UNIVERSIDADE DE SAO PAULOFACULDADE DE FILOSOFIA, LETRAS E CIENCIAS HUMANAS

DEPARTAMENTO DE FILOSOFIAPROGRAMA DE POS-GRADUACAO EM FILOSOFIA

Felipe de Souza Salvatore

Topics in Modal Quantification Theory

Sao Paulo2015

Felipe de Souza Salvatore

Topics in Modal Quantification Theory

Dissertacao apresentada ao Programade Pos-Graduacao em Filosofia do De-partamento de Filosofia da Faculdadede Filosofia, Letras e Ciencias Hu-manas da Universidade de Sao Paulo,para obtencao do tıtulo de Mestre emFilosofia sob a orientacao do Prof. Dr.Rodrigo Bacellar da Costa e Silva.

Sao Paulo2015

Autorizo a reprodução e divulgação total ou parcial deste trabalho, por qualquer meioconvencional ou eletrônico, para fins de estudo e pesquisa, desde que citada a fonte.

Catalogação na PublicaçãoServiço de Biblioteca e Documentação

Faculdade de Filosofia, Letras e Ciências Humanas da Universidade de São Paulo

S182tSalvatore, Felipe de Souza Topics in Modal Quantification Theory / Felipe deSouza Salvatore ; orientador Rodrigo Bacellar daCosta e Silva. - São Paulo, 2015. 94 f.

Dissertação (Mestrado)- Faculdade de Filosofia,Letras e Ciências Humanas da Universidade de SãoPaulo. Departamento de Filosofia. Área deconcentração: Filosofia.

1. filosofia. 2. lógica. 3. lógica modal deprimeira ordem. 4. lógica da justificação . 5.interpolação. I. Costa e Silva, Rodrigo Bacellar da,orient. II. Título.

Acknowledgements

I would first like to acknowledge the academic and personal support of my

supervisor Rodrigo Bacellar (a.k.a. Roderick Batchelor). He introduced me to modal

logic, and (more important) his enthusiasm in this area influenced me completely.

Also, his comments and suggestions were very significant for the final version of this

text.

Professor Rodrigo Freire is responsible for a great part of my education in

mathematical logic. Not only his classes, but also the conversations that I have had

with him and the books that he recommended, all of this has profoundly molded

my technical knowledge.

I would like to thank professor Melvin Fitting for receiving me as a visiting

research scholar at the Graduate Center at the City University of New York (CUNY).

He not only made me feel very welcome at CUNY, but he also had the patience to

hear all my comments on first-order justification logic, answer all my questions on

di↵erent logic related topics; and sent me material when I needed it. I was very

inspired by his writings before I went to New York; after meeting him personally I

discovered that he is the model researcher of logic that I myself would one day like

to be.

I am grateful to Sergei Artemov in allowing me to present what is now

Chapter 5 of this dissertation in his seminar, and his comments were very important

to mature my understanding of justification logic.

I would also like to thank all my colleagues and friends from USP, UNI-

CAMP and CUNY; especially: Alfredo Roque, Bruno Ramos, Edgar Almeida, Hen-

rique Meretti, Julio de Rizzo and Konstantinos Pouliasis; thank you for all the

discussion on logic and for all the laughs.

A special thanks also goes to David Gilbert for all the advice that he gave

me. I would like to thank the remainder of my thesis committee, Edelcio de Souza

1

and Marcelo Finger.

I am grateful to Mike Knight for helping me to correct the English in this

thesis.

I owe a heartfelt thanks to Mariana Bardelli for her unconditional support

and kindness. Her presence was fundamental to the realization of this project.

Eu gostaria de agraceder meus pais e meu irmao por todo o amor, apoio e

compreensao na minha trajetoria academica.

Finally, I would like to thank the Sao Paulo Research Foundation (Fundacao

de Amparo a Pesquisa do Estado de Sao Paulo, FAPESP) for the financial support

both in Brazil and abroad.

2

Resumo

SALVATORE, F. S. Topicos em Teoria da Quantificacao Modal. 2015. 94 f. Dis-

sertacao (Mestrado) – Faculdade de Filosofia, Letras e Ciencias Humanas. Depar-

tamento de Filosofia, Universidade de Sao Paulo, Sao Paulo, 2015.

A logica modal S5 nos oferece um ferramental tecnico para analizar algumas

nocoes filosoficas centrais (por exemplo, necesidade metafısica e certos conceitos

epistemologicos como conhecimento e crenca). Apesar de ser axiomatizada por

princıpios simples, esta logica apresenta algumas propriedades peculiares. Uma das

mais notorias e a seguinte: podemos provar o Teorema da Interpolacao para a versao

proposicional, mas esse mesmo teorema nao pode ser provado quando adicionamos

quantificadores de primeira ordem a essa logica. Nesta dissertacao vamos estudar a

falha dos Teoremas da Definibilidade e da Interpolacao para a versao quantificada

de S5. Ao mesmo tempo, vamos combinar os resultados da logica da justificacao

e investigar a contraparte da versao quantificada de S5 na logica da justificacao (a

logica chamada JT45 de primeira ordem). Desse modo, vamos explorar a relacao

entre logica modal e logica da justificacao para ver se a logica da justificacao pode

contribuir para a restauracao do Teorema da Interpolacao.

Palavras-chave: logica, logica modal de primeira ordem, logica da justificacao,

interpolacao.

3

Abstract

SALVATORE, F. S. Topics in Modal Quantification Theory. 2015. 94 f. Thesis

(Master Degree) – Faculty of Philosophy, Languages and Literature, and Human

Sciences. Department of Philosophy, University of Sao Paulo, Sao Paulo, 2015.

The modal logic S5 gives us a simple technical tool to analyze some main notions

from philosophy (e.g. metaphysical necessity and epistemological concepts such as

knowledge and belief). Although S5 can be axiomatized by some simple rules, this

logic shows some puzzling properties. For example, an interpolation result holds for

the propositional version, but this same result fails when we add first-order quanti-

fiers to this logic. In this dissertation, we study the failure of the Definability and

Interpolation Theorems for first-order S5. At the same time, we combine the results

of justification logic and we investigate the quantified justification counterpart of

S5 (first-order JT45). In this way we explore the relationship between justification

logic and modal logic to see if justification logic can contribute to the literature

concerning ‘the restoration of the Interpolation Theorem’.

Keywords: logic, first-order modal logic, justification logic, interpolation.

4

Contents

1 Introduction 7

1.1 Modal quantification theory: adding first-order quantifiers to modal

logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Preliminaries 11

2.1 Syntactical considerations . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Models: basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 First-order S5: two versions . . . . . . . . . . . . . . . . . . . . . . . 17

3 Interpolation and Definability 21

3.1 Models: isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Interpolation and definability as properties . . . . . . . . . . . . . . . 25

3.3 Failure of Interpolation and Beth’s Definability Theorems in FOS5V . 29

3.4 Failure of Interpolation and Beth’s Definability Theorems in FOS5 . . 31

3.5 Inner and outer quantifiers . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Justification Logic: a very short introduction 37

4.1 History and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 The propositional case: language and axiom system . . . . . . . . . . 43

4.3 From propositional logic to first-order . . . . . . . . . . . . . . . . . . 45

5 First-order JT45 47

5.1 Language and axiom system . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Semantics: basic definitions . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 Semantics: non-validity . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5

5.4 Soundness and Completeness . . . . . . . . . . . . . . . . . . . . . . . 55

5.4.1 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4.2 An obstacle in the proof of the Completeness Theorem . . . . 55

5.4.3 Language extension . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4.4 Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.5 Using templates for Henkin-like theorems . . . . . . . . . . . . 75

5.4.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6 Conclusion and future research 82

6.1 An axiomatic system for FOS5 . . . . . . . . . . . . . . . . . . . . . . 82

6.2 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3 Justification logic and interpolation . . . . . . . . . . . . . . . . . . . 88

7 Appendix 90

Bibliography 93

6

Chapter 1

Introduction

1.1 Modal quantification theory: adding first-order

quantifiers to modal logic

The aim of the present thesis is to present a study of some relevant topics

concerning first-order modal logic. To make our intentions precise, some restrictions

must be made.

It is ambiguous to write ‘first-order modal logic’, because unlike in classical

logic we can use alternative propositional logics to be the background logic, and even

when we choose one specific propositional logic, there are di↵erent choices that can

be made to construct the first-order version of modal logic. Among the variety of

propositional logics, in this thesis we are only concerned with S5. This choice is not

arbitrary, because among the other propositional modal logics S5 is more interesting

for the researcher in philosophy. To be more precise, S5 gives us a plausible way to

deal with main philosophical concepts such as the concept of metaphysical necessity

and the epistemological notions of knowledge and belief.

Although use of first-order S5 is made mainly by philosophers, from the

technical point of view, this logic is a very intriguing one. That is the richness of

first-order S5 as a research subject: by studying this logic, we can study a complex

technical subject and at the same time we can stay connected with deep philosophical

problems.

The failure of the Interpolation Theorem is a good example. Also known

as Craig’s interpolation theorem, this theorem was first proved for classical logic. It

7

states that if a formula ' implies a formula then there is a formula ✓, referred to

as the interpolant between ' and , such that every nonlogical symbol in ✓ occurs

both in ' and , ' implies ✓, and ✓ implies .

Investigating if this theorem holds in other logics (like modal logic) is, from

the technical point of view, interesting in itself. The richness that we mentioned is

that we can show that this theorem fails for first-order S5 and moreover from this

failure we can conclude some philosophical implications.

In the metaphysical debate on necessity and existence there are two main

positions: one claims that necessarily everything is necessarily something, i.e. ex-

istence is necessary ; the other claims that possibly something is possibly nothing,

i.e. existence is contingent. Following Timothy Williamson [20], we call the first

position Necessitism and the second Contingentism.

Suppose we assume the Contigentism thesis. Then, sometimes di↵erent

possible worlds have di↵erent inhabitants. In this setting, it makes sense to define

two kinds of quantifiers: the inner quantifiers 9 and 8; and the outer quantifiers ⌃

and ⇧. Without entering into the technicalities, we can say, very informally, that

the formula 9x' is true at the world w, if ' is true at w for a specific interpretation

that interprets x into an inhabitant of w. On the other hand, ⌃x' is true at the

world w, if ' is true at w for a specific interpretation that interprets x into an

inhabitant of any possible world w0. As usual, we define 8 as the dual of 9 and ⇧

as the dual of ⌃.

Saul Kripke pointed out in [19], that from the failure of the Interpolation

Theorem for first-order S5, it follows that, for this modal logic, the outer quantifiers

are not definable in the usual modal language with the inner quantifiers. And so a

philosophical result follows: some metaphysical notions that can be expressed using

quantification over possible entities cannot be emulated by a restricted language

which has only quantification over actual entities. Putting it another way: if we

assume the Contigentism thesis and if we assume that the only meaningful discourse

is the one that only speaks about actual entities, then there are some metaphysical

notions that we are not going to be able to express.

Going beyond this problem of metaphysical modality, the reason for the

failure of the Interpolation Theorem is understood as a lack of expressiveness of

the quantified version of S5. This lack of expressive power had left a natural ques-

tion open: if we add more machinery to first-order S5 are we able to restore the

8

Interpolation Theorem? The answer is not an easy one. There are many ways to

extend the expressiveness of modal logic. In the literature around ‘the restoration of

the Interpolation Theorem’ we find examples where the restoration of this theorem

can be obtained (e.g., when we use hybrid logics [1], or when we use propositional

quantifiers [11]), but there is not a general argument to show how to extend the

expressive power of modal logic in order to guarantee the Interpolation Theorem.

That is why, from a theoretical point of view, it is significant to investigate

how the Interpolation Theorem behaves in di↵erent extensions of modal logic.

Justification logic is a term used to classify a relatively new kind of modal-

like logics. The first justification logic, LP (Logic of Proofs), was originated from a

question in provability logic (the logic that arises when we interpret the modal for-

mulas with arithmetical semantics). Nowadays we work with an extensive family of

propositional justification logics. And, for the philosophical discussion, the interest

in justification logic lies in the connection between this logic and some epistemic

notions: as the name indicates, justification logic enables us to introduce the notion

of justification into the setting of epistemic logic.

Although justification logic is now a well-studied subject, the main focus is

on the propositional case. There are only a few papers in quantified justification

logic, and the majority of those papers investigate the justification counterpart of

first-order S4.

In this thesis we present the failure of the Definability and Interpolation

Theorems for first-order S5. We establish the basic setting for the justification

counterpart of the first-order version of S5. And we indicate how we can relate the

failure of the Interpolation Theorem to the research agenda of justification logic.

In Chapter 2 we give an introduction to the basic subjects that are present

when modal operators and quantifiers come to the discussion. In Chapter 3 we

present the now classical proofs of the failure of Interpolation and Beth’s Definability

theorems. In Chapter 4 we give a brief presentation of justification logic. In Chapter

5 we present the justification counterpart of quantified S5 (called first-order JT45).

And in Chapter 6 we comment on how all the topics presented in this thesis can be

combined in order to advance the research on modal logic.

9

1.2 Notation

In this text we abbreviate ‘if and only if’ with ‘i↵’, and we use the following

set-theoretical notation:

• P(A) denotes the power-set of A.

• A\B denotes the set {x | x 2 A ^ x /2 B}.

• We write A ✓fin B to say that A ✓ B and A is a finite set.

• If f is a function, we write Dom(f), Rng(f) and Field(f) to denote the sets

{x | 9y((x, y) 2 f)}, {y | 9x((x, y) 2 f)} and Dom(f)[Rng(f), respectively.

We also write f ⇠ A and f [A] to denote the sets {(x, y) | x 2 A ^ (x, y) 2 f}and {f(x) | x 2 Dom(f) \ A}, respectively.

• If f and g are functions, f � g denotes the composition of functions, i.e., f � gdenotes the set {(x, z) | 9y((x, y) 2 g ^ (y, z) 2 f)}. And if f is an injective

function, we write f�1 to denote the function {(y, x) | (x, y) 2 f}.

10

Chapter 2

Preliminaries

2.1 Syntactical considerations

Definition 1. A language L is a set of symbols. Throughout this dissertation we

are going to work only with relational laguages; in some specific moments we will

add constants to the language, but we will be explicit when we are doing so. We use

P,Q, P 0, Q0, . . . to denote relation symbols. It is assumed that each relation symbol

P of L is an n-ary relation symbol for n 2 !. We call a 0-ary relation symbol a

propositional letter, and we use p, q, p0, q0, . . . to denote propositional letters (also

called propositional variables).

We use L,L0,L00, . . . as variables for languages. If L ✓ L0, we say that L0 is

an expansion of L, and that L is a reduction of L0.

Definition 2. Together with L we define the following logical symbols :

• x0, x1, x2, . . . (variables);

• ¬,_ (not, or);

• 9 (there exists);

• ⌃ (possibility symbol);

• = (equality symbol);

• ), ( (parentheses).

11

We use x, y, z, . . . as syntactical variables for variables.

Definition 3. The set Fml(L) of formulas of L is defined by the following rules:

• If x, y are variables, then x = y is a formula of L.

• If x1, . . . , xn (n � 0) are variables and P is an n-ary relation symbol of L, thenPx1 . . . xn is a formula of L.

• If ' is a formula of L, then ¬' is a formula of L.

• If ' and are formulas of L, then (' _ ) is a formula of L.

• If ' is a formula of L, then ⌃' is a formula of L.

• If ' is a formula of L and x a variable, then 9x' is a formula of L.

We assume the standard syntactical notions of atomic formula, free variable,

bound variable, sentence, formula complexity and proof (definition) by induction on

formulas. We are going to employ the usual abbreviations:

(' ^ ) := ¬(¬' _ ¬ )('! ) := (¬' _ )

(' ! ) := ('! ) ^ ( ! ')

8x' := ¬9x¬'⇤' := ¬⌃¬'

We write '(x1, . . . , xn) to denote that the free variables of ' are among

{x1, . . . , xn}. Where y1, . . . , yn are variables, we write '(y1/x1, . . . , yn/xn) to de-

note the formula obtained by substitution of y1, . . . , yn for all the free occurrences

of x1, . . . , xn in ', respectively. When it is clear from the context which vari-

ables are free in ' we simply write '(y1, . . . , yn) instead of '(y1/x1, . . . , yn/xn).

We use ~x, ~y, . . . for sequence of variables; and we write 8~x'(~x) in the place of

8x1 . . . 8xn'(x1, . . . , xn).

12

2.2 Models: basic notions

Definition 4. A frame is a tuple hW ,Ri in which:

• W 6= ;.

• R ✓W ⇥W .

Definition 5. A skeleton1 is a quadruple hW ,R,D, Di in which hW ,Ri is a frame

and:

• D 6= ;.

• D : W ! P(D), and for every w of W , D(w) 6= ;.

• D =S

w2W Dw.

The intuition behind the notion of skeleton is the same as in [18]: W is the

set of all ‘possible worlds’; R is the accessibility relation between worlds; D is a

function which gives to each world a domain of individuals, and D is the set of all

possibles individuals.

We use w, v, u, w0, w0, w1, . . . as variables for worlds. From now on we write

Dw instead of D(w). In the cases where D is a constant function we write hW ,R,Diinstead of hW ,R,D, Di.

Definition 6. A (modal) model for L is a quintuple M = hW ,R,D, D, Ii in which

hW ,R,D, Di is a skeleton and I is an interpretation function, i.e., a function as-

signing to each n-ary relational symbol P of L and each possible world w an n-ary

relation I(P,w) on D.

We use M,N ,M0, . . . as variables for models.

Definition 7. Let L and L0 be languages such that L0 ✓ L, M = hW ,R,D, D, Iibe a model for L and M0 = hW 0,R0,D0, D0, I 0i be a model for L0. We call M0 a

reduct of M (and M an expansion for M0) i↵ W = W 0, R = R0, D = D0, D = D0,

and I and I 0 agree on the symbols of L0. We write M0 = M|L0 .

1Sometimes called augmented frame.

13

Definition 8. A valuation in a model M = hW ,R,D, D, Ii is a function h from

the set of variables to D.2 We say that h0 is an x-variant of h if the two valuations

agree on all variables except possibly x. Similarly, we say that a valuation h0 is an

x-variant of h at w if h0 is an x-variant of h and h0(x) 2 Dw.

Definition 9. Let M = hW ,R,D, D, Ii be a model for L, ' a formula of L, h a

valuation in M and w 2W . The notion ' is true at world w of M with respect to

valuation h, in symbols M, w |=h ', is defined recursively as follows:

M, w |=h x = y i↵ h(x) = h(y).

M, w |=h Px1 . . . xn i↵ hh(x1), . . . , h(xn)i 2 I(P,w).

M, w |=h ¬ i↵ M, w 6|=h .

M, w |=h _ ✓ i↵ M, w |=h or M, w |=h ✓.

M, w |=h ⌃ i↵ there is a w0 2W such that wRw0 and M, w0 |=h .

M, w |=h 9x i↵ there is an x-variant h0 of h at w such that M, w |=h0 .

