Intensional and higher-order modal logic: With applications to Montague semantics (North-Holland mathematics studies)

INTENSIONAL AND HIGHER-ORDER MODAL LOGIC

In memory of Richard Montague

NORTH-HOLLAND MATHEMATICS STUDIES

19

Intensional and Higher-Order Modal Logic With Applications to Montague Semantics

DANIEL GALLIN Department of Mathematics University of San Francisco San Francisco, California, USA

1975

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK

0 NORTH-HOLLAND

PUBLISHING COMPANY - AMSTERDAM, 1975

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North-Holland ISBN: 0 7204 0360 X American Elsevier ISBN: 0 444 11002 X

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PRINTED IN THE NETHERLANDS

PREFACE

In a series of papers written during the period 1967-1971, Richard Montague outlined a highly original approach to the problem of providing a precise account of natural language syntax and semantics. In a sharp departure from the linguistic methods of the Chomsky school, Montague introduced a powerful body of techniques from the field of mathematical logic, principally the set-theoretic semantical methods pioneered by his teacher Tarski. Montague's tragic death in 1971 cut short what was certainly the most ambitious research undertaking o f his career, and one for which he was uniquely qualified. Although he completed only three papers dealing specifically with natural language, the ideas they contain have provided the basis for an entire branch of current linguistic research, and the interest in his work continues to grow among philosophers, linguists and logicians. The present work attempts to provide the technical background necessary for a thorough understanding of Montague Semantics, at the same time exploring some of the mathematically interesting applications of higherorder modal logic. The focus of Part I is the logic of intensions, denoted by IL, which Montague introduced in his paper "Universal Grammar.'' This system extends Church's functional theory of types by the addition of two operators, corresponding roughly to intension and extension. Montague's formalized English fragments admit translation into IL, which is given a "possible worlds'' semantics along the lines of Carnap-Kripke. Following a brief introduction to the Montague program in Chapter 1, the syntax and semantics of IL are set out in detail. A natural axiomatization is provided, and Henkin's generalized completeness theorem for the theory of types is extended to the Montague system. This leads to a standard completeness theorem for a restricted class of "persistent" formulas, a result which has applications to certain "extensional" fragments of Eng 1ish .

vi

PREFACE

In Chapter 2 some natural axiomatic extensions of I L are considered and normal forms are obtained f o r formulas of IL. In addition, Montague's system is compared with

a

two-sorted extensional theory of types.

Part 11, which is essentially self-contained, deals with an alternative formulation of higher-order modal logic, denoted by MLp. This system takes quantifiers and the necessity operator as primitives and allows only predicate types, in distinction to the arbitrary functional types of IL. Although equivalent to Montague's system, MLp is perhaps more natural to the logician, and it has a number of interesting applications of its own in modal logic and set theory. Bressan has shown that such systems are also of interest in connection with the foundations of physics.

In Chapter 3, generalized completeness is proved for MLP and for the theory MLP+C obtained by adding a natural axiom schema of comprehension. A related principle of extensional comprehension, first proposed by Bressan, is shown to be equivalent in ML P +C to an axiom of atomic propositions considered by Kaplan and Fine. Every general model of MLp is shown in $10 to be homomorphic, in a truth-preserving sense, to one in which any two indices (possible worlds) a r e distinguishable by a formula. In $ 1 2 a general theory of propositional operators is developed within MLp which includes "axiomatically" defined classes of operators and those arising from Kripke-type relevance relations as special cases. I n Chapter 4 a Boolean semantics is defined which validates every

theorem of MLp+C . T h i s semantics is applied to show the independence of the extensional comprehension principle from the axioms of MLp+C , and to obtain a number of other independence results in higher-order modal logic. Topological models, in the sense of McKinsey and Tarski, a r e explored in $16, and in $17 the Boolean semantics for MLP is combined with the earlier generalized semantics to reconstruct the Scott-Solovay proof of Cohen's result on the independence of the continuum hypothesis. In this application of higher-order modal logic to set theory, certain modal sentences function as "interpolants" which express in formal terms various properties of the underlying Boolean algebra. Except for minor revisions, the present work constituted my doctoral dissertation in mathematics, submitted to the University of California, Berkeley, in September 1 9 7 2 . I began working with Professor Montague

vii

PREFACE

in July 1970, investigating several questions related to his system I L . Our work was interrupted by his death in March 1971, and Professor Dana Scott generously agreed to supervise the completion of my dissertation, for which I am deeply appreciative. I am a l s o greatly indebted to the other members of my doctoral committee, Professors Leon Iienkin and Robert Vaught, for their consistent direction and advice.

