as one is inclined to put it—the individual l1

represents as my sister is the very individual that l2 represents as a cellist. What is

needed for linking is that s1 and s2 be counterparts: that they be similar in the right

sorts of respects.

The same maneuver can be used when it comes to a-worlds. Like the Lewisian,

we shall use counterparthood amongst representations to capture linking. For Lewis,

representations are counterparts just in case they are similar in the right sorts of respects.

From the present perspective, we shall say that representations are counterparts just in

case they are identical (though other ways of defining the counterpart relation could be

used as well). Here is an example. We have seen that the a-world 〈D1, I1〉 represents a

possibility whereby I have a sister who is a philosopher. Now consider a-worlds 〈D2, I2〉

and 〈D3, I3〉 from figure 7.1. Each of them represents a possibility whereby someone is

a cellist rather than a philosopher. But only 〈D2, I2〉 is linked to 〈D1, I1〉. For 〈D1, I1〉

and 〈D2, I2〉 both employ 〈Socrates, ‘nonactual’〉 as a representation, and it is this that

guarantees that—as one is inclined to put it—the individual who 〈D1, I1〉 represents as my

sister is the very individual that 〈D2, I2〉 represents as a cellist. On the other hand, since

156 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

〈D3, I3〉 represents a possibility whereby someone is a cellist by using 〈Plato, ‘nonactual’〉

rather than 〈Socrates‘nonactual’〉, what one gets is that—as one is inclined to put it—

the individual who 〈D1, I1〉 represents as my sister is distinct from the individual that

〈D2, I2〉 represents as a cellist.

An a-worlds-semantics is not a way of improving on the informal characterization of

linking that I supplied a few paragraphs back (i.e. representations are linked if—as one

is inclined to put it—they concern the same individual, even if the individual in question

doesn’t exist). In particular, it is not a way of dispensing with the qualifying phrase

‘as one is inclined to put it’. What an a-worlds-semantics delivers is an (actualistically

acceptable) device for representing possibilities which makes clear when two representa-

tions are to be counted as linked. The reason this is helpful is that, as we shall see below,

much of the theoretical work that can be carried out by quantifying over possibilities

can be carried out by quantifying over representations of possibilities instead. So an

a-worlds-semantics puts the actualist in a position to get on with the theoretical work

without having to worry about giving a proper characterization of linking.

It is worth emphasizing that by availing oneself of an ontology of Lewisian worlds and

a Lewisian counterpart relation one does not immediately do any better. What one gets

is a way of making clear when two Lewisian worlds are to be counted as linked, not a

characterization of linking. A friend of the Lewisian ontology can, however, give a proper

characterization of linking by also subscribing to a foundationalist claim: that for it to

be possible that p just is for there to be a Lewisian world at which (i.e. representing

that) p. One could then claim for two possibilities to concern the same individual just is

for objects in the relevant Lewisian worlds to be each other’s counterparts.

7.3. ADMISSIBILITY 157

7.3 Admissibility

There are a-worlds according to which someone is a married bachelor, and a-worlds

according to which there might have been a human who wasn’t essentially human. Such

representations need to be excluded from our semantics, on pain of getting the result that

‘♦(∃x(Married(x)∧Bachelor(x)))’ or ‘♦(∃x(Human(x)∧♦(∃y(y = x∧Human(x))))’ are

true. What we need is a notion of admissibility. Armed with such a notion, one can say

that p♦φq is true just in case φ is true at some admissible a-world, and that p�φq is true

just in case φ is true at every admissible a-world. (Here and throughout I assume that

the accessibility relation is trivial.)

It is important to distinguish between the semantic project of explaining what an

(actualistically acceptable) Kripke-semantics for modal languages might consist in, on

the one hand, and the foundational project of identifying grounds for admissibility, on

the other. A semantics based on a-worlds is meant to address the semantic project, but

not the foundational project. Accordingly, the notion of admissibility should be thought

of as a placeholder for whatever limits on the metaphysically possible turn out to be

uncovered by the foundational project.

My preferred answer to the foundational question is spelled out in chapter 2, and a

formal characterization of the ensuing accessibility relation is supplied in appendix C.

But—on reasonable assumptions—one can show that a suitable notion of admissibility is

guaranteed to exist however the foundational question is answered.2 A semantics based

2More precisely, what one can show is this: provided there is a determinate fact of the matter aboutwhich sentences of L♦ are true, there is a notion of admissibility relative to which an a-worlds semanticsassigns the right truth-value to every sentence in L♦. (I assume that the domain of L is a set.) Proof:Where S is the set of true sentences in L♦, use Kripke’s completeness theorem for modal languagesto construct a Kripke-model for S in which the domain consists of equivalence-classes of terms. Thentransform the Kripke-model into an a-world semantics by substituting the pair 〈x, ‘actual’〉 for eachequivalence class in the domain of the actual world of the Kripke-semantics containing a standard namefor x, and the pair 〈x, ‘nonactual’〉 for each object x in the domain of some non-actual world of theKripke-semantics but not in the domain of the actual world of the Kripke-semantics. (The proof relieson the reasonable assumption that S is consistent relative to a normal logic. To avoid talking aboutaccessibility relations, I have also assumed that S is consistent relative to S5. It is worth noting that theCompleteness Theorem assumes a weak version of the Axiom of Choice, so the resulting characterization

158 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

on a-words is compatible with a brutalist answer to the foundational project, according

to which: (a) for it to be possible that φ just is for one’s specialized modal ontology to

include a possible world at which φ, and (b) there is no explaining why one’s ontology

includes the possible worlds that it does contain. But there are other avenues available.

One’s account might appeal to a Principle of Recombination, for example, or to a set of

basic modal truths.3

Even if one’s answer to the foundational question commits one to an ontology of pos-

sible worlds, one might have good reasons for using a-worlds rather than possible worlds

for the purposes of semantic theorizing. The easiest way to see this is by distinguish-

ing between sparse and abundant conceptions of possible worlds.4 A sparse conception

countenances worlds according to which there are objects that don’t actually exist, but

not worlds according to which it is true of specific non-existent objects that they exist.

There is, for example, a possible world w1 according to which I have a sister who is a

philosopher and might have been a cellist rather than a philosopher, but no possible

world according to which it is true of the specific individual who would have been my

sister had w1 obtained that she exists. (Not even w1 is such a world, for even though

w1 is a world according to which I have a sister, it is not a world according to which it

is true of some specific individual that she is my sister.) On an abundant conception of

possible worlds, on the other hand, there are possible worlds according to which it is true

of specific non-existent objects that they exist. There is, for instance, a possible world

w2 according to which it is true of the very individual who would have been my sister

had w1 been actualized that she is a cellist rather than a philosopher.

On the sparse conception of possible worlds, the existence of a world is conditional

on the existence of the objects the world represents as existing, in the same sort of

of admissibility is non-constructive.)3On the Principle of Recombination, see Lewis (1986) §1.8. For other approaches to grounding

admissibility see Fine (1994) and Peacocke (1999), ch. 4.4For a sparse conception of possible worlds, see Stalnaker’s ‘On what there isn’t (but might have

been)’. Stalnaker makes clear that he does not see sparseness as an obstacle for doing Kripke-semantics.

7.3. ADMISSIBILITY 159

way that the existence of a set is conditional on the existence of its members. Had w1

been actualized, I would have had a sister, and all manner of sets containing that very

individual would have existed. But as things stand, my sister doesn’t exist, and neither

do sets having her as a member. Similarly—the story would go—had w1 been actualized,

I would have had a sister, and a world according to which that very individual is a cellist

would have existed. But as things stand, my sister doesn’t exist, and neither do possible

worlds according to which she herself exists.

The absence of w2 does not prevent a defender of the sparse conception from using

possible worlds to determine a truth-value for Sister. For, on the assumption that

possible worlds track metaphysical possibility, the existence of w1 is enough to guarantee

that Sister is true. But the absence of w2 does mean that the sparse worlds do not by

themselves deliver the ontology that would be needed to give a Kripke-semantics for a

sentence like Sister. For a Kripke-semantics will only count Sister as true if the range

of one’s metalinguistic quantifiers contains both w1 and w2.

More generally, a sparse ontology of possible worlds is enough to guarantee the ex-

istence of a notion of admissibility relative to which an a-worlds semantics assigns the

right truth-value to every sentence in the language (see footnote 2). So it is open to

the sparse theorist to use admissible a-words, rather than possible worlds, for the pur-

poses of semantic theorizing. The upshot is not, of course, that one has done away with

one’s specialized modal ontology, since possible worlds may be needed to pin down the

crucial notion admissibility. But by using a-worlds as the basis of one’s semantics, the

requirements on one’s modal ontology are confined to needs of the foundational project.

And a far as the foundational project is concerned, an abundant conception of worlds is

unnecessary.

A related point can be made with respect to mere possibilia. By using a-worlds

rather than a specialized modal ontology as the basis of one’s semantics, there is no need

to postulate mere possibilia, or specialized surrogates, such as Plantinga’s individual

160 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

essences. (See Plantinga (1976); for a critique of Plantinga, see Fine (1985).)

7.4 Interlude: The Principle of Representation

In previous sections I have made informal remarks about the ways in which a-worlds

represent possibilities. The purpose of this interlude is to be more precise. (Uninterested

readers may skip ahead to section 7.5.)

When 〈D, I〉 is considered in isolation from other a-worlds, anything that can be said

about the possibility represented by 〈D, I〉 is a consequence of the following principle:

Representation Principle (Isolated-World Version)

Let s be a sentence of L, and suppose s says that p. Then:

according to the possibility represented by 〈D, I〉, p

if and only if

s is true at 〈D, I〉.

Accordingly, when considered in isolation from other a-worlds, 〈D2, I2〉 and 〈D3, I3〉 rep-

resent the same possibility. It is the possibility that there be exactly one thing and that

it be a cellist but not a philosopher.

We shall normally assume that L (and therefore L♦) contains a name for every object

in the domain of L. With this assumption is in place, the following is a consequence of

the Representation Principle:

Suppose z is in the domain of L. Then z exists according to the possibility

represented by 〈D, I〉 just in case 〈z, ‘actual’〉 is in D.

In particular, one gets the result that none of the objects in the domain of L exists

according to the possibility represented by 〈D2, I2〉 (since p∃x(x = c)q is false at 〈D2, I2〉

7.4. INTERLUDE: THE PRINCIPLE OF REPRESENTATION 161

for any constant c in L), and that I exist according to the possibility represented by

〈D1, I1〉 (since ‘∃x(x = ar)’ is true at 〈D1, I1〉).

So much for considering a-worlds in isolation. When they are considered in the context

of a space of a-worlds, linking plays a role. So there is slightly more to be said about

the possibilities that they represent. Let A be a space of a-worlds and let 〈D, I〉 be in

A. Then anything that can be said about the possibility represented by 〈D, I〉 in the

context of A is a consequence of the following principle:

Representation Principle (Official Version)

1. Let s be a sentence of L♦, and suppose s says that p. Then:

according to the possibility represented by 〈D, I〉 in the context

of A, p

if and only if

s is true at 〈D, I〉 in the Kripke-model based on A.

2. Let 〈D∗, I∗〉 be an arbitrary a-world in A. Let p∃~x φ(~x)q and p∃~x γ(~x)q

be sentences of L♦ which say, respectively, that the ~x are F and that

the ~x are G. Assume that p∃~x φ(~x)q is true at 〈D, I〉 in A and that

p∃~x γ(~x)q is true at 〈D∗, I∗〉 in A. Then:

as one is inclined to put it, some of the individuals that are F

according to the possibility represented by 〈D, I〉 in the context

of A are the very same individuals as some of the individuals

that are G according to the possibility represented by 〈D∗, I∗〉

in the context of A

if and only if

one of the sequences of pairs that witnesses p∃~x φ(~x)q at 〈D, I〉

in A is identical to one of sequences of pairs that witnesses

162 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

p∃~x γ(~x)q at 〈D∗, I∗〉 in the Kripke-model based on A.

When A includes both 〈D1, I1〉 and 〈D2, I2〉, the first clause yields the result that, ac-

cording to the possibility represented by 〈D2, I2〉 in the context of A, there is a cellist

who might have been my sister. And the two clauses together yield the slightly stronger

result that—as one is inclined to put it—the individual who is a cellist according to the

possibility represented by 〈D2, I2〉 in the context of A is the very same object as the

individual who is my sister according to the possibility represented by 〈D1, I1〉 in the

context of A.

A consequence of the Principle of Representation is that the possibilities represented

by a-worlds are not maximally specific. Suppose, for example, that the property of tallnes

is not expressible in L. Then the possibility represented by 〈D1, I1〉 is compatible with a

more specific possibility whereby my sister is tall and it is compatible with a more specific

possibility whereby my sister is not tall. On the other hand, the possibilities represented

by a-worlds are maximally specific as far as the language is concerned: one can only add

specificity to the possibility represented by an a-world by employing distinctions that

cannot be expressed in L♦.