This definition enables us to speak of the truth of a formula at a world in a

model without mentioning the valuation. We write M, w |= ' if for every valuation

h, M, w |=h '; and when that is the case we say that ' is true in M at w. We

write M |= ' if for every world w of M, M, w |= '. And we say that a formula '

is valid in a class of models, if for every model M of this class, M |= '.

Let � be a set of formulas of L (we also call � a theory); then M, w |= � if

for every ' 2 �, M, w |= '. In this case, we say that the pair M, w is a model for

�. Two formulas ' and are equivalent if for every model M, M |= ' i↵ M |= .

There are some basic propositions about the relation |=. Since their proofs

are straightforward and they can be found in many di↵erent textbooks, we are going

to state these propositions without proof.

Proposition 1. Suppose that M and M0 are models for L and L0, respectively;

that L ✓ L0; and that M is the reduct of M0 to L. Then for every world w of M,

for every valuation h in M, if ' is a formula of L then:

2Although is more natural to use v to denote a valuation, it is easy to get lost in the proofswhen we use v for valuations and w and u for worlds.

14

M, w |=h ' i↵ M0, w |=h '.

Proposition 2. Let M be a model for L, w a world of M, h1 and h2 valuations in

M and ' a formula of L. If h1 and h2 agree on all the free variables of ', then

M, w |=h1 ' i↵ M, w |=h2 '.

Definition 10. Let M = hW ,R,D, D, Ii be a model for L:

• M is an S5-model i↵ R is an equivalence relation.

• M is an universal model i↵ R = W ⇥W .

• M is a constant domain model i↵ for every w, v 2W , Dw = Dv.

• M is a monotonic model i↵ for every w, v 2W , if wRv, then Dw ✓ Dv.

• M is an anti-monotonic model i↵ for every w, v 2W , if wRv, then Dv ✓ Dw.

• M is a locally constant domain model i↵ for every w, v 2 W , if wRv, then

Dw = Dv.

Very often, in di↵erent books and papers on first-order modal logic, there

is the mentioning of the ‘Barcan Formula’. The following explains the connection

between locally constant domain models and this formula.

Definition 11. Let M = hW ,R,D, D, Ii be a model for L:

• We say that M satisfies the Barcan Formula i↵ for every ' 2 Fml(L) of theform 8x⇤ ! ⇤8x , we have that M |= '.

• We say that M satisfies the Converse Barcan Formula i↵ for every ' 2Fml(L) of the form ⇤8x ! 8x⇤ , we have that M |= '.

By well-know equivalences of first-order modal logic, we have:

M satisfies the Barcan Formula i↵ for every ' 2 Fml(L) of the form ⌃9x !9x⌃ , we have that M |= '.

M satisfies the Converse Barcan Formula i↵ for every ' 2 Fml(L) of the form9x⌃ ! ⌃9x , we have that M |= '.

15

Proposition 3. Let M = hW ,R,D, D, Ii be a model for L:

(a) M is an anti-monotonic model i↵ M satisfies the Barcan Formula.

(b) M is a monotonic model i↵ M satisfies the Converse Barcan Formula.

(c) M is a locally constant domain model i↵ M satisfies the Barcan Formula and

its converse.

Proof. (a) ()) Let ' 2 Fml(L) be a formula of the form 8x⇤ ! ⇤8x , w 2W and h a valuation. If M, w |=h 8x⇤ , then for every x-variant h0 of h at w

M, w |=h0 ⇤ . Let v be a member of W such that wRv. By hypothesis, Dv ✓ Dw,

so every x-variant h0 of h at v is an x-variant h0 of h at w, thus M, v |=h 8x . Sincev was arbitrarily chosen, M, w |=h ⇤8x , and hence M, w |=h '.

(() Suppose that M satisfies the Barcan Formula and M is not an anti-

monotonic model; then there are w, v 2 W such that wRv and Dv * Dw. Hence,

there is an a 2 D such that a 2 Dv and a /2 Dw. Then for a valuation h such

that h(y) = a, M, v |=h 9x(x = y); and, since wRv, M, w |=h ⌃9x(x = y). By

hypothesis, M, w |= ⌃9x(x = y) ! 9x⌃(x = y). So, in particular, M, w |=h

⌃9x(x = y) ! 9x⌃(x = y); hence, M, w |=h 9x⌃(x = y). Then, there is an x-

variant h0 of h at w such that M, w |=h0 ⌃(x = y); so there is a w0 2 W such that

wRw0 and M, w0 |=h0 (x = y), hence h0(x) = h0(y). Since h0(x) 2 Dw, a 2 Dw;

a contradiction. Therefore, if M satisfies the Barcan Formula, then M is an anti-

monotonic model.

(b) ()) Let ' 2 Fml(L) be a formula of the form ⇤8x ! 8x⇤ , w 2Wand h a valuation. If M, w |=h ⇤8x , then let v be a member of W such that

wRv; so M, v |=h 8x . Then, for every x-variant h0 of h at v, M, v |=h0 . By

hypothesis, Dw ✓ Dv; therefore every x-variant h0 of h at w is an x-variant h0 of

h at v, thus M, v |=h0 for every x-variant h0 of h at w. Since v was arbitrarily

chosen, M, w |=h ⇤ for every x-variant h0 of h at w. So M, w |=h 8x⇤ and hence

M, w |=h '.

(() Suppose that M satisfies the Converse Barcan Formula and M is not

a monotonic model; then there are w, v 2W such that wRv and Dw * Dv. Hence,

there is an a 2 D such that a 2 Dw and a /2 Dv. So for a valuation h such that

h(x) = a, M, v |=h 8y(y 6= x), and since wRv, M, w |=h ⌃8y(y 6= x) and so

M, w |= 9x⌃8y(y 6= x). By hypothesis, M, w |= 9x⌃8y(y 6= x) ! ⌃9x8y(y 6=

16

x). So, M, w |= ⌃9x8y(y 6= x). Thus there is a w0 2 W such that wRw0 and

M, w0 |= 9x8y(y 6= x); this clearly implies a contradiction. Therefore, if M satisfies

the Converse Barcan Formula, then M is a monotonic model.

(c) The result follows directly from (a) and (b).

Strictly speaking, both the Barcan Formula and the Converse Barcan For-

mula are not formulas, they are formula schemes. So it is natural to ask if there

is a formula which has the same ‘expressive power’ as the Barcan Formula and the

Converse Barcan Formula. In fact, dealing with S5-models we can find this formula.

Proposition 4. Let M = hW ,R,D, D, Ii be an S5-model for L, then:

M |= ⇤8x⇤9y(y = x) i↵ M satisfies the Barcan Formula and its converse.

Proof. ()) Let w and v be members of W such that wRv. If a 2 Dw, then since

M, w |= ⇤8x⇤9y(y = x) and wRw, we have that M, w |= 8x⇤9y(y = x). In

particular, for a valuation h such that h(x) = a, M, w |=h ⇤9y(y = x). Then,

M, v |=h 9y(y = x). So there is an x-variant h0 of h at v such that M, v |=h0 y = x,

thus h0(y) = h0(x) and so a 2 Dv. Hence, Dw ✓ Dv. We can prove that Dv ✓ Dw in

a similar way. Therefore, M is a locally constant domain model; by Proposition 3,

M satisfies the Barcan Formula and its converse.

(() Suppose that M satisfies the Barcan Formula and its converse and

there is a w 2 W such that M, w 6|= ⇤8x⇤9y(y = x). So, for some valuation

h, M, w 6|=h ⇤8x⇤9y(y = x). By Proposition 3, M is a locally constant domain

model; and by equivalences of first-order modal logic, M, w |=h ⌃9x⌃8y(y 6= x).

Hence there is a v 2W such that wRv and M, v |=h 9x⌃8y(y 6= x). So there is an

x-variant h0 of h at v such that M, v |=h0 ⌃8y(y 6= x). Then there is a w0 2W such

that vRw0 and M, w0 |=h0 8y(y 6= x). Therefore, h0(x) 2 Dv\Dw0 ; contradicting the

assumption that M is a locally constant domain model.

2.3 First-order S5: two versions

Before we advance, we need to address some technical details concerning S5-

models. In order to save time we are going to skip the proofs of the propositions in

this section.

17

First, since R is an equivalence relation in an S5-model, all the di↵erent

notions of monotonic, anti-monotonic and locally constant domain model become

equivalent when we work with an S5-model. Therefore, we shall only distinguish

between locally constant domain models and varying domain models (models with

no restriction on the domains).

Second, the distinction between constant domain models and locally con-

stant domain models can be dropped. Of course, as mathematical structures con-

stant domain models and locally constant domain models are very di↵erent objects.

But from the point of view of modal formulas they are the same. The following

proposition states this fact more clearly:

Proposition 5. Let ' be a formula of L. ' is valid in the class of constant domain

models for L i↵ ' is valid in the class of locally constant domain models for L.

Third, sometimes both for technical and theoretical reasons it is more useful

to deal with universal models instead of S5-models. And as before, although they

are di↵erent mathematical structures, from the point of view of the valid formulas

we can take them as the same:

Proposition 6. Let ' be a formula of L. ' is valid in the class of universal models

for L i↵ ' is valid in the class of S5-models for L.

We can now define the two main kinds of models that we are going to work

with. Propositions 5 and 6 serve to show the non-arbitrariness of the following

definition and to connect it with the results of the previous section.

Definition 12. For a fixed language L we say that:

• a model for first-order S5 with constant domains, denoted FOS5-model, is a

universal and constant domain model.

• a model for first-order S5 with varying domains, denoted FOS5V-model, is a

universal and varying domain model.

The following definitions apply both to FOS5 and FOS5V models; to avoid

duplication of definitions we use L as a variable for FOS5 and FOS5V. From now

on, when dealing with FOS5V-models we omit the accessibility relation, and when

working with FOS5-models we omit the D function too.

18

Definition 13. Let L be a language and let {'}, { } and � be sets of sentences of

L:

• ' is L-valid, in symbols |=L ', i↵ ' is valid in the class of L-models. We say

that ' is L-satisfiable i↵ there is an L-model M and a world w of M such

that M, w |= '. And we say that ' is L-unsatisfiable i↵ ¬' is L-valid.

• ' is a consequence of � in L, in symbols � |=L ', i↵ for every pair M, w, if

M, w is an L-model for �, then M, w |= '. Instead of { } |=L ' we write

|=L '.

For example, from propositional modal logic it is well-known that:

|=L ⇤('! )! (⇤'! ⇤ )|=L ⇤'! '

|=L ⇤'! ⇤⇤'|=L ¬⇤'! ⇤¬⇤'

And using what we have seen so far, we have:

|=FOS5V ⇤8x⇤9y(y = x)! (⇤8xPx ! 8x⇤Px)

|=FOS5 ⇤8xPx ! 8x⇤Px

These last two examples are just instances of a more general fact that is an

immediate consequence of Propositions 3 and 4.

Proposition 7. A sentence ' of L is FOS5-valid i↵ ⇤8x⇤9y(y = x)! ' is FOS5V-

valid.

Now we have all the ingredients to present a notion of logic.

Definition 14. The logic L is a tuple hLan, |=Li where Lan is a function which

associates to every language L a set sen(L), the set of sentences of L; and |=L is

the relation as defined above.

A last basic topic worth noticing is that we can define an unary relation

symbol E such that Ex expresses that the individual denoted by x exists in the

world in question. The definition of this relation, often called existence predicate, is:

19

Ex := 9y(y = x)

Obviously, M, w |=h Ex i↵ h(x) 2 Dw. The following proposition states

some useful facts about the relation |= and Ex.

Proposition 8. For a formula ' of L such that fv(') = {x1, . . . , xn}, let E~x be an

abbreviation of Ex1 ^ . . . ^ Exn, then:

• |=FOS5V 8~x' i↵ |=FOS5V (E~x! ').

• |=FOS5 8~x' i↵ |=FOS5 '.

• If |=FOS5V ', then |=FOS5V 8~x'.

20

Chapter 3

Interpolation and Definability

This chapter is completely based on the paper [7] by Kit Fine. Only the last

section is based on other material, the already mentioned review by Saul Kripke [19].

3.1 Models: isomorphism

Definition 15. Let M = hW ,D, D, Ii be a model for L and w 2W .

• The external model of M at w is the triple Mw = hD, Dw, Iwi where Iw is a

function on L such that Iw(P ) = {ha1, . . . , ani 2 Dn | ha1, . . . , ani 2 I(P,w)},for every n-ary relation symbol P 2 L.

• The internal model of M at w is the tuple Mw = hDw, Iwi where Iw is a

function on L such that Iw(P ) = {ha1, . . . , ani 2 Dnw | ha1, . . . , ani 2 I(P,w)},

for every n-ary relation symbol P 2 L.

Definition 16. Let M, Mw and Mw be as in the previous definition. We can easily

define a notion of isomorphism for models of the form Mw and Mw. For the former,

the notion is the same as in the classical case. For the latter, let N = hV ,B, B,J ibe a model for L, v 2 V and Nv = (B, Bv,Jv). Let � be an one-one function

from D onto B; we say that � is an isomorphism between Mw and Nv, in symbols

� : Mw⇠= Nv, i↵:

• for every a1, . . . , an 2 D, for every n-ary relation symbol P 2 L, ha1, . . . , ani 2Iw(P ) i↵ h�(a1), . . . , �(an)i 2 Jv(P ).

21

• �[Dw] = Bv.

Definition 17. Let M = hW ,D, D, Ii and N = hV ,B, B,J i be models for L. We

say that � is an isomorphism from M onto N , in symbols � : M ⇠= N , i↵ � is an

one-one function from D onto B such that:

(i) For every w 2W there is a v 2 V such that � : Mw⇠= Nv.

(ii) For every v 2 V there is a w 2W such that � : Mw⇠= Nv.

Let M and N be models for L, and let � be a function from D to B. If h

is a valuation in M we write h� to denote the valuation � � h in N .

Lemma 1. Let M = hW ,D, D, Ii and N = hV ,B, B,J i be models for L, w 2W ,

v 2 V and � : D ! B such that � : M ⇠= N and � : Mw⇠= Nv . Then for every

valuation h and every formula ' of L:

M, w |=h ' i↵ N , v |=h� '

Proof. Induction on '.

(' is x = y)

M, w |=h x = y

i↵ h(x) = h(y)

i↵, since � is injective, �(h(x)) = �(h(y))

i↵ h�(x) = h�(y)

i↵ N , v |=h� x = y.

(' is Px1 . . . xn)

M, w |=h Px1 . . . xn

i↵ hh(x1), . . . , h(xn)i 2 I(P,w)i↵ hh(x1), . . . , h(xn)i 2 Iw(P )

i↵, by hypothesis, h�(h(x1)), . . . , �(h(xn))i 2 Jv(P )

i↵ h�(h(x1)), . . . , �(h(xn))i 2 J (P, v)

i↵ hh�(x1), . . . , h�(xn)i 2 J (P, v)

22

i↵ N , v |=h� Px1 . . . xn.

If ' is ¬ or _ ✓, then the result follows from the induction hypothesis.

(' is ⌃ )

If M, w |=h ⌃ , then there is a w0 2 W such that M, w0 |=h . Since

� : M ⇠= N , then, by condition (i) of Definition 17, there is a v0 2 V such that

� : Mw0 ⇠= Nv0 . By induction hypothesis,

M, w0 |=h i↵ N , v0 |=h�

So, N , v0 |=h� , and hence N , v |=h� ⌃ . The converse implication follows

from the condition (ii) of Definition 17 and the induction hypothesis.

(' is 9x )

On the one hand, if M, w |=h 9x , then for an x-variant h0 of h at w,

M, w |=h0 . By induction hypothesis, N , v |=h0� . Since h0(x) 2 Dw and �[Dw] =

Bv, then h0�(x) 2 Bv. So, h0� is an x-variant of h� at v. Therefore N , v |=h� 9x .On the other hand, if N , v |=h� 9x , then for some x-variant h0 of h� at

v, N , v |=h0 . Since h0(x) 2 Bv and �[Dw] = Bv, there is an a 2 Dw such that

�(a) = h0(x). Let h⇤ be a valuation in M such that for every variable y

h⇤(y) =

(h(y) if y 6= x

a if y = x

Clearly, h⇤� = h0 and h⇤ is an x-variant of h at w . Since N , v |=h⇤� , then,

by induction hypothesis, M, w |=h⇤ , and so M, w |=h 9x .

Lemma 2. Let M = hW ,D, D, Ii and N = hV ,B, B,J i be models for L, w 2W ,

v 2 V and ⇢ : Dw ! Bv such that ⇢ : Mw⇠= Nv and for every ⇢0 ✓fin ⇢ there is a �

such that ⇢0 ✓ � and � : M ⇠= N . In these conditions, for every formula ' of L and

for every valuation h such that h[fv(')] ✓ Dw:

M, w |=h ' i↵ N , v |=h⇢ '

23

Proof. Induction on '.

(' is x = y)

M, w |=h x = y

i↵ h(x) = h(y)

i↵, since ⇢ is injective, ⇢(h(x)) = ⇢(h(y))

i↵ h⇢(x) = h⇢(y)

i↵ N , v |=h⇢ x = y.

(' is Px1 . . . xn)

M, w |=h Px1 . . . xn

i↵ hh(x1), . . . , h(xn)i 2 I(P,w)i↵ hh(x1), . . . , h(xn)i 2 Iw(P )

i↵, by hypothesis, h⇢(h(x1)), . . . , ⇢(h(xn))i 2 Jv(P )

i↵ h⇢(h(x1), . . . , ⇢(h(xn))i 2 J (P, v)

i↵ hh⇢(x1), . . . , h⇢(xn)i 2 J (P, v)

i↵ N , v |=h⇢ Px1 . . . xn.

If ' is ¬ or _ ✓, then the result follows from the induction hypothesis.

(' is ⌃ )

If M, w |=h ⌃ , then there is a w0 2W such that M, w0 |=h . Since there

is only a finite number of free variables occurring in , if ⇢0 = ⇢ ⇠ h[fv(')], then

⇢0 ✓fin ⇢. By hypothesis, there is a � such that ⇢0 ✓ � and � : M ⇠= N . By

condition (i) of Definition 17, there is a v0 2 V such that � : Mw0 ⇠= Nv0 . So all the

conditions of Lemma 1 are fulfilled; then for every valuation h0 and every formula ✓

of L:

M, w0 |=h0 ✓ i↵ N , v0 |=h0� ✓.

In particular we have,

24

M, w0 |=h i↵ N , v0 |=h� .

And since M, w0 |=h , we have N , v0 |=h� .

Now, by the definition of �, � and ⇢0 agree on all the elements of h[fv(')].

So, if y 2 fv('), then:

h�(y) = �(h(y))

= ⇢0(h(y))

= ⇢(h(y))

= h⇢(y)

Therefore, h� and h⇢ agree on all the free variables of ; then, by Proposi-

tion 2, N , v0 |=h⇢ . And so, N , v |=h⇢ ⌃ . The converse implication follows from

the condition (ii) of Definition 17 and Lemma 1.

If ' is 9x , then the result follows from the induction hypothesis and the

fact that ⇢[Dw] = Bv.

3.2 Interpolation and definability as properties

In this section we will assume that some countable language L is fixed.