I must thank in addition Nuel Belnap, Harry Deutsch, Haim Gaifman, David Kaplan, Uwe Monnich, Barbara Partee and Robert Solovay for helpful conversations and correspondence, my wife Janet for her patience, and the National Science Foundation for providing financial support during 19701971 under N.S.F. Science Faculty Fellowship No. 60068. Montague's semantical methods are coming to seem less formidable, thanks largely to the efforts of Barbara Partee and others to bridge the separate disciplines of linguistics, philosophy of language, and mathematical logic. One is encouraged to hope that the work of Richard hlontague may eventually bring these disciplines closer to their common goal, the understanding of language. Daniel Gallin University of San Francisco June 1975

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CONTENTS PART I . CHAPTER 1 .

INTENSIONAL LOGIC

INTENSIONAL LOGIC

.

$1 Natural Language and Intensional Logic ...................... $2. The Logic IL ............................................... $3

. .

$4

Generalized Completeness of IL ............................. Persistence in IL ..........................................

3

10 17 37

CHAPTER 2 . ALTERNATIVE FORMULATIONS OF IL $5. Modal T-Logic

$6. Extensions of $7 .

.

$8

Normal Forms

.............................................. IL and MLT ................................... ...............................................

Two-Sorted Type Theory

.....................................

41 44

53 58

PART I1 . HIGHER-ORDER MODAL LOGIC CHAPTER 3 . HIGHER-ORDER MODAL LOGIC $9. $10.

.

$11

$12. $13. CHAPTER

. $15. $14

$16.

.

$17

...................................... Propositions in MLp ........................................ Atomic Propositions and EC ................................. Propositional Operators .................................... Relative Strength o f IL and MLp ............................ 4 . ALGEBRAIC SEMANTICS Boolean Models of MLp ..................................... Modal Independence Results ................................ Topological Models of MLp ................................. Modal Predicate Logic

Cohen's Independence Results

Bibliography

..............................

.......................................................

67 79 84

89 98

106

112 122

132 144

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PART I .

INTENSIONAL LOGIC

This Page Intentionally Left Blank

CHAPTER 1. INTENSIONAL LOGIC

$1. Natural Language and Intensional Logic When we speak of a theory of meaning for a natural language such as English, we have in mind an analysis which obeys the functionality principle of Frege, according to which the meaning of a given expression should be a function of the meanings of its constituents.' Philosophers of language since Frege have accepted the distinction, in discussions of meaning, between the extension o r denotation of an expression , and its




With the exception of the modal operators, these definitions follow Henkin [1963]. We write [A A B] instead of [ [ A A ] B] where A and B are formulas; similarly for the other binary connectives.

16

INTENSIONAL LOGIC

3X

-VX

A =

- A ,

[A = B ] a a

=

rl

= T] ,

[A

A =

[^A

^B] ,

I

--O-A.

+ A =

I t i s r e a d i l y v e r i f i e d t h a t t h e c o n n e c t i v e s and q u a n t i f i e r s have t h e i r u s u a l meanings i n any model ple,

i, a

bl,

bl, i , a

sat

f o r every

A

[A v B]

sat

B , and

sat

j

c

j u s t in case either

M, i , a

.

X C Ma

sat

Vxa A

0A

sat

i, a

bf,

j u s t in case

The n e c e s s i t y o p e r a t o r

M, i, a

f i e r over i n d i c e s : f o r every

under t h e s e d e f i n i t i o n s ; hence, f o r exam-

h1

sat

or

A

M , i , a(x/X)

