Tel Aviv University

Faculty of Humanities

Mathematical Necessity Seminar on Logic and Necessity

Lecturer: Professor H. Putnam

Logic and Necessity

Author: Yoav Aviram

Date: July 2004

- 2 -

Logic and Necessity

Table of Contents

Abstract................................................................................................................................3

Rethinking Mathematical Necessity – An Exegesis............................................................7

The Unique Status of Logical Truths...............................................................................8

A Suggestion for a New Distinction................................................................................9

A Relative Notion of Truth............................................................................................11

Unrevisability of Some a Posteriori Statements...........................................................12

Giving Sense to Riddles.................................................................................................13

Speaking of Logic..........................................................................................................14

How a Change of Context Affects Statements..............................................................15

Is There Absolute Necessity?........................................................................................17

The Need for Justification..............................................................................................18

Discussion..........................................................................................................................20

Revising Absolute a Priori............................................................................................20

Conclusions........................................................................................................................25

Works Cited.......................................................................................................................27

- 3 -

Logic and Necessity

Abstract

A major concern of analytic philosophy in the last century has been whether or not we

can say that there are true statements that are necessary, and if so, on what grounds. It has

been argued that the notion of such absolute necessity is a dogma, and that statements

that we call a priori are more similar to empirical hypothesis. A reason for this can be

given by analogy; since historically some statements considered at one time to be

necessary were later revised, it is possible that some future experience will motivate us to

revise statements that we consider to be necessary today. But can we really make sense of

the notion of revising a statement such as "7+5=12"?

Hillary Putnam suggests that even though the notion of absolute necessity is

compromised by the argumentation offered against it, there are statements we consider to

be necessary relative to our "conceptual schema". Einstein's theory of relativity showed

that Euclidian geometry dose not correctly describe physical space. But in the 18 th

century statements of Euclidian geometry had an epistemic status quite different than that

of empirical hypothesis. By analogy, this may be the case with statements within our

present conceptual schema, for example - statements of arithmetic.

This essay begins with an historical introduction to the discussion of the analytic

synthetic distinction. The historical overview in the introduction is brief, and represents

one of several possible narratives. The introduction is followed by an exegesis of

Putnam's paper titled "Rethinking Mathematical Necessity". The arguments in that paper

are explained and major points are discussed in detail. Views, examples and opinions that

are not in the original paper or in other referenced works are remarked as my own. The

paper concludes with my own suggestion for an alternative view of the matter, and a

conclusion to the entire essay.

- 4 -

Logic and Necessity

Introduction

The discussion about the distinction between analytic and synthetic knowledge has its

roots deep within the history of Western philosophy. The very justification of a

philosophical discussion might be seen as one that depends on there being an analytic, a

priori way of knowing facts about the world. This paper deals with a narrow aspect of

this discussion - the status of mathematical knowledge. Understanding the issues at hand

requires an understanding of the context of the philosophical debate in which the issues

were introduced. In this introduction I hope to briefly explain this context, limiting the

discussion to the major views expressed in the Twentieth Century. It is important to note

that these views are based on earlier views which are also important, such as the

philosophy of Kant, which I will only discuss briefly.

Kant explained that truths of logic are analytic a priori and that they are trivial.

He explains that logical laws govern relationships between concepts. He also explains

that logic is the vary structure of thought. After Kant, the next important milestone was

the work that began with Gottlob Frege. Frege elaborated on the ideas of Kant by

identifying the logical structure of an ideal language with the nature of thought.

The status of mathematical truths, in terms of the analytic-synthetic discussion,

was up until this point controversial1. In his works, Frege showed that it is possible to

base all mathematical concepts on logical ones. This attempt was later improved in the

works of Russell and Whitehead and was called logicism. Showing that the truths of

mathematics are truths based on logic alone situated mathematics for the first time clearly

within the analytic domain, at least according the common view that considered logic to

be analytic. Logicism reinstated the analytic-synthetic discussion in the twentieth century.

For the first time it seemed possible to classify all truths as either analytic or synthetic.

The logical positivist movement attempted such a classification of truths. Rudolf

Carnap explains that "it became possible for the first time to combine the basic tenet of

empiricism with a satisfactory explanation of the nature of logic and mathematics"

(Carnap 1963). The positivist philosophers attempted to reduce all knowledge to a basis

1 Kant argued that there is a third realm of synthetic a priori truths to which mathematics belonged, while

Mill suggested that mathematical knowledge was empirical. Wittgenstein explained in his early writings

that all analytic truths are tautologies, but did not include mathematics amongst them.

- 5 -

Logic and Necessity

of certainty. For this task they chose to adopt a phenomenological language whose basic

sentences were about sense impressions. More complex sentences had to be reducible to

these simpler ones, and a statement was considered true if there was a viable method of

confirming it.

By formalizing language in such a way, the hope was to create a basis for sciences

to thrive on. Other than the class of empirical sentences and sentences that were reducible

to empirical claims, there was also a class of analytic sentences. The positivists explained

that analytic statements were true by virtue of convention or stipulation. As a result of the

adoption of logicism, the class of analytic sentences now included all of mathematics.

This attempt sharpened the analytic-synthetic distinction creating a large class of analytic

statements that were also thought to be a priori, and therefore immune from revision.