Definition 18. Let L be a logic and let � be a set of sentences of L. Then:

• L has the Interpolation property (or the Interpolation Theorem holds for L) i↵

for any sentences ' and of L, if |=L ' ! , then there is a formula ✓ such

that |=L '! ✓, |=L ✓ ! and the non-logical symbols that occur in ✓ occur

both in ' and .

• Let L be a language such that the n-ary relation symbol P belongs to L. LetP 0 be a new n-ary relation symbol not occurring on L, L0 = (L\{P}) [ {P 0}and �0 be the result of replacing each occurrence of P in the sentences of �

with P 0. � implicitly defines P in L if �[�0 |=L 8~x(P~x ! P 0~x). � explicitly

defines P in L if � |=L 8~x(P~x ! ✓) for some formula ✓ 2 Fml(L\{P}).

25

We say that the logic L has the Definability property (or Beth’s Definability

Theorem holds for L) i↵ whenever � defines P implicitly in L, also � defines

P explicitly in L.

Proposition 9. If L has the Interpolation property then L has the Definability

property.

Proof. Here we shall present the proof only for FOS5V. We do that because the

proof for FOS5 is very close to the proof for the classical case.

Suppose that � implicitly defines P in FOS5V, i.e.

� [ �0 |=FOS5V 8~x(P~x ! P 0~x)

Hence, by Proposition 8,

� [ �0 |=FOS5V E~x! (P~x ! P 0~x)

And by propositional logic,

� [ �0 |=FOS5V E~x! (P~x! P 0~x).

By Compactness1 there is �0 ✓fin �[�0 such that �0 |=FOS5V E~x! (P~x!P 0~x). Let ' be the conjunction of all sentences of � \ �0 and be the conjunction

of all sentences of �0 \ �0. So,

' ^ |=FOS5V E~x! (P~x! P 0~x)

It is easy to check that for every sentence ' and , ' |=FOS5V i↵ |=FOS5V

'! . Thus, using this fact we have

|=FOS5V ' ^ ! (E~x! (P~x! P 0~x))

By propositional logic,

|=FOS5V (E~x ^ ' ^ P~x)! ( ! P 0~x)

By hypothesis, FOS5V has the Interpolation property; so there is a ✓ such

that ✓ 2 Fml(L \ L0), |=FOS5V (E~x ^ ' ^ P~x)! ✓ and |=FOS5V ✓ ! ( ! P 0~x).

1A proof of the Compactness Theorem for first-order modal logic can be found in [8].

26

By propositional logic,

|=FOS5V E~x! ('! (P~x! ✓))

|=FOS5V ! (✓ ! P 0~x)

Let ⇤ 2 Fml(L) be the sentence obtained from by replacing every oc-

currence of P 0 by P . It can be easily seen that |=FOS5V ⇤ ! (✓ ! P~x).

So, by Proposition 8 and by the fact that both ' and ⇤ are sentences, we

have

|=FOS5V '! 8~x(P~x! ✓)

|=FOS5V ⇤ ! 8~x(✓ ! P~x)

Now, from the choice of �0, both ' and ⇤ are conjunctions of sentences of

�, so we have � |=FOS5V ' and � |=FOS5V ⇤. Hence,

� |=FOS5V 8~x(P~x! ✓)

� |=FOS5V 8~x(✓ ! P~x)

And so,

� |=FOS5V 8~x(P~x ! ✓)

Directly from the construction of L0 it follows that ✓ 2 Fml(L\{P}). There-fore, � explicitly defines P in FOS5V.

We are going to focus our attention on some aspects regarding propositional

letters, because in the next section both counterexamples to the Definability prop-

erty for FOS5V and FOS5 use propositional letters. So it is useful to point out some

details.

First, if P is a propositional letter p, we have � |=FOS5V 8~x(p ! ✓).

And this implies � |=FOS5V p ! 8~x✓. So, when working with propositional

letters, we say that � explicitly defines p in L if � |=L p ! ✓ for some sentence

✓ 2 Fml(L\{p}).Second, let M = hW ,D, D, Ii and w 2W . Clearly, M, w |= p i↵ I(p, w) =

Iw(p) = Iw(p) = {hi} and M, w 6|= p i↵ I(p, w) = Iw(p) = Iw(p) = ;.

27

Definition 19. Let L be a language such that p 2 L. We say that � preserves

p in L i↵ for all L-models for � M, w and N , w with the same set of worlds and

possible individuals and with respective interpretation functions I and J , if for

every j 2 (L\{p}) and every v 2W I(j, v) = J (j, v), then Iv(p) = Jv(p).

Proposition 10. Let L be a language such that p 2 L and � ✓ sen(L). � preserves

p in L i↵ � implicitly defines p in L.

Proof. ()) Let M = hW ,D, D, Ii be an L-model for L [ L0, w 2 W and M, w |=�[�0. Let M|L = hW ,D, D, I 0i and M|L0 = hW ,D, D, I 00i. Hence, by Proposition

1, M|L, w |= � and M|L0 , w |= �0. Let N = hW ,D, D, I⇤i be an L-model for Lsuch that for every j 2 (L\{p}) and every v 2W , I⇤(j, v) = I 0(j, v) and I⇤(p, v) =

I 00(p, v). It is evident that N , w |= �.

Now, suppose that M, w 6|= p ! p0. Then, either M, w |= p and M, w 6|=p0 or M, w 6|= p and M, w |= p0. In the first case, by Proposition 1, M|L, w |= p

and M|L0 , w 6|= p0. Then, by the definition of I⇤, I 0w(p) = {hi} and I⇤

w(p) = ;.Since both M|L, w and N , w are L-models for � and for every j 2 (L\{p}) and

every v 2 W , I⇤(j, v) = I 0(j, v); then, by hypothesis, I 0v(p) = I⇤

v (p), in particular,

I 0w(p) = I⇤

w(p); a contradiction. In the second case we can deduce a contradiction

in a similar way. Therefore, M, w |= p ! p0, and so � [ �0 |=L p ! p0.

(() Let M, w and N , w be L-models for � such that M = hW ,D, D, Ii,N = hW ,D, D,J i and for every j 2 (L\{p}) and every v 2 W , I(j, v) = J (j, v).

Let N 0 be an L-model for L0 such that N 0 = (W ,D, D,J 0), N 0|(L\{p}) = N|(L\{p})and for every v 2W , J 0(p0, v) = J (p, v). It is evident that N 0, w |= �0.

Let M0 be an L-model for L [ L0 such that M0|L = M and M0|L0 = N 0.

Hence, by Proposition 1, M0, w |= � and M0, w |= �0, thus M0, w |= � [ �0, By

hypothesis, M0, w |= p ! p0, i.e.

M0, w |= p i↵ M0, w |= p0

By Proposition 1,

M0|L, w |= p i↵ M0|L0 , w |= p0

By definition,

M, w |= p i↵ N 0, w |= p0

28

By the construction of N 0,

M, w |= p i↵ N , w |= p

Hence,

Iw(p) = Jw(p)

Therefore, � preserves p in L.

3.3 Failure of Interpolation and Beth’s Definabil-

ity Theorems in FOS5V

Proposition 11. Let L = {P, p} and � = {⇤8x⇤(Px ! p),⌃9x⇤(p ! Px)};then:

(a) � implicitly defines p in FOS5V.

(b) � does not explicitly define p in FOS5V.

Proof. (a) In view of Proposition 10, we have to show only that � preserves p in

FOS5V. Let M, w and N , w be FOS5V-models for � such that M = hW ,D, D, Ii,N = hW ,D, D,J i and for every w0 2W , I(P,w0) = J (P,w0).

Suppose that hi 2 Iw(p); then M, w |= p. Since M, w is a model for �,

M, w |= ⌃9x⇤(p ! Px), so there is a w0 2 W such that M, w0 |= 9x⇤(p ! Px).

Then, for some valuation h such that h(x) 2 Dw0 , we have that M, w0 |=h ⇤(p !Px). So, for every w00 2 W , M, w00 |=h p ! Px. In particular, M, w |=h p ! Px.

Since M, w |=h p, then M, w |=h Px, i.e. hh(x)i 2 I(P,w). So, by hypothesis,

hh(x)i 2 J (P,w).

Now, since N , w is a model for �, N , w |=h ⇤8x⇤(Px ! p). So, for every

w00 2 W , N , w00 |=h 8x⇤(Px ! p). In particular, N , w0 |=h 8x⇤(Px ! p). So for

every x-variant h0 of h, N , w0 |=h0 ⇤(Px! p). In particular, N , w0 |=h ⇤(Px! p).

Hence, we have N , w |=h Px ! p. Since hh(x)i 2 J (P,w); N , w |=h Px, and so

N , w |=h p, i.e. hi 2 Jw(p).

Therefore, Iw(p) ✓ Jw(p). We can show with a similar argument, that

Jw(p) ✓ Iw(p). Hence, Iw(p) = Jw(p).

(b) Let M = hW ,D, D, Ii be an FOS5V-model for {P} where:

29

• W = {w, v, u};

• D = {a, b};

• Dw = Dv = {a}, Du = {a, b};

• I(P,w) = {hbi}, I(p, w) = {hi} and

I(P, v) = I(P, u) = I(p, v) = I(p, u) = ;.

It can be easily seen that for a valuation h such that h(x) = b, we have:

M, w |=h p! Px

M, v |=h p! Px

M, u |=h p! Px.

So, M, u |= 9x⇤(p ! Px). Hence, M, w |= ⌃9x⇤(p ! Px) and M, v |=⌃9x⇤(p! Px).

In a similar way, we have for every valuation h:

M, w |=h Px! p

M, v |=h Px! p

M, u |=h Px! p.

Hence,

M, w |=h ⇤(Px! p)

M, v |=h ⇤(Px! p)

M, u |=h ⇤(Px! p).

Then,

M, w |= 8x⇤(Px! p)

M, v |= 8x⇤(Px! p)

M, u |= 8x⇤(Px! p).

So, M, w |= ⇤8x⇤(Px! p) and M, v |= ⇤8x⇤(Px! p). Therefore, both

M, w and M, v are FOS5V-models for �.

Now, let M0 = hW ,D, D, I 0i be a model for {P} such that M0 = M|{P};

then,

30

M0w = h{a}, I 0

wiM0

v = h{a}, I 0vi.

It is evident that I 0w(P ) = I 0

v(P ) = ;. Let ⇢ be the identity function on

{a} and � the identity function on {a, b}; then clearly ⇢ : M0w⇠= M0

v and for every

⇢0 ✓fin ⇢, � is a function such that ⇢0 ✓ � and � : M0 ⇠= M0. Since all the conditions

of Lemma 2 have been established, it follows that for every ' 2 sen({P})

M0, w |= ' i↵ M0, v |= '.

So, by this fact and by Proposition 1, we have

(+) M, w |= ' i↵ M, v |= ', for every ' 2 sen({P}).

Now, suppose that � explicitly defines p in FOS5V. So there is a ✓ 2sen({P}) such that � |=FOS5V p ! ✓. Since both M, w and M, v are FOS5V-

models for �, M, w |= p ! ✓ and M, v |= p ! ✓. By the definition of M,

M, w |= p, so M, w |= ✓. By (+), M, v |= ✓, hence M, v |= p; a contradiction.

Therefore, � does not explicitly define p in FOS5V.

Theorem 1. Beth’s Definability Theorem and the Interpolation Theorem fail for

FOS5V.

Proof. By Proposition 11, FOS5V does not have the Definability property, hence,

by Proposition 9, FOS5V does not have the Interpolation property.

3.4 Failure of Interpolation and Beth’s Definabil-

ity Theorems in FOS5

Before continuing, we are going to state some basic facts about permutations

without proof.

Definition 20. Let ⌧ be a permutation on A. We say that ⌧ is an essentially finite

permutation on A i↵ D⌧ = {a 2 A | ⌧(a) 6= a} is a finite set.

Proposition 12. If ⌧ and � are essentially finite permutations on A, then � � ⌧ is

an essentially finite permutation on A.

31

Proposition 13. If � is an essentially finite permutation on A, then ��1 is an

essentially finite permutation on A.

Proposition 14. Let ⌧ be a permutation on A. If ⌧ 0 ✓fin ⌧ , then there is a � such

that ⌧ 0 ✓ � and � is an essentially finite permutation on A.

Proposition 15. Let L = {P, p} and � = {p ! ⌃8x(Px ! ⇤(p ! ¬Px)),¬p !⇤9x(Px ^⇤(¬p! Px))}; then:

(a) � implicitly defines p in FOS5.

(b) � does not explicitly define p in FOS5.

Proof. (a) We proceed exactly like in Proposition 11. Let M, w and N , w be FOS5-

models for � such that M = hW ,D, Ii, N = hW ,D,J i and for every v 2 W ,

I(P, v) = J (P, v).

Suppose that Iw(p) 6= Jw(p); then either Iw(p) = {hi} and Jw(p) = ;or Iw(p) = ; and Jw(p) = {hi}. In the first case, since M, w is a model for �,

M, w |= ⌃8x(Px ! ⇤(p ! ¬Px)). Then, there is a w0 2 W such that M, w0 |=8x(Px! ⇤(p! ¬Px)). And since N , w is a model for �, then N , w |= ⇤9x(Px^⇤(¬p ! Px)). In particular, N , w0 |= 9x(Px ^ ⇤(¬p ! Px)). So, there is a

valuation h such that N , w0 |=h Px ^ ⇤(¬p ! Px). So hh(x)i 2 J (P,w0) and

N , w0 |=h ⇤(¬p! Px). Thus, N , w |=h ¬p! Px. And since Jw(p) = ;, N , w |=h

Px, i.e. hh(x)i 2 J (P,w).

Since M, w0 |= 8x(Px ! ⇤(p ! ¬Px)), we have that M, w0 |=h Px !⇤(p! ¬Px). By hypothesis, hh(x)i 2 I(P,w0) and hh(x)i 2 I(P,w). So M, w0 |=h

⇤(p ! ¬Px). In particular, M, w |=h p ! ¬Px. Hence, M, w |=h ¬Px, i.e.

hh(x)i /2 I(P,w); a contradiction. In the second case, we can deduce a contradiction

in a similar manner. Therefore, Iw(p) = Jw(p).

(b) Let M = hW ,D, Ii be an FOS5-model for {P} where:

• W = {hk, ⌧i | k 2 {0, 1, 2} and ⌧ is an essentially finite permutation on Z};

• D = Z;

• Let N,O and E be the sets of the natural numbers, odd natural numbers and

even natural numbers, respectively. If a 2 Z, then:

32

hai 2 I(P, h0, ⌧i) i↵ a 2 ⌧ [N ]

hai 2 I(P, h1, ⌧i) i↵ a 2 ⌧ [0]hai 2 I(P, h2, ⌧i) i↵ a 2 ⌧ [E]

Let i be the identity function on Z; w0 = h0, ii, w1 = h1, ii and w2 = h2, ii.Let Mw0 = hZ,Z, Iw0i, Mw1 = hZ,Z, Iw1i and ⇢ be any permutation on Z such

that ⇢[N ] = O. Then, for every a 2 Z:

hai 2 Iw0(P ) i↵ hai 2 I(P,w0) i↵ a 2 i[N ] i↵ a 2 N i↵ ⇢(a) 2 O i↵ ⇢(a) 2 i[O] i↵

h⇢(a)i 2 I(P,w1) i↵ h⇢(a)i 2 Iw1(P ).

Thus, ⇢ : Mw0⇠= Mw1 . Now, consider the following:

(+) For every ⇢0 ✓fin ⇢, there is a � such that ⇢0 ✓ � and � : M ⇠= M.

(Proof of (+)) If ⇢0 ✓fin ⇢, then, by Proposition 14, there is a � such that

⇢0 ✓ � and � is an essentially finite permutation on Z. Let w = hk, ⌧i be a member

of W ; by Proposition 12, hk, � � ⌧i is a member of W . Let M 2 {N,O,E}; then:On the one hand, if hai 2 Ihk,⌧i(P ), then a 2 ⌧ [M ], so there is a b 2 M

such that a = ⌧(b). Thus, �(a) = �(⌧(b)) = � � ⌧(b), then �(a) 2 � � ⌧ [M ], and so

h�(a)i 2 Ihk,��⌧i(P ).

On the other hand, if h�(a)i 2 Ihk,��⌧i(P ), then �(a) 2 � � ⌧ [M ], so there is

a b 2M such that �(a) = ��⌧(b), i.e. �(a) = �(⌧(b)). Since � is injective, a = ⌧(b),

thus a 2 ⌧ [M ], and so hai 2 Ihk,⌧i(P ).

Hence � : Mhk,⌧i ⇠= Mhk,��⌧), i.e. the condition (i) of Definition 17 is

satisfied.

Let w = hk, ⌧i be a member of W ; by Propositions 12 and 13, hk, ��1 � ⌧iis a member of W . Let M 2 {N,O,E}, then:

On the one hand, if hai 2 Ihk,��1�⌧i(P ), then a 2 ��1 � ⌧ [M ], so there is a

b 2 M such that ��1 � ⌧(b) = a, i.e. ��1(⌧(b)) = a . Thus, �(��1(⌧(b))) = �(a),

and so ⌧(b) = �(a), then �(a) 2 ⌧ [M ] and so h�(a)i 2 Ihk,⌧i(P ).

On the other hand, if h�(a)i 2 Ihk,⌧i(P ), then �(a) 2 ⌧ [M ], so there is a

b 2 M such that ⌧(b) = �(a). Thus, ��1(⌧(b)) = ��1(�(a)), i.e. ��1 � ⌧(b) = a,

then a 2 ��1 � ⌧ [M ], and so hai 2 Ihk,��1�⌧i(P ).

Hence � : Mhk,��1�⌧i ⇠= Mhk,⌧i, i.e. the condition (ii) of Definition 17 is

satisfied. Therefore, � : M ⇠= M. ⇤

33

Now, since ⇢ : Mw0⇠= Mw1 , it is evident that ⇢ : Mw0

⇠= Mw1 . By this fact

and by (+), all the conditions of Lemma 2 have been established, it follows that:

(++) M, w0 |= ✓ i↵ M, w1 |= ✓, for every ✓ 2 sen({P}).

Let M0 = hW ,D, I 0i be the expansion for M to L where hi 2 I 0(p, w) i↵

w 6= w0. And let M00 = hW ,D, I 00) be the expansion for M to L where hi 2 I 00(p, w)

i↵ w = w1.

(+++) M0, w0 and M00, w1 are FOS5-models for �.

(Proof of (+++)) First, it is clear that M0, w0 |= p ! ⌃8x(Px ! ⇤(p !¬Px)). Now, let w = hk, ⌧i be a member of W , and M 2 {N,O,E}. Clearly, M is

an infinite set andM ✓ N . Suppose that ⌧ [M ] ✓ Z\N , thenM ✓ D⌧ ; contradicting

the assumption that ⌧ is an essentially finite permutation on Z. Therefore, there is

an a 2 Z such that a 2 ⌧ [M ] and a 2 N . Let h be valuation such that h(x) = a.

Since for every w0 2W\{w0}, M0, w0 |=h p and M0, w0 |=h Px, then:

M0, w |=h ⇤(¬p! Px).