The Principle of Representation can be used to determine which properties of an a-

world are essential to its representing the possibility that it represents, and which ones are

merely artifactual. It entails, for example, that 〈D2, I2〉 and 〈D3, I3〉 represent the same

possibility when considered in isolation, so any differences between them are merely arti-

factual. In particular, the use of 〈Socrates, ‘nonactual’〉 in 〈D2, I2〉 is merely artifactual.

On the other hand, 〈D2, I2〉 and 〈D3, I3〉 represent different possibilities when considered

in the context of {〈D1, I1〉, 〈D2, I2〉, 〈D3, I3〉}. For whereas according to 〈D2, I2〉 there

is a cellist who might have been my sister, according to 〈D3, I3〉 there is a cellist who

couldn’t have been my sister. So the use of 〈Socrates, ‘nonactual’〉 in 〈D2, I2〉 is essential

in the context of {〈D1, I1〉, 〈D2, I2〉, 〈D3, I3〉}. This is not to say, however, that a possi-

bility whereby there is a cellist who might have been my sister can only be represented

7.5. THE DOT-NOTATION 163

by an a-world if the a-world contains 〈Socrates, ‘nonactual’〉. For the possibilities repre-

sented by 〈D1, I1〉, 〈D2, I2〉, and 〈D3, I3〉 in the context of {〈D1, I1〉, 〈D2, I2〉, 〈D3, I3〉}

are precisely the possibilities represented by {〈D4, I4〉, 〈D3, I3〉, 〈D2, I2〉} in the context

of {〈D2, I2〉, 〈D3, I3〉,〈D4, I4〉}.

When I speak of the possibility that an a-world represents what I will usually have

in mind is the possibility that is represented in the context of the space of all admissible

a-worlds.

7.5 The dot-notation

I would like to introduce a further piece of notation: the dot. Intuitively, the dot may

be thought of as a function that takes objects represented to the objects doing the

representing. Suppose I am seeing a play according to which I have a sister; applying the

dot-function is like shifting my attention from a character in the play—my sister—to the

actress who is representing my sister.

Consider the following two formulas:

[F(x)]w [F(x)]w

For w a fixed representation, the undotted formula is satisfied by all and only objects

z such that w represents a possibility whereby z is an F ; the dotted formula, on the

other hand, is satisfied by all and only objects z such that z is used by w to represent

something as being an F . Thus, if π is a performance of a play according to which I have

a sister, the actress playing my sister satisfies ‘[Sister(ar, x)]π’ but not ‘[Sister(ar, x)]π’

(since the performance uses the actress to represent someone as being my sister, but the

performance does not represent a scenario whereby I have that actress as my sister). And

I satisfy ‘[∃y Sister(x, y)]π’ but not ‘[∃y Sister(x, y)]π’ (since the performance represents

164 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

a scenario whereby I have a sister, but—unlike the actors and props—I am not used by

the performance to represent anything).

Now consider how the dot-notation might be cashed out from the perspective of a

Lewisian. Let l1 be a Lewisian world representing a possibility whereby I have a sister.

Accordingly, l1 contains an individual a1, who is my counterpart, and an individual s1,

who is a1’s sister. Now consider the following two formulas:

[Sister(ar, x)]l1 [Sister(ar, x)]l1

From the perspective of the Lewisian, no inhabitant of the actual world satisfies the

undotted formula. For no inhabitant of the actual world could have been my sister; so—

on the assumption that Lewisian worlds track metaphysical possibility—no inhabitant of

the actual world is such that l1 represents a possibility whereby she is my sister. The

dotted formula, on the other hand, is satisfied by s1, since she is used by l1 to represent

something as being my sister.

Here is a second pair of examples:

[∃y Sister(x, y)]l1 [∃y Sister(x, y)]l1

The undotted formula is satisfied by me, since l1 represents a possibility whereby I have

a sister. But it is not satisfied by a1. For although it is true that a1 has a sister in l1, l1

represents a possibility whereby I have a sister, not a possibility whereby my counterpart

has a sister. The dotted formula, on the other hand, is satisfied by a1, since a1 is used

by l1 to represent something as having a sister (i.e. me). But the dotted formula is not

satisfied by me, since it is only the inhabitants of l1 that do any representing for l1, and

I am an inhabitant of the actual world.

Let me now illustrate how the dot-notation works from the perspective of the modal

actualist, with a-worlds in place of Lewisian worlds. (A detailed semantics is given in

7.5. THE DOT-NOTATION 165

Appendix A.) Here is the first pair of examples (where w1 is the a-world 〈D1, I1〉 from

section 7.2):

[Sister(ar, x)]w1 [Sister(ar, x)]w1

Since I1(‘Sister’) = {〈〈Agustın, ‘actual’〉 , 〈Socrates, ‘nonactual’〉〉}, w1 represents a possi-

bility whereby I have a sister who doesn’t actually exist. Accordingly, from the perspec-

tive of a modal actualist, there is no z such that w1 represents the possibility that z is

my sister. So, from the perspective of the modal actualist, nothing satisfies the undotted

formula. The dotted formula, on the other hand, is satisfied by 〈Socrates, ‘nonactual’〉,

since 〈Socrates, ‘nonactual’〉 is used by w1 to represent something as being my sister.

Now consider the second pair of examples:

[∃y Sister(x, y)]w1 [∃y Sister(x, y)]w1

Since the pair 〈〈Agustın, ‘actual’〉 , 〈Socrates, ‘nonactual’〉〉 is in I1(‘Sister’), w1 represents

a possibility whereby I have a sister. The undotted formula is therefore satisfied by me.

But it is not satisfied by 〈Agustın, ‘actual’〉 because w1 does not represent a possibility

whereby any ordered-pairs have sisters. The dotted formula, on the other hand, is satis-

fied by 〈Agustın, ‘actual’〉, since 〈Agustın, ‘actual’〉 is used by w1 to represent something

as having a sister (i.e. me). But it is not satisfied by me, since it is only ordered-pairs

that do any representing in w1, and I am not an ordered-pair.

7.5.1 Inference in a language with the dot-notation

The semantics for a-worlds that is supplied in Appendix A guarantees the truth of every

instance of the following schemas:

1. Validity

[ψ]w (where ψ is valid in a negative free logic)

166 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

2. Conjunction

[ψ ∧ θ]w ↔ ([ψ]w ∧ [θ]w)

3. Negation

[¬ψ]w ↔ ¬[ψ]w

4. Identity

Where v may occur dotted or undotted,

[v = v]w ↔ [∃y(y = v)]w

(This makes clear that identity is being used in the stronger of the two senses

mentioned in section 1.2, and therefore that self-identity may be used as an existence

predicate.)

x = y → ([φ(x)]w → [φ(y)]w)

[x = y]w → ([x = x]w ∧ x = y)

5. Quantification

[∃y(φ(y))]w ↔ ∃y([y = y]w ∧ [φ(y)]w) (where y is an ordinary variable)

[∃w′(φ)]w ↔ ∃w′([φ]w) (where w′ is a world-variable)

6. Trivial accessibility

[[φ]w]w′ ↔ [φ]w

(If the accessibility relation is non-trivial, one gets the following instead: [[φ]w]w′ ↔

([φ]w ∧ Acccessible(w′, w)).)

7. Atomic Predication

[Fnj (v1, . . . , vn)]w → ([v1 = v1]w ∧ . . . ∧ [vn = vn]w)

(where the vi may occur dotted or undotted)

7.5. THE DOT-NOTATION 167

8. Names

[ψ(c)]w ↔ ∃x(x = c ∧ [ψ(x)]w) (for c a non-empty name)

Schemas 2–5 are enough to guarantee that any sentence in the actualist’s language is

equivalent to a sentence in which only atomic formulas occur within the scope of ‘[. . .]w’.

For instance, the actualist rendering of ‘♦(∃x(Phil(x) ∧ ♦(¬Phil(x))))’:

∃w[∃x(Phil(x) ∧ ∃w′([¬Phil(x)]w′))]w

is equivalent to

∃w∃x([Phil(x)]w ∧ ∃w′(¬[Phil(x)]w′)).

As a result, the dot-notation allows a language containing the modal operator ‘[. . .]w’ to

have the inferential behavior of a (non-modal) first-order language.

7.5.2 The expressive power of the dot-notation

In this section we shall see that a suitably qualified version of the following claim is

true: anything the Lewisian can say, the modal actualist can say too—by using the

dot-notation.

Here is an example. The Lewisian can use her mighty expressive resources to capture

a version of the following thought:

Linking

There are possible worlds w1 and w2 with the following properties: according

to w1, there is an individual who is a philosopher; according to w2, that very

individual is a cellist.

It is done as follows:

∃w1∃w2∃x1∃x2(I(x1, w1) ∧ I(x2, w2) ∧ Phil(x1) ∧ Cellist(x2) ∧ C(x1, x2))

168 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

(Read: There are Lewisian worlds w1 and w2 and individuals x1 and x2 such

that: (a) x1 is an inhabitant of w1 and x2 is an inhabitant of w2, (b) x1 is a

philosopher and x2 is a cellist, and (c) x1 and x2 are counterparts.)

How might this be emulated by a modal actualist equipped with the dot-notation? Con-

sider what happens when one treats the variables in the Lewisian rendering of Linking

as ranging over (admissible) a-worlds rather than Lewisian worlds, and carries out the

following replacements:

I(xn, wn) −→ [xn = xn]wn

Phil(xn) −→ [Phil(xn)]wn

Cellist(xn) −→ [Cellist(xn)]wn

C(xn, xm) −→ xn = xm

The result is this:

∃w1∃w2∃x1∃x2([x1 = x1]w1 ∧ [x2 = x2]w2 ∧ [Phil(x1)]w1 ∧ [Cellist(x2)]w2 ∧ x1 = x2)

(Read: There are admissible a-worlds w1 and w2 and objects x1 and x2 such

that: (a) x1 is used by w1 to represent something and x2 is used by w2 to

represent something, (b) x1 is used by w1 to represent a philosopher and x2

is used by w2 to represent a cellist, and (c) x1 = x2.)

or equivalently:

∃w1∃w2∃x([Phil(x)]w1 ∧ [Cellist(x)]w2)

(Read: There are admissible a-worlds w1 and w2 and an object x such that:

x is used by w1 to represent a philosopher and x is used by w2 to represent a

cellist.)

7.5. THE DOT-NOTATION 169

What gives the actualist’s method its punch is the fact that it generalizes: one can show

that there is a systematic transformation of arbitrary Lewisian sentences into dotted actu-

alist sentences which preserves truth-values and inferential conections.5 (See Appendix B

for details.)

The actualist’s transformation-method does not preserve meaning—where Lewisian

sentences quantify over Lewisian possibilia, their actualist transformation quantify over

a-worlds and ordered-pairs. But meaning-preservation is not what the actualist wants,

since she doesn’t want to countenance Lewisian possibilia. What she wants is a way

of enjoying the theoretical benefits of quantification over Lewisian possibilia within the

sober confines of an actualist framework. Here are two examples of ways in which she is

able to do so:

1. Firstorderizing Modal Sentences

By quantifying over Lewisian possibilia, the Lewisian is able to render any sentence

in the language of first-order modal logic in (non-modal) first-order terms. The

sentence

♦(∃x(Phil(x) ∧ ♦(Cellist(x))))

(read: there might have been a philosopher who might have been a cel-

list),

5When I say that the transformation preserves truth-value what I mean is that there is a notionof a-world admissibility which guarantees that the actualist transformation of an arbitrary Lewisiansentence is true just in case the original Lewisian sentence would count as true from the perspectiveof the Lewisian. When I say that the transformation preserves inferential role, what I mean is that aLewisian sentence φ follows from a set of Lewisian sentences Γ just in case φ’s transformation followsfrom the the transformations of sentences in Γ.

The result assumes that atomic predicates in the Lewisian’s language other than ‘I’, ‘C’ and ‘=’ (andany set-theoretic vocabulary) be projectable. For a monadic predicate P to be projectable is for it tobe the case that a Lewisian world represents a possibility whereby something is P by containing aninhabitant who is P . (And similarly for many-place predicates.) Thus, ‘Philosopher’ is projectablebecause a Lewisian world represents a possibility whereby something is a philosopher by containing aninhabitant who is a philosopher; but ‘inhabits a Lewisian world which is part of a pluriverse containingmany Lewisian worlds’ is not projectable because a Lewisian world does not represent a possibilitywhereby something inhabits a Lewisian world which is part of a pluriverse containing many Lewisianworlds by containing an inhabitant who inhabits a Lewisian world which is part of a pluriverse containingmany Lewisian worlds.

170 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

for example, gets rendered as the (non-modal) first-order sentence:

∃w1∃w2∃x1∃x2(I(x1, w1) ∧ I(x2, w2) ∧ Phil(x1) ∧ Cellist(x2) ∧ C(x1, x2))

And Lewis (1968) shows that it can be done in general.6

The (non-modal) firstorderizability of modal sentences brings two immediate advan-

tages. The first is that it allows one to think of the inferential connections amongst

modal sentences in terms of the inferential connections amongst the corresponding

non-modal sentences; the second is that it allows one to read off a semantics for

modal sentences from the semantics of the corresponding non-modal sentences.