In 1951 Quine set out to criticize the views of Carnap and the positivists in his

paper "Two Dogmas of Empiricism" (Quine 1951). The first dogma of which Quine

speaks is that the distinction between analytic and synthetic statements really exists. The

second dogma is that every meaningful sentence in our language is structured from

logical components that refer to immediate experience. The argument against the first

dogma is that the idea of analyticity rests on other notions, such as that of synonymy 2,

which are just as unclear as the notion of analyticity, and that any attempt at explaining

analyticity using these notions inherits this unclarity. What is implicit in Quine's

arguments is that not only is there no meaningful distinction, but that all statements

considered analytic are, in fact, synthetic3. Since Quine, like the positivists, identifies in

his writings the notion of analyticity with the notion of a priori, the conclusion is that no

statement is immune from revision.

As for the second dogma, Quine explains that only a holistic theory of meaning is

an adequate one. He draws a picture of what he calls our "web of believes"; our

knowledge about the world is organized so that some statements take a more central role

amongst our beliefs. Only statements in the periphery are statements about immediate

2 Quine refers to a linguistic notion of analyticity: A sentence is analytic if it can be obtained from a logical

truth by substituting synonyms for synonyms. A logical truth is one in which only words of formal logic

occur essentially.3 Quine does agrees to one notion of analytic he calls "stimulus analytic". This is a relativised notion of

analyticity, much like Putnam's notion of quasi-necessity.

- 6 -

Logic and Necessity

experience and all other statements are logically constructed of peripheral ones. Any

statement in a language might be revised, even a central statement of the kind we call

logical laws. A revision of such a statement is likely to require us to make extensive

modification to our web of beliefs. Similarly, it is possible to hold a statement about

immediate experience immune from revision by altering neighboring statements or even

more central statements4.

Quine's account of the distinction triggered numerous responses. Some followed in

his footsteps while others objected to Quine's radical views. In a paper titled "In Defense

of a Dogma", Grice and Strawson (1956) argued that Quine's claim that we have not yet

clarified the distinction in a rigorous manner is also true of many other distinctions that

we use in philosophy and are not willing to reject. Hillary Putnam argues in his

philosophy that although there is a lot of sense to Quine's theses, its radical form misses

out on some important methodological distinctions. The claim that every statement is

revisable does not account well enough to the difference between empirical statements

and the statements once considered a priori. In this paper I will attempt to follow

Putnam's investigation of this matter, and towards the end introduce some views of my

own.

4 This can be done, as Quine suggests, by pleading hallucination.

- 7 -

Logic and Necessity

Rethinking Mathematical Necessity – An Exegesis

The problematic nature of discussing the epistemic status of mathematical truths may be

described by the following claim: A good justification we can give for the necessity of

mathematical truths is that they are true, without exception. Unfortunately it is this vary

claim that situates mathematical truths as something to be justified in the light of use and

experience. Similarly, the ontological status of mathematical entities as real existing ones

seems to be justified well by the claim that mathematics is simply so successful and

useful as a prediction tool, that we do not know how to explain its success other than to

say that it deals with real entities. Unfortunately this type of justification also situates

mathematics on par with empirical science.

Putnam begins the discussion of mathematical necessity in his paper "Rethinking

Mathematical Necessity" (Putnam 1994) with an overview of Quine's view. Putnam calls

our attention to the type of reasoning Quine employs when he points to the successes of

mathematics as a reason for the existence of mathematical entities. This type of argument,

Putnam explains, unavoidably reduces mathematical statements to the same level of

necessity as that of statements of physics, making the difference between the two one of

degree and not of kind. Mathematical statements by Quine's account are subject to

revision in the light of experience just the same as statements of physics, except

mathematical statements are closer to the center of our "web of beliefs" and thus we are

far less likely to revise them.

Quine's attack on the views of Carnap and the positivists leaves mathematical

knowledge indistinguishable from scientific knowledge from the epistemic point of view.

Putnam claims that even though there is a lot of sense to Quine's theses, it misses a

critical distinction that exists between the necessities of the two kinds of statements. This

distinction exists even if we accept that ultimately no statement is immune from change.

Putnam intends to introduce a different way of viewing the situation, one in the light of

the views of Kant and Frege. I will now attempt to explain his view in relation to the

views of Quine and Carnap.

- 8 -

Logic and Necessity

The Unique Status of Logical Truths

Putnam shares with us some thoughts he had about the writings of Wittgenstein on logic

which he claims to have found a little puzzling at first glance. In the Tractatus

(Wittgenstein 1922) it seems that for Wittgenstein it is clear that logical truths are without

sense, and at the same time are not nonsense5. I think that Wittgenstein spends a large

amount of time (relative to his succinct style) on this fine distinction. He explains that "A

logical picture of facts is a thought" and that "Thought can never be of anything

illogical". Propositions for Wittgenstein express thoughts, and thoughts do not actually

contain their sense, but only the possibility of expressing it6. This fine distinction between

containment and expression is the line Wittgenstein draws between logic that is present in

our thoughts, but which we cannot exactly speak about7. We cannot speak about logic

because it does not have a sense, but since logic is thought, we cannot say that it is

nonsense. Putnam's puzzlement has to do with this ambivalence in Wittgenstein's attitude

towards logic.

While trying to clarify Wittgenstein's notion of logic, Putnam is reminded of what

Kant has to say about it. In the Critique of Pure Reason (Kant 1787) Kant identifies logic

with the structure of rational thought. He introduces the notion that the ways we view the

world have something in common that we can investigate. These common features of

thought are the only type of objectivity to which Kant is willing to commit8. Since logic

is a feature of human thought, it is incorrect to say that it provides us with any positive

descriptive knowledge of the world. This notion of logic differs greatly from the Platonist

metaphysical claim that the laws of logic are the most general laws there are. Even

though it is less metaphysically ambitious, there is still quite a lot that we can say about

Kant's view of logic and thought. No thought is allowed to violate the laws of logic and

any thought that does so is considered to be irrational. For Kant any sort of talk, event

talk of "noumena", must conform to the laws of logic. This must be so because talk is a

5 This line of thought is still noticeable in Wittgenstein's later works.6 Tractatus: 3 – 3.13.7 In an earlier section of the Tractatus Wittgenstein uses a different analogy when he explains that "A

picture cannot, however, depict its pictorial form: it displays it." (2.172)8 This objectivity rests on certain assumptions that Kant must make, such as the unity of thought and the

existence of god.