Since h(x) 2 ⌧ [M ],

M0, w |=h Px ^⇤(¬p! Px)

and so

M0, w |= 9x(Px ^⇤(¬p! Px)).

Since w was arbitrarily chosen, M0, w0 |= ⇤9x(Px^⇤(¬p! Px)), and so,

M0, w0 |= ¬p! ⇤9x(Px ^⇤(¬p! Px)). Therefore, M0, w0 |= �.

Second, it is clear that M00, w1 |= ¬p ! ⇤9x(Px ^ ⇤(¬p ! Px)). Now,

let h be a valuation. If h(x) 2 E, then for every w 2 W\{w1}, M00, w 6|=h p and

M00, w1 |=h ¬Px, then:

M00, w2 |=h ⇤(p! ¬Px).

Hence,

M00, w2 |= 8x(Px! ⇤(p! ¬Px)).

34

Thus M00, w1 |= ⌃8x(Px ! ⇤(p ! ¬Px)), and so M00, w1 |= p !⌃8x(Px! ⇤(p! ¬Px)). Therefore, M00, w1 |= �. ⇤

Now, suppose that � explicitly defines p in FOS5. So there is a ✓ 2 sen({P})such that � |=FOS5 p ! ✓. So, by (+++),M0, w0 |= p ! ✓ andM00, w1 |= p !✓. Since M00, w1 |= p, then M00, w1 |= ✓. Hence, by Proposition 1, M, w1 |= ✓. By

(++), M, w0 |= ✓. So, again by Proposition 1, M0, w0 |= ✓. Thus M0, w0 |= p, a

contradiction. Therefore, � does not explicitly define p in FOS5.

Theorem 2. Beth’s Definability Theorem and the Interpolation Theorem fail for

FOS5.

Proof. Direct from Propositions 9 and 15.

3.5 Inner and outer quantifiers

Fix a language L. We are going to add the new logical symbols ⌃ and ⇧.

Together with these symbols, we will define new kinds of modal formulas. For the

sake of brevity, from now on we are only going to indicate the structure of the

formulas, we are going to skip the full recursive definition.

Definition 21. (Extended formulas)

' ::= x = y | Px1 . . . xn | ¬ | _ ✓ | 9x | ⇤ | ⌃x | ⇧x

9 and 8 are called inner quantifiers ; ⌃ and ⇧ are called outer quantifiers.

We write Fml(L)+ to denote the set of all extended formulas.

Definition 22. Let M = hW ,D, D, Ii be an FOS5V-model for L, ' 2 Fml(L)+,h a valuation in M and w 2 W . The notion M, w |=h ' is defined as before; the

new clauses are:

M, w |=h ⌃x i↵ there is an x-variant h0 of h such that M, w |=h0 .

M, w |=h ⇧x i↵ for every x-variant h0 of h M, w |=h0 .

It can be easily seem that for every '(x) 2 Fml(L)+:

|=FOS5V ⌃x'(x) ! ¬⇧x¬'(x)|=FOS5V ⇧x'(x) ! ¬⌃x¬'(x)

35

Definition 23. We say that the outer quantifiers ⌃ and ⇧ are definable in FOS5V

i↵ for every '(x) 2 Fml(L)+ there are sentences , ✓ 2 Fml(L) such that and ✓

have exactly the same non-logical symbols occurring in '(x) and

|=FOS5V ⌃x'(x) !

|=FOS5V ⇧x'(x) ! ✓

Proposition 16. The outer quantifiers ⌃ and ⇧ are not definable in FOS5V.

Proof. By the equivalences stated above, is enough to show that ⌃ is not definable

in FOS5V.

Let � = {⇤8x⇤(Px! p),⌃9x⇤(p! Px)}. First, we shall show that

� |=FOS5V p ! ⌃xPx

Let M, w be a FOS5V-model for �. First, suppose M, w |= p. Since

M, w |= ⌃9x⇤(p ! Px), then for some w0 2 W , M, w0 |= 9x⇤(p ! Px). So for

some valuation h such that h(x) 2 Dw0 , M, w0 |=h ⇤(p ! Px). Hence, M, w |=h

p! Px, so M, w |=h Px. Thus, M, w |= ⌃xPx.

Second, suppose M, w |= ⌃xPx. Then there is a valuation h and a w0 2Wsuch that M, w |=h Px and h(x) 2 Dw0 . Since M, w |= ⇤8x⇤(Px ! p), then

M, w0 |= 8x⇤(Px ! p). So, M, w0 |=h ⇤(Px ! p). Hence, M, w |=h Px ! p.

Thus, M, w |=h p, i.e., M, w |= p.

Now, suppose that ⌃ is definable in FOS5V. Then there is a 2 sen({P})such that

|=FOS5V ⌃xPx !

Hence, � |=FOS5V p ! . Thus, � explicitly defines p in FOS5V. But this

contradicts Proposition 11.

36

Chapter 4

Justification Logic: a very short

introduction

4.1 History and motivation

Justification logic is one of those few subjects in which a historical introduction

is more fruitful than a plain exposition of the syntax and the semantics of the logic.

In the debate around foundations of mathematics one of the philosophical

positions that arose was Brouwer’s intuitionism. Briefly, intuitionism says that

the truth of a mathematical statement should be identified with the proof of that

statement. Summarizing the core idea of this position in a slogan: truth means

provability. Starting from this core idea an informal semantics was created. Now,

this semantics is known as Brouwer–Heyting–Kolmogorov (BHK) semantics. It gives

an informal meaning to the logical connectives ?,^,_,!,¬ in the following way:

• ? is a proposition which has no proof (an absurdity, e.g. 0 = 1).

• A proof of ' ^ consist of a proof of ' and a proof of .

• A proof of ' _ is given by exhibiting either a proof of ' or a proof of .

• A proof of '! is a construction which, given a proof of ', returns a proof

of .

37

• A proof of ¬' is a construction which transforms any proof of ' into a proof

of a contradiction. 1

Using this semantics we can give an informal argument to show that some

formulas are intuitionistic validities (formulas like ' ! ', ' ! ( ! ') and

? ! ') and show that some formulas that are classical validities are not validities

by this interpretation (formulas like ' _ ¬' and ¬¬' ! '). More important than

to decide whether some formula is a validity or not, this semantics gives us a way

to grasp the intended reasoning that intuitionistic logic (Int) wants to capture.

The first step toward a formalization of this semantics was given by Godel

in 1933 [14]. He added a new unary operator B to classical logic; B' should be

read as ‘' is provable’ (B stand for ‘beweisbar’, the German word for ‘provable’).

This new operator was added in order to express the notion of provability in classi-

cal mathematics. To describe the behavior of this operator Godel constructed the

following calculus:

All tautologies

B'! '

B('! )! (B'! B )

B'! BB'

(Modus Ponens) ` ', ` '! ) `

(Internalization) ` ' ) ` B'

Since this axiom system is equivalent to Lewis’s S4 when we translate B'

by ⇤', we will refer to this calculus of provability in classical mathematics simply

as S4.

Based on the intuitionistic notion of truth as provability, it is possible to

define the following translation from formulas of intuitionistic logic to formulas of

S4:1By this definition, we can clearly treat ¬' as an abbreviation of '! ?.

38

• pB = Bp;

• ?B = ?;

• (' ^ )B = ('B ^ B);

• (' _ )B = ('B _ B);

• ('! )B = B('B ! B).

It was shown by Godel, McKinsey and Tarski (for all the references see [4])

that this translation ‘makes sense’, i.e., that the following theorem holds:

For every formula ', Int ` ' i↵ S4 ` 'B.

The next step is to give a formal interpretation of the B operator. One

natural interpretation is the following: fix a first-order version of Peano Arithmetic

(PA); B should be interpreted as the predicate 9yProof(y, x) which asserts that

there exits a proof (in PA) with Godel number y for a formula with Godel number

x. This predicate has the following property:

For every sentence ' in the language of PA, PA ` ' i↵ Proof(n, p'q) holds forsome n.

For simplicity, we will use Prov(x) as an abbreviation of 9yProof(y, x). Let

⇤ be a bijection between the sentences of PA and the propositional variables. We

can extend the mapping ⇤ to give an arithmetical interpretation of all S4 formulas

as follows:

• ?⇤ = ?;

• (' ^ )⇤ = ('⇤ ^ ⇤);

• (' _ )⇤ = ('⇤ _ ⇤);

• ('! )⇤ = ('⇤ ! ⇤);

• (B')⇤ = Prov(p'⇤q).

39

On the one hand, it was straightforward how to interpret the modal formulas

in the language of PA; on the other hand it was not clear how to give a formal

interpretation of this provability calculus (S4) in PA. In [14] Godel pointed out that

S4 does not correspond to the calculus of the predicate Prov(x) in PA. Simply

because S4 proves the formula B(B(?) ! ?). Using the above translation this

formula correspond to Prov(pProv(p?q)! ?q). And since the following sentences

are equivalent in PA:

Prov(p?q)! ?¬Prov(p?q)Consist(PA),

Prov(pProv(p?q) ! ?q) means that the consistency of PA is internally provable

in PA, which contradicts Godel’s Second Incompleteness Theorem.

In a lecture in 1938 [15] Godel suggested a way to remedy this problem.

Instead of using the implicit representation of proofs by the existential quantifier

in the formula 9yProof(y, x) one can use explicit variables for proofs (like t) in

the formula Proof(t, x). In these lines, Godel proposed expanding the language

of classical propositional logic with variables for proofs and adding the following

ternary operator tB(', ) which should be read as ‘t is a derivation of from '’.

Using tB(') as an abbreviation of tB(>,'), Godel formulated the following axiom

system:

All tautologies

tB(')! '

tB(', )! (sB( , ✓)! f(t, s)B(', ✓))2

tB(')! t0B(tB('))

(Modus Ponens) ` ', ` '! ) `

(Internalization) ` ' ) ` tB(') (where t is an derivation of ').

2To understand the motivation behind this function f consider the following. Suppose t is aderivation of from ' and s is a derivation of ✓ from . Then it can be easily seen that theconcatenation of t and s, tas, is a derivation of ✓ from '. So, if t is a derivation of from ' ands is a derivation of ✓ from , then f(t, s) = t

as.

40

Godel just formulated this system but he did not give a proof of how this

system could be used to be a bridge between Int and PA. Independently of Godel’s

system presented in [15] (the lecture was published only in 1998), Sergei Artemov

(in [3]) proposed the use of explicit variables and constants for proofs and some

basic operations between proofs (Application ‘·’, Sum ‘+’ and Verifier ‘!’). Instead

of having B' (or the more modern notation of provability logic ⇤'), the non-

classical formulas are of the form t:' (which should be read as ‘t is a proof of '’);

where t is a simple or complex term composed of proof variables or constants. With

this new language Artemov stipulated the following axiom system to capture the

behavior of this explicit provability:

All tautologies

t:'! '

t:('! )! (s:'! [t · s]: )

t:'! !t:t:'

t:'! [t+ s]:'

s:'! [t+ s]:'

(Modus Ponens) ` ', ` '! ) `

(axiom necessitation) ` c:', where ' is an axiom and c is a justification

constant.

This logic was called Logic of Proofs (LP) and it was the first example of

justification logic.

If ' is a S4 formula, there is a mapping r (called a realization) from the

occurrences of B’s (or boxes) into terms. The result of this mapping on ' is denoted

'r. The following theorem express the connection between S4 and LP:

(Realization Theorem between S4 and LP) For every ' in the language of

S4, there is a realization r such that

S4 ` ' i↵ LP ` 'r

41

There is a way to define an interpretation ⇤ of the LP formulas into the

sentences of PA (for details see [3]). And with all this machinery Artemov was able

to prove the following result:

(Provability Completeness of Intuitionistic Logic) For every ', for every

interpretation ⇤, there is a realization r such that

Int ` ' i↵ S4 ` 'B i↵ LP ` ('B)r i↵ PA ` (('B)r)⇤

This result shows that instead of the philosophical attitude of understanding

intuitionistic logic as a reasoning di↵erent from the reasoning that classical logic

wants to capture, we can interpret intuitionistic logic as provability in classical

mathematics. Thus, the primitive notions that appear in the BHK semantics (‘proof’

and ‘construction’) can have a formal meaning in a classical setting.

Going beyond the specific problem of the formalization of BHK semantics,

justification logic can be seen as a new tool to introduce the notion of justifications

in the well-established discussion of epistemic logic (for a more detailed discussion

see [2]). Instead of using modal formulas like ⇤' to express:

For a given agent, ' is known,

we use justification formulas like t:' to express:

For a given agent, ' is known for the reason t.

Informally, we can see the terms t, s, . . . as justifications and the operators

+, ·, ! can be seen as means of epistemic action. In fact, this point of view enables us

to see justification logic as something bigger than the logic of the explicit provability;

justification logic can be seen as a logic of explicit knowledge.

Our main interest in justification logic lies in this connection with epistemic

logic. We are not going to focus on the arithmetical interpretation of this logic,

instead we are going to work only with the Kripke-style semantics introduced by

Melvin Fitting for this logic. But it is important to have the provability interpre-

tation in mind because some of the choices made to formulate specific aspects of

justification logic are directly motivated by the relationship with provability logic

and the arithmetical interpretation.

42

4.2 The propositional case: language and axiom

system

Definition 24. (Basic vocabulary)

• p, q, p0, q0, . . . (propositional variables);

• !,? (boolean connectives);

• x, y, z, . . . . . . (justification variables);

• a, b, . . . , with indices, 1, 2, . . . (justification constants);

• +, · (justification operators);

• ), ( (parentheses).

Definition 25. (Justification terms)

t ::= x | c | (t1 · t2) | (t1 + t2)

Definition 26. (Justification formulas)

' ::= p | ? | ( ! ✓) | t:

We define ¬,^, ! and _ as usual. Sometimes, to help readability, we use

the brackets ‘], [’ together with ‘), (’.

The minimal justification logic J0 is axiomatized by the following axiom

schemes and inference rules:

All tautologies

(Application Axiom) t:('! )! (s:'! [t · s]: )

(Sum Axioms) t:'! [t+ s]:', s:'! [t+ s]:'

(Modus Ponens) ` ', ` '! ) `

43

(axiom necessitation) ` c:', where ' is an axiom and c is a justification constant.

The notion of derivation in this system, J0 ` ', is defined as usual.

Definition 27. Let C be a non-empty set of formulas. We say that C is a constant

specification, if for every ' 2 C, ' = c: where c is a justification constant and

is an axiom. A proof meets constant specification C provided that whenever the

inference rule ‘axiom necessitation’ is used to introduce c: , then c: 2 C.We say that a constant specification C is axiomatically appropriate if i) for

every axiom ' there is a justification constant c1 such that c1:' 2 C; and ii) if

cn, cn�1, . . . , c1:' 2 C, then cn+1, cn, . . . , c1:' 2 C, for each n � 1.

For a constant specification C, by JC we mean J0 plus formulas from C as

additional axioms.

Theorem 3. (Internalization) Suppose C is an axiomatically appropriate constant

specification. In these conditions, JC satisfies internalization. That is, if JC ` ' then

JC ` t:', for some justification term t.

There are some well-know examples of justification logic other than J0; in

this thesis we are going to mention only two of them. The first one is the already

mentioned Logic of Proofs (LP): it extends the language of J0 with the unary justi-

fication operator ! and has the following additional axiom schemes:

(Factivity Axiom) t:'! '

(Positive Introspection Axiom) t:'! !t:t:'

The second one is called JT45, it extends the language of LP with the unary

justification operator ? and has the following additional axiom scheme:

(Negative Introspection Axiom) ¬t:'! ?t:¬t:'

We have stated the Internalization Theorem above for J0, but this theorem

also holds for LP and JT45. Because of the Positive Introspection Axiom we can

prove this result for LP and JT45 with a weaker notion of axiomatically appropriate

44

constant specification C. In both of these logics we just say that a constant speci-

fication C is axiomatically appropriate if for every axiom ' there is a justification

constant c such that c:' 2 C. It should be noted that the Internalization Theorem

is just an explicit form of the necessitation rule.

Informally speaking, the forgetful projection of a justification formula ',

denoted '�, is the result of replacing every subformula t: with ⇤ . We also com-

mented on the notion of realization. With these two notions we can state more

clearly the relationship between modal logic and justification logic.

Definition 28. Suppose KL is a normal modal logic and let JL be a justification

logic mentioned above. We say that JL is a counterpart of KL if the following holds:

• If JL ` ', then KL ` '�.

• If KL ` ', then there is a realization r such that JL ` 'r.

It can be proved that for an axiomatically appropriate constant specification

C:

JC is a counterpart of K

LP is a counterpart of S4

JT45 is a counterpart of S5

4.3 From propositional logic to first-order

Before we start presenting the first-order version of JT45 we need to remember

some properties of derivations in classical first-order logic. It is useful to remember

these details, because first-order justification logic tries to mirror some aspects of

the individual variables in classical first-order derivations.

Let '(x) be any tautology, and let t be the following derivation:

1. '(x)

2. 8x'(x) (generalization)

3. 8x'(x)! (Px! 8x'(x)) (tautology)

45

4. Px! 8x'(x) (Modus Ponens)

Although x is free in the formula Px! 8x'(x), if c is an individual term we

cannot substitute c for x in t in order to obtain a derivation t(c/x) of Pc! 8x'(x)(if we do that we ruin the derivation at 2).

Now, let s be the following derivation:

1. '(x)

2. 8x'(x) (generalization)

3. 8x'(x)! (Py ! 8x'(x)) (tautology)

4. Py ! 8x'(x) (Modus Ponens)

y is free in the formula Py ! 8x'(x) and moreover for every individual

term c the result of substituting c for y in s, s(c/y), is a derivation of Pc! 8x'(x).These examples show us that there are two di↵erent roles of variables in a

derivation: a variable can be a formal symbol that can be subjected to generalization

or a place-holder that can be substituted for. In t, x is both a formal symbol and a

place-holder. And in s, x is a formal symbol and y is a place-holder.

This consideration motivates the following definition:

x is free in the derivation t of the formula ' i↵ for every individual term c, t(c/x)

is a derivation of '(c/x).

In propositional justification logic we write t:' to express that t is a deriva-

tion of '. In order to represent the distinct roles of variables in first-order justifica-

tion logic, we are going to write formulas of the form:

t:Px! 8x'(x)s:{y}Py ! 8x'(x)

The role of {y} in s:{y}Py ! 8x'(x) is to point out that y is free in the

derivation s of Py ! 8x'(x).

46

Chapter 5

First-order JT45

This chapter is based on three di↵erent texts. We have used [5] and [13] to lay

down the basic syntax and semantics of first-order JT45. To prove completeness we

have used an unpublished paper by Melvin Fitting. The first time Sergei Artemov

constructed the quantified version of LP, it could support a constant domain seman-

tics. In the unpublished paper Fitting proved completeness for that early version of

first-order LP. Since Artemov changed the construction of the quantified version of

LP, Fitting left that paper unpublished. The Completeness Theorem presented in

this chapter is just an adaptation of the proof strategy presented in that paper (the

use of templates) for first-order JT45.