0 acts a s a quanti-

i f and o n l y i f

M, j , a

sat

A

I .6

Many of t h e u s u a l p r i n c i p l e s of t y p e t h e o r y

--

t a u t o l o g i e s , laws of

rewrite f o r bound v a r i a b l e s , etc. - - c o n t i n u e t o h o l d i n I L . However, t h e r e a r e some accustomed laws which t u r n o u t n o t t o be v a l i d i n t h e i n t e n s i o n a l s e t t i n g , i n p a r t i c u l a r [ a s i s t y p i c a l o f modal q u a n t i f i c a t i o n t h e o r i e s ) unr e s t r i c t e d laws o f u n i v e r s a l i n s t a n t i a t i o n and s u b s t i t u t i v i t y o f e q u a l s . For example, t h e formulas (i)

vxe ]Ye

(ii)

c

e

~d

be

Ye]

[ c = c e e

+

e

3Ye [Ce

-+

-t

c

e

,

Ye]

5

= d

e

]

a r e not v a l i d i n IL.7 Various r e s t r i c t e d formulations of these p r i n c i p l e s a r e v a l i d i n I L , however: Let and d e n o t e by

in

A[xa)

A(x )

b e a term i n v o l v i n g t h e v a r i a b l e

t h e r e s u l t of r e p l a c i n g a l l f r e e o c c u r r e n c e s o f

A(Ba)

Ba , r e w r i t i n g bound v a r i a b l e s i n A(x,)

by t h e term

xu,

xa

i f neces-

s a r y t o avoid c l a s h e s . Then t h e f o l l o w i n g schemata a r e v a l i d i n IL: (i)

Vx, At[xa) t h e s c o p e of

A (B )

-t

t

A

a

in

, p r o v i d e d no f r e e o c c u r r e n c e o f At(xu)

xu

lies i n

,

Thus, n e c e s s i t y i s an S5 o p e r a t o r ; see Kripke [1963b].

' The f a i l u r e of

u n i v e r s a l i n s t a n t i a t i o n i s due t o t h e t r e a t m e n t of cons t a n t s i n IL. I n t h e modal p r e d i c a t e l o g i c of $9, which can b e viewed as an a l t e r n a t i v e f o r m u l a t i o n o f IL, a n u n r e s t r i c t e d p r i n c i p l e o f u n i v e r s a l i n s t a n t i a t i o n i s v a l i d . Cf. a l s o t h e t w o - s o r t c d t y p e t h e o r y of $ 8 .

GENERAL1 ZED COMPLETENESS OF IL

At(xa)

, if

At(Ba)

17

(ii)

'dx,

(iii)

B = C A (B,) = A ( C a ) , provided no free occurrence of a a P B lies in the scope of in .tp(xa) ,

(iv)

B = C

(v)

B

--t

is modally closed,

Ba

--f

a

a

C

a

a

A (B,)

P

--f

A

--t

P

(B ) a

E

A (Ca)

, if Ba

2

11 ( C a )

.

P P

and

Ca

x

are modally closed,

Generalized Completeness of IL

$3.

Since IL incorporates the theory of types, its valid formulas are not recursively enumerable, and therefore no complete axiomatization exists. In this section we prove a generalized completeness theorem for an axiomatic formulation of IL, based on the corresponding result in tlenkin [ 1 9 5 0 ] . Generalized Semantics.l based on D

I

we

Let

D and

be non-empty sets. By a

I

understand an indexed family

(bI,)aCT

frame

of sets,

where D ,

(i)

Me

(ii)

Mt = 2 = {O,l} ,

(iii) E.1 (iv) A

aP

Msa

=

is a non-empty subset of F1

F'U

P

is a non-empty subset of \la

' I

.

general model (g-model) of IL based on D

(bla,

(i) (ii)

I

is a system M =

mIacT , where is a frame based on D

and

I

,

m (the meaning function) is a mapping which assigns to each constant

ca a function from

I

into PIa ,

The terminology employed here is that of Henkin [1950].

18

INTENSIONAL LOGIC

(iii)

M There exists a function V (the value function) which assigns, to each i 6 I , a 6 A s @ ) , and Au E Tm, , a value V.M (A,) C hlu , I ,a in such a way as to satisfy the recursive conditions (1) through (7) on page 13.

remarked in Ilenkin [1950], this notion of general model is impredicative, as a result of clause (iii). The difficulty in attempting to define

As

value in an arbitrary frame is caused by the recursive conditions ( 4 ) and ( 6 ) corresponding to X and * , since the functions described there may simply fail to belong to the appropriate domain Ma

. We must

therefore add

clause (iii), which stipulates the existence of VM ; the uniqueness of VM follows immediately from the recursive conditions (1) through ( 7 ) . In $6 we provide a more direct characterization of a large class of g-models. The notion of satisfaction of a formula in a g-model is exactly as before, with model replaced by g-model. The formula A i s a g-semantical consequence, in IL, of a set r of formulas, and we write

r

)=

g

A

in IL,

if M, i, a sat A whenever M and M , i, a and we write

sat

r .