- 9 -

Logic and Necessity

rational human activity. Therefore, there could not possibly be talk of anything illogical,

since this sort of talk would not be talk at all but rather meaningless mumbling.

Putnam explains that in his reading of Frege he identifies a tension between a

Platonist and a Kantian view. On the one hand there are many reasons why one may

claim that Frege takes the Platonist approach. It may be said that for Frege the laws of

logic are quantifications over all objects and concepts. On the other hand since Frege

identifies the structure of thought with the structure of an ideal language, and since the

laws of logic govern that ideal language, it seems that he also adopts the Kantian view

whereby the laws of logic have a different status than empirical laws. It is now easy to

see the similarities between the Kantian view, where there cannot be an illogical thought,

and Wittgenstein's claim that statements of logic have no sense but are not nonsense.

Interpreting Wittgenstein in this way allows Putnam to adopt an alternative

explanation to the necessity of logical statements than the one Quine attacked in "Two

Dogmas". Quine directed his attack towards the explanation of logical necessity provided

by Carnap. Carnap explained that logical statements are necessary because they are no

more than conventions. A change of the logical laws would be nothing more than a

change to the meaning of the words. Wittgenstein's view allows Putnam to claim that the

necessity of logical statements needs no explanation, nor could we ever provide such an

explanation, since they do not describe things in the world but only the way we think.

A Suggestion for a New Distinction

When Quine rejected the analytic synthetic distinction in "Two Dogmas" he explained

that the reason for this was the lack of clarity in the terms that we use to define

analyticity. This unclarity in his opinion was so vast that it distracted us from seeing that

analytic statements are not epistemologically deferent than empirical hypothesis. One of

the main arguments used by Quine in that paper was that we take logically true sentences

(sentences in which only the logical words effect the truth value of the sentence), which

one may view as a narrow definition of the class of analytic statements, and extend it into

a much wider definition of analyticity by substitution of synonymous words. This

substitution attempt, Quine argues, is fallacious since the notion of synonymy itself

depends on the notion of analyticity.

- 10 -

Logic and Necessity

We are now faced with two definitions of analyticity. The first is a narrow definition

which only includes logical truths and the second is a wider definition which also

includes sentences synonymous with logical truths. Putnam points out that Quine's

famous argument is only valid when referring to the wider definition. The narrow

definition escapes Quine's claws unharmed, but this is still not enough to establish that

logical truths are necessary and counter Quine's main theses regarding necessity. Even

with the narrow attempt at a definition of analyticity we are left with logical truths that

are "contextually a priori"9, a notion that is vary alien to the line of though of Kant and

Frege that Putnam is attempting to promote.

The notion of revising logical truths is a hard one to swallow, even for Quine himself.

After suggesting that the revision is possible, it seems like Quine attempts to defend the

special status of logical truths. He gives two arguments; the first is that even though

truths of logic are theoretically revisable, they are located at the far end of the continuum

of revisability. This continuum has, on its one end, empirical statements of the kind we

revise every day and on its other end statements like the truths of logic which we choose

to hold come what may. The second argument is that the revision of logical truths only

occurs during attempts at translation, and that in those cases it is only a change of

meaning and not a revision in a deeper sense10.

Putnam's dissatisfaction has to do with how Quine's arguments might be interpreted.

Quine describes the space between the synthetic and the analytic as a continuum rather

than a clear cut distinction. But it seems that for all intents and purposes he abandons the

traditional notion of analytic a priori in favor of classifying all statements as synthetic a

posteriori ones. The notion of "reluctance to give up" does not capture well the difference

between what makes logical truths true, and what makes empirical hypothesis correct.

To illustrate the matter, Putnam asks us to consider three sentences:

1. It is not the case that the Eiffel Tower vanished mysteriously last night and in

its place there has appeared a log cabin.

9 The term "contextually a priori" is not the original term used by Putnam. I use it regarding stamens that

we can revise, but do not know how to give sense to their revision. This is similar the class of statements

Putnam terms "quasi-necessary relative to a conceptual schema" or "necessary relative to a body of

knowledge". I think that the term captures well the problematic nature of such truths.10 This type of revision is similar to Carnap's change in convention.

- 11 -

Logic and Necessity

2. It is not the case that the entire interior of the moon consists of Roquefort

cheese.

3. For all statements p, '~(p•~p)' is true11.

Putnam explains that the difference between the first two statements and the third one

does not only have to do with how much one is reluctant to give up each statement. It is a

real distinction that has to do with the fact that one knows how statements one and two

may turn out to be false, while one does not even understand what it means for the third

statement to be false.

A Relative Notion of Truth

This difference between statements is significant, Putnam explains, not because it shows

something new about us, but because it exposes something about the methodology

involved. At first glance it seems like the notion that we cannot imagine something, is a

psychological fact and that this kind of reasoning is psychologism. But what is crucial for

understanding the significance is to understand what is needed for the realization that the

third statement is false. And what is needed is a different method of reasoning, an

alternative logic. When dealing with quasi-necessary truths, an entire alterative theory

must be available before we can give a sense to the revision.