5.1 Language and axiom system

For this whole chapter we set L = {P,Q, P 0, Q0, . . . } to be a countable relationallanguage with no propositional letters.

Definition 29. (Basic vocabulary)

• x0, x1, x2, . . . (individual variables);

• !,? (boolean connectives);

• 8 (universal quantifier);

• p0, p1, p2, . . . (justification variables);

47

• c0, c1, c2A, . . . (justification constants);

• +, ·, !, ?, genx (justification operators – for every individual variable x, there

is an operator genx)1;

• (·) :X (·),(for every finite set of individual variables X);

• ), ( (parentheses).

Definition 30. (First-order justification terms)

t ::= pi | c | (t1 · t2) | (t1 + t2) | !s | ?s | genx(s)

Definition 31. (First-order justification formulas)

' ::= Px1 . . . xn | ? | ( ! ✓) | 8x | t:X

The set of all formulas is denoted by FmlJ . We are assuming that the

set of individual variables, justification variables and justification constants are all

countable sets. Thus, it is easy to check that FmlJ itself is a countable set.

Definition 32. We define the notion of free variables of ', fv('), recursively as

follows:

• If ' is atomic, then fv(') is the set of all variables occurring in '.

• If ' is ( ! ✓), then fv(') is fv( ) [ fv(✓).

• If ' is 8x , then fv(') is fv( )\{x}.

• If ' is t:X , then fv(') is X.

Similarly as in the classical case, we must define the notion of an individual

variable y being free for x in the formula '. The definition is the same as in

the classical case, we only add the following clause: y is free for x in t:X' if two

conditions are met, i) y is free for x in ' (in the classical sense), ii) if y 2 fv('),

then y 2 X.

1To be precise, there is a operator geni for each i 2 !. We identify each operator geni with theindividual variables xi. There is no occurrence of a variable in a justification operator, it is just alabel.

48

We write Xy instead of X [{y}; in this case it is assumed that y /2 X. And

we use t:' as an abbreviation for t:;'

The first-order JT45, FOJT45, is axiomatized by the following axiom schemes

and inference rules:

A1 classical axioms of first-order logic

A2 t:Xy'! t:X', provided y does not occur free in '

A3 t:X'! t:Xy'

B1 t:X'! '

B2 t:X('! )! (s:X'! [t · s]:X )

B3 t:X'! [t+ s]:X', s:X'! [t+ s]:X'

B4 t:X'! !t:Xt:X'

B5 ¬t:X'! ?t:X¬t:X'

B6 t:X'! genx(t):X8x', provided x /2 X

R1 (Modus Ponens) ` ', ` '! ) `

R2 (generalization) ` ' ) ` 8x'

R3 (axiom necessitation) ` c:', where ' is an axiom and c is a justification

constant.

We use �,�,⇥, . . . as variables for sets of formulas. The notion of � ` ' is

defined as usual. The only thing that should be noted is that, if � deduces ' using

the generalization rule, then this rule was not applied to a variable which occurs

free in the formulas of �.

49

Since derivations depend on the constant specification being considered,

we sometimes write `C ' to point out that the proof of ' meets the constant

specification C.

Lemma 3. (Deduction) �,' ` i↵ � ` '! .

Proof. A similar proof as the one from the classical case.

Theorem 4. (Internalization) Let C be an axiomatically appropriate constant spec-

ification; p0, . . . , pk be justification variables; X0, . . . , Xk be finite sets of individual

variables, and X = X0 [ · · ·[Xk. In these conditions, if p0:X0'0, . . . , pk:Xk'k `C ,

then there is a justification term t(p0, . . . , pk) such that

p0:X0'0, . . . , pk:Xk'k `C t:X .

Proof. The same proof as presented in [5, p. 7].

Proposition 17. (Explicit counterpart of the Barcan Formula and its converse) Let

y be an individual variable. For every finite set of individual variables X such that

y /2 X, for every formula '(y) and every justification term t, there are justification

terms CB(t) and B(t) such that:

` t:X8y'(y)! 8yCB(t):Xy'(y)

` 8yt:Xy'(y)! B(t):X8y'(y)

Proof. In Appendix.

Proposition 18. Let y be an individual variable. For every finite set of individual

variables X such that y /2 X, for every formula '(y) and every justification term t,

there is a justification term s(t) such that:

` 9yt:Xy'(y)! s(t):X9y'(y)

Proof. In Appendix.

50

5.2 Semantics: basic definitions

In Chapters 2 and 3 we have used valuation functions to define the relation

|=. In the present case it is more convenient to define the semantic notions adding

constants to the basic language. That is the path that we take here. So, for any

non-empty set D we are going to use the elements of D as constants. And we are

going to use ~a,~b, . . . to denote sequences of constants.

Definition 33. Let D be a non-empty set. The set of all D-formulas, D-FmlJ , is

defined as follows:

D-FmlJ = {'(~a) | '(~x) 2 FmlJ and ~a 2 D}.

As usual, for a D-formula ', we say that ' is closed if ' has no free variables.

Definition 34. A Fitting model is a structureM = hW ,R,D, I, Ei where hW ,R,Diis a skeleton, R is an equivalence relation2, I is an interpretation function and:

• E is an evidence function, i.e., for any justification term t and D-formula ',

E(t,') ✓W .

Definition 35. Evidence Function Conditions. Let M = hW ,R,D, I, Ei be a

Fitting model. We require the evidence function to meet the following conditions:

· Condition E(t,'! ) \ E(s,') ✓ E([t · s], ).

+ Condition E(s,') [ E(t,') ✓ E([s+ t],').

! Condition E(t,') ✓ E(!t, t:X '), where X is the set of constant occurring

in '.

? Condition W\E(t,') ✓ E(?t,¬t:X '), where X is the set of constants

occurring in '.

R Closure Condition If w 2 E(t,') and wRw0, then w0 2 E(t,').

Instantiation Condition If w 2 E(t,'(x)) and a 2 D, then w 2 E(t,'(a)).2Of course, we can define a Fitting model more generally for any kind of relation R, but for

our purposes we are going to use this restricted definition.

51

genx Condition E(t,') ✓ E(genx(t), 8x').

We say that a model M = hW ,R,D, I, Ei meets constant specification C i↵

whenever c:' 2 C, then E(c,') = W .

Definition 36. Let M = hW ,R,D, I, Ei be a Fitting model, ' a closed D-formula

and w 2W . The notion that ' is true at world w of M, in symbols M, w |= ', is

defined recursively as follows:

• M, w |= P (~a) i↵ h~ai 2 I(P,w).

• M, w 6|= ?.

• M, w |= ! ✓ i↵ M, w 6|= or M, w |= ✓.

• M, w |= 8x (x) i↵ for every a 2 D, M, w |= (a).

• Assume t:X (~x) is closed and ~x are all the free variables of . Then, M, w |=t:X (~x) i↵

(a) w 2 E(t, (~x)) and

(b) for every w0 2W such that wRw0, M, w0 |= (~a) for every ~a 2 D.

Definition 37. Let ' 2 FmlJ be a closed formula. We say that ' is valid in the

Fitting model M = hW ,R,D, I, Ei provided for every w 2 W , M, w |= '. A

formula with free individual variables is valid if its universal closure is valid.

Definition 38. A Fitting model for FOJT45 is a Fitting modelM = hW ,R,D, I, Eiwhere E is a strong evidence function, i.e., for every term t and D-formula ',

E(t,') ✓ {w 2 W | M, w |= t:X'} where X is the set of constant occurring in

'.

For a formula ' and constant specification C, we write |=C ' if for every

Fitting model for FOJT45 M meeting C, ' is valid in M.

5.3 Semantics: non-validity

Before we deal with soundness and completeness, it is useful to know some

examples of non-validity in order to see that the provisions of some axioms make

52

sense. There is only a minor problem, we require that Fitting models for FOJT45

have a strong evidence function, and it is not so easy to construct models with that

property. The following proposition helps us to circumnavigate this issue.

Proposition 19. If M = hW ,R,D, I, Ei is a Fitting model such that for every

justification term t and D-formula ', E(t,') = W , then there is a Fitting model for

FOJT45 M⇤ = hW ,R,D, I, E⇤i such that for every w 2 W and every formula ',

M, w |= ' i↵ M⇤, w |= '.

Proof. Let M⇤ = hW ,R,D, I, E⇤i where for every justification term and D-

formula ',

E⇤(t,') = {w 2W | M, w |= t:X'}

where X is the set of constants occurring in '.

It is straightforward to check that M⇤ is indeed a Fitting model. Now con-

sider the following:

(+) For every w 2W and every closed D-formula ', M, w |= ' i↵ M⇤, w |= '.

(Proof of (+)) Induction on '. Crucial case, ' is t:X . For simplicity, let us

assume that ' is t:{a} (a, y).

()) If M, w |= t:{a} (a, y), then by definition w 2 E⇤(t, (a, y)) and for

every w0 2 W , if wRw0, then M, w0 |= (a, b) for every b 2 D. By the induction

hypothesis, for every w0 2 W , if wRw0, then M⇤, w0 |= (a, b) for every b 2 D.

Thus, M⇤, w |= t:{a} (a, y).

(() If M⇤, w |= t:{a} (a, y), then w 2 E⇤(t, (a, y)). By definition, M, w |=t:{a} (a, y). ⇤

By (+) we have that,

E⇤(t,') = {w 2W | M, w |= t:X'} = {w 2W | M⇤, w |= t:X'}

Hence, E⇤ is a strong evidence function and M and M⇤ agree on all D-

formulas. Therefore, M⇤ is a Fitting model for FOJT45 and M and M⇤ agree on

all formulas.

53

With this proposition we can construct non-validity examples similar to

those presented in [13].

Example 1: the restriction on axiom A2 is needed. Take, for example, the

formula t:{x,y}Qxy ! t:{x}Qxy; let M = hW ,R,D, I, Ei be a Fitting model where:

• W = {w0, w1};

• R = W ⇥W ;

• D = {a, b};

• I(w0, Q) = I(w1, Q) = {ha, bi};

• E(t,') = W , for every term t and formula '.

Clearly, M, w0 |= t:{a,b}Qab and M, w0 6|= t:{a}Qay. Hence, M, w0 6|=t:{x,y}Qxy ! t:{x}Qxy. By Proposition 19, t:{x,y}Qxy ! t:{x}Qxy is not valid in

every Fitting model for FOJT45.

Example 2: The proviso of axiom B6 is necessary. Take, for example, the

formula t:{x}Qx ! genx(t):{x}8xQx; let M = hW ,R,D, I, Ei be a Fitting model

where:

• W = {w0};

• R = W ⇥W ;

• D = {a, b};

• I(w0, Q) = {a};

• E(t,') = W , for every term t and formula W .

Clearly, M, w0 |= t:{a}Qa and since M, w0 6|= Qb, then M, w0 6|= 8xQx, and

so M, w0 6|= genx(t):{a}8xQx. Hence, M, w 6|= t:{x}Qx ! genx(t):{x}8xQx. Again

by Proposition 19, t:{x}Qx ! genx(t):{x}8xQx is not valid in every Fitting model

for FOJT45.

54

5.4 Soundness and Completeness

5.4.1 Soundness

Theorem 5. (Soundness) Let C be a constant specification. For every formula

' 2 FmlJ , if `C ', then |=C '.

Proof. The proof is by induction on the theorems of the axiom system using the

constant specification C. The argument is exactly the same as presented in [13, pp.

9-10]. We are going to show validity for the specific axiom of FOJT45.

Suppose ' is an instance of B5, i.e., ' is ¬t:X ! ?t:X¬t:X . For sim-

plicity, assume X = {x} and = (x, y). So, we have that `C ¬t:{x} (x, y) !?t:{x}¬t:{x} (x, y).

Let M = hW ,R,D, I, Ei be a Fitting model for FOJT45 meeting C, w 2Wand a 2 D. Suppose M, w |= ¬t:{a} (a, y). Then, M, w 6|= t:{a} (a, y). By

the definition of the strong evidence function, w /2 E(t, (a, y)). By the ? con-

dition, w 2 E(?t,¬t{a}: (a, y)). Again, by the strong evidence function M, w |=?t:{a}¬t:{a} (a, y).

5.4.2 An obstacle in the proof of the Completeness Theorem

There are two ways that we can prove the Completeness Theorem, one simple

and the other more complex. Here we shall present the complex version. Although

we are going to have much more work (if compared to the simple version) it is

worthwhile because, we believe that the methods that we are going to use in the next

subsections can be used to prove the semantical version of the Realization Theorems

for FOJT45 (in Chapter 6 we give a more detailed exposition of that theorem).

The general strategy is the same as presented in [16, pp. 256-265]. Let us

just briefly comment on what is the obstacle that we find when trying to adapt the

proof from the modal case to the justification case. In one step of the proof [16, pp.

259-260] we need to establish the following:

(+) There is an individual variable y⇤ such that �# [ {�n ^ (�(y⇤/x)! 8x�)} is

consistent,

55

where �# = {' | ⇤' 2 �} and � is a maximal consistent set. We begin proving (+)

with the following argument. Suppose (+) is false.

(1) Then for every individual variable y, �# [ {�n ^ (�(y/x) ! 8x�)} is

inconsistent. Hence, for some �1, . . . , �k 2 �# we have that

` (�1 ^ . . . ^ �k)! (�n ! ¬(�(y/x)! 8x�));

by the usual reasoning in modal logic,

` (⇤�1 ^ . . . ^⇤�k)! ⇤(�n ! ¬(�(y/x)! 8x�))

Since ⇤�1, . . . ,⇤�k 2 �, then ⇤(�n ! ¬(�(y/x)! 8x�)) 2 �.

(2) It is assumed that � has the ‘8-property’, i.e., for every formula '(x)

there is an individual variable y⇤ such that '(y⇤/x)! 8x' 2 �.

Now, using these two facts we can conclude the following: let z be a variable

that does not occur in �n and �. By (2), there is a variable y⇤ such that

⇤(�n ! ¬(�(y⇤/x)! 8x�))! 8z⇤(�n ! ¬(�(z/x)! 8x�)) 2 �

And by (1) for the particular case when y = y⇤,

⇤(�n ! ¬(�(y⇤/x)! 8x�)) 2 �.

So, by the maximal consistency of � we can conclude that 8z⇤(�n !¬(�(z/x) ! 8x�)) 2 �. The rest of the proof of (+) is not important for our

point here.

The adaptation of this step for the first-order justification logic is problem-

atic because justification terms internalize Hilbert-style derivations.

It should be noted that for two di↵erent individual variables y and y0 if

�# [ {�n ^ (�(y/x)! 8x�)} and �# [ {�n ^ (�(y0/x)! 8x�)} are inconsistent sets,

then there are two finite subsets of �#, {�1, . . . , �k} and {�01, . . . , �

0k0} such that

` (�1 ^ . . . ^ �k)! (�n ! ¬(�(y/x)! 8x�))` (�0

1 ^ . . . ^ �0k0)! (�n ! ¬(�(y0/x)! 8x�))

and we cannot assume that {�1, . . . , �k} = {�01, . . . , �

0k0}. So, for each variable y we

may have a di↵erent derivation.

If we adopt the argument (1) for first-order justification logic we would have

that for each individual variable y

56

ty:X(�n ! ¬(�(y/x)! 8x�)) 2 �,

where ty is a term constructed by the Internalization Theorem, the axiom B2 and

the fact that �# [ {�n ^ (�(y/x)! 8x�)} is inconsistent. Hence, ty depends on the

individual variable y.

Now, let us try to continue the argument. Let z be a variable that does not

occur in �n and �. If we adapt (2) for justification logic, we would have that for

every individual variable y there is an individual variable y⇤ such that

ty:X(�n ! ¬(�(y⇤/x)! 8x�))! 8zty:X(�n ! ¬(�(z/x)! 8x�)) 2 �

But from this adapted version of (2) we cannot conclude that there is a

variable y⇤ such that

ty⇤:X(�n ! ¬(�(y⇤/x)! 8x�))! 8zty⇤ :X(�n ! ¬(�(z/x)! 8x�)) 2 �

So we cannot use (1) to conclude that 8zty⇤ :X(�n ! ¬(�(z/x)! 8x�)) 2 �.

A way to remedy this problem is to make the ‘8-property’ stronger. If

'(y⇤/x) ! 8x' 2 � we say that y⇤ instantiates the formula 8x'. We want that

the same individual variable is used to simultaneously instantiate an infinite list of

formulas of the same form. In order to guarantee this feature we are going to use the

notion of templates. But in doing so we need to stablish some facts about templates.

That makes the proof bigger than it should be, and that is why we divided the proof

of the Completeness Theorem into di↵erent subsections.

5.4.3 Language extension

The basic idea is to extend the language in order to prove a Henkin-style Com-

pleteness Theorem. Instead of using constants to construct our canonical model we

shall add a new kind of variable called ‘witness variable’. We do that because when

working with maximal consistent sets we need to be able to do formal derivations

and so bind some witness variables.

Definition 39. Two formulas are variable variants provided each can be turned

into the other by a uniform renaming of free individual variables, bound individual

variables and labels of justification terms. We are always assuming that the renam-

ing is safe, i.e., the new variables that are being introduced do not occur in the

original formula.

57

Definition 40. A constant specification C is variant closed i↵ whenever ' and

are variable variants, then c:' 2 C i↵ c: 2 C.

Definition 41. Fix a countable set V = {a0, a1, a2, . . . } of additional individual

variables that are not in the original language. We define a new set of formulas

FmlJ(V) in the same fashion as FmlJ . It should be noted that variables of V can

be bound. We add every finite subset of V [ {x0, x1, . . . } to the language; and for

every a 2 V we add the justification operator gena.3 It can be easily checked that

FmlJ(V) is a countable set.

Until the end of this chapter we write ‘individual variables’ to denote the

members of V[{x0, x1, . . . }, ‘basic variables’ to denote the members of {x0, x1, . . . }and ‘witness variables’ to denote the members of V.

We are interested in using V as the domain D of the canonical model, so

from now on we shall call a D-formula a formula of FmlJ(V) where the members

of V occur only free (not bound, nor as labels of justification terms). And we say

that a D-formula is closed if no basic variable occurrences are free.

Together with this new language we construct a new axiomatic system for

FOJT45 based on the formulas from FmlJ(V).

Definition 42. Let C be a variant closed constant specification for the basic system.

C(V) is the smallest set satisfying the following:

If ' 2 C, 2 FmlJ(V) and ' and are variable variants, then 2 C(V).

From this definition we can make some observations:

• C ✓ C(V).

• C(V) is variant closed.

• If C is axiomatically appropriate, then C(V) is axiomatically appropriate.

• We can prove the Deduction Lemma, the Internalization Theorem, Proposi-

tions 17 and 18 for the new axiom system.

3To be precise, we add gen!+i for each i 2 !. And we identify each operator gen!+i with ai.

58

Proposition 20. Let C be a variant closed constant specification for the basic

system and C(V) its extension for FmlJ(V). In these conditions, for every ' 2FmlJ , if `C(V) ', then `C '.

Proof. Let 1, 2, . . . , n = ' be a FOJT45 proof in the language of FmlJ(V) using

C(V). Let a1, . . . , ak be all the witness variables that occur free, bound or as a label

in the proof. Let y1, . . . , yk be basic variables that do not appear free, bound or

as a label in the proof. And let ( i)� be the result of replacing each aj with yj

throughout.