The actualist transformation-method allows actualists equipped with the dot-notation

to enjoy both of these advantages.

2. Characterizing Intensions

On a standard way of doing intensional semantics for natural languages, charac-

terizing the semantic value of an expression calls for quantification over possibilia.

(For a representative textbook, see Heim and Kratzer (1998), ch. 12.) Oversimpli-

fying a bit, the semantic value of, e.g. ‘philosopher’ might be taken to be the set

of pairs 〈w, z〉 where w is a possible world and z is an (actual or merely possible)

individual who is a philosopher at w.

As emphasized in Lewis (1970), the Lewisian is able to do the job by quantifying

over Lewisian possibilia:

J‘philosopher’K = {〈w, z〉 : I(z, w) ∧ Phil(z)}6As Lewis observes, a feature of the 1968 translation is that ‘∀x�(∃y(x = y))’ turns out to be true.

For this reason, I prefer a modification of the translation whereby (�φ)β is ∀β1(W (β1)→ (φ)β1), (♦φ)β

is ∃β1(W (β1)∧ (φ)β1) and (P (x1, . . . , xn))β is ∃γ1 . . . ∃γn(I(γ1, β)∧C(γ1, x1)∧ . . . I(γn, β)∧C(γn, xn)∧P (γ1 . . . , γn) (for P atomic). The modified translation delivers the same truth-values as a version ofKripke-semantics in which atomic formulas (including identity-statements) can only be satisfied at aworld by objects that exist at that world.

7.5. THE DOT-NOTATION 171

But the actualist transformation-method allows actualists armed with the dot-

notation to follow suit, by quantifying over a-worlds and ordered pairs:

J‘philosopher’K = {〈w, z〉 : [z = z]w ∧ [Phil(z)]w}

or, equivalently,

J‘philosopher’K = {〈w, z〉 : [Phil(z)]w}.

Since the actualist transformation-method preserves inferential connections, the ac-

tualist semantics is guaranteed to deliver the same theorems as its Lewisian coun-

terpart. And since the transformation-method preserves truth-value the actualist’s

axioms will be true just in case their Lewisian counterparts would count as true

from the perspective of the Lewisian.

In particular, one can expect the actualist semantics to deliver every instance of

the (world-relative) T-schema. For instance:

True(‘∃xPhil(x)’, w)↔ ([∃xPhil(x)]w)

or, equivalently,

True(‘∃xPhil(x)’, w)↔ ∃x([Phil(x)]w)

(Read: The object-language sentence ‘∃xPhil(x)’ is true at admissible a-

world w just in case there is an individual which is used by w to represent

a philosopher.)

7.5.3 Limitations of the proposal

A-worlds are subject to an important limitation. Whereas differences between a-worlds

are no more fine-grained than is required to make distinctions expressible in one’s lan-

172 CHAPTER 7. A-WORLDS AND THE DOT NOTATION

guage, there might be differences amongst Lewisian worlds too fine-grained to be ex-

pressed in one’s language.7

Because of this limitation, some of the metaphysical work that the Lewisian gets out

of Lewisian possibilia cannot be replicated by an actualist equipped with the dot-notion.

Here is an example. Lewis (1986) treats properties as sets of worldbound individuals. Up

to a certain point, the actualist is able to follow suit. When the Lewisian claims that the

property of being a philosopher is to be identified with the set of philosophers inhabiting

actual or non-actual Lewisian worlds, for instance, the actualist could claim that the set

{〈z, w〉 : [Phil(z)]w} is to be used as a surrogate for the property of being a philosopher.

But the strategy breaks down when it comes to properties making finer distinctions than

can be expressed in one’s language.8

In general, whether or not the actualist’s limitation turns out to get in the way will

depend on whether the job at hand calls for using possibilia to make finer distinctions

than can be expressed in one’s language. When the job at hand is a piece of semantic

theorizing the extra resources are unnecessary: since a semantic theory is ultimately an

effort to explain how language is used, it need not be concerned with distinctions too

fine-grained to figure in our explanations. But when the job at hand is metaphysical

reduction, matters are otherwise.

7More precisely, there might be different Lewisian worlds such that every inhabitant of the one worldis a counterpart of an inhabitant of the other, and every predicate in the language which is projectablein the sense of footnote 5 is satisfied by inhabitants of one world just in case it is satisfied by thecounterparts of those individuals at the other. The distinct possibilities represented by such Lewisianworlds would both be compatible with the less-specific possibility represented by an a-world in whichthe behavior of the predicates mirrors the behavior of the predicates at the Lewisian worlds.

8Any set of worldbound individuals containing an inhabitant of one of the Lewisian worlds describedin footnote 7 but not its counterpart at the other corresponds to a Lewisian property with no actualistsurrogate.

Chapter 8

Translation

A trivialist paraphrase-function is a mapping that takes each arithmetical sentence φ

to a sentence whose truth-conditions are uncontroversially the truth-conditions that the

trivialist semantics associates with φ—uncontroversially, in the sense that whether or

not one takes φ’s paraphrase to have the right truth-conditions doesn’t depend on what

identity-statements one accepts.

We will investigate the question of whether it possible to specify a trivialist paraphrase-

function for the language of arithmetic. Attention will be restricted to paraphrase-

functions that can be characterized algorithmically. Without this restriction, it is obvi-

ously true that there is trivialist paraphrase-function for the language of pure arithmetic:

map every true sentence to a tautology and every false sentence to a contradiction.

The answer to our question will depend on the expressive resources of the language

in which paraphrases are given.

Higher-Order Languages

Suppose one takes the paraphrase-language to be an nth-order language, for some fi-

nite n. Then, assuming the Church-Turing Thesis, it is impossible to specify a trivi-

alist paraphrase-method for the language of arithmetic. (For a proof of this result, see

173

174 CHAPTER 8. TRANSLATION

Rayo (2008).)

There are a few paraphrase-methods that have the right flavor, but don’t quite give

us what we’re after. Consider, for example, the method of universal Ramseyfication.

If φ is a sentence in the language of pure arithmetic, its universal Ramseyfication is

the universal closure of p(D → φ)∗q, where D is the conjunction of the (second-order)

Dedekind-Peano Axioms and (. . .)∗ is the result of uniformly substituting variables for

mathematical vocabulary in (. . . ).

The method of universal Ramseyfication only delivers the right results on the assump-

tion that the world is infinite. To see this, consider an arithmetical falsehood, such as

‘0 = 1’. If the world is finite, the Dedekind-Peano Axioms, D , will turn out to be false. So

‘0 = 1’s universal Ramseyfication (i.e. the universal closure of p(D → 0 = 1)∗q) will turn

out to be true, which is contrary to what we want. Of course, a trivialist will think that

finite worlds are de mundo unintelligible, and therefore that commitment to infinitely

many objects is no commitment at all. So a trivialist will think that the method of

universal Ramseyfication does, after all, deliver the right assignment of truth-conditions.

But such a conclusion won’t count as uncontroversial, since it won’t be acceptable to the

typical non-trivialist.

Other higher-order paraphrase-methods that have the right flavor but depend on

infinity-assumptions to deliver the right results include Hodes (1984), Fine (2002a) II.5,

and Rayo (2002a).

Languages of Very High Order

If one’s paraphrase-language includes variables of transfinite order, then a trivialist

paraphrase-method for the language of arithmetic is available.

For α an arbitrary ordinal, let L α∈ be a version of the language of set-theory in

which each occurrence of a quantifier is restricted by some Vβ (β < α). As it turns out,

any sentence in the language of first-order arithmetic can be ‘translated’ as a sentence

175

of L ω+1∈ . What one does is replace arithmetical quantifiers by set-theoretic quantifiers

restricted to finite ordinals, and replace each occurrence of arithmetical vocabulary by

its counterpart in ordinal arithmetic. (This procedure presupposes that all the relevant

set-theoretic notions can be adequately characterized in L ω+1∈ , but it is straightforward

to check that this is indeed the case.)

Linnebo and Rayo (typescript) shows that, for arbitrary α, every sentence of L α∈ can

be paraphrased as a sentence in a language of order α + 2 (or order α, if α is a limit

ordinal). From this it follows that one can use the ‘translation’ of first-order arithmetical

sentences into L ω+1∈ to characterize a trivialist paraphrase-method from the language of

first-order arithmetic into a language of order ω + 3.

Infinitary Languages

Another way of characterizing a trivialist paraphrase-function is by allowing for infinite

conjunctions and disjunctions.

Because we will be dealing with infinite sentences, our paraphrase-function won’t be

algorithmic in the standard sense (i.e. it won’t be computable in finite time). But it will

still count is algorithmic in a derived sense: one could write a finite computer program

that approximates the paraphrase asymptotically, and outputs it after infinitely many

steps.

Our paraphrase-function will consist of a four-step transformation. We shall assume

that the input formula is a sentence of the language of pure first-order arithmetic, exclud-

ing mixed identities. (To make life simpler, we shall also assume that the input formula

is in prenex normal form, and that only atomic formulas are in the scope of a negation

sign.)

The basic strategy is due to Yablo (2002):

Step 1

Replace each subformula p∃nφ(n)q by the infinite disjunction pφ(0) ∨ φ(1) ∨ φ(2) ∨ . . .q,

176 CHAPTER 8. TRANSLATION

and each subformula p∀nφ(n)q by the infinite conjunction pφ(0) ∧ φ(1) ∧ φ(2) ∧ . . .q.

For instance, the result of applying Step 1 to ‘∀n∃m(n+1 = m)’ is an infinite conjunction

of infinite disjunctions:

(0+1 = 0∨0+1 = 1∨0+1 = 2∨ . . .)∧ (1+1 = 0∨1+1 = 1∨1+1 = 2∨ . . .)∧ (2+1 = 0∨2+1 = 1∨2+1 = 2∨ . . .)∧ . . .

In general, the result of applying Step 1 is a (possibly infinite) sentence in which ev-

ery atomic formula is an identity statement pt1 = t2q, where t1 and t2 contain no free

variables.

We must now deal with atomic formulas. Yablo suggests that we do so by substituting

a suitable non-arithmetical paraphrase for each occurrence of pt1 = t2q. An unnegated

occurrence of pk + l = mq, for example, gets replaced by:

[∃!kxFx ∧ ∃!lxGx ∧ ∀x¬(Fx ∧Gx)]→ ∃!mx(Fx ∨Gx)

(Read: if there are exactly k Fs, there are exactly l Gs and nothing is an

F-and-G, then there are exactly m F-or-Gs.)

This procedure succeeds in eliminating all mathematical vocabulary from the original

arithmetical sentence. But what one gets is a formula that is only guaranteed to have

the right truth-value on the assumption that the world is infinite. Consider, for example,

the arithmetical falsehood ‘∃n(n+ 1 = n)’. The result of applying Step 1 is an infinite

disjunction:

(0 + 1 = 0) ∨ (1 + 1 = 1) ∨ (2 + 1 = 2) ∨ . . .

But now suppose that there are exactly k objects, and consider the (k+ 1)th disjunct in

the series: pk + 1 = kq. According to Yablo’s procedure, this disjunct should be replaced

by

[∃!kxFx ∧ ∃!1xGx ∧ ∀x¬(Fx ∧Gx)]→ ∃!kx(Fx ∨Gx)

which is guaranteed to be true in a world with k objects (since its antecedent is guaranteed

177

to be false). From this it follows that the result of applying Yablo’s procedure is true,

even though ‘∃n(n+ 1 = n)’ is false.

Fortunately, there is a way around the problem. As Santos (typescript) points out,

one can make the following changes to the output of Step 1:

Step 2

Replace each occurrence of pk + lq by ‘SS . . . S︸ ︷︷ ︸k+l

(0)’, and each occurrence of

pk × lq by ‘SS . . . S︸ ︷︷ ︸k×l

(0)’. (Make sure you apply the procedure to subordinate

formulas first.)

Step 3

Replace occurrences of ‘SS . . . S︸ ︷︷ ︸k

(0) = SS . . . S︸ ︷︷ ︸l

(0)’ by > whenever k = l;

replace them by ⊥ whenever k 6= l.

(Here> is a logical truth from a finitary language containing no mathematical vocabulary,

and⊥ is a logical falsity from a finitary language containing no mathematical vocabulary.)

The result is a (possibly infinite) sentence that contains no arithmetical vocabulary

and can be shown to have the right truth-value regardless of the identity statements one

presupposes in the metatheory. In fact, every truth of pure arithmetic gets mapped onto

a truth of infinitary logic and every falsity of pure arithmetic gets mapped onto a falsity

of infinitary logic.