In his paper "It Ain't Necessarily So" (Putnam 1975) Putnam gives several great

examples of this sort of revision. Euclidian geometry was overthrown as the geometry

that describes physical space when Einstein's relativity theory showed that space has

attributes that are inconsistent with it. Proving that statements of Euclidian geometry are

false may be viewed just like any other falsification of an empirical hypothesis in the

light of contradicting observations. Putnam argues that prior to their overthrow truths of

Euclidian geometry did not have a status of empirical hypothesis. In order to explain this

argument Putnam introduces a relative notion of truth - “true relative to a conceptual

schema”

A statement true relative to a conceptual schema is necessary according to everything

the scientific community knows at the time (in case we are dealing with scientific truths). 11 In his paper "There is at least one a priori truth" (Putnam 1978), Putnam argues that a minimal law of

contradiction, the statement that “Not every statement is both true and not true.” is the most certain a priory

truth.

- 12 -

Logic and Necessity

An empirical statement by this account is a statement that it is known how to falsify,

based on the knowledge inherent in that conceptual schema alone. Alternatively,

statements for which the knowledge inherent in the conceptual schema is not enough to

falsify are statements necessary relative to that conceptual schema. Since these statements

might still be falsified by knowledge outside of the conceptual schema (for example

future knowledge), they are only quasi-necessary and not absolutely necessary truths.

Since Euclidian geometry was the only geometry available during the 18th century, it

was a necessary truth relative to the conceptual schema of the time. Only when

alternative geometries were developed by Lobachevski and Reimann, and the theory of

relativity applied such geometry to physical space, was it possible to understand what it

means for space to be non-Euclidian. For example, while Euclidian geometry asserted

that space was infinite, Einstein's physics claimed it was not. Similar things might be said

regarding the necessity of the law of contradiction, and the possibility of revising it. The

motivation behind Putnam's introduction of a relativised notion of truth is not to salvage

analyticity or to revive the analytic synthetic discussion. It is to point out an important

distinction we can make between different types of knowledge that was eclipsed as a

result of the collapse of the analytic-synthetic dichotomy.

Unrevisability of Some a Posteriori Statements

Wittgenstein narrows the gap between analytic and synthetic statements from a different

direction. There are, he explains, statements that we commonly classify as empirical, but

whose revision we fail to give sense to. Much like the analytic statements that we dealt

with before, which we thought are necessary until history showed to be revisable, these

new "quasi-empirical"12 statements we tend to think are revisable but do not know how

that revision is possible. It seems that what makes the revision of this type of statements

inconceivable is that changing them requires altering some of our fundamental

assumptions about the world. Change those assumptions and language looses its

orientation.

One example of such a case would be for one to wake up one day and discover that

his name is not really what he thought it was all his life. We can treat this matter like an

12 The term "quasi-empirical" is not in the original text.

- 13 -

Logic and Necessity

empirical discovery that one just made about who he really is, and say that he was

mistaken about his name. But from a subjective point of view, when such a fundamental

assumption about the world is lost, it makes just as much sense to think that the events

after the discovery are factitious as it is to think that past experiences are. When we are

faced with the discovery that a fundamental scientific assumption we have about the

world is false, we are collectively faced with a similar subjective situation.

An example of such a situation would be to discover that the statement "water has

boiled in the past" is false. Even though this is agreed to be an empirical statement, so

much of our other knowledge depends on it being true that the discovery of its falsity

causes language to loose its reference point. Such a discovery would require such

extensive modifications elsewhere to our web of believes that we would not know what

to keep constant and what to change. Putnam explains that what Wittgenstein's examples

show is that even though it is true that in a certain sense we can revise any statement

(even in a trivial way, by altering the meaning of words), in another sense there are cases

where it is not rational to do so.

Giving Sense to Riddles

When Quine examines the possibility of revising logical truths he limits his discussion to

talk of statements and avoids talk of beliefs and meanings because of the problematic

nature he attributes to them13. Even from Quine's angle it is possible to see why the

statements described above are problematic. Any attempt at a translation of a language

which contains notions such as "water has never boiled" or "7+5=13" is not something

that we can presently understand. The unique situation that arises when examining such

statements is that we cannot say of them that they are unrevisable, but at the same time

we are unable to attach a clear sense to there revision.

History shows that eventually, a sense is given to the revision of such statements after

all. The non-Euclidian statement that "there is only a finite number of places to go to

travel as you will" today makes perfect sense. The trick to understanding how it was

possible to give such a revision a sense, may be shown by a riddle. What makes a riddle

13 In "Two Dogmas" Quine explains that this problematic nature has to do with the terms meaning,

synonymy and analyticity having interdependent definitions.

- 14 -

Logic and Necessity

intriguing is exactly that we do not know what sense to give to it. Only after we have

arrived at a solution do we understand the sense of the question. If the riddle-question had

only one clear sense, it would not be a riddle at all (or at least not an interesting one).

Putnam explains that prior to knowing the answer any attempt at a translation of the

riddle would be futile since the translator would be unable to preserve its intended sense.

Similarly, the sense of revising a contextual a priori truth is something that is only

understood after the fact.

Speaking of Logic

Looking back at the positions of Kant, Frege and early Wittgenstein described earlier,

there are several important distinctions to be made. The idea that logical truths are true by

virtue of the nature of thought is a metaphysical idea that Putnam would like to jettison.

Kant describes not two, but three types of knowledge; synthetic a posteriori, synthetic a

priori and analytic a priori truths. While the distinction between a priori and a posteriori

truths is clear, the distinction between the status of analytic and synthetic a priori is a

more subtle one. Synthetic a priori truths for Kant are true because they describe the

structure of reason. They do not say something about the world, but rather what they say

is that any experience is "filtered" according to the innate conceptual structures of reason

common to us all. Analytic truths on the other hand are true because they are a result of

the nature of thought. The difference between structure of reason and nature of thought is

an important one for Kant. Synthetic a priori truths are truths that we can rationalize

about, for example by explaining why they are necessary or by thinking of a world where

they do not exist. We cannot rationalize, negate or explain analytic truths precisely

because they are reason.