We shall show that ( 1)�, ( 2)�, . . . , ( n)� is a FOJT45 proof in the lan-

guage of FmlJ using C. And so `C ( n)�, i.e., `C '.If i is an axiom, since we are using axiom schemes and the introduced

variables are new (to prevent that any proviso be violated), then ( i)� is also an

axiom.

If i is a member of C(V), then there is a � 2 C such that i and � are

variable variants. Now, ( i)� and � may not be variable variants, because they may

have some basic variable in common. But we can construct a formula ✓ 2 FmlJ

such that ✓ has no variable in common with ( i)� and �, ✓ and ( i)� are variable

variants, and ✓ and � are variable variants. Since � 2 C and C is variant close,

( i)� 2 C.If i is deduced from i1 and i2 = i1 ! i by modus ponens, then ( i2)

�

is ( i1)� ! ( i)�. So ( i)� also follows from ( i2)

� and ( i1)� by modus ponens.

If i is deduced from l by generalization, then i is 8x l. If x is a basic

variable, then 8x( l)� is deduced from ( l)� by generalization. If x = aj, then

8yj( l)� is deduced from ( l)� by generalization.

Proposition 21. (Controlled Internalization) Let C be a constant specification

variant closed and axiomatically appropriate, C(V) its expansion to FmlJ(V) and

' 2 FmlJ(V). If ' is a D-formula and `C(V) ', then there is a justification term t

of FmlJ such that

`C(V) t:'

Proof. Let a1, . . . , an be the witness variables occurring free in '. So we can write '

as '(a1, . . . , an). Let x1, . . . , xn be basic variables that do not occur in the proof of

59

'(a1, . . . , an). By an argument similar to the one presented in the proof of Propo-

sition 20, we have that

`C '(x1, . . . , xn)

Since C is axiomatically appropriated, by the Internalization Theorem there

is a justification term s of FmlJ such that

`C s:'(x1, . . . , xn)

Let ‘gen~x(s)’ be the abreviation of ‘genx1(genx2 . . . (genxn(s)))’. By repeated

use of the axiom B6,

`C gen~x(s):8x1 . . . 8xn'(x1, . . . , xn)

Now, since the axiom system in the language of FmlJ(V) using C(V) is an

extension of the basic axiom system using C, we have that

`C(V) gen~x(s):8x1 . . . 8xn'(x1, . . . , xn)

By the fact that C(V) is axiomatically appropriate, we have that the fol-

lowing formulas are elements of C(V):

c1:[8x18x2 . . . 8xn'(x1, x2, . . . , xn)! 8x2 . . . 8xn'(a1, x2, . . . , xn)]

c2:[8x28x3 . . . 8xn'(a1, x2, x3, . . . , xn)! 8x3 . . . 8xn'(a1, a2, x3, . . . , xn)]...

cn:[8xn'(a1, . . . , an�1, xn)! '(a1, . . . , an�1, an)].

Hence, by repeated use of axiom B2 and modus ponens,

`C(V) [cn· . . . ·[c1 · gen~x(s)]]:'(a1, . . . , an)

Take t as [cn· . . . ·[c1 · gen~x(s)]].

It should be noted that in the proofs of Proposition 17 and 18 we can use

Proposition 21 in the place of the Internalization Theorem. So if '(y) is a D-formula

and t is a term of FmlJ , then the terms constructed by Propositions 17 and 18 –

CB(t), B(t) and s(t) – are also justification terms of FmlJ .

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Definition 43. Let C be a variant closed constant specification for the basic lan-

guage and � ✓ FmlJ . We say that � is C-inconsistent i↵ � `C ?. By the Deduction

Lemma, � is C-inconsistent i↵ there is a finite subset { 1, . . . , n} of � such that

`C ( 1 ^ . . . ^ n) ! ?. A set � is C-consistent if it is not C-inconsistent. And we

say that � is C-maximal consistent whenever � is C-consistent and � has no proper

extension that is C-consistent. We have similar notions for C(V).

It follows from Proposition 20 that for every set of basic formulas �, if � is

C-consistent, then � is C(V)-consistent.

Proposition 22. (Lindenbaum) Let C be a constant specification variant closed and

C(V) its extension. If � ✓ FmlJ(V) is C(V)-consistent then there is a �0 ✓ FmlJ(V)

such that � ✓ �0 and �0 is a C(V)-maximal consistent set.

Proof. A similar proof as the one from the classical case.

5.4.4 Templates

Definition 44. (Template vocabulary)

• p0, p1, p2, . . . (propositional variables);

• ¬,_,^ (boolean connectives);

• ⇤ (necessity);

• ), ( (parentheses).

We are going to use p, q and r as meta-variables for propositional variables.

Similarly, we write ~p to denote a sequence of propositional variables.

Definition 45. We define the notions of template F and the occurrence set of F ,

occ(F ), recursively as follows:

a) • p is a template.

• occ(p) = {p}.

b) • If F is a template, then ¬F is a template.

• occ(F ) = occ(¬F ).

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c) • If F and G are templates and if occ(F ) \ occ(G) = ;, then F _ G is a

template.

• occ(F _G) = occ(F ) [ occ(G).

d) • If F and G are templates and if occ(F ) \ occ(G) = ;, then F ^ G is a

template.

• occ(F ^G) = occ(F ) [ occ(G).

e) • If F is a template, then ⇤F is a template.

• occ(F ) = occ(⇤F ).

Similarly as in the case when we work with formulas, we can define the

notion of complexity of a template (the number of occurrences of boolean and modal

connectives). So we shall define some notions recursively based on the complexity

of templates and prove some facts by induction on the complexity of templates.

Definition 46. Let ~p be an n-ary sequence of propositional variables, ~' be an n-

ary sequence of D-formulas and F (~p) a template. We define the instantiation set

kF (~')k recursively as follows:

a) If F (~p) is pi, then kF (~')k = {'i}.

b) If F (~p) is ¬G(~p), then kF (~')k = {¬ | 2 kG(~')k}.

c) If F (~p) is G(~p) _H(~p), then kF (~')k = { _ ✓ | 2 kG(~')k and ✓ 2 kH(~')k}.

d) If F (~p) is G(~p) ^H(~p), then kF (~')k = { ^ ✓ | 2 kG(~')k and ✓ 2 kH(~')k}.

e) If F (~p) is ⇤G(~p), then kF (~')k = {t:X | 2 kG(~')k}; where t is a justification

term of FmlJ and X is the set of all witness variables occurring in .

Clearly, for every template F (~p) and every sequence ~' ofD-formulas, kF (~')kis a set of D-formulas.

Definition 47. We say that the template F is positive if all the boolean connectives

that occur in F are ^ and _. Similarly, we say that F is disjunctive if all the boolean

connectives that occur in F are _.

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From now to the end of this subsection we shall prove some facts about

templates. We are always assuming that there is a fixed constant specification

variant closed and axiomatically appropriate C for the basic language, and that

C(V) is its extension. To make things simple, we will not refer to this assumption in

every proposition and, in this subsection only, we shall write ‘`’ to denote ‘`C(V)’,

‘consistent’ to denote ‘C(V)-consistent’, ‘inconsistent’ to denote ‘C(V)-inconsistent’

and ‘maximal-consistent’ to denote ‘C(V)-maximal consistent’.

Proposition 23. (Semi-Replacement) Let F (~p,q) be a positive template, ' and

D-formulas, and ~' a sequence of D-formulas. In these conditions, if ` '! , then

for every � 2 kF (~',')k there is a ✓ 2 kF (~', )k such that

` �! ✓

Proof. (Induction on the complexity of F (~p,q)).

(F (~p,q) is atomic)

i) F (~p,q) = pi. Then for any � 2 kF (~',')k = {'i}, � = 'i. Since

'i 2 kF (~', )k = {'i}, take ✓ as 'i.

ii) F (~p,q) = q. Then for any � 2 kF (~',')k = {'}, � = '. Since

2 kF (~', )k = { }, take ✓ as .

(F (~p,q) is G(~p,q) _H(~p,q))

Let � 2 kF (~',')k. So � is �0_�00 where �0 2 kG(~',')k and �00 2 kH(~',')k.By the induction hypothesis, there are ✓0 2 kG(~', )k and ✓00 2 kH(~', )k such that

` �0 ! ✓0 and ` �00 ! ✓00

hence,

` �0 _ �00 ! ✓0 _ ✓00.

Since ✓0 _ ✓00 2 kF (~', )k, take ✓ as ✓0 _ ✓00.

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If F (~p,q) is G(~p,q)^H(~p,q), then the argument is similar as the previous

one.

(F (~p,q) is ⇤G(~p,q))

Let � 2 kF (~',')k. So � is t:X�0 where �0 2 kG(~',')k. By the induction

hypothesis, there is a ✓0 2 kG(~', )k such that ` �0 ! ✓0. By Proposition 21, there

is a justification term s of FmlJ such that

` s:(�0 ! ✓0)

by repeated use of axiom A3 and classical reasoning

` s:X(�0 ! ✓0)

by axiom B2 and modus ponens

` t:X�0 ! [s · t]:X✓0.

Let Y be the set of all witness variables that occur in ✓0. By repeated use

of axioms A2 and A3, we have that

` [s · t]:X✓0 ! [s · t]:Y ✓0

hence,

` t:X�0 ! [s · t]:Y ✓0.

Since [s · t]:Y ✓0 2 kF (~', )k, take ✓ as [s · t]:Y ✓0.

Corollary 1. (Variable Change) Let � ✓ FmlJ(V), F (~p,q) a positive template,

~' a sequence of D-formulas, 8x'(x) a D-formula, and y a basic variable that does

not occur free in 8x'(x). In these conditions, if � [ k¬F (~', 8x'(x))k is consistent,then � [ k¬F (~', 8y'(y))k is consistent.

Proof. Suppose that �[ k¬F (~', 8x'(x))k is consistent and �[ k¬F (~', 8y'(y))k isinconsistent. Then, there are 1, . . . , n 2 kF (~', 8y'(y))k such that

� ` 1 _ . . . _ n

64

by classical logic,

` 8y'(y)! 8x'(x).

Hence by Proposition 23, for each i there is a ✓i 2 kF (~', 8x'(x))k such

that

` i ! ✓i

thus,

� ` ✓1 _ . . . _ ✓n.

And since each ¬✓i 2 k¬F (~', 8x'(x))k, �[k¬F (~', 8x'(x))k is inconsistent;a contradiction.

Proposition 24. (Vacuous Quantification) Let F (~p) be a disjunctive template,

and ~' a sequence of D-formulas none of which contain free occurrences of the basic

variable y. In these conditions, for each 2 kF (~')k there is some ✓ 2 kF (~')k suchthat

` 9y ! ✓

Proof. (Induction on the complexity of F (~p))

(F (~p) is pi)

For each 2 kF (~')k = {'i}, = 'i. Since y does not occur free in 'i,

` 9y'i ! 'i. We can take ✓ as 'i.

(F (~p) is G(~p) _H(~p))

Let 2 kF (~')k. So is 0 _ 00 where 0 2 kG(~')k and 00 2 kH(~')k. Bythe induction hypothesis, there are ✓0 2 kG(~')k and ✓00 2 kH(~')k such that

` 9y 0 ! ✓0 and ` 9y 00 ! ✓00

by classical logic,

` 9y( 0 _ 00) ! (9y 0 _ 9y 00)

65

hence,

` 9y( 0 _ 00)! ✓0 _ ✓00.

Since ✓0 _ ✓00 2 kF (~')k, take ✓ as ✓0 _ ✓00.

(F (~p) is ⇤G(~p))

Let 2 kF (~')k. So is t:X� where � 2 kG(~')k. By the axiom A3,

` t:X�! t:Xy�

by classical logic,

` 9yt:X�! 9yt:Xy�.

By definition, X is a set of witness variables and since y is a basic variable

we have that y /2 X; so by Proposition 18,

` 9yt:Xy�! s(t):X9y�

By induction hypothesis, there is a ✓0 2 kG(~')k such that ` 9y� ! ✓0. By

Proposition 21 and by the axiom A3, there is a justification term s0 of FmlJ such

that

` s0:X(9y�! ✓0)

by axiom B2,

` s(t):X9y�! [s0 · s(t)]:X✓0.

Let Y be the set of all witness variables that occur in ✓0. By repeated use

of axioms A2 and A3, we have that

` [s0 · s(t)]:X✓0 ! [s0 · s(t)]:Y ✓0

hence,

` 9yt:X�! [s0 · s(t)]:Y ✓0.

66

Since [s0 · s(t)]:Y ✓0 2 kF (~')k, take ✓ as [s0 · s(t)]:Y ✓0.

Proposition 25. (Generalized Barcan) Let F (~p,q) be a disjunctive template, y a

basic variable, '(y) a D-formula, and ~' a sequence of D-formulas none of which

contain free occurrences of y. In these conditions, for each 2 kF (~','(y))k thereis some ✓ 2 kF (~', 8y'(y))k such that

` 8y ! ✓

Proof. (Induction on the complexity of F (~p,q))

If F (~p,q) is atomic, then the result is trivial.

(F (~p,q) is G(~p,q) _H(~p,q))

By the definition of template, the propositional variable q can occur at

most once in F (~p,q). So either it does not occur in G(~p,q) or it does not occur in

H(~p,q). Assume that it does not occur in H(~p,q) (the other case is symmetric);

then we can assume that H(~p,q) is H(~p).

Let 2 kF (~','(y))k. So is �0 _ �00 where �0 2 kG(~','(y))k and �00 2kH(~')k. By classical logic, we have that

` 8y(�0 _ �00)! (8y�0 _ 9y�00)

Since y does not occur free in any formula of ~', then by Proposition 24 there

is some ✓00 2 kH(~')k such that

` 9y�00 ! ✓00

By the induction hypothesis, there is ✓0 2 kG(~', 8y'(y))k such that

` 8y 0 ! ✓0

hence,

` 8y(�0 _ �00)! ✓0 _ ✓00.

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And so we can take ✓ as ✓0 _ ✓00.

(F (~p,q) is ⇤G(~p,q))

Let 2 kF (~','(y))k. So is t:X� where � 2 kG(~','(y))k. By definition,

X is a set of witness variables, then y /2 X. So, by Proposition 17

` 8yt:Xy�! B(t):X8y�

by axiom A3,

` t:X�! t:Xy�

by classical logic,

` 8yt:X�! 8yt:Xy�

so,

` 8yt:X�! B(t):X8y�.

By the induction hypothesis, there is a ✓0 2 kG(~', 8y'(y))k such that `8y� ! ✓0. By Proposition 21 and by the axiom A3, there is a justification term s

of FmlJ such that

` s:X(8y�! ✓0)

by axiom B2,

` B(t):X8y�! [s · B(t)]:X✓0.

Let Y be the set of all witness variables that occur in ✓0. By repeated use

of axioms A2 and A3, we have that

` [s · B(t)]:X✓0 ! [s · B(t)]:Y ✓0

hence,

` 8yt:X�! [s · B(t)]:Y ✓0.

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Take ✓ as [s · B(t)]:Y ✓0.

Proposition 26. (Formula Combining) Let F (~p) be a disjunctive template, and ~'

a sequence of D-formulas. In these conditions, for any 1, . . . , k 2 kF (~')k there issome formula ✓ 2 kF (~')k such that

` ( 1 _ . . . _ k)! ✓

Proof. (Induction on the complexity of F (~p).)

If F (~p) is atomic, then the result is trivial.

(F (~p) is G(~p) _H(~p))

Let 1, . . . , k 2 kF (~')k. So there are �01, . . . ,�

0k 2 kG(~')k and �00

1 , . . . ,�00k 2

kH(~')k, such that i = �0i_�

00i . By the induction hypothesis, there are ✓

0 2 kG(~')kand ✓

00 2 kH(~')k such that

` (�01 _ . . . _ �0

k)! ✓0

` (�001 _ . . . _ �00

k)! ✓00

hence,

` ((�01 _ . . . _ �0

k) _ (�001 _ . . . _ �00

k))! ✓0 _ ✓00

and so,

` ((�01 _ �

001) _ . . . _ (�

0k _ �

00k))! ✓0 _ ✓00

i.e.,

` ( 1 _ . . . _ k)! ✓0 _ ✓00.

Take ✓ as ✓0 _ ✓00.

69

(F (~p) is ⇤G(~p))

Let 1, . . . , k 2 kF (~')k. So there are justification terms t1, . . . , tk and

�1, . . . ,�k 2 kG(~')k such that i = ti:Xi�i. By the induction hypothesis, there is

✓0 2 kG(~')k such that

` (�1 _ . . . _ �k)! ✓0

hence, by classical reasoning, for each i,

` �i ! ✓0

So, by Proposition 21 and by the axiom A3 there are justification terms

s1, . . . , sk of FmlJ such that for each i,

` si:Xi(�i ! ✓0)

by axiom B2,

` ti:Xi�i ! [si · ti]:Xi✓0

by an appropriate use of axiom B3, we have that for each i,

` [si · ti]:Xi✓0 ! [[s1 · t1]+ . . . +[sk · tk]]:Xi✓

0.

Let Y be the set of all witness variables that occur in ✓0. By repeated use

of axioms A2 and A3, we have that

` [[s1 · t1]+ . . . +[sk · tk]]:Xi✓0 ! [[s1 · t1]+ . . . +[sk · tk]]:Y ✓0

hence, for each i,

` ti:Xi�i ! [[s1 · t1]+ . . . +[sk · tk]]:Y ✓0

so,

` (t1:X1�1 _ . . . _ tk:Xk�k)! [[s1 · t1]+ . . . +[sk · tk]]:Y ✓0.

Since [[s1 · t1]+ . . . +[sk · tk]]:Y ✓0 2 kF (~')k, we can take ✓ as [[s1 · t1]+ . . .

+[sk · tk]]:Y ✓0.

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Proposition 27. (Existential Instantiation) Let F (~p,q) be a disjunctive template,

� ✓ FmlJ , ~' a sequence ofD-formulas, 8x'(x) aD-formula, and a a witness variable

that does not occur free in 8x'(x) and in any member of ~'. In these conditions, if

� [ k¬F (~', 8x'(x))k is consistent, then � [ k¬F (~','(a))k is consistent.

Proof. Suppose that �[k¬F (~', 8x'(x))k is consistent and �[k¬F (~','(a))k is in-consistent. Then, there are 1, . . . , n 2 � and ¬�1(a), . . . ,¬�k(a) 2 k¬F (~','(a))ksuch that

` ( 1 ^ . . . ^ n) ^ (¬�1(a) ^ . . . ^ ¬�k(a))! ?

hence,

` ( 1 ^ . . . ^ n)! (�1(a) _ . . . _ �k(a)).

By Proposition 26 there is a (a) 2 kF (~','(a))k, such that

` (�1(a) _ . . . _ �k(a))! (a)

hence,

` ( 1 ^ . . . ^ n)! (a)

by generalization (remember, a is a variable in the new language),

` 8a[( 1 ^ . . . ^ n)! (a)].