By adding a final step to the process, one can get the transformation to output a

finite sentence:

Step 4

Substitute > for infinite conjunctions all of whose conjuncts are true and

infinite disjunctions some of whose conjuncts are true; substitute ⊥ for infinite

disjunctions all of whose conjuncts are false and infinite disjunctions some of

178 CHAPTER 8. TRANSLATION

whose conjuncts are false. (Make sure you apply the procedure to subordinate

formulas first.)

When the original input sentence is true, the result of applying steps 1–4 will be a logical

truth from a finitary language with no mathematical vocabulary; otherwise, the result

will be a logical falsehood from a finitary language with no mathematical vocabulary.

A nice feature of this procedure is that it delivers a completeness theorem for an in-

finitary version of the Dedekind-Peano Axioms, in which the Induction Axiom is replaced

by rules stating that a universal quantifier is equivalent to the conjunction of its numeri-

cal instances and an existential quantifier is equivalent to the disjunction of its numerical

instances. All one needs to do to prove the result is carry out the transformation, and

note that each of its four steps can be justified on the basis of our axioms. (To justify

the fourth step one needs to assume that one is working with a suitable infinitary logic.)

Intensional Operators

The presence of intensional operators makes it much easier to define a trivialist paraphrase-

function. (See Hellman (1989), Yablo (2001) and Dorr (2007).) A particularly straight-

forward strategy would be to map each sentence of pure arithmetic to the result of adding

a box in front of its universal Ramseyfication. This will deliver the right results on the

assumption that there might have been infinitely many objects. Perhaps it is uncontro-

versial that commitment to the possibility of infinitely many objects is no commitment

at all. If so, the method of necessitated universal Ramseyfication counts as a trivialist

paraphrase-function for the special case of pure arithmetic.

If one wishes to accommodate applied arithmetic, something more elaborate is needed.

Here are two familiar proposals:

• Counterfactualism

Paraphrase an arbitrary arithmetical sentence φ as:

179

If the world contained an infinity of extra objects playing the role of

numbers, it would be the case that φ.

• Fictionalism

Paraphrase an arbitrary arithmetical sentence φ as:

According to a fiction whereby the world is as it actually is except for

the addition of an infinity of extra objects playing the role of numbers,

φ.

Properly construed, both of these strategies can be made to deliver the desired assignment

of correctness-conditions. But they are both potentially misleading. For it is easy to

overestimate their ability to shed light on the truth-conditions that a trivialist would

associate with arithmetical sentences. In fact, neither of them is any more illuminating

than the following:

• The No-Frills Strategy

Paraphrase an arbitrary arithmetical sentence φ as:

φ, except for all that stuff about numbers.

Any appearance of progress in the more familiar strategies comes from the fact that

we have a satisfying story to tell about what it takes for φ to be true at a world (or

fiction): one gives a standard compositional semantics for the language in question, and

relativizes it to worlds (or fictions). But that masks the real problem. The real problem

is not to explain what it takes for a world (or fiction) to verify φ, but to explain what

it would take for a world (or fiction) to be like a world (or fiction) that verifies φ in

all non-mathematical respects. For it is the latter that is needed to understand what it

would take for φ’s paraphrase to be true at a world (or fiction). And about this the

paraphrase-strategies under consideration have nothing illuminating to say.

180 CHAPTER 8. TRANSLATION

One could, of course, rehearse the familiar set of toy examples. One could insist, for

example, that what it takes for a world to be like a world which verifies ‘the number of the

planets is Eight’ in all non-mathematical respects is for it to contain exactly eight planets.

But the real challenge is to explain how the proposal is supposed to work in general—and

that task is no easier than the original task of supplying a trivialist paraphrase method

for the language of arithmetic.

Moral

Characterizing a trivialist paraphrase-method for the language of arithmetic is not as

straightforward as one might have thought. One must either avail oneself of poten-

tially controversial logical resources—such as variables of very high type or infinitary

operations—or make use of intensional operators.

Intensional operators should be used with care. They can give the appearance of

progress where there is none. It is not clear, in particular, that they are effective as a

device for shedding light on the truth-conditions that a trivialist would associate with

arithmetical sentences. To my mind, the most illuminating way of getting clear on

the trivialist’s truth-conditions is by setting forth a compositional semantics of the sort

discussed in section 4.4.

Chapter 9

Introducing Mathematical

Vocabulary

9.1 Linguistic Stipulation for Anti-Tractarians

A familiar way of introducing mathematical vocabulary is by setting forth an axiom

system, and stipulating the the new vocabulary is to be understood in such a way that

the axioms turn out to be true (or in such a way that they turn out to be necessarily

true).

For a Tractarian the effect of such stipulations is relatively straightforward. One

proceeds against the background of a fixed domain of objects and properties: those carved

out by the world’s metaphysical structure. So whenever there is exactly one assignment

of objects to names and properties to predicates that would make the axioms true (or

necessarily true), the effect of the stipulation—assuming it gives rise to the right sort of

linguistic practice—is to pair each name with the relevant object and each predicate with

the relevant property. (If there is more than one such assignment, Tractarians might fret

about how to proceed. They might learn to live with the ensuing indeterminacy, or insist

that the stipulation has been unsuccessful.)

181

182 CHAPTER 9. INTRODUCING MATHEMATICAL VOCABULARY

For an anti-Tractarian, the situation is somewhat more nuanced. This is because

different ways of carving up the world yield different domains of objects and properties.

And although a stipulation may be intended to work within an existing carving, it might

also be intended introduce a new carving altogether, with its own range of objects and

properties.

As it is understood here, a carving of the world is a compositional system of rep-

resentation for describing the world. (See section 3.2.) So to say that a stipulation is

meant to work within an existing carving is to say that the new lexical items are to be

counted as building blocks of a compositional language that is already in place—and,

in particular, that they should yield meaningful sentences when combined with extant

vocabulary of the right semantic category. If, for instance, the new lexical items include

a singular term, the result of using it to fill the argument place of an extant predicate

should result in a meaningful atomic sentence.

Suppose one introduces the new name ‘Muncle’ by stipulating that it is to be under-

stood in such a way that sentence ‘Muncle is Marcus’s favorite uncle’ turns out to be

true at the time of the stipulation. One would expect such a stipulation to be intended

to work within an extant way of carving the world into objects and properties. In par-

ticular, one would expect it to deliver the result that sentences like ‘Muncle isn’t much

of a conversationalist’ and ‘Muncle = Julius Caesar’ have well-defined truth-conditions.

In contrast, when a stipulation is intended to introduce a new way of carving up the

world, there should be no general expectation that the new vocabulary will yield mean-

ingful sentences when combined with extant vocabulary of the right semantic category.

Suppose, for example, that the language of arithmetic is introduced as a new way of

carving up the world. Then there should be no expectation that ‘8 = Julius Caesar’ will

be meaningful. This is not to say that every combination of old and new vocabulary

should be counted as meaningless: the sentence ‘23 = 8 ∧ Caesar is a Roman’, for exam-

ple, would be totally unproblematic, as would ‘#xPlanet(x) = 8’. (For more on mixed

9.1. LINGUISTIC STIPULATION FOR ANTI-TRACTARIANS 183

identities, see section 4.3.1.)

Just because different fragments of one’s discourse correspond to different carvings of

the world, it doesn’t mean that one is barred from using a single language to formalize

one’s theorizing. One can use terms and predicates of different sorts to talk about objects

corresponding to different carvings. Thus, one might use variables of one sort to range

over material objects, variables of a second sort to range over numbers, variables of a

third sort to range over sets, variables of a fourth sort to range over classes, and so forth.

An advantage of the multi-sorted strategy is that one can give a clear statement of

which sentences of the language should be expected to have well-defined truth-conditions

and which ones shouldn’t. Consider a language containing vocabulary of the four sorts

just mentioned. One could impose the following restrictions on the well-formedness of

formulas:

• An identity statement pt = τq where t and τ are terms of different sorts can never

be well-formed.

• Material-object-predicates and function-letters, such as ‘Volcano(. . . )’ or ‘Father-

Of(. . . )’, can only take material-object-terms as arguments.

• Arithmetical predicates and function-letters can only take arithmetical terms as ar-

guments, with the exception of ‘#v(φ(v))’, in which v can be either an arithmetical

variable or a material-object-variable.

• The sole set-theoretic predicate ‘∈s’ must take a set-theoretic term in its second

argument place, but it can take either a set-theoretic term or a material-object-term

in its first argument-place.

• The sole class-theoretic predicate ‘∈c’ must take a class-theoretic term in its second

argument place, but it can take either a class-theoretic term or a material-object-

term in its first argument-place.

184 CHAPTER 9. INTRODUCING MATHEMATICAL VOCABULARY

• If the language contains second-order variables, a second-order variable of a given

sort can only take as arguments first-order terms of that same sort.

Such a language would allow one to express just about any arithmetical, set-theoretic

or class-theoretic statement that plays a role in non-philosophical theorizing, including

statements including vocabulary of more than one sort. One can say, for example, that

there are more primates than even primes,

∀n∀m(#x(n,Primate(x)) ∧#r(m,EvenPrime(r))→ n > m)

(Where ‘x’ is a material-object-variable, and ‘n’, ‘m’ and ‘r’ are arithmetical

variables.)

and even that there are more primes than primates:

∀m∀n(#r(m,Prime(r) ∧#x(n,Primate(x)))→ m > n)

(Where ‘x’ is a material-object-variable, and ‘n’, ‘m’ and ‘r’ are arithmetical

variables.)

One can also state versions of the claim that there is a set of urelements,

∃α(∀x(x ∈s α))

(Where ‘x’ is a material-object-variable, and ‘α’ is a set-theoretic variable.)

and the claim that there is a class V of all non-proper classes:

∃V (∀x(x ∈c V ) ∧ ∀C(∃D(C ∈c D)→ C ∈c V ))

(Where ‘x’ is a material-object-variable, and ‘V ’, ‘C’ and ‘D’ are class-

theoretic variables.)

On the other hand, there is no way of expressing the general claim that there are more

sets than classes (or that there are more classes than sets). If one wanted to state such

9.2. SUCCESS CONDITIONS 185

claims, one would have to introduce a new sort of variable to range over collections which

can have both sets and classes as elements.

In such a language it can be strictly and literally true to say—using fully unrestricted

set-theoretic quantifiers—that there are inaccessibly many objects, and also strictly and

literally true to say—using fully unrestricted class-theoretic quantifiers—that there are

successor-of-an-inaccessible-many objects. But this is not because each sort of quantifier

ranges over a different fragment of a single ‘maxi-domain’. (See section 3.2.) It is

because quantifiers of different sorts presuppose different carvings of the world. One

of the carvings delivers inaccessibly many objects; the other delivers successor-of-an-

inaccessible many.

On the most natural way of developing this picture, the feature of reality that is fully

and accurately described by the set theoretic sentence ‘∃α(Socrates ∈s α)’ is also fully and

accurately described by the class theoretic sentence sentence ‘∃C(Socrates ∈c C)’ (and

by the material-object-sentence ‘∃x(Socrates = x)’). Similarly, a set-theoretic sentence

stating that there are inaccessibly many objects and a class-theoretic sentence stating that

there are successor-of-an-inaccessible-many objects both have trivial truth-conditions. So

the (trivial) feature of reality that is fully and accurately described by one of them is

also fully and accurately described by the other (and by any truth of logic). Cardinality

claims like these are not incompatible de mundo: it is not the case that what the truth

of them requires of the world is incompatible with what the truth of the other requires

of the world. They are only incompatible in the sense that they presuppose systems of

representation that cannot be happily combined.

9.2 Success Conditions

The goal of a linguistic stipulation is to specify an assignment of truth-conditions to

sentences involving the newly introduced vocabulary. A Tractarian will think that the

186 CHAPTER 9. INTRODUCING MATHEMATICAL VOCABULARY

only way of producing such an assignment is by pairing each singular term with one of

the objects carved out by the world’s metaphysical structure, and pairing each predicate

with one of the properties carved out by the world’s metaphysical structure. The anti-

Tracatrian will disagree. She will claim that correspondence with metaphysical structure

is no part of what it takes for the new vocabulary to be in good order.

The anti-Tractarian will agree that a successful stipulation results in a pairing of lex-

ical items to objects and properties. But she won’t see this as a substantial constraint.

For a carving of the world to be in place just is for a compositional system of repre-

sentation to be in place. So when a stipulation succeeds in specifying truth-conditions

for whichever sentences one wishes to make available for use it thereby makes available

a way of carving up the world with respect to which one could pair each of the new

lexical items with a suitable object or property. Such pairings can be made explicit by

enriching one’s metalanguage with the newly introduced vocabulary and setting forth

a homophonic semantics for one’s object language. (Does it follow that there isn’t an

objective, language-idependent fact of the matter about which objects exist? Absolutely

not. For an extended discussion of such issues, see chapter 3.)