For Wittgenstein the opposition between synthetic a priori and analytic truths is no

longer a concern. In his view the opposition is between a posteriori (empirical) truths and

a priori truths. Wittgenstein follows Frege's footsteps in identifying the nature of thought

with the structure of an ideal language. The resulting view is that truths are either logical

truths, now identified with the structure of an ideal language, or they are about the world.

In the first case their revision is without a sense, just as they are unthinkable for Kant. In

- 15 -

Logic and Necessity

the second case the revision of judgments about the world is thinkable and subject to

confirmation.

What Putnam hopes to retain from the Kant-Frege-early Wittgenstein idea is not the

metaphysical principle that guarantees the unrevisability of logic due to it being the

nature of thought. What he hopes to retain is the notion that a revision of logic is

something that we cannot give sense to in principle at the time of the revision. It is not

the case that logical truths are absolutely unrevisable or absolute a priori. But it is the

case that they hold quite a different status than empirical hypothesis.

How a Change of Context Affects Statements

Before we move on to consider how the discussion we had so far influences arithmetic,

Putnam offers some important clarifications. In the previous paragraphs we came to the

conclusion that there is a type of questions that we can only give sense to after we have

the answer at hand. Putnam points out that what we mean when we talk about giving

question a sense is something much broader then to simply understand what the person

who asked it meant. Understanding sense in this way is something much closer to

understanding the full context of the answer. Since I think that this point is crucial for the

debate, I will try to clarify it further by examining the way the Chinese philosophy dealt

with the similar questions.

In Chinese philosophy it is said that a question that Western philosophy considers to

be a yes/no question may be answered by a third alternative. When a question is

answered by the term 'Mu'14 it is meant that the context in which the question was asked

is too narrow to contain the answer. This is a way of explaining that the fact that the

question was asked indicates a lack of fundamental knowledge. There is a story of a

student who asked his master: "It is said that the nature of the Buddha is present in all

things. In what way is it present in a dog?" - To which the master replied 'Mu'. Taking the

case of Chinese philosophy as an analogy, I think that it illustrates well the point that

there are some questions that are asked at a time when the context is too narrow to

contain the answer.14 The Chinese term 'Mu' has several philosophical uses. The one I am referring two is taken from Zen

Buddhism and is, naturally, subject to interpretations. The particular interpretation brought here is the

popular one.

- 16 -

Logic and Necessity

After the context is broadened, usually by scientific development, we are able to

understand words in both the new and old contexts. This is the case with the terms of

physics after Einstein's theory succeeded Newton's. Putnam points out that when we shift

from an old context to a new one, we notice that terms change their reference (a change

in extension). We thought that momentum is the product of mass and velocity and today

we know that this is not the case. Rather than saying that only the meaning of the word

momentum has changed, we say that the old theory was wrong and that we have

discovered something genuinely new. I think that this indicates a broader change than a

change of reference; it indicates a change in intension.

My opinion on how a change of context is related to a change of meaning differs from

Putnam's. In his paper "It Ain't Necessarily So"15 he asks what we would say about the

statement "all cats are animals" if one day we were to discover that all cats really are, and

always were, automata. Putnam point out that the problem in this case is not with the use

of the word cat (we will keep calling them cats) as much as it is with the claim that cats

are animals. He explains that in this case to say that "cats turned out not to be animals"

would keep the meaning of the words 'cat' and 'animal' unchanged. I think that Putnam is

correct if we understand meaning as extension. But if we understand meaning as

intension, than both the meaning of 'cat', and to a lesser degree 'animal' has changed.

Intension, taken as a term that captures the sum of all the different uses of a word, is

capable, in my opinion, of explaining how a change in context affects statements.

Another clarification Putnam suggests has to do with the status of contradictions. It

was noted earlier that for Frege and Wittgenstein who identified logic with the structure

of an ideal language, a thought about contradiction is not really a thought at all. Putnam

explains that a thought that is a contradiction, such as thinking that '(p•~p)' is what those

philosophers considered to be a thought without a sense. That does not mean that we

cannot make use of a contradiction in order to express something else, such as the truth

value 'false' with in a sentence. We may be unable to understand a sentence even if it is

"well formed" if present theories prevents us from doing so.

15 The original example is taken from Donnellan (1962).

- 17 -

Logic and Necessity

Is There Absolute Necessity?

At least one major question still seems to stand despite all that was discussed so far;

whether we can determine if a proposition of the kind that we have talked about is only

quasi-necessary or if it is truly necessary. Putnam argues that that is simply not a good

question. The reason for this is that the question assumes a point of view external to

human knowledge from which the matter can be judged. Such an objective point of view

does not exist. We can not give sense of the demand to justify or explain a contextual a

priori statement because we accept those statements since we do not doubt them.

The absolute-necessity or quasi-necessity of a proposition is not something to be

discovered about the world but a fact about us. In order to revise a proposition of the type

discussed so far, there must be an alternative, confirmable theory present. Since the

invention of such an alternative theory is something that is entirely a human activity, we

are the ones who determine whether a proposition will ultimately be revised or not. If we

fail to conceive some alternative theory, and as a result never revise some propositions,

can it truly be said that those propositions are objectively necessary? It cannot, and that

is why any attempt to claim that a proposition is either absolutely necessary, or

conversely that all propositions are revisable, is a dogma.