Let y be a basic variable that does not occur in 1, . . . , n, 8x'(x), ~', '(a)and (a). By classical logic,

` 8a[( 1 ^ . . . ^ n)! (a)]! [( 1 ^ . . . ^ n)! (a)](y/a).

Since � is a set of basic formulas, a does not occur in any formula of �;

in particular, a does not occur in any i. Hence [( 1 ^ . . . ^ n) ! (a)](y/a) is

( 1 ^ . . . ^ n)! (y). So, by modus ponens and generalization,

` 8y[( 1 ^ . . . ^ n)! (y)].

Since y does not occur in any i, by classical reasoning,

` ( 1 ^ . . . ^ n)! 8y (y)

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Since a does not occur free in any formula of ~', it can be easily checked

that for every formula (a),

If (a) 2 kF (~','(a))k, then (y) 2 kF (~','(y))k.

By this fact, we have that (y) 2 kF (~','(y))k. Now since y does not occur

in ~', then by Proposition 25 there is a ✓ 2 kF (~', 8y'(y))k such that

` 8y ! ✓

thus,

` ( 1 ^ . . . ^ n)! ✓.

Since ¬✓ 2 k¬F (~', 8y'(y))k, it follows that � [ k¬F (~', 8y'(y))k is incon-sistent. By Corollary 1, � [ k¬F (~', 8x'(x))k is inconsistent, a contradiction.

Definition 48. If � ✓ FmlJ(V), then let �# be the set of all formulas 8~y' such

that t:X' 2 �, where t:X' is a closed D-formula with X being the set of witness

variables in ', and ~y are the free basic variables of '.

Proposition 28. (Up and Down Consistency) Let F (~p) = ⇤G(~p) be a template,

� ✓ FmlJ(V), and ~' a sequence of D-formulas.

1) Suppose � is maximal consistent. In these conditions, if �# [k¬G(~')k is consis-tent, then � [ k¬F (~')k is consistent.

2) Suppose G(~p) is a disjunctive template. In these conditions, if � [ k¬F (~')k isconsistent, then �# [ k¬G(~')k is consistent.

Proof. 1) Suppose �# [ k¬G(~')k is consistent and � [ k¬F (~')k is inconsistent.

Then, for some ¬t1:X1✓1, . . . ,¬tk:Xk✓k 2 k¬F (~')k (where ✓1, . . . , ✓k 2 kG(~')k)

� ` (¬t1:X1✓1 ^ . . . ^ ¬tk:Xk✓k)! ?

hence,

� ` t1:X1✓1 _ . . . _ tk:Xk✓k.

Now, since � is maximal consistent set, for some i, ti:Xi✓i 2 �. And since

ti:Xi✓i is a closed D-formula, 8~x✓i 2 �#. By classical logic,

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` ¬✓i ! ¬8~x✓i

Since ¬✓i 2 k¬G(~')k, we have that �# [k¬G(~')k is inconsistent, a contra-

diction.

2) Suppose � [ k¬F (~')k is consistent and �# [ k¬G(~')k is inconsistent.

Then, there are 8~x1 1, . . . , 8~xn n 2 �# (where t1:X1 1, . . . , tn:Xn n 2 �) and

¬✓1, . . . ,¬✓k 2 k¬G(~')k such that

` (8~x1 1 ^ . . . ^ 8~xn n) ^ (¬✓1 ^ . . . ^ ¬✓k)! ?

so,

` (8~x1 1 ^ . . . ^ 8~xn n)! (✓1 _ . . . _ ✓k).

Since ✓1, . . . , ✓k 2 kG(~')k and G(~p) is a disjunctive template, then by

Proposition 26 there is a ✓ 2 kG(~')k such that

` (✓1 _ . . . _ ✓k)! ✓

by classical logic,

` 8~x1 1 ! . . .! 8~xn n ! ✓.

Now, for each i any member of the sequence ~xi does not occur in the set Xi

(the set Xi is a set of witness variables). So, by repeated use of axiom B6 we have

that for each i

` ti:Xi i ! gen~xi(t):Xi8~x i

It should be noted that ‘gen~xi(t)’ is not a justification term, it is just an

abbreviation that we use to help readability. Let X = X1[ . . . [Xn, by axiom A3,

` ti:Xi i ! gen~xi(t):X8~x i

By Proposition 21 and axiom A3 there is a justification term s of FmlJ

such that

` s:X(8~x1 1 ! . . .! 8~xn n ! ✓)

and by repeated use of axiom B2

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` gen~x1(t):X8~x 1 ! . . .! gen~xn(t):X8~x n ! [s · gen~x1(t)· . . . ·gen~xn(t)]:X✓.

Let Y be the set of all witness variables of ✓; by axioms A2 and A3

` [s · gen~x1(t)· . . . ·gen~xn(t)]:X✓ ! [s · gen~x1(t)· . . . ·gen~xn(t)]:Y ✓

by classical reasoning,

` (t1:X1 1 ^ . . . ^ tn:Xn n)! [s · gen~x1(t)· . . . ·gen~xn(t)]:Y ✓.

Since each ti:Xi i 2 � and [s · gen~x1(t)· . . . ·gen~xn(t)]:Y ✓ 2 kF (~')k we have

that � [ k¬F (~')k is inconsistent; a contradiction.

Definition 49. A set of formulas � admits instantiation provided for each dis-

junctive template F (~p,q), for each sequence ~' of D-formulas, and each universally

quantified D-formula 8x'(x), if � [ k¬F (~', 8x'(x))k is consistent, then for some

witness variable a, � [ k¬F (~','(a))k is consistent.4

Proposition 29. Suppose � is maximal consistent and � admits instantiation. For

every D-formula 8x'(x), if ¬8x'(x) 2 �, then there is a witness variable a such

that ¬'(a) 2 �.

Proof. If ¬8x'(x) 2 �, then S [ {¬8x'(x)} is consistent. Let q be a propositional

letter; F (q) = q is a disjunctive template. Since k¬F (8x'(x))k = {¬8x'(x)}, then� [ k¬F (8x'(x))k is consistent. Since � admits instantiation, there is a witness

variable a such that � [ k¬F ('(a))k is consistent, i.e., � [ {¬'(a)} is consistent.

By the maximality of �, ¬'(a) 2 �.

Proposition 30. Let � ✓ FmlJ(V). If � is maximal consistent and admits instan-

tiation, then �# also admits instantiation.

Proof. Suppose � is maximal consistent, � admits instantiation, F (~p,q) is a dis-

junctive template, ~' is a sequence of D-formulas, 8x'(x) is a D-formula, and �# [k¬F (~', 8x'(x))k is consistent. By item 1) of Proposition 28, �[k¬⇤F (~', 8x'(x))kis consistent. ⇤F (~p,q) is also a disjunctive template. Then, since � admits instan-

tiation, for some witness variables a, � [ k¬⇤F (~','(a))k is consistent. By item 2)

of Proposition 28, �# [ k¬F (~','(a))k is consistent.4This is the stronger version of the ‘8-property’ that we mentioned in subsection 5.4.2.

74

5.4.5 Using templates for Henkin-like theorems

Since the set of all templates is a countable set, the set of all disjunctives

templates is also a countable set. By the same set-theoretical considerations, since

FmlJ(V) is countable, the set of all sequences ~' of D-formulas is also countable.

Hence, the set of all pairs hF (~p), ~'i is countable, where F is a disjunctive template, ~p

is n-ary sequence of propositional variables and ~' is a n-ary sequence of D-formulas.

For this whole subsection we shall assume that the members of the set of

pairs hF (~p), ~'i are arranged in a sequence

hF1(~p1), ~'1i, hF2(~p2), ~'2i, hF3(~p3), ~'3i, . . .

From now on we shall refer to this sequence as the ‘initial sequence’. This

sequence of pairs determines a corresponding sequence of instantiation sets:

kF1(~'1)k, kF2(~'2)k, kF3(~'3)k, . . .

It should be noted that for two di↵erent pairs hFi(~pi), ~'ii, hFj(~pj), ~'jithe corresponding instantiation sets may be the same. For example, the pairs

hp0, h8x'(x)ii, hp1, h8x'(x)ii determine the same set {8x'(x)}. Hence there are

some repetitions in the sequence of instantiation sets, but this will not cause any

trouble.

Proposition 31. (Basic expansion) Let C be a variant closed and axiomatically

appropriate constant specification for the basic language, C(V) its extension and let

� ✓ FmlJ be a C-consistent set. In these conditions, there is a �0 ✓ FmlJ(V) such

that � ✓ �0, �0 is C(V)-maximal consistent set and �0 admits instantiation.

Proof. We define a sequence of sets of FmlJ(V) formulas �1,�2,�3, . . . so that:

• �n is C(V)-consistent.

• �n is either � or � [ k¬Fi1(~'i1)k[ . . . [k¬Fik(~'ik)k.

First of all, �1 = �. By the remark at the end of subsection 5.4.3, �1 is

C(V)-consistent.

Now, suppose �n is constructed and it is of the form � [ k¬Fi1(~'i1)k[ . . .

[k¬Fik(~'ik)k (the other case has a similar proof). Let hFn(~pn), ~'ni be the nth pair

75

of the initial sequence. If the last term of the sequence ~'n is not a universal formula,

let �n+1 = �n. Otherwise, consider the following. ~'n is of the form ~ , 8x'(x). AndFn(~pn) is the disjunctive template G(~q, r) and so k¬Fn(~'n)k = k¬G(~ , 8x'(x))k.

If �n [ k¬G(~ , 8x'(x))k is not C(V)-consistent, then take �n+1 as �n.

If �n [ k¬G(~ , 8x'(x))k is C(V)-consistent, we shall show that for some

witness variable a, �n [ k¬G(~ ,'(a))k is C(V)-consistent.

First, we can assume that there is no overlap between the propositional

variables ~pi1 , . . . , ~pik,~q, r because from the point of view of the instantiation sets it

does not matter if there is an overlap or not, and we are going to work only with

the instantiation sets. Hence, by the definition of template

Fi1(~pi1) _ . . . _ Fik(~pik) _G(~q, r)

is a disjunctive template.

Second, from the definition of instantiation set and from classical reasoning,

it can be easily checked that the sets

� [ k¬Fi1(~'i1)k [ · · · [ k¬Fik(~'ik)k [ k¬G(~ , 8x'(x))k� [ k¬Fi1(~'i1) ^ . . . ^ ¬Fik(~'ik) ^ ¬G(~ , 8x'(x))k� [ k¬(Fi1(~'i1) _ . . . _ Fik(~'ik) _G(~ , 8x'(x)))k

have the same consequences. Thus, �[k¬(Fi1(~'i1)_ . . ._Fik(~'ik)_G(~ , 8x'(x)))kis C(V)-consistent.

Third, let a be the first witness variable that does not occur in �, ~'i1 , . . . , ~'ik ,~

and 8x'(x) (remember � is a set of formulas from the basic language). Then, by

Proposition 27, � [ k¬(Fi1(~'i1) _ . . . _ Fik(~'ik) _ G(~ ,'(a)))k is C(V)-consistent.

As before, it can be seen that

� [ k¬Fi1(~'i1)k [ · · · [ k¬Fik(~'ik)k [ k¬G(~ ,'(a))k

is C(V)-consistent. That is:

�n [ k¬G(~ ,'(a))k

is C(V)-consistent. So, take �n+1 as �n [ k¬G(~ ,'(a))k.It can be easily checked that

Sn2! �n is C(V)-consistent. So, by Proposition

22 there is a set �0 such thatS

n2! �n ✓ �0 and �0 is C(V)-maximal consistent.

Clearly, � ✓S

n2! �n ✓ �0. Now we show that �0 admits instantiation.

76

Let ~' be a sequence of D-formulas, 8x'(x) a D-formula and F (~p,q) a

disjunctive template. Suppose that �0 [ k¬F (~', 8x'(x))k is C(V)-consistent. So,

for some k 2 !, hF (~p,q), h~', 8x'(x)ii is the kth term of the initial sequence. Since

�k ✓S

n2! �n ✓ �0, �k [ k¬F (~', 8x'(x))k is C(V)-consistent. By construction,

for some witness variable a, �k+1 = �k [ k¬F (~','(a))k is C(V)-consistent. Thus

k¬F (~','(a))k ✓ �0. Hence, �0 [ k¬F (~','(a))k is C(V)-consistent.

Lemma 4. Suppose � is a set of formulas that admits instantiation, F (~p) is a

disjunctive template, and ~' is a sequence of D-formulas. Then, � [ k¬F (~')k alsoadmits instantiation.

Proof. Let ~ be a sequence of D-formulas, 8x'(x) a D-formula and G(~q, r) a dis-

junctive template. Suppose (� [ k¬F (~')k) [ k¬G(~ , 8x'(x))k is C(V)-consistent.

As before, we can assume that occ(F (~p)) \ occ(G(~q, r)) = ;. So F (~p) _G(~q, r) is a disjunctive template. And as before, the sets

(� [ k¬F (~')k) [ k¬G(~ , 8x'(x))k� [ k¬F (~') ^ ¬G(~ , 8x'(x))k� [ k¬(F (~') _G(~ , 8x'(x)))k

have the same consequences. Thus, �[k¬(F (~')_G(~ , 8x'(x)))k is C(V)-consistent.

Since � admits instantiation, there is a witness variable a such that � [ k¬(F (~') _G(~ ,'(a)))k is C(V)-consistent. Hence, (� [ k¬F (~')k) [ k¬G(~ ,'(a))k is C(V)-

consistent.

Proposition 32. (Secondary expansion) Let C be a variant closed and axiomatically

appropriate constant specification for the basic language, C(V) its extension and

� ✓ FmlJ(V) a C(V)-consistent set that admits instantiation. In these conditions,

there is a �0 ✓ FmlJ(V) such that � ✓ �0, �0 is C(V)-maximal consistent set and

�0 admits instantiation.

Proof. The proof is very similar to the proof of Proposition 31.

We define a sequence �1,�2, . . . of C(V)-consistent sets that admit instan-

tiation. First, �1 = �.

Now, suppose �n is already constructed. Let hFn(~pn), ~'ni be the nth pair of

the initial sequence. If the last term of the sequence ~'n is not a universal formula,

let �n+1 = �n. Otherwise, consider the following. ~'n is of the form ~ , 8x'(x). And

77

Fn(~pn) is the disjunctive template G(~q, r) and so k¬Fn(~'n)k = k¬G(~ , 8x'(x))k.If �n [ k¬G(~ , 8x'(x))k is not C(V)-consistent, then take �n+1 as �n.

If �n [ k¬G(~ , 8x'(x))k is C(V)-consistent, then, since �n admits instanti-

ation, there is a witness variable a such that �n[k¬G(~ ,'(a))k is C(V)-consistent.

By Lemma 4, �n [ k¬G(~ ,'(a))k admits instantiation. So, take �n+1 as �n [k¬G(~ ,'(a))k.

As before, it can be checked thatS

n2! �n is a C(V)-consistent set that

admits instantiation. By Proposition 22 there is a set �0 such thatS

n2! �n ✓ �0

and �0 is C(V)-maximal consistent. It is easy to see that �0 admits instantiation.

5.4.6 Completeness

Definition 50. A canonical model M = hW ,R,D, I, Ei, using constant specifica-

tion C, is specified as follows.

• W is the set of all C(V)-maximally consistent sets that admit instantiation.

• Let �,� 2W . �R� i↵ �# ✓ �.

• D = V.

• For an n-place relation symbol P and for � 2W , let I(P,�) be the set of all

~a where ~a 2 V and P (~a) 2 �.

• For � 2W , set � 2 E(t,') i↵ t:X' 2 �, where t:X' is a closed D-formula and

X is the set of witness variables in '.

First we need to check that M is indeed a Fitting model meeting C. Sincethe argument is similar to the one presented in [13, pp. 13-14] we are only going to

show that R is an equivalence relation and that the ? Condition holds.

R is reflexive. Let � 2W , and let t:X' = t:X'(~y) be a closed D-formula in

� such that ~y is an n-ary sequence of basic variables, say y1, . . . , yn and, of course,

~y /2 X. By repeated use of axiom B6 and classical reasoning:

`C(V) t:X'(~y)! geny1(geny2 . . . (genyn(t))):X 8~y'(~y)

by axiom B1,

78

`C(V) geny1(geny2 . . . (genyn(t))):X 8~y'(~y)! 8~y'(~y)

hence, by the maximal consistency of �, 8~y'(~y) 2 �. Thus �# ✓ �, i.e., �R�.

R is transitive. Let �,�,⇥ 2W such that �R� and �R⇥; and let ' 2 �#,

i.e., ' = 8~y (~a, ~y) (~a is a sequence of witness variables and ~y is a sequence of basic

variables) and t:{~a} (~a, ~y) 2 �.

By the axiom B4 and by the maximal consistency of �, !t:{~a}t:{~a} (~a, ~y) 2 �.

Since t:{~a} (~a, ~y) has no free basic variables and �R�, then t:{~a} (~a, ~y) 2 �. And

since �R⇥, then 8~y (~a, ~y) 2 ⇥, i.e., ' 2 ⇥. Thus, �# ✓ ⇥, i.e., �R⇥.

R is symmetric. Let �,� 2 W . Suppose that �R� and suppose it is not

the case that �R�. Then �# * �. So for some term t, some set of witness variables

X and some D-formula '(~y), t:X'(~y) 2 � and 8~y'(~y) /2 �. By the maximal consis-

tency of �, ¬8~y'(~y) 2 �. Now, assume that t:X'(~y) 2 �. Then by repeated use of

axiom B6, geny1(geny2 . . . (genyn(t))):X 8~y'(~y) 2 �. By axiom B1, 8~y'(~y) 2 �, a

contradiction. Hence, t:X'(~y) /2 �, by the maximal consistency of �, ¬t:X'(~y) 2 �.

By axiom B5, ?t:X¬t:X'(~y) 2 �. Since �# ✓ �, then ¬t:X'(~y) 2 �, a contradic-

tion. Therefore, if �R�, then �R�.

? Condition. Suppose � 2 W\E(t,'); and let X be the set of all witness

variables occurring in '. Thus, by the definition of E , t:X' /2 �. By the max-

imal consistency of �, ¬t:X' 2 �. By the axiom B5, ?t:X¬t:X' 2 �. Hence,

� 2 E(?t,¬t:X').

We have shown that the canonical model is a Fitting model meeting C.Now, to show that the canonical model is a Fitting model for FOJT45, we need to

show that E is a strong evidence function. This is going to be a consequence of the

following Lemma:

Lemma 5. (Truth Lemma). Let M = hW ,R,D, I, Ei be a canonical model. For

each � 2W and for each closed D-formula ',

M,� |= ' i↵ ' 2 �

79

Proof. Induction on the complexity of '. The crucial cases are when ' is t:X and

when ' is 8x (x).

(' is t:X )

()) Suppose t:X /2 �. Let X 0 ✓ X be a set where X 0 contain exactly the

witness variables that occur in . It is not the case that t:X0 2 �. Otherwise, by

axiom A3 and by the maximal consistency of �, t:X 2 �. So by the definition of

E , � /2 E(t, ), thus M,� 6|= t:X .