On the anti-Tractarian picture, what does it take for a linguistic stipulation to be

successful? Let us begin with the simplest case. Say that a pure mathematical stipulation

is a stipulation in which:

1. one introduces new vocabulary, and makes clear that it is to be regarded as falling

under a new sort;

2. one makes clear that any atomic formula that mixes vocabulary of old and new

sorts is to be counted as ill-formed;

3. one uses the new vocabulary to build an axiom system in which all the variables

and non-logical constants are of the new sort;

9.2. SUCCESS CONDITIONS 187

4. one stipulates that the new vocabulary is to be understood in such a way that the

axiom system turns out to be necessarily true.

From the perspective of an anti-Tractarian, all it takes for a pure mathematical stipu-

lation to succeed—assuming it gives rise to the right sort of linguistic practice—is for the

relevant axiom system to be internally coherent (i.e. for it not to have a logical absurdity

as a logical consequence).

The reason internal coherence is sufficient for success is that it is enough to guarantee

that one can construct a stable assignment of truth-conditions whereby the axioms are all

counted as necessarily true. The assignment in question can be specified as follows. Let

Lnew be the fragment of the language in which all the variables and non-logical constants

are of the new sort. Then: (1) a sentence of Lnew is taken to have trivial truth-conditions

if it is a logical consequence of the axioms; (2) a sentence of Lnew is taken to have trivially

unsatisfiable truth-conditions if its negation is a logical consequence of the axioms; and

(3) other sentences of Lnew are taken to lack well-defined truth-conditions.

The fact that truth-conditions are specified on the basis of logical consequence (and

therefore logical form) guarantees that syntactic structure is semantically significant,

and that a sentence’s truth-conditions are determined by the inferential behavior of its

constituent parts. In this sense, the proposed specification of truth-conditions is compo-

sitional. It is also stable, in the following sense:

• Since the axiom system is internally coherent, the procedure is guaranteed not to

deliver more than one assignment of truth conditions to a given sentence of Lnew .

[Here I assume that the background logic is not paraconsistent. Note, however, that if one’s

background logic is paraconsistent, one need not see contradictory assignments of truth-conditions

as inherently problematic.]

• Since the axiom system is internally coherent, and since atomic formulas that mix

vocabulary of old and new sorts are counted as ill-formed, the sentences of Lnew float

188 CHAPTER 9. INTRODUCING MATHEMATICAL VOCABULARY

free, inferentially speaking, from sentences that contain no new vocabulary. This

means, in particular, that the new axiom system is (strongly) conservative over

sentences with no new vocabulary. So the proposed assignment of truth-conditions

is guaranteed to have no effect on sentences that contain no new vocabulary.

[For axiom system A to be strongly conservative over sentences containing no new vocabulary is for

the following condition to be satisfied: let O be a set of sentences containing no new vocabulary,

and let φ be a sentence containing no new vocabulary; then φ is only a logical consequence of O

and A if it is a logical consequence of O. For A to be weakly conservative is for a weaker conditions

to be satisfied: φ is only a logical consequence of O and A if it is either a logical consequence of

O or lacks well-defined truth-conditions.

These definitions can be used to define notions of semantic conservativeness or syntactic conser-

vativeness, depending on whether logical consequence is understood syntactically or semantically.

(When the original language is governed by a logic that is syntactically complete—such as first-

order logic—semantic conservativeness entails syntactic conservativeness.) Which of these two

kinds of conservativeness is relevant to the present discussion depends on whether one thinks that

being a semantic consequence of the newly introduced axioms is sufficient for having well-defined

truth-conditions. If on thinks semantic consequence is enough, one should work with semantic

conservativeness; if one thinks that syntactic consequence is needed, one should work with syn-

tactic conservativeness. (The claims I make in this chapter are independent of whether one thinks

of logical consequence syntactically or semantically, but see footnote 1 from chapter 4.)]

The Tractarian will think that something important has been left out. She will agree

that we have found a principled way of adding the label ‘necessary’ to some sentences

and the label ‘impossible’ to others. But she will think that this is not enough to show

that the axioms are true—let alone necessarily true. For it ignores reference. One cannot

say true things with names unless the world contains referents for those names, and one

cannot say true things with quantifiers unless the world supplies a suitable domain for

the quantifiers to range over. The problem, according to the Tractarian, is that adding

9.3. INTERNAL COHERENCE 189

labels to sentences—even when it is done in a principled way—does nothing to show that

the world contains enough objects for such referential pairings to be possible.

The anti-Tractarian would agree that one cannot say true things with names unless

the world contains referents for those names, and that one cannot say true things with

quantifiers unless the world supplies a suitable domain for the quantifiers to range over.

But she would insist that the right sort of assignment of truth-conditions to sentences

shows that the world does contain enough objects for such referential pairings to be

available. For a suitable assignment of truth-conditions results in a compositional system

of representation, and for a carving of the world to be in place just is for a compositional

system of representation to be in place. So in specifying a suitable assignment of truth-

conditions one carves up the world into enough objects for the referential pairings in

question to be available.

9.3 Internal Coherence

We have seen that, from the perspective of the anti-Tractarian, all it takes for a pure

mathematical stipulation to succeed is internal coherence (assuming the stipulation gives

rise the the right sort of linguistic practice).

It would be a mistake to conclude from this, however, that it is generally a straight-

forward mater to determine whether a pure mathematical stipulation will succeed. For

it is not, in general, a straightforward matter to determine whether an axiom system

is internally coherent. Suppose, for example, that one sets forth a pure mathematical

stipulation based on Quine’s NF. The consistency of NF remains an open question.1 So

it is by no means clear that knowledge that the stipulation would succeed is within our

reach.

In general, acquiring a warrant for the internal coherence of an axiom system is

1There is, on the other hand, a consistency proof for NFU (New Foundations with Urelements). SeeJensen (1969).

190 CHAPTER 9. INTRODUCING MATHEMATICAL VOCABULARY

a delicate matter. One can sometimes prove the consistency of one formal system in

another. But it is a consequence of Godel’s Second Theorem that (when the systems in

question are consistent and sufficiently strong) the system in which the proof is carried

out cannot be a subsystem of the system the proof is about. There is therefore reason

to expect that one’s warrant for the internal coherence of a formal system will turn on

more than just consistency proofs. It might turn on whether one has a good feel for the

sorts of things that can be proved in the system, or on whether one has a good feel for

the sorts of things that can be proved in formal systems within which one has been able

to produce a consistency proof. It might also turn on whether one has a good feel for

what a model for the axiom system in question would look like.

As a result, one should expect one’s warrant for the internal coherence of an axiom

system to be defeasible (whether or not it is also a priori). And since one’s warrant for

the truth of an axiom system that has been set forth as a pure mathematical stipulation

will be no better than one’s warrant for the system’s internal coherence, one should also

expect one’s warrant for the truth of the system to be defeasible.

The picture of mathematical knowledge that we are left with is a messy one. One’s

warrant for the truth of the axioms of pure mathematics will be typically defeasible. It

will sometimes turn on informal considerations, such as whether one has a good feel for

the sorts of things that can be proved on the basis of the axioms, or for what a model

for the axioms would look like. And it will sometimes depend on one’s warrant for the

truth of further mathematical theories.

This is as it should be. One wouldn’t want one’s account of mathematical knowledge

to yield the result that mathematical knowledge is easier to come by than the practice

of mathematicians would suggest.

9.4. APPLIED MATHEMATICS 191

9.4 Applied Mathematics

The claim that internal coherence is enough for success depends on the assumption that

the mathematical stipulation in question is pure. To see this, imagine a case in which we

introduce arithmetical vocabulary by way of a (second-order) axiom system which entails

(1) that every number is finite, (2) that any material objects can be numbered, and (3)

that the number of the Fs is n just in case there are precisely n Fs. Such an axiom

system is internally coherent. But one certainly wouldn’t want to count it as necessarily

true, since it entails that there are only finitely many stars, and one shouldn’t be able

to settle the question of whether there are finitely many stars on purely mathematical

grounds.

The problem with this axiom system is that it fails to be conservative. In the case

of pure mathematical stipulations conservativeness is guaranteed by the fact that mixed

atomic formulas are always counted as ill-formed. But when it comes to applied arith-

metic we have no choice but to allow for mixed atomic formulas, and therefore counte-

nance substantial inferential interactions between sentences built from new vocabulary

and sentences built from old vocabulary.

Fortunately, the role that was played by internal coherence in the case of pure math-

ematics can be played by conservativeness in the case of applied mathematics. Say that

a generalized mathematical stipulation is a stipulation in which:

1. one introduces new vocabulary, and makes clear that it is to be regarded as falling

under a new sort;

2. one makes clear which mixed atomic formulas are to be counted as ill-formed;

3. one uses the new vocabulary to build an axiom system which may or may not

include vocabulary of the old sort;

4. one stipulates that the new vocabulary is to be understood in such a way that the

192 CHAPTER 9. INTRODUCING MATHEMATICAL VOCABULARY

axiom systems turns out to be necessarily true.

From the perspective of an anti-Tractarian, all it takes for a generalized mathematical

stipulation to succeed—assuming it gives rise to the right sort of linguistic practice—is

for the relevant axiom system to be (weakly) conservative.

The reason conservativeness is sufficient for success is that it is enough to guarantee

that one can construct a stable assignment of truth-conditions whereby the axioms are

all counted as necessarily true. The assignment in question can be specified as follows.

Let A be the relevant axiom system, φ be a sentence containing new vocabulary, w be

a world and O be the set of sentences containing no new vocabulary that are true at w.

Then: (1) φ is counted as true at w just in case φ is a logical consequence of A∪O; (2)

φ is counted as false at w just in case p¬φq is a logical consequence of A∪O; and (3) φ

is otherwise counted as lacking well-defined truth-conditinons.

As in the pure case, the fact that truth-conditions are specified on the basis of logical

consequence (and therefore logical form) guarantees that syntactic structure is semanti-

cally significant, and that a sentence’s truth-conditions are determined by the inferential

behavior of its constituent parts. In this sense, the proposed specification of truth-

conditions is compositional. It is also stable, in the following sense:

• since the axiom system is conservative, it is guaranteed to be internally coher-

ent, and therefore guaranteed not to deliver more than one assignment of truth

conditions to sentences containing vocabulary of the new sort;

• If the axiom system is strongly conservative, the procedure is guaranteed to have no

effect on sentences including no vocabulary of the new sort. If the axiom system is

weakly conservative, the procedure will only have an effect on sentences including

no vocabulary of the new sort when the sentences in question previously lacked

well-defined truth-conditions.

9.5. COMPOSITIONAL SEMANTICS 193

As in the pure case, the Tractarian will think that something important has been left

out, on the grounds that we have been given no guarantee that the world contains an

appropriate domain of objects for quantifiers of the new sort to range over. As in the

pure case, the anti-Tractarian will reply that the right assignment of truth-conditions to

sentences shows that the world does contain the requisite domain. For in specifying a

suitable assignment of truth-conditions one carves up the world into enough to objects

for the needs of the theory to be satisfied.

9.5 Compositional Semantics

When a set of new mathematical vocabulary has been introduced by stipulation and

everything goes well, it is straightforward to construct a compositional semantics for

the extended language. For the success of the stipulation makes the new vocabulary

available for use. And once the vocabulary is available it can be incorporated into the

metalanguage, and used to give a homophonic semantics for the object-language.

A homophonic semantics is guaranteed to give an accurate statement of truth-conditions

for every sentence in the object-language. But it won’t shed any light on the question

of what the world would have to be like in order for those truth-conditions to be sat-

isfied. Suppose, for example, that one is fluent with arithmetical vocabulary, but isn’t

sure whether ‘2 + 2 = 4’ has trivial truth-conditions. Being told that ‘2 + 2 = 4’ is true

at a world just in case the world is such that 2 + 2 = 4 won’t help. One will regard the

statement as intelligible, and recognize it as true. But it won’t deliver an elucidation of

the sort one was after.

In contrast, a trivialist semantics of the sort described in section 4.4.1 can supply a

non-trivial elucidation of what the world would have to be like in order for the truth-

conditions of an object-language sentence to be satisfied. In the special case of a pure

mathematical stipulation, it is straightforward to construct such a semantics. All one

194 CHAPTER 9. INTRODUCING MATHEMATICAL VOCABULARY

needs to do is ‘outscope’ mathematical vocabulary from the homophonic semantic clauses.

Just how illuminating the result turns out to be, however, will depend on how much

mathematics one is able to prove in the metatheory. All the semantics delivers on its

own is the result that the truth-conditions of, say, ‘2+2=4’ are either trivial or trivially

unsatisfiable depending on whether 2 + 2 = 4. This is more than one gets from a

homophonic semantics, but it’s not much. In order to get more specific results, one

would have to add the axioms that were used in the stipulation to the metatheory, and

use them to prove metalinguistic versions of the object-language sentences whose truth-

conditions one is hoping to shed light on.