There is a serious objection to consider to what was described in the last paragraph.

We have concluded that the necessity of a proposition is dependent on the way in which

human knowledge develops and not on some metaphysical principle. The problem arises

when we consider Gödel's incompleteness theorems. What those theorems say is that all

logical systems of any complexity (including arithmetic) are, by definition, incomplete;

each of them contains, at any given time, more true statements than it can possibly prove

according to its own defining set of rules. What makes these statements true has to be

independent of human knowledge, since they cannot be proven from within our

arithmetic system. These are arithmetical truths which are not conceptual truths. Does

that not mean that the necessity of the truth of these statements is independent of human

epistemology? To this claim Putnam replies that the fact that there are currently

undecided mathematical propositions that are true does not carry any metaphysical

weight. Bivalence is not a principle according to which we may draw metaphysical

conclusions.

- 18 -

Logic and Necessity

The Need for Justification

It is time to consider what we are left with at the end of the discussion. Regarding

Quine’s theses in “Two Dogmas”, no objections were made against the main point of that

essay; the concept of analyticity, in the sense it had for Carnap, ought to be disposed of.

In other words, the attempt to base knowledge on a priori grounds has not withstood the

test of time. Instead, a new distinction was offered between empirical statements and

“contextually a priori” statements. The intention was not to identify this new distinction

with a new metaphysical category of truths but to describe how statements face historical

changes. This conclusion has ramifications for the idea that the existence of mathematical

entities needs justification.

There are different senses in which we can speak of the existence of mathematical

objects. When we do mathematics we often come across statements such as: “There exist

prime numbers greater then a million” or even “Numbers exist”. In his paper “On What

There is” (Quine 1948) Quine explains that when a speaker states that “something

exists”, he is ontologically committed to its existence and that in the case of mathematics

objects are "intangible". Even if we adhere to the idea that we are committed to the

existence of numbers as a result of doing mathematics, just as we were unable to give

sense to the notion that arithmetic is wrong, so is there no sense in giving a reason why

arithmetic is right. Arithmetic needs no justifications.

In empirical sciences there is a strong connection between the usefulness of a theory

and its truth; this is not the case in arithmetic. For example, the existence of electrons is a

theory of physics conceived based on numerous observations. The fact that this theory

has been able to produce accurate predictions and explain otherwise unexplained

observations consistently, persuades us to except that there really are electrons. In

mathematics on the other hand, it is possible for a theory to be completely useless and

still be inarguably correct. The difference between empirical sciences and mathematics is

quite large. The existence of mathematical objects has nothing to do with how

extensively the theory is applied. It may never be applied and still this would not change

a thing concerning the discussion we have about the existence of mathematical objects.

- 19 -

Logic and Necessity

The usefulness of a mathematical theory is an empirical issue completely separate from

the unempirical question –“is the theory correct?”

Quine's battle against the a priori has ended in his claim that the base of all

knowledge is empirical. This attempt at "naturalizing epistemology" is problematic due to

the construct of our language. Notions such as sense, truth and confirmation – in other

words the majority of the terms used when the issue is debated - are normative ones. The

sense of the attempt to naturalize these terms is unclear. Putnam suggests that there is a

better way of describing the role of arithmetic in scientific discovery than to naturalize it,

a way that does not depend on the notion of a priori. Arithmetic is the tool with which we

justify; in itself it does not need justification.

- 20 -

Logic and Necessity

Discussion

In the following section I would like to discuss some of my own thoughts on the issues

expressed in this seminar. These thoughts are concerned with the general discussion

about logic and necessity and not particularly with the subject of mathematical necessity.

The main works that I reference are Putnam's "Rethinking Mathematical Necessity" and

"It' Ain't Necessarily So", and Quine's "Two Dogmas".

Revising Absolute a Priori

In the aftermath of Quine's "Two Dogmas" several views surfaced in an attempt to

salvage something of the notion of a priori. The reason for these attempts, the same

reason described earlier in this essay, is that it seems that Quine's account does not

accurately describe some of our every day experiences with knowledge, although it is

convincing in most other areas. Perhaps the most objectionable of Quine's statements is

the one claiming that every statement is revisable. What differentiates between statements

of physics and statements of logic, he explains, is their different location within our web

of believes. Putnam agrees that the old notion of absolute a priori has crumbled, and

offers us a rehabilitated notion of necessity; one which he calls quasi-necessary (and I

call "contextual a priori"). For this he relies heavily on the study of events from the

history of science. The concept of contextual a priori is an attempt to explain the

phenomenon whereby statements that are considered necessary at one time are later

revised. In this section I will attempt to provide a different explanation for this

phenomenon.

What I wish to explain is how statements change the degree of necessity attributed to

them by the scientific community over the course of time. This explanation may be used

to clarify how the apparent revision of statements that are considered absolute-a priori is

possible. However, I do not intend for this it to be a justification of the belief in the

existence in such a priori. This explanation may also be applied towards a relativised

notion of a priori of the kind Putnam promotes. I agree with Putnam that the question

whether there is an absolute notion of truth or only a relativised one is unanswerable.

This is so because we will never be able to determine if some truths remain constant

- 21 -

Logic and Necessity

because they are absolutely necessary or because we have not discovered an alternative

yet16. Any attempt to give an answer to this question assumes an external point of view.