If

r

is a g-model of IL, i E I

is empty we say that A

, a

E As(M)

,

is g-valid in IL,

in IL.

A

Z of formulas is g-satisfiable in IL if M, i, a sat Z for some g-model M , index i , and assignment a .

A set

It should be remarked that the sentential connectives, quantifiers and modal operators continue to have their usual meaning even in general mod-

els. The valid schemata (i) through (v) given at the end o f $ 2 continue to be g-valid, and since every standard model of IL is a g-model it is immediate that

r

)=

I=g

A

g

A

r f

in IL implies in I L implies

I=

A

A

in IL,

in IL,

Z satisfiable in IL implies Z g-satisfiable in IL.

GENERALIZED COMPLETENESS OF IL

19

We g i v e a d e d u c t i v e s t r u c t u r e t o t h e language o f I L i n

The Theory IL.

t h e u s u a l way, f i r s t s p e c i f y i n g a r e c u r s i v e s e t o f axioms and i n f e r e n c e r u l e s and t h e n d e f i n i n g a theorem o f I L t o b e any formula o b t a i n a b l e from

e u s e t h e term IL t o ret h e axioms by r e p e a t e d a p p l i c a t i o n o f t h e r u l e s . W f e r t o b o t h t h e language and t h i s d e d u c t i v e t h e o r y w i t h i n t h e language; no confusion should a r i s e . Axioms of I L .

=

Al.

Stt T

A2.

x a =-y ,

A3.

vx

AS 4

(Ax, A p ( x ) ) Ba

A

gtt F f

+

at

vx t [gxl

x = f

at

y ,

1 = Cf

[ fp, x 5 g

Ap(Ba)

s

9

E

91

9

, where

A (B,)

replacing a l l f r e e occurrences of f r e e occurrence of A(x)

in

A(x)

x

comes from

by t h e term

Ap(xa)

l i e s w i t h i n t h e s c o p e of

A

, or e l s e ( i i ' )

by

B , and ( i ) no

l i e s within a part

Xy C

B , and e i t h e r ( i i ) no f r e e o c c u r r e n c e o f

i s free i n

y

x

P

B

where x

in

i s modally

closed,

13 [ " f s a

A5.

AS 6

" A

A

z A

a

E

a

=

"gsa]

[f

P

g] ,

.

Rule o f I n f e r e n c e . From

R.

B'

and t h e f o r m u l a

Aa D A '

comes from

B

a t e l y p r e c e d e d by

B

t o i n f e r t h e formula

by r e p l a c i n g one o c c u r r e n c e o f h ) by t h e term

A'

B'

, where

A ( n o t immedi-

.

This axiomatization f o r I L corresponds very c l o s e l y t o t h e axiomatization f o r t h e t h e o r y o f p r o p o s i t i o n a l t y p e s g i v e n i n Henkin [1963], a s s i m p l i f i e d i n Andrews [1963]. AS4 i s j u s t H e n k i n ' s Axiom Schema 7, s u i t a b l y m o d i f i e d f o r IL. The new axioms A5 and AS6 a r e a n a l o g u e s o f A3 and AS4, w i t h t h e intensional abstractor

X

.