A trivialist semantics can be more illuminating in the case of a generalized mathe-

matical stipulation. When one is told, for example, that ‘#x(Planet(x)) = 8’ is true at a

world w just in case w satisfies the following metalinguistic formula:

The number of the zs such that [z is a planet]w = 8

one immediately gets a non-trivial elucidation of the sentence’s truth-conditions. For

one immediately learns that whether or not they are satisfied depends entirely on how

matters stand with the planets. If, in addition, one has access to the relevant axioms in

the metatheory, and is able to prove that the number of the planets is Eight if and only

if there are exactly eight planets, then one will also have learned that what it takes for

‘#x(Planet(x)) = 8’s truth-conditions to be satisfied is for there to be eight planets.

In the case of generalized mathematical stipulations, however, it is not always straight-

forward to give a trivialist semantics. One reason problems can arise in the case of applied

mathematics is that there is no general recipe for constructing trivialist semantic clauses

for atomic formulas that mix mathematical and non-mathematical vocabulary.

Things work out for the language of applied arithmetic because we have the following

equivalence:

[#x(F(x)) = n]w ↔ #x([F(x)]w) = n.

9.5. COMPOSITIONAL SEMANTICS 195

Read: at w (the number of the Fs = n) just in case (the number of the xs

such that at w (x is an F)) = n.

(Or an actualist version thereof—see section 4.4.1.) It is because of this equivalence that

we are able to ‘outscope’, and go from the homophonic semantic clause for the mixed

atomic formula ‘#x(F(x)) = n’,

‘#x(F(x)) = n’ is true at w ↔ [#x(F(x)) = n]w,

to a trivialist semantic clause,

‘#x(F(x)) = n’ is true at w ↔ #x([F(x)]w) = n.

But such equivalences are not guaranteed to be available in general. For instance, I know

of no general way of defining a trivialist semantics for a plural language that has been

enriched with non-logical atomic plural predicates. (For discussion of PFO+ languages,

see Rayo (2002b).) Happily, one can give a trivialist semantics for the language of set-

theory with urelements. (See section 4.6.1.)

Moral: There is no free lunch. If you want an illuminating account of truth-conditions,

you’ll have to do work to get it.

Intended Interpretations

Up to now we have been focusing on the project of giving a compositional semantics for a

language whose component parts have been rendered meaningful by linguistic stipulation.

We have seen that a homophonic semantics is always available, and that it some cases

it may be possible to modify a homophonic semantics to get a trivialist semantics. We

have also seen that the ability of the trivialist semantics to deliver useful information

about a sentence’s truth conditions will depend, in part, on our ability to prove results

in the metatheory, by using the axiom system on which the stipulation was based.

196 CHAPTER 9. INTRODUCING MATHEMATICAL VOCABULARY

There is, however, an alternate method for arriving at a trivialist semantics for math-

ematical discourse. On the alternate method, one needn’t presuppose that the object

language has been previously rendered meaningful by stipulating that a given axiom sys-

tem is to count as necessarily true. One needn’t even have access to an axiom system

on which such a stipulation might have been based. Instead, one relies on the ability to

identify an intended interpretation for the new vocabulary.

One could, for example, construct a trivialist assignment of truth-conditions to arith-

metical sentences by working on the assumption that number-terms should be interpreted

as standing for finite ordinals. On the most natural way of spelling out the details, a

trivialist semantics would yield the result that ‘#x(Planet(x)) = 8’ is true at a world w

just in case w satisfies the following metalinguistic formula:

The ordinal in one-one correspondence with the set of zs such that [z is a

planet]w = 8ord

The thing to note is that even though the semantic values of arithmetical terms are based

on ordinals rather than numbers, one gets the very same assignment of truth-conditions

that are delivered by the number-based semantics of section 4.4.1. In both cases, one

gets the result that ‘#x(Planet(x)) = 8’ is true at w just in case there are eight planets

at w. And the point generalizes: assuming restrictions on the well-formedness of mixed

atomic formulas, one gets identical assignments of truth-conditions for every sentence in

the language.

A Tractarian would insist that an assignment of truth-conditions to arithmetical

sentences can only be accurate if it uses numbers rather than sets to build the semantic

values of arithmetical terms. But, from the point of view of an anti-Tractarian, the

only legitimate constraint on an assignment of semantic values is that it deliver the right

specification of truth-conditions.

According to the anti-Tractarian, the real difference between a trivialist semantics

9.5. COMPOSITIONAL SEMANTICS 197

based on numbers and a trivialist semantics based on ordinals is that different metathe-

oretical resources need to be deployed before one can use the semantics to make illumi-

nating statements about the truth-conditions of object-language sentences. When one

uses numbers to build the semantic values of arithmetical terms, one needs to derive the

formula ‘eiπ = 1’ in the metatheory in order to ascertain whether the object-language

formula ‘eiπ = 1’ has trivial (rather than trivially unsatisfiable) truth-conditions; when

one uses ordinals, one needs to derive a set-theoretic formula instead.

198 CHAPTER 9. INTRODUCING MATHEMATICAL VOCABULARY

Part III

Appendices

199

Appendix A

A Semantics for a Language with

the Dot Notation

I give a formal semantics for a language Lw which allows for empty names and contains

both the intensional operator [. . .]w (read ‘according to w, . . . ’) and the dot-notation.

Lw consists of the following symbols:

1. for n > 0, the n-place (non-modal) predicate letters: pFn1q, pF

n2q, . . . (each with an

intended interpretation);

2. for n > 0, the one-place modal predicate letters: pB1q, pB2q, . . . (each with an

intended interpretation);

3. the identity symbol ‘=’;

4. for n > 0, the individual non-empty constant-letter pcnq (each with an intended

referent);

5. for n > 0, the individual empty constant-letter penq;

6. the individual constant ‘α’

201

202APPENDIX A. A SEMANTICS FOR A LANGUAGE WITH THE DOT NOTATION

7. the dot ‘˙’;

8. the monadic sentential operator ‘[. . .]’;

9. the monadic sentential operator ‘¬’;

10. the dyadic sentential operators ‘∧’,

11. the quantifier-symbol ‘∃’;

12. the modal variables: ‘w’, ‘v’ ‘u’ with or without numerical subscripts;

13. the non-modal variables: ‘x’, ‘y’, ‘z’ with or without numerical subscripts;

14. the auxiliaries ‘(’ and ‘)’.

Undotted terms and formulas are defined as follows:

1. any modal variable is an undotted modal term;

2. ‘α’ is an undotted modal term;

3. any non-modal variable or individual constant-letter is an undotted non-modal

term;

4. if τ1, . . . , τn are undotted non-modal terms, then pFni (τ1, . . . , τn)q is an undotted

formula;

5. if τ1 and τ2 are either both undotted non-modal terms or both undotted modal

terms, then pτ1 = τ2q is an undotted formula;

6. if w is an undotted modal term, then pBi(w)q is an undotted formula;

7. if φ is an undotted formula and w is an undotted modal term, then p[φ]wq is an

undotted formula;

203

8. if v is an undotted (modal or non-modal) variable and φ is an undotted formula,

then p∃v(φ)q is an undotted formula;

9. if φ and ‘ψ’ are undotted formulas, then p¬φq, p(φ ∧ ψ)q, p(φ ∨ ψ)q and p(φ ⊃ ψ)q

are undotted formulas;

10. nothing else is an undotted term or formula.

A non-modal term is either an undotted non-modal term or the result of dotting a

non-modal variable; a modal term is an undotted modal term; a formula is the result of

dotting any free or externally bounded occurrences of non-modal variables in an undotted

formula.1

Next, we characterize the notion of an a-world and of a variable assignment. An

a-world is a pair 〈D, I〉 with the following features:

1. D is a set of individuals in the range of the non-modal variables, each of which is

either of the form 〈x, ‘actual’〉 or of the form 〈x, ‘nonactual’〉.2

2. I is a function assigning a subset of D to each 1-place predicate-letter, and a subset

of Dn to each n-place predicate letter (for n < 1). In addition, if e is an empty

name, I may or may not assign a referent to e (and if a referent is assigned, it may

or may not be in D).

The actualized a-world 〈Dα, Iα〉 will be singled out for special attention. Dα is the set of

pairs 〈z, ‘actual’〉 for z an individual in the range of the non-modal variables; and Iα(‘Fnj ’)

is the set of sequences 〈〈z1, ‘actual’〉 , . . . 〈zn, ‘actual’〉〉 such tha z1 . . . zn are in the range

of the non-modal variables and satisfy F .

A variable assignment is a function σ with the following features:

1An occurrence of a non-modal variable in an undotted formula is free iff it is not bound by aquantifier; an occurrence of a non-modal variable in an undotted formula is externally bounded iff it isbound by a quantifier which is not within the scope of ‘[. . .]’.

2I assume that D is a set for the sake of simplicity. The assumption can be avoided by characterizingthe notion of an a-world in second-order terms. This can be done by employing the technique in Rayoand Uzquiano (1999).

204APPENDIX A. A SEMANTICS FOR A LANGUAGE WITH THE DOT NOTATION

1. σ assigns an a-world to each modal variable.

2. σ assigns an individual to each non-modal variable.

This puts us in a position to characterize notions of quasi-denotation and quasi-satisfaction.

(Denotation and satisfaction proper will be characterized later). For v a non-modal vari-

able, σ a variable assignment, φ a formula and w an a-world, we characterize the quasi-

denotation function δσ,w(v) and the quasi-satisfaction predicate Sat(φ, σ). In addition,

we characterize an auxiliary (a-world-relative) quasi-satisfaction predicate Sat(φ, σ, w).

We proceed axiomatically, by way of the following clauses:

• If v is a (modal or non-modal) variable, δσ,w(v) is σ(v);

• If v is a non-modal variable, w is an a-world and σ(v) is an ordered pair of the

form 〈z, ‘actual’〉, then δσ,w(pvq) is the first member of σ(v); otherwise δσ,w(pvq) is

undefined;

• if c is a non-empty constant-letter and w is an a-world, δσ,w(c) is the intended

referent of c.

• if e is an empty constant-letter and w is an a-world, δσ,w(e) is the w-referent of e if

there is one, and is otherwise undefined;

• δσ,w(‘α’) is 〈Dα, Iα〉;

• if τ1 and τ2 are terms (both of them modal or both of them non-modal) and neither

of them is an empty constant-letter, then Sat(pτ1 =, τ2)q, σ) if and only if δσ,w(τ1) =

δσ,w(τ2) for arbitrary w;

• if τ1 and τ2 are non-modal terms at least one of which is an empty constant-letter,

then not-Sat(pτ1 =, τ2)q, σ);

205

• if τ1, . . . τn are non-modal terms none of which is an empty constant-letter, then

Sat(pFni (τ1, . . . , τn)q, σ) if and only if F n

i (δσ,w(τ1), . . . , δσ,w(τn)), where w is arbi-

trary and pFni q is intended to express F n

i -ness;

• if τ1, . . . τn are non-modal terms at least one of which is an empty constant-letter,

then not-Sat(pFni (τ1, . . . , τn)q, σ);

• if v is a modal variable, Sat(pBi(v)q, σ) if and only if Bi(δσ,w(v)) for arbitrary w,

where Bi is intended to express Bi-ness;

• if v is a non-modal variable, Sat(p∃v(φ)q, σ) if and only if there is an individual z

in the range of the non-modal variables such that Sat(pφq, σv/z), where σv/z is just

like σ except that it assigns z to v;

• if v is a modal variable, Sat(p∃v(φ)q, σ) if and only if there is an a-world z such

that Sat(pφq, σv/z), where σv/z is just like σ except that it assigns z to v;

• Sat(p¬φq, σ) if and only if it is not the case that Sat(φ, σ);

• Sat(pφ ∧ ψq, σ) if and only if Sat(φ, σ) and Sat(ψ, σ);

• Sat(p[φ]wq, σ) if and only if Sat(φ, σ′, σ(w)), where σ′(x) = σ(x) for x a modal

variable and σ′(x) = 〈σ(x), ‘actual’〉 for x a non-modal variable;

• if τ1 and τ2 are non-modal terms neither of which is an empty constant-letter

without a w-reference, then Sat(pτ1 =, τ2)q, σ, w) if and only if δσ,w(τ1) is in the

domain of w and is identical to δσ,w(τ2);

• if τ1 and τ2 are non-modal terms at least one of which is a constant-letter without

a w-reference, then not-Sat(pτ1 =, τ2)q, σ, w);

• if τ1 and τ2 are modal terms, then Sat(pτ1 =, τ2)q, σ, w) if and only if δσ,w(τ1) =

δσ,w(τ2) for arbitrary w;

206APPENDIX A. A SEMANTICS FOR A LANGUAGE WITH THE DOT NOTATION

• if τ1, . . . τn are non-modal terms none of which is a constant-letter without a w-

reference, then Sat(pFni (τ1, . . . , τn)q, σ, w) if and only if 〈δσ,w(τ1), . . . , δσ,w(τn)〉 is in

the w-extension of pFni q;

• if τ1, . . . τn are non-modal terms at least one of which is a constant-letter without

a w-reference, then not-Sat(pFni (τ1, . . . , τn)q, σ, w);

• if v is a modal term, Sat(pBi(v)q, σ, w) if and only if Bi(δσ,w(v)), where pBiq is

intended to express Bi-ness;

• if v is a non-modal variable, Sat(p∃v(φ)q, σ, w) if and only if there is an individual

z in the domain of w such that Sat(pφq, σv/z, w), where σv/z is just like σ except

that it assigns z to v;

• if v is a modal variable, Sat(p∃v(φ)q, σ, w) if and only if there is an a-world z such

that (a) any empty constant-letter which is assigned a referent by w is assigned the

same referent by z, and (b) Sat(pφq, σv/z, w), where σv/z is just like σ except that

it assigns z to v;

• Sat(p¬φq, σ, w) if and only if it is not the case that Sat(φ, σ, w);

• Sat(pφ ∧ ψq, σ, w) if and only if Sat(φ, σ, w) and Sat(ψ, σ, w);

• Sat(p[φ]uq, σ, w) if and only if Sat(φ, σ, σ(u)).