As a case study I wish to examine how Einstein's physics revised notions regarded as

necessary by Newtonian physics. For example, the notion of absolute simultaneity - that

the statement "either two events occur at the same time or they do not" is true. This is a

good notion to examine since even today laymen’s intuition is inconsistent with the way

such basic concepts are presented by the theory of general relativity. Let us briefly

describe a thought experiment17; in the diagram bellow the long arrow represents a train

traveling at velocity v in the direction of the arrow and the dashed line represents the

railway embankment. For people traveling aboard, the train is a frame of reference - they

regard all events in reference to the train. Every event which takes place along the

embankment also takes place at a particular point of the train. The definition of

simultaneity can be given relative to the train just as it can be given in respect to the

embankment.

The following question arises: if two strokes of lightning emanating from points A

and B are simultaneous with reference to the railway embankment, are they also

simultaneous relatively to the train? Before the time of the theory of relativity it had

always been assumed in physics that a statement of time had an absolute significance, i.e.

that it is independent of the state of motion of the body of reference. But we can define

simultaneity in this case by saying that the two lightning strokes occur at the same time if

they both reach point M, located in the middle of the distance AB, at the same time (since

they both travel at the speed of light).

16 A reason for not discovering such an alternative maybe that we are incapable of conceiving it.17 This thought experiment was originally described by Einstein (1920).

- 22 -

Embankment

TrainV

A M B

M'

Logic and Necessity

Events A and B also correspond to positions A and B on the train. When the flashes

of lightning occur, point M' coincides with point M, but it moves towards the right in the

diagram with the velocity v of the train. An observer sitting in the position M’ in the train

is traveling towards the beam of light coming from B and away form the beam coming

from A. If the train is traveling vary fast, the observer will see the beam of light emitted

from B earlier than he will see that emitted from A. Observers who take the railway train

as their frame of reference must therefore come to the conclusion that the lightning flash

at B took place earlier than the lightning flash at A. The conclusion is that events which

are simultaneous with reference to the embankment are not simultaneous with respect to

the train, and vice versa. Every reference frame has its own particular time. Unless we are

told the frame of reference to which the statement of time refers, there is no meaning to a

statement about the time of an event.

Let us now discuss how the theory of general relativity was developed by Einstein.

Historically, it is said that the first time that Einstein's theory met with empirical

confirmation was at an experiment conducted by Sir Arthur Eddington at 1919. Critics of

that experiment claim that an empirical conformation of Einstein's theories was only

achieved at a later time. What I would like to point out is that before 1919 there was no

empirical confirmation for relativity. The significance of this fact is that it shows that the

methods used by Einstein when developing the theory of relativity must not have been

empirical methods18.

It is reasonable to say that the development of the theory of relativity was done

mainly by mathematical means. Sure, there were empirical observations involved and, as

a consequence of those observations, inconsistencies in Newtonian physics were

discovered. These inconsistencies were probably a major reason why an alternative

theory was needed. But we know that those observations alone do not account for the

development of the body of knowledge we call the theory of relativity. We know this

because it would not have been an issue to empirically confirming the theory of relativity,

even when it was first conceived, if its development was a result of empirical

observations.

18 Note that I use the terms empirical and non-empirical here as if a clear distinction exists between the two.

For now my intention is only to point out our linguistic behavior. I will return to this issue shortly.

- 23 -

Logic and Necessity

One more point for clarification: We have already suggested that the truths of logic

escape Quine's criticism. This is so because Quine's criticism is directed at the notion of

synonymy when it is employed to expand the class of logically true sentences into other

forms of analyticity. Logically true sentences19 are not dependant on any of the terms

Quine showed to be unclear. In the description I gave so far of how the development of

the theory of relativity caused the revision of statements considered to be a priori until

then, I have used the terms empirical and a priori as though there is a clear distinction

between them. I would now like to clarify my intentions.

Suppose that there really is a distinction between a priori and a posteriori knowledge.

Suppose also that a priori knowledge consists in its core of the truths of logic that have

escaped Quine's judgments. We can also suppose that a priori knowledge consists of the

entire class of mathematical statements in addition to statements of pure logic, but this is

not essential to the point. At some point in history, someone arranged the class of

statement considered a priori (according to the definition we have just supposed) by

some theory he conceived. How can we explain that later on a new theory was devised

and that class of a priori statement changed?

A simple, almost trivial, explanation is human error. During the course of time, a

person reviewing the notes left by past scholars may discover that they were wrong,

perhaps by employing some deductive method in an incorrect way. This might seem like

a weak explanation for the revision of a priori statements when the class of statements is

small. But the larger we allow the class of a priori statements to be the more sense it

makes. When that class contains the whole of mathematics, a revision of a priori

statements due to human error is a common event. The class of a priori statements

changes over the course of time because we continually correct mistakes in it.

If what Einstein did when he developed the theory of relativity was to review the

mathematical formulas governing Newtonian physics, correct and further develop them,

sometimes by devising whole new mathematical methods, then in a sense what he did

was to correct Newton's mistakes. The result of these mistakes was that some statements,

such as the one about absolute simultaneity, were included in the class of a priori

statements by mistake. When Lobachevski and Reimann developed alternative

19 For a definition of logical truth see footnote 2.

- 24 -

Logic and Necessity

geometries, they did not do so in order to correct a mistake in Euclidian geometry20. But

the development of alternative geometries alone was not enough to demonstrate how

space is non-Euclidean. That was only accomplished by the theory of relativity.

Can this explanation really account for the claim that every statement is revisable? I

think that the best we can say is that it does a good job at describing our struggle with

knowledge. It gives a reason for why we revise knowledge of any type, empirical or

other. The difference between this explanation and the other ones discussed in this essay

is that it allows us to keep our belief in the existence of a priori truths (relative or not)

and still explain the history of science. One problematic aspect of this explanation is that

calling the theory of Newtonian physics a mistake hardly does it justice. Perhaps the word

mistake (or error) is not the right one to use in this context.