(() First, suppose t:X 2 �. Again, let X 0 ✓ X be as above. So, by the

axiom A2 and by the maximal consistency of �, t:X0 2 �. Hence, � 2 E(t, ).Second, let � 2W such that �R�. So 8~y 2 � where ~y are the free basic variables

of . Thus, by the classical axioms and by the maximal consistency of �, for every

~a 2 V, (~a) 2 �. By the induction hypothesis, for every ~a 2 V, M,� |= (~a).

Therefore, M,� |= t:X0 , and so M,� |= t:X .

(' is 8x (x))

()) Suppose 8x (x) /2 �. By the maximal consistency of �, ¬8x (x) 2 �.

Since � admits instantiation, then by Proposition 29 there is an a 2 V such that

¬ (a) 2 �. By the consistency of �, (a) /2 �. By the induction hypothesis,

M,� 6|= (a), thus M,� 6|= 8x (x).(() Suppose 8x (x) 2 �. By the classical axioms and by the maximal

consistency of �, for every a 2 V, (a) 2 �. By the induction hypothesis, M,� |= (a), for every a 2 V. Therefore, M,� |= 8x (x).

By the Truth Lemma, we have the following:

� 2 E(t,')) t:X' 2 �)M,� |= t:X') � 2 {w 2W | M, w |= t:X'}

Hence E is a strong evidence function, and so M is a Fitting model for FOJT45

meeting C.

Theorem 6. (Completeness) Let C be a constant specification. For every closed

formula ' 2 FmlJ , if |=C ', then `C '.

80

Proof. Suppose 6`C '. Then {¬'} is C-consistent. By Proposition 31, there is a

C(V)-maximal consistent � such that � admits instantiation and {¬'} ✓ �. By the

Truth Lemma, M,� |= ¬', so M,� 6|= '. Hence, 6|=C '.

Definition 51. A model M = hW ,R,D, I, Ei is fully explanatory if the following

condition is fulfilled. Let ' be a formula with no free individual variables, but with

constants from the domain of the model. Let w 2W . If for every v 2W such that

wRv, M, v |= ', then there is a justification term t such that M, w |= t:X', where

X is the set of domain constants appearing in '.

Theorem 7. The canonical model is fully explanatory.

Proof. Let M = hW ,R,D, I, Ei be a canonical model, � 2 W , ' a closed D-

formula and X the set of the witness variables occurring '. We shall show that if

M,� 6|= t:X' for every justification term t of FmlJ , then there is a � 2 W such

that �R� and M,� 6|= '.

If M,� 6|= t:X' for every justification term t of FmlJ , then by the Truth

Lemma, ¬t:X' 2 � for every justification term t of FmlJ . The template G(p) = p

is a disjunctive template. Let F (p) = ⇤G(p). Hence, k¬F (')k ✓ �. And so,

� [ k¬F (~')k is C(V)-consistent. By item 2) of Proposition 28, �# [ k¬G(')k is

C(V)-consistent, i.e., �# [ {¬'} is C(V)-consistent. By Proposition 30, �# admits

instantiation. By Lemma 4, �# [ {¬'} admits instantiation. By Proposition 32,

there is a C(V)-maximal consistent set � such that � admits instantiation and

�# [ {¬'} ✓ �. Since �# ✓ �, �R�. And since ¬' 2 �, by the Truth Lemma,

M,� 6|= '.

81

Chapter 6

Conclusion and future research

6.1 An axiomatic system for FOS5

To prove the results of Chapters 2 and 3 it was convenient to state things in

terms of ¬,_, 9,⌃ and =. But to stay connected with the formulations of the last

chapter consider now the version of first-order modal logic defined using ?, !, 8and ⇤ (without equality). Let L be the same language fixed in Chapter 5. To make

things simple, we write Fml instead of Fml(L).Since we have started working only with semantical notions, we have de-

fined the logic FOS5 as the set of all valid sentences relative to the class of all

FOS5-models. Alternatively we can study the logic FOS5 using a simple and ele-

gant axiomatic system composed of the following axiom schemes and inference rules:

A01 classical axioms of first-order logic

A02 ⇤'! '

A03 ⇤'! ⇤⇤'

A04 ¬⇤'! ⇤¬⇤'

A05 ⇤('! )! (⇤'! ⇤ )

82

R01 (Modus Ponens) ` ', ` '! ) `

R02 (generalization) ` ' ) ` 8x'

R03 (necessitation) ` ' ) ` ⇤'.

As in the case for FOJT45 we make use of the standard notion of � ` '.Here the restriction on the generalization rule is the same as stated for FOJT45, and

the necessitation rule is allowed only when � = ;. We write FOS5 ` ' to denote

that in this axiomatic system ; ` '.In the seminal paper by Kripke [17] the Completeness Theorem for this logic

was shown, and so the semantical and the syntactical characterization of FOS5 are

equivalent. To be more precise, for every sentence ' 2 Fml,

FOS5 ` ' i↵ |=FOS5 '.

6.2 Realization

Definition 52. Let ' be a formula of FOS5. We define the realization of ' in the

language of FOJT45, 'r, as follows:

• If ' is atomic, then 'r is '.

• If ' is ! ✓, then 'r is r ! ✓r

• If ' is 8x , then 'r is 8x r

• If ' is ⇤ and fv(') = {x1, . . . , xn}, then 'r is t:{x1,...,xn} r

A realization is normal if all negative occurrences of ⇤ are assigned justifi-

cation variables. It can easily be checked that for every ' 2 Fml, fv(') = fv('r).

Definition 53. Let ' be a formula of FOJT45. The forgetful projection of ', '�,

is defined as follows:

• If ' is atomic, then '� is '.

83

• If ' is ! ✓, then '� is � ! ✓�

• If ' is 8x , then '� is 8x �

• If ' is t:X , then '� is ⇤8~y �

where ~y 2 fv( )\X.

As before, it can easily be checked that for every ' 2 FmlJ , fv(') = fv('�).

Proposition 33. For every constant specification C and for every ' 2 FmlJ ,

If FOJT45 `C ', then FOS5 ` '�.

Proof. Induction on the theorems of FOJT45 with C. In this proof only we shall

use ` to denote FOS5 `. And for simplicity we are going to deal only with a rep-

resentative special case of each axiom. These special cases are simpler versions of

each axiom; the argument can be easily generalized.

(' is an instance of A2)

Suppose ' is

t:{x,y} (x, z)! t:{x} (x, z)

since y /2 fv( (x, z)),

{z} = fv( (x, z))\{x, y} = fv( (x, z))\{x}

thus, '� is

⇤8z �(x, z)! ⇤8z �(x, z).

Clearly, ` '�.

(' is an instance of A3)

Suppose ' is

t:{x} (x, y, z)! t:{x,y} (x, y, z)

84

then, '� is

⇤8y8z �(x, y, z)! ⇤8z �(x, y, z)

by classical axioms,

` 8y8z �(x, y, z)! 8z �(x, y, z)

by necessitation and the distributivity of ⇤ over !,

` ⇤8y8z �(x, y, z)! ⇤8z �(x, y, z).

(' is an instance of B1)

Suppose ' is

t:{x} (x, y)! (x, y)

then, '� is

⇤8y �(x, y)! �(x, y)

by A02,

` ⇤8y �(x, y)! 8y �(x, y)

and by classical axioms,

` 8y �(x, y)! �(x, y)

so,

` ⇤8y �(x, y)! �(x, y).

(' is an instance of B2)

Suppose ' is

t:{x,x0}( (x, y)! ✓(x0, z))! (s:{x,x0} (x, y)! [t · s]:{x,x0}✓(x0, z))

then, '� is

85

⇤8y8z( �(x, y)! ✓�(x0, z))! (⇤8y �(x, y)! ⇤8z✓�(x0, z))

by classical reasoning,

` 8y8z( �(x, y)! ✓�(x0, z))! (8y8z �(x, y)! 8y8z✓�(x0, z))

since z /2 fv( �(x, y)) and y /2 fv(✓�(x0, z)), we have that

` 8y8z �(x, y) ! 8y �(x, y)

` 8y8z✓�(x0, z) ! 8z✓�(x0, z)

hence,

` 8y8z( �(x, y)! ✓�(x0, z))! (8y �(x, y)! 8z✓�(x0, z))

by necessitation and the distributivity of ⇤ over !,

` ⇤8y8z( �(x, y)! ✓�(x0, z))! (⇤8y �(x, y)! ⇤8z✓�(x0, z)).

(' is an instance of B3)

If ' is t:{x} (x, y)! [t+s]:{x} (x, y), then '� is⇤8y �(x, y)! ⇤8y �(x, y).

Clearly, ` '�. The same argument holds when ' is s:{x} (x, y)! [t+ s]:{x} (x, y).

(' is an instance of B4)

If ' is t:{x} (x, y)! !t:{x}t:{x} (x, y), then '� is⇤8y �(x, y)! ⇤⇤8y �(x, y),

which is an instance of axiom A03; hence ` '�.

(' is an instance of B5)

If ' is ¬t:{x} (x, y) ! ?t:{x}¬t:{x} (x, y), then '� is ¬⇤8y �(x, y) !⇤¬⇤8y �(x, y), which is an instance of axiom A04; hence ` '�.

86

(' is an instance of B6)

Suppose ' is

t:{y} (x, y, z)! genx(t):{y}8x (x, y, z)

so '� is

⇤8x8z �(x, y, z)! ⇤8z8x �(x, y, z)

by classical reasoning,

` 8x8z �(x, y, z)! 8z8x �(x, y, z)

by necessitation and the distributivity of ⇤ over !,

` ⇤8x8z �(x, y, z)! ⇤8z8x �(x, y, z).

If ' is derived by using the rules R1 or R2 the result easily follows from

the induction hypothesis.

Suppose ' is derived using the rule R3. So ' is c: (x) where (x) is an

axiom. By the argument above, ` �(x). By generalization and necessitation,

` ⇤8x �(x), i.e., ` '�.

As usual in the study of justification logic, the proof of Proposition 33 is a

trivial induction on the theorems of the justification logic in question (in this case

FOJT45). What is a more significant result is the following:

(Realization Theorem) If FOS5 ` ', then FOJT45 `C 'r for a constant spec-

ification C and a normal realization r.

Right now we believe that the best path to try to prove this theorem is to

apply all the notions and results presented in this thesis in order to adapt the proof

of the Realization Theorem using semantical tools (as presented in [12], [10] and [9])

for FOJT45. But we also consider di↵erent ways. Another strategy is to study the

constructive argument using nested sequent calculus (as presented in [6]) and see

how this argument can be used for this case. We want to consider these two paths

in future research.

87

6.3 Justification logic and interpolation

When studying justification logic it is natural to investigate the relationship

between this logic and modal logic. The Realization Theorem gives us a tool to see

this relationship. Although we have left the proof of this theorem for future work,

it is worthwhile to see one easy conclusion of the Realization Theorem. To do so we

need to state one definition:

The Interpolation Theorem holds for FOJT45 i↵ for every constant specifica-

tion C and sentences ' and , if `C ' ! , then there is a formula ✓ such

that `C ' ! ✓, `C ✓ ! and the non-logical symbols and the justification

terms that occur in ✓ occur both in ' and .

Proposition 34. If the Realization Theorem holds between FOS5 and FOJT45,

then the Interpolation Theorem fails for FOJT45.

Proof. Suppose that the Interpolation Theorem holds for FOJT45. By Theorem 2

and by the Completeness Theorem for FOS5, let ' and be sentences such that

FOS5 ` ' ! and there is no interpolant between them. By the Realization

Theorem, there is a normal realization r such that

FOJT45 `C 'r ! r

By hypothesis, there is a formula ✓ such that the non-logical symbols and

the justification terms that occur in ✓ occur both in 'r and r. Moreover, we have

that

FOJT45 `C 'r ! ✓

FOJT45 `C ✓ ! r

by the forgetful projection:

FOS5 ` ('r ! ✓)�

FOS5 ` (✓ ! r)�

i.e.,

FOS5 ` '! ✓�

FOS5 ` ✓� !

88

Now, since there is no interpolant between ' and , then there is no relation

symbol occurring in ✓�. Hence, ✓� is a formula such that ? is the only atomic formula

that occur in ✓�. Thus, either ✓� is FOS5-valid or ✓� is FOS5-unsatisfiable.

On the one hand, if ✓� is FOS5-valid, then, since |=FOS5 ✓� ! , is

FOS5-valid. And so, '! has an interpolant, contradicting our hypothesis.

On the other hand, if ✓� is FOS5-unsatisfiable, then, since |=FOS5 ' ! ✓�,

' is FOS5-unsatisfiable. And so, ' ! has an interpolant, contradicting our

hypothesis.

We hope that the topics presented in this thesis fulfilled two objectives: i)

give a brief introduction to first-order S5; ii) clarify the connections between first-

order modal logic and first-order justification logic.

About the last objective it is important to stress that, as Proposition 34

shows, the failure of the Interpolation Theorem for FOJT45 is just a straightforward

consequence of the Realization Theorem. And so to prove this theorem for FOJT45

will not be only a subject of interest for the researchers involved in justification

logic, but will be a result of interest for the broader modal logic community.

89

Chapter 7

Appendix

Proof of Proposition 17

Proof of the explicit version of the converse Barcan Formula:

1. 8y'(y)! '(y) (classical axiom)

2. c1:(8y'(y)! '(y)) (axiom necessitation)

3. c1:Xy(8y'(y)! '(y)) (A3 + Modus Ponens)

4. c1:Xy(8y'(y)! '(y))! (t:Xy8y'(y)! [c1 · t]:Xy'(y)) (Axiom B2 )

5. t:Xy8y'(y)! [c1 · t]:Xy'(y) (Modus Ponens)

6. t:X8y'(y)! t:Xy8y'(y) (Axiom A3 )

7. t:X8y'(y)! [c1 · t]:Xy'(y) (classical reasoning)

8. 8y(t:X8y'(y)! [c1 · t]:Xy'(y)) (generalization)

9. t:X8y'(y)! 8y[c1 · t]:Xy'(y) (y /2 X + classical reasoning)

90

Proof of the explicit version of the Barcan Formula:

1. 8yt:Xy'(y)! t:Xy'(y) (classical axiom)

2. c1:(8yt:Xy'(y)! t:Xy'(y)) (axiom necessitation)

3. c1:X(8yt:Xy'(y)! t:Xy'(y)) (A3 + Modus Ponens)

4. geny(c1):X8y(8yt:Xy'(y)! t:Xy'(y)) (y /2 X + B6 + Modus Ponens)

5. geny(c1):X8y(8yt:Xy'(y)! t:Xy'(y))! 8y[c2·geny(c1)]:Xy(8yt:Xy'(y)! t:Xy'(y))

(Converse Barcan Formula1)

6. 8y[c2 · geny(c1)]:Xy(8yt:Xy'(y)! t:Xy'(y)) (Modus Ponens)

7. [c2 · geny(c1)]:Xy(8yt:Xy'(y)! t:Xy'(y)) (classical axiom + Modus Ponens)

8. c3:Xy((8yt:Xy'(y) ! t:Xy'(y)) ! (¬t:Xy'(y) ! ¬8yt:Xy'(y))) (tautology +

axiom necessitation + A3 )

9. [c3 · [c2 · geny(c1)]]:Xy(¬t:Xy'(y)! ¬8yt:Xy'(y)) (B2 + 7,8 + Modus Ponens)

10. [c3 · [c2 · geny(c1)]]:Xy(¬t:Xy'(y)! ¬8yt:Xy'(y))! (?t:Xy ¬t:Xy'(y)! [[c3 · [c2 ·geny(c1)]]·?t]:Xy¬8yt:Xy'(y)) (Axiom B2 )

11. ?t:Xy ¬t:Xy'(y)! [[c3 · [c2 · geny(c1)]]·?t]:Xy¬8yt:Xy'(y) (Modus Ponens)

12. ¬[[c3 · [c2 · geny(c1)]]·?t]:Xy¬8yt:Xy'(y)! ¬?t:Xy ¬t:Xy'(y) (classical reasoning)

13. ¬(?t:Xy ¬t:Xy'(y))! '(y) (JT45 theorem)

14. ¬[[c3 · [c2 · geny(c1)]]·?t]:Xy¬8yt:Xy'(y)! '(y) (classical reasoning)

15. ¬[[c3 · [c2 · geny(c1)]]·?t]:X¬8yt:Xy'(y)! ¬[[c3 · [c2 · geny(c1)]]·?t]:Xy¬8yt:Xy'(y)

(A2 + classical reasoning)

16. ¬[[c3 · [c2 · geny(c1)]]·?t]:X¬8yt:Xy'(y)! '(y) (classical reasoning)

17. 8y(¬[[c3 · [c2 · geny(c1)]]·?t]:X¬8yt:Xy'(y)! '(y)) (generalization)

18. ¬[[c3 · [c2 · geny(c1)]]·?t]:X¬8yt:Xy'(y)! 8y'(y) (y /2 X + classical reasoning)

1where c2:8y(8yt:Xy'(y)! t:Xy'(y))! (8yt:Xy'(y)! t:Xy'(y)) 2 C.

91

19. r: (¬[[c3 · [c2 · geny(c1)]]·?t]:X¬8yt:Xy'(y)! 8y'(y)) (Internalization)

20. r:X(¬[[c3 · [c2 · geny(c1)]]·?t]:X¬8yt:Xy'(y)! 8y'(y)) (A3 + Modus Ponens)

21. ?[[c3 · [c2 · geny(c1)]]·?t]:X¬[[c3 · [c2 · geny(c1)]]·?t]:X¬8yt:Xy'(y)!! [r·?[[c3 · [c2 · geny(c1)]]·?t]]:X8y'(y) (B2 + 20 + Modus Ponens )

22. 8yt:Xy'(y)!?[[c3 · [c2 · geny(c1)]]·?t]:X¬[[c3 · [c2 · geny(c1)]]·?t]:X¬8yt:Xy'(y)

(JT45 theorem)

23. 8yt:Xy'(y)! [r·?[[c3 · [c2 · geny(c1)]]·?t]]:X8y'(y) (classical reasoning)

Proof of Proposition 18

1. 9y'(y)! 9y'(y) (tautology)

2. 8y('(y)! 9y'(y)) (classical reasoning)

3. r:(8y('(y)! 9y'(y))) (Internalization)

4. r:X(8y('(y)! 9y'(y))) (A3 + Modus Ponens)

5. 8yCB(r):Xy('(y)! 9y'(y)) (Converse Barcan formula + Modus Ponens)

6. CB(r):Xy('(y)! 9y'(y)) (classical axioms + Modus Ponens)

7. t:Xy'(y)! [CB(r) · t]:Xy9y'(y) (B2 + 6 + Modus Ponens )

8. [CB(r) · t]:Xy9y'(y)! [CB(r) · t]:X9y'(y) (Axiom A2 )

9. t:Xy'(y)! [CB(r) · t]:X9y'(y) (classical reasoning)

10. 8y(t:Xy'(y)! [CB(r) · t]:X9y'(y)) (generalization)

11. 9yt:Xy'(y)! [CB(r) · t]:X9y'(y) (y /2 X + classical reasoning).

92

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