Finally, we say that a formula φ is quasi-true if and only if Sat(φ, σ) for any variable

assignment σ.

With a suitable notion of admissibility is on board, one can characterize truth and

satisfaction for Lw. Satisfaction is the special case of quasi-satisfaction in which attention

is restricted to admissible a-worlds, and truth is the special case of quasi-truth in which

attention is restricted to admissible a-worlds.

Appendix B

If Lewis Can Say It, You Can Too

In section 7.5.2 I described a transformation from Lewisian sentences to dotted actualist

sentences. The purpose of this appendix is to explain how the transformation works in

general. I shall assume that the Lewisian language is a two-sorted first-order languages,

with world-variables ‘w1’, ‘w2’, etc. ranging over Lewisian worlds, and individual-variables

‘x1’, ‘x2’, etc. ranging over world-bound individuals in the Lewisian pluriverse. The lan-

guage contains no function-letters; there is a world-constant ‘α’ referring to the actual

Lewisian world and individual-constants ‘c1’, ‘c2’, etc. referring to world-bound indi-

viduals. The only atomic predicates are ‘I’ (which takes an individual-variable and

a world-variable), ‘C’ (which takes two individual-variables), ‘=’ (which takes (i) two

world-variables, (ii) two individual-variables, (iii) a world-constant and a world-variable,

or (iv) an individual-constant and an individual-variable) and, for each j, pPnj q (which

takes n individual-variables). (If one likes, one can also take the language to contain

set-theoretic vocabulary.) Finally, I shall assume that universal quantifiers are defined in

terms of existential quantifiers in the usual way, and that logical connectives other than

‘∧’ and ‘¬’ are defined in terms of ‘∧’ and ‘¬’ in the usual way.

The plan is to proceed in two steps. The first is to get the Lewisian sentence into

a certain kind of normal form; the second is to convert the normal-form sentence into

207

208 APPENDIX B. IF LEWIS CAN SAY IT, YOU CAN TOO

a dotted actualist sentence. Here is a recipe for getting an arbitrary Lewisian sentence

into normal form:

1. Start by relabeling variables in such a way that no world-variable in has the same

index as an individual-variable;

2. next, replace each occurrence of pI(xj, wk)q by pwj = wkq;

3. then replace each occurrence of the atomic formula pPnj (xk1 , . . . , xkn)q by

(Pnj (xk1 , . . . , xkn) ∧ wk1 = wk2 ∧ . . . ∧ wk1 = wkn),

each occurrence of pcj = xkq by p(cj = xk ∧ α = wk)q, and each occurrence of

pxj = xkq by p(xj = xk ∧ wj = wk)q.1

4. finally, replace each occurrence of p∃xj(. . .)q by p∃xj∃wj(I(xj, wj) ∧ . . .)q.

On the assumption that pPnj q is projectable (and, hence, that pPn

j (xk1 , . . . , xkn)q can only

be true if xk1 , . . . , xkn are world-mates), it is easy to verify that this procedure respects

truth-value. (For a characterization of projectability, see footnote 5 of chapter 7.)

Here is an example. The Lewisian sentence

∃w17∃x2∃x5(I(x2, w17) ∧ Sister(x2, x5))

(Read: There is a Lewisian world w17 an individual x2 and an individual w5

such that x2 is an inhabitant of w17 and x2 has x5 as a sister.)

gets rewritten as:

∃w17∃x2∃w2(I(x2, w2)∧∃x5∃w5(I(x5, w5)∧w2 = w17∧Sister(x2, x5)∧w2 = w5))

(Read: There is a Lewisian world w17, an individual x2 inhabiting Lewisian

world w2, and an individual x5 inhabiting Lewisian world w5 such that w2 is

identical to w17, x2 has x5 as a sister and w2 is identical to w5.)

1These three replacements are needed to secure the base clause in the induction below.

209

Once one has a Lewisian sentence in normal form, it can be transformed into a dotted

actualist sentence by carrying out the following replacements:

I(xj, wj) −→ [∃y(y = xj)]wj

Pnj (xk1 , . . . , xkn) −→ [Pn

j (xk1 , . . . , xkn)]wk1

C(xj, xk) −→ xj = xk

cj = xk −→ [cj = xk]wk

Here is an example. The Lewisian rendering of ‘♦(∃x1∃x2 Sister(x1, x2))’ is

∃w3∃x1∃x2(I(x1, w3) ∧ I(x2, w3) ∧ Sister(x1, x2))

whose normal form

∃w3∃x1∃w1(I(x1, w1) ∧ ∃x2∃w2(I(x2, w2) ∧ w1 = w3 ∧ w1 = w3∧

Sister(x1, x2) ∧ w1 = w2))

gets transformed by the actualist into

∃w3∃x1∃w1([∃y(y = x1)]w1 ∧ ∃x2∃w2([∃y(y = x2)]w2 ∧ w1 = w3 ∧ w1 = w3∧

[Sister(x1, x2)]w1 ∧ w1 = w2))

which boils down to:

∃w3∃x1∃x2([Sister(x1, x2)]w3)

(Read: There is are objects x1 and x2 and an admissible a-world w such that

x1 and x2 are used by w to represent someone’s having a sister.)

210 APPENDIX B. IF LEWIS CAN SAY IT, YOU CAN TOO

and is guaranteed by the Appendix A semantics to be equivalent to

∃w3([∃x1∃x2 Sister(x1, x2)]w3)

(Read: There is an admissible a-world w according to which there are indi-

viduals x1 and x2 such that x1 has x2 as a sister.)

which is the actualist’s rendering of ‘♦(∃x1∃x2 Sister(x1, x2))’.

Here is a slightly more complex example. The actualist’s rendering of

There are possible worlds w1 and w2 with the following properties: according

to w1, there is an individual who is my sister and a philosopher; according to

w2, that very individual is a cellist rather than a philosopher.

is

∃w4∃w5∃x1∃x2∃x3(

I(x1, w4)∧I(x2, w4) ∧ I(x3, w5)∧

∃x6(ar = x6 ∧ C(x6, x1)) ∧ C(x2, x3)∧

Sister(x1, x2) ∧ Phil(x2) ∧ Cellist(x3) ∧ ¬Phil(x3))

whose normal form

∃w4∃w5∃x1∃w1(I(x1, w1) ∧ ∃x2∃w2(I(x2, w2)∧∃x3∃w3(I(x3, w4)∧

w1 = w4 ∧ w2 = w4 ∧ w3 = w5∧

∃x6∃w6(I(x6, w6) ∧ ar = x6 ∧ α = w6 ∧ C(x6, x1)) ∧ C(x2, x3)∧

Sister(x1, x2) ∧ Phil(x2) ∧ Cellist(x3) ∧ ¬Phil(x3))))

211

gets transformed by the actualist into

∃w4∃w5∃x1∃w1([∃y(y = x1)]w1 ∧ ∃x2∃w2([∃y(y = x2)]w2 ∧ ∃x3∃w3([∃y(y = x3)]w3∧

w1 = w4 ∧ w2 = w4 ∧ w3 = w5∧

∃x6∃w6([∃y(y = x6)]w6 ∧ [ar = x6]w6 ∧ α = w6 ∧ x6 = x1) ∧ x2 = x3∧

[Sister(x1, x2)]w1 ∧ w1 = w2 ∧ [Phil(x2)]w2 ∧ [Cellist(x3)]w3 ∧ ¬[Phil(x3)]w3)))

which boils down to

∃w4∃w5∃x2([Sister(ar, x2)]w4 ∧ [Phil(x2)]w4 ∧ [Cellist(x2)]w5 ∧ [¬Phil(x2)]w5)

(Read: there are admissible a-worlds w4 and w5 and an individual x2 such that: (i) x2 is

used by w4 to represent my sister and a philosopher, and (ii) x2 is used by w5 to represent

a cellist rather than a philosopher.)

Finally, we show that there is a notion of a-world admissibility which guarantees that

the actualist transformation of an arbitrary Lewisian sentence has the truth-value that

the Lewisian sentence would receive on its intended interpretation. (The proof relies on

the assumption that the Lewisian language is rich enough—and the space of Lewisian

worlds varied enough with respect to predicates occurring in the language—that the set of

true sentences has a model in which any two worlds are such that some atomic predicate

satisfied by a sequence of objects inhabiting one of the worlds, but not by the result of

replacing each object in the sequence by its counterpart in the other world.) Start by

enriching the Lewisian language with a standard name for each inhabitant of the actual

world, and let S be the set of true sentences in the extended language. One can use

the Completeness Theorem to generate a model M of S in which the domain consists of

‘=’-equivalence classes of terms and in which the assumption above is satisfied. If a1 is

an object in the individual-variable domain of M , let a∗1 be the set of individuals a2 such

212 APPENDIX B. IF LEWIS CAN SAY IT, YOU CAN TOO

that ‘C(x1, x2)’ is satisfied by a1 and a2. (I assume that ‘C’ is an equivalence relation, and

therefore that ∗ partitions the individual-variable domain of M into equivalence classes.)

If a1 is an object in the individual-variable domain of M , let a† be 〈z, ‘actual’〉 if the

standard name of z is in the transitive closure of a∗, and 〈a∗, ‘nonactual’〉 otherwise.

For each c in the world-variable domain of M , we construct an a-world wc, as follows:

the domain of wc is the set of a† such that ‘I(x1, w1)’ is satisfied by a and c in M ; the

wc extension of pPnj q is the set of sequences

⟨a†1, . . . , a

†n

⟩such that ‘I(x1, w1) ∧ . . . ∧

I(xn, w1) ∧ Pnj (x1 . . . , xn)’ is satisfied by a1, . . . , an, c in M . In addition we let individual

constants receive their intended interpretations, and let ‘α’ be wc, where c is the M -

referent of ‘α’. Say that an a-world is admissible just in case it is a wc for some c in

the world-variable domain of M . If σ is a variable-assignment function in M , let σ† be

such that σ†(pxjq) is σ(pxjq)†, σ†(pwjq) is wc if pxjq occurs in φ (where c is such that

‘I(x1, w1)’ is satisfied by σ(pxjq) and c in M), and σ†(pwjq) is wσ(pwjq) if pxjq does not

occur in φ. An induction on the complexity of formulas shows that a Lewisian formula φ

is satisfied by an assignment function σ in M just in case the actualist transformation of

its normal form is satisfied by σ† when the world-variables range over admissible a-worlds

and the individual-variables range over the union of the domains of admissible a-worlds.

Appendix C

The Canonical Space of Worlds

The Construction

I shall assume that L has a set-sized vocabulary and a set-sized domain. An essence is

a set of constitutive predicates of L such that possession of every predicate in the set is

consistent with the set of true identity-statements. Since the vocabulary of L is set-sized,

the essences must form a set. Let the cardinality of this set be η.

We construct the canonical space of worlds in stages. At each stage, we do two things:

(1) we introduce a set of objects and assign each of them an essence, and (2) we introduce

a set of a-worlds for L♦.

At stage 0 we introduce the set of objects 〈x, ‘actual’〉 for x in the domain of L. For

each x, we let the essence of 〈x, ‘actual’〉 be the set of constitutive predicates x actually

satisfies. The only world we introduce at stage 0 is the actualized a-world for L♦ (i.e. the

a-world whose domain consists of pairs 〈x, ‘actual’〉 for x in the domain of L, and in

which every predicate receives its intended interpretation, corrected for the fact that the

domain consists of ordered pairs 〈x, ‘actual’〉 instead of their first components).

For k a natural number, stage k + 1 is constructed as follows. First, we introduce an

object 〈αk+1, ‘non-actual’〉 for each α ∈ 2η. The newly introduced objects are assigned

essences in such a way that every essence gets represented infinitely many times. Finally,

213

214 APPENDIX C. THE CANONICAL SPACE OF WORLDS

we introduce an a-world w just in case it meets the following conditions:

C1 Every object in the domain of w was introduced at some stage ≤ k + 1.