But perhaps the gravity of the situation has to do with the immense changes that have

to occur to the scientific body of knowledge before someone can correct these "mistakes".

Maybe we have talked about revisions for so long that we have forgotten that we are the

ones making them. And the reason we usually have in mind for doing so when we make

them is that we do so to correct a mistake, no matter what type of knowledge we are

dealing with. It is possible that the truths of logic are the most basic laws there are. It is

also possible that the laws of logic describe the nature of thought. But as any student of

arithmetic or logic would testify, this does not prevent us from making mistakes when we

apply them.

20 When Lobachevski first published his theories he called his work "imaginary geometry".

- 25 -

Logic and Necessity

Conclusions

Quine introduced the notion of naturalizing epistemology in a paper called

"Epistemology Naturalized" (Quine 1969). He did so by both showing that traditional

epistemology has failed and arguing that natural science can succeed in its place. Quine

thinks of traditional epistemology as being a doctrinal project concerned with identifying

the foundation and deducting from it beliefs about the physical world. Quine also refers

to the conceptual project of Carnap and the logical positivists, which is concerned with

providing definitions for translating talk about physical bodies into talk about sense

impressions. Quine thinks that not only did Carnap fail; it is in principle impossible to

succeed (to prove this he describes a thought experiment about translation of the word

"gavagai" to "rabbit").

Quine asserts that the quest for certainty will not succeed. The alternative he provides

is for us to study the relation between sense impressions and theories about the world. He

feels that we should reject both the need for justification and the quest for certainty.

Rather we must study scientifically the natural phenomenon in the human brain. In

essence, he has removed normativity from epistemology.

In "Rethinking Mathematical Necessity" Putnam argues that Quine's account of

knowledge implicitly situates mathematics as empirical and as a result misses out on

some important distinctions. First, Quine's denial of a priori truths mistakenly classifies

all statements as empirical. The study of arithmetical statements shows that there is an

important methodological distinction between contextual a priori truths and empirical

truths. This distinction is evident throughout the entire history of scientific discovery. The

new distinction Putnam offers is not meant to be a new metaphysical device, a second

realm of truths. Instead the distinction offers a better description of scientific behavior.

The second point Putnam criticizes in Quine is the notion that the attempt at

justification is useless. The criticism goes much further than to say that we do not know

what sense to attach to the notion that logical truths are wrong. The criticism is that our

language (and in a deeper sense the way we think) cannot account for a revision of

logical truths. Normative notions dominate the way we speak about epistemology and

those notions cannot be naturalized, reduced to empirical means. Borrowing from Kant,

Frege and the early Wittgenstein Putnam argues that mathematics escapes the need for

- 26 -

Logic and Necessity

justification. Again, not meant as a metaphysical statement, but as a statement that better

describes our relation with knowledge.

From a different prospective, Putnam points out that a verificationist theory of

meaning fails at places where a theory of meaning as uses succeeds. The first reason for

this is that a verificationist theory of meaning fails to apply to many scientifically

decidable truths where verification is impossible due to technical reasons. We should not

be forced to give up scientific knowledge we can justify but cannot verify. Another

reason is that in some cases we do not grasp the meaning of certain statements until there

is a change in use. This was the case with the statements like "there are finitely many

places to go to, travel as you will" before Euclidian geometry was overthrown. Only

when this statement was actually used in scientific discourse we were able to give it

sense. Realism cannot be settled with a verificationist point of view; that was the problem

with the positivist project. But if we naturalize epistemology we miss interpret the history

of science; that is the problem with Quine's view.

- 27 -

Logic and Necessity

Works Cited

1. Carnap, R. (1963), The philosophy of Rudolf Carnap, P. A. Schilpp (ed.), pp. 44-

67; 868-877; 915-922.

2. Donnellan, K. (1962), "Necessity and Criteria", the Journal of Philosophy, LIX,

No. 2, pp. 647—58.

3. Einstein, A. (1920) "The Relativity of Simultaneity", Relativity: The Special and

General Theory. New York: Henry Holt, pp. 168; Bartleby.com, 2000.

http://www.bartleby.com/173/9.html.

4. Grice, H.P. and Strawson P.F. (1956), “In defense of a dogma”, Philosophical

Review, AP 1956; 65

5. Kant, Immanuel (1787), Critique of pure reason, translated and edited by Paul

Guyer, Allen W. Wood, Cambridge University Press, 1998

6. Putnam, H. (1975), “It ain’t necessarily so”, Mathematics, Matter and Method

(Philosophical papers, vol. 1), Cambridge University Press.

7. Putnam, H. "There is at least one a priori truth", orig. published in 1978; reprinted

in Putnam's Realism and Reason, Philosophical Papers vol. 3 (Cambridge Univ.

Press, 1983)

8. Putnam, H. (1994), “Rethinking mathematical necessity”, Words and life, J.

Conant (ed.), Harvard University Press.

9. Quine, W.V. (1948) “On What There Is”, The Review of Metaphysics 2, pp. 21-

28. Reprint in many places including Quine, From a Logical Point of View 2nd

ed. (Cambridge: Harvard University Press, 1980).

10. Quine, W.V. (1951), “Two dogmas of empiricism”, Philosophical Review JA

1951; 60

11. Quine, W.V. (1969), "Epistemology Naturalized" reprinted in Naturalising

Epistemology, ed. H. Kornblith, 1985. Cambridge MIT press pp. 23-24.

12. Wittgenstein L. (1922), Tractatus Logico-Philosophicus, translated by D.F. Pears

and B.F. McGuinness (Routledge and Kegan Paul, London 1961).

- 28 -