C2 If pφ(x1, . . . , xn) ≡x1,...,xn ψ(x1, . . . , xn)q is a true identity-statement,

p∀x1 . . . ∀xn(φ(x1, . . . , xn)↔ ψ(x1, . . . , xn))q is true at w.

C3 If z is in the domain of w, every predicate in the essence of z is satisfied by z at w.

We let the cannonical space of worlds, M , be the set of a-worlds introduced at some stage

of this process, and designate the actualized a-world as M ’s center. It is straightforward

to verify that the Principle of Identity is satisfied in M .

Possibility

Let a possibility statement be a sentence of L♦ of the following form:

∃ ~x1(φ1( ~x1) ∧ ♦(∃ ~x2(φ2( ~x1, ~x2) ∧ ∃ ~x3(φ3( ~x1, ~x2, ~x3) ∧ . . .)))

where none of the φi contain boxes or diamonds. A possibility statement will be said to

be good just in case there is a way of assigning an essence to each variable, and of deciding

which variables will be treated as correferential, such that the following condition is met:

Modify the possibility statement so as to get the following:

∃ ~x1(E1( ~x1) ∧ I1( ~x1) ∧ φ1( ~x1) ∧ ♦(∃ ~x2(E2( ~x2) ∧ I2( ~x1, ~x2) ∧ φ2( ~x1, ~x2) ∧ . . .))

where Ei(. . .) applies every predicate in the essence assigned to a given vari-

able to that variable, and Ii(. . .) is the conjunction consisting of a conjunct

‘∃y∃z(xi = y ∧ xj = z) → xi = xj’ whenever xi and xj are treated as coref-

erential and a conjunct ‘∃y∃z(xi = y ∧ xj = z) → xi 6= xj’ whenever xi and

215

xj are treated as referring to different objects. Then each of the following

conditions is satisfied:

1. ∃ ~x1(E1( ~x1) ∧ I1( ~x1) ∧ φ1( ~x1)) is true.

2. For each i > 1, there is an a-world w and a family of interpreted con-

stants cα (which may or may not refer to objects in the domain of w)

such that: (1) w satisfies C2, and (2) ∃~xi(Ei(~xi) ∧ Ii(~c1, . . . , ~ci−1, ~xi) ∧

φi(~c1, . . . , ~ci−1, ~xi)) is true at w.

One can then prove the following:

Theorem 1 Every good possibility statement is true in M .

Proof: Let ψ be a good possibility statement containing n diamonds. We know that

there are assignments of essences and coreferentiality such that a version of ψ modified

as above satisfies conditions 1 and 2 above. Since modified-ψ entails ψ, it will suffice to

show that modified-ψ is true in M .

We proceed by reductio. For k ≤ n + 1, let the k-truncation of modified-ψ be the

result of eliminating from modified-ψ the subformula beginning with the kth diamond

and the conjunction sign that precedes it (or doing nothing, if k = n+ 1). Suppose that

modified-ψ is false in M . It follows from condition 1 of the definition of goodness, and the

fact that the M ’s center is the actualized a-world, that the 1-truncation of modified-ψ is

true in M . So there must be some k ≤ n such that the kth-truncation of modified-ψ is

true in M and the k + 1th truncation is false in M .

Since the kth-truncation of modified-ψ is true, there must be a sequence of a-worlds

w1, . . . wk in M and a sequence of objects ~a1, . . . , ~ak such that: (1) w1 is the actualized

a-world, (2) each of the ~ai is in the domain of wi, and (3) wi verifies the subformula of the

kth-truncation of modified-ψ that follows the string of existential quantifiers that binds ~xi

when each variable in ~xj is assigned the corresponding ~aj as a value (j ≤ i). By condition

216 APPENDIX C. THE CANONICAL SPACE OF WORLDS

2 of the definition of goodness, there is an a-world w verifying every identity-statement

and such that the subformula ∃ ~xk+1(Ek+1( ~xk+1) ∧ Ik+1( ~x1, . . . ~xk+1) ∧ φk+1( ~x1, . . . ~xk+1))

of ψ is true when the ~xj (j < k+ 1) are replaced by suitable new constants. Where m is

the maximum of the stages at which w1, . . . wk were introduced to M , it is easy to verify

that there is a world w∗ which is isomorphic to w and is introduced to M at stage m+ 1.

To see this, we assume with no loss of generality that w has a countable domain, and

replace each object z in the domain of w by z∗, where (. . .)∗ is defined as follows:

1. If cij is the new constant introduced to take the place of the ith member of ~xj, and

if the referent z of cij is in the domain of w, then z∗ is ith member of ~aj.

(This assignment is guaranteed to be one-one because we know that the result of

substituting new constants for variables in Ik+1( ~x1, . . . ~xk+1) is true at w.)

2. To each remaining object z in the domain of w, (. . .)∗ assigns a distinct m+1-stage

object whose essence matches the distribution of constitutive predicates that are

satisfied by z in w.

The construction of M guarantees that some a-world introduced to M at stage m+ 1 is

isomorphic to w under (. . .)∗.

This can be used to show that the k + 1th truncation of modified-ψ is true in

M , and therefore to complete our reductio. It suffices to check that the subformula

∃ ~xk+1(Ek+1( ~xk+1)∧Ik+1( ~x1, . . . ~xk+1)∧φk+1( ~x1, . . . ~xk+1)) of modified-ψ is true in w∗ when

the values of the ~x1, . . . , ~xk are taken to be ~a1, . . . , ~ak. Start by fixing referents ~bk+1 for

~xk+1 in the domain of w that witness the truth of ∃ ~xk+1(Ek+1( ~xk+1)∧Ik+1(~c1, . . . ~ck, ~xk+1)∧

φk+1(~c1, . . . ~ck, ~xk+1)) in w. Let σ be an assignment function for L♦ that assigns the refer-

ent of a new constant replacing a given variable to that variable and assigns ~bk+1 to ~xk+1;

let σ∗ be an assignment function for L♦ that assigns ~a1, . . . , ~ak to ~x1, . . . , ~xk, assigns z∗

to a variable v whenever σ(v) = z and z is in the domain of w, and assigns an object

outside the domain of w∗ to v whenever σ(v) is an object outside the domain of w.

217

To prove the result, it suffices to show that φk+1( ~x1, . . . ~xk+1) is true in w relative to

σ just in case it is true in w∗ relative to σ∗. We proceed by induction on the complexity

of φk+1:

• φk+1( ~x1, . . . ~xk+1) is P ( ~x1, . . . ~xk+1) for P atomic.

I am assuming that atomic formulas (including formulas of the form ‘y = z’) are

false at an a-world whenever they involve empty terms. So the empty-term case

follows from the observation that the result of applying σ to ~x1, . . . , ~xk+1 is outside

the domain of w just in case the result of applying σ∗ to ~x1, . . . , ~xk+1 is outside the

domain of w∗.

When there are no empty terms, the result follows from the observation that w and

w∗ are isomorphic under (. . .)∗.

• φk+1( ~x1, . . . ~xk+1) is ∃z(θ( ~x1, . . . ~xk+1, z)). Suppose ∃z(θ( ~x1, . . . ~xk+1, z)) is true in

w relative to σ, then there is some y in the domain of w such that θ( ~x1, . . . ~xk+1, z)

is true in w relative to σ[y/z]. By inductive hypothesis, θ( ~x1, . . . ~xk+1, z) is true in

w∗ relative to σ[y∗/z]. So ∃z(θ( ~x1, . . . ~xk+1, z)) is true in w∗ relative to σ∗. The

converse is analogous.

• The remaining cases are trivial.

218 APPENDIX C. THE CANONICAL SPACE OF WORLDS

Appendix D

Modal Sentences and Modal Facts

As described in section 2.2, the foundational problem is the problem of explaining what

it takes for modal truths to be true. I have argued that it can be addressed by appeal to

identity statements, on the grounds that the set of true identity statements can be used

to state the truth-conditions of sentences in L♦ without appealing to vocabulary outside

L. But perhaps you think that this is not enough: to fully address the foundational

problem one would have to explain what it takes for an arbitrary ‘modal fact’ to obtain,

regardless of whether the fact happens to be expressible in L♦.

It is not clear to me that one can make sense of this more ambitious project without

committing oneself to potentially controversial assumptions. But let me suggest a way

of spelling it out. Suppose that one is a realist about properties, and takes them to

form a Boolean Algebra. One believes, in particular, that properties are closed under

complements and intersections: if the property of being F exists and the property of

being G exists, then so do the property of being not-F and the property of being F-and-

G. It follows that one can talk about one property’s being a ‘part’ of another: F-ness is

a part of G-ness just in case G-ness = F-and-G-ness. (Thus, the property of containing

hydrogen is a part of the property of being composed of water.) And since the space of

properties forms a Boolean Algebra there must an impossible property, which is identical

219

220 APPENDIX D. MODAL SENTENCES AND MODAL FACTS

to the intersection of any property and its complement. (Thus, the impossible property

= the property of containing hydrogen and not containing hydrogen = the property of

having a mustache and not having a mustache.)

Within this framework, one can ask a version of the foundational question: what

grounds the difference between property-instantiations that are (metaphysically) possible

and those that are not? And the answer is as simple as can be: for an instantiation to be

possible just is for it to not to include an instantiation of the impossible property when

one assumes closure under intersections.

The reason the answer is so straightforward is that one can use identities amongst

properties to play the same sort of role that identity statements have been playing in

our discussion so far. Consider, for example, an assignment of two properties to a given

object x: the property of being composed of water and the property of not containing

hydrogen. Is this assignment possible? The property of being composed of water = the

property of being composed of H2O, and the property of containing hydrogen is part

of the property of being composed of H2O. So the property of containing hydrogen is

part of the property of being composed of water. It follows that the intersection of the

property of being composed of water and the property of not containing hydrogen is the

impossible property. So when one assumes that the property-assignment is closed under

intersections, one gets the result that x instantiates the impossible property. From this

we may conclude that the property-assignment in question is not possible, and therefore

that it is not possible for something to be composed of water and not contain hydrogen.

Similar arguments could be given even if the properties in question were described

using explicitly modal vocabulary. For instance, by assuming that the property of being

human = the property of being a mammal that might have been human, one can show

that property-assignments whereby the same object instantiates both the property of

not being human and the property of being a mammal that might have been human are

impossible. A more extreme case is the property of, e.g. being such that there might have

221

been a purple elephant. If you take the truth-conditions of ‘There might have been a

purple elephant’ to be trivial, you will think this property is the trivial property (i.e. the

Boolean Algebra’s >), and therefore think that an assignment of the property would

be possible. If, on the other hand, you take the truth-conditions to be impossible, you

will think the property is the impossible property (i.e. the Boolean Algebra’s ⊥), and

therefore think that an assignment would be impossible.

This underlines the generality of the proposal, but also the fact that the proposal

is of limited interest. When one focuses on the space of properties, and abstracts away

from the predicates that might be used to express these properties, possibility is a pretty

boring concept: a possible assignment of properties to objects is simply one that steers

clear from the impossible property. The real action comes from the project of ascertaining

identities amongst properties (e.g. ascertaining whether the property of being composed

of water is identical to the property of being composed H2O). But this is only interesting

insofar as language enters into the picture—otherwise all there is to be said about identity

is that every property is identical to itself and nothing else. The interest comes from

ascertaining whether the satisfaction conditions of ‘is composed of water’ are any different

from the satisfaction conditions of ‘is composed of H2O’ or, equivalently, whether one

should accept ‘To be composed of water just is to be composed of H2O’. (If you believe

in concepts, you might wish to make the point at the level of thought, rather than

language. You might wish to claim that the interest comes from ascertaining whether

the satisfaction conditions of the concept composed of water are any different from the

satisfaction conditions of the concept composed of H2O.)

Metaphysically minded philosophers may nonetheless think it important to develop

a language-independent answer to the foundational problem. (They might claim, for

example, that a metaphysics of modality is needed to supply ‘truth-makers’ for modal

truths.) Postulating a space of properties such as the one we have been discussing

could be the first step in such a project. But it is worth emphasizing that we’re not

222 APPENDIX D. MODAL SENTENCES AND MODAL FACTS

there yet. Up to now, attention has been restricted to monadic properties, and there

is no reason to think that every modal fact can be cashed out as an assignment of

monadic properties. A successful proposal would presumably have to encompass polyadic

properties (e.g. the relation of being a sister of), zero-place properties (e.g. that a marriage

take place) and higher-level properties (e.g. being infinite in number), and this gives rise

to a complication. One needs it to turn out to be impossible for the relation of being

a sister of to be instantiated without there also being instantiations of the monadic

property of being a sister. My guess, for what it’s worth, is that one would be be able

to solve the problem by claiming that some properties have others as ‘constituents’, and

postulating the right sort of connections between the instantiation of a property and the

instantiation of its constituents.

For the purposes of this book, however, the language-dependent version of the pro-

posal is all we need. We will always be able to pick a first-order language L expressive

enough for the task at hand.

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