A

playing t h e r o l e of t h e functional a b s t r a c t o r

We remark i n p a s s i n g t h a t AS4 can b e r e p l a c e d by t h e f o l l o w i n g schema-

t a , c o r r e s p o n d i n g t o Henkin's schemata 7 . 1 t h r o u g h 7 . 5 : AS4.1

( h a A ) Ba

AS4.2

(Ax, xa) Ba

AP '

P

G

Ba

,

if

i s not f r e e i n

A

P '

20

I N T E N S I O N A L LOGIC

(Ax,

AS4.3 AS4.4 AS4.5

PY

Ba

(Xx, [AP

E

(Axa Xy

A 1 Ba

y

Ba

Cg])

Ba

C ) B]

, ,

(AX C) B ]

5

, if

Xy [ ( X x A ) B ]

x

and

y

a r e d i s t i n c t and

,

I3

A) B] ,

'[(Ax

z

B] [(Xx

A)

[ (Ax A ) B

e

E

ii Y )

[(Xx

i

i s not f r e e i n

(Xx, "A

AS4.6

1) '0

[A C

SP

(Axa

AS4.7

Ha

^Ap)

E

^[(Ax

,

A) H ]

if

i s modally c l o s e d

B

S i m i l a r l y , Rule R can be r e p l a c e d by t h e e i g h t r u l e s below: K1.

From

R2.

From

R3.

From

A

kal)

A

and

A;

f

t

A&

t

t o Infer

t o infer to infer

:A '

a

A

a

A Ila

B

aP

R4.

From

AB

E

A;

to infer

Ax

R5.

From

A

z A'

t o infer

*A

1Cnj(A.) I

I

A A

A ]

-

and using

,

A ]

,

j # i } , and from this it easily follows that

GENERALIZED COMPLETENESS OF IL

-

r 1- 0 Cnj(Ai) , so that contradicting Lemma 3 . 2 . 2 .

{

0 Cnj(Aj) 1

zi

free for x

c

0

}

is inconsistent in IL,

and each formula B(x,) , we have E Ti for some variable y which is

LEMMA 3 . 2 . 7 . For each i E w if and only if B(y,)

3 x B(x) E

j

29

B(x) .

in

Proof: The implication from right to left follows from Lemma 3 . 2 . 5 and _ _ T26. For the implication from left to right, let A be the formula 3 x B(x) and suppose A C

F. . Let

be the pair

k

is relatively i-consistent with C k , since it is relatively i-consistent with 2 (i,A)

(ik,A ) ; then A

by Lemma 3 . 2 . 2 . Therefore by the construction of Zk+' , we have Ck"

1

c

-

Ti

free for x

for some variable y

LEMMA 3 . 2 . 8 .

For each

i E w

1

o

6

.

and each formula B , we have

c F. for some j c

if and only if H

in B(x)

B(y,)

0B

E

g.

.

Proof: F o r the implication from right to left, suppose B C r. but 1 0 B 1 Fi . Then i # j , in view of T41 and Lemma 3 . 2 . 5 , and by Lemma 3 . 2 . 6 is we have -,0B E Ti and therefore 0 B E Fi by T46. But then

-

relatively inconsistent, since the set

{ o n - B , OB

}

is inconsistent

in IL by T49. This contradicts Lemma 3 . 2 . 2 . The implication from left to right follows as in the proof of Lemma 3 . 2 . 7 . This completes the proof of Lemma 3 . 2 . Suppose Z = ( Zi )iEw satisfies (i) through (iv) of Lemma 3 . 2 . Then it satisfies a l s o :

REMARK 3 . 2 . 9 .

(v)

For each i C w and each formula B(x,) , we have Vx B(x) C Ti if and only if B(y,) C for every variable y which is free for x in B(x) .

xi

(vi)

For each

only if R

i

€ w

and each formula B , we have 0 B E

c F. for every j 1

E w

ci

if and

.

We omit the proof, which is straightforward. Lemma 3 . 2 , it should be noted, depends for its proof only on the fact that the theorems of IL include the ordinary laws of sentential and predi-

30

INTENSIONAL LOGIC

cate logic, together with the S5 modal laws T40, T42, T43 and T44. The lemma will therefore also hold for any theory in which these laws obtain, e.g. the first-order extensions of S5 described in Kripke [1959], Bayart [1958], o r Hughes and Cresswell [1968]. Lemma 3.2 considerably simplifies the Henkin-type completeness proofs which have been given for these logics, and it is not difficult to modify the lemma and its proof to apply to quantified

extensions of certain weaker modal logics than S5. We are now ready to prove: THEOREM 3.3 (Generalized Completeness Theorem for IL) (i)

in I L implies

A

(ii)

r I= g

(iii)

Z

A

A

in I L implies

r I-

in IL, A

in IL,

consistent in IL implies 2 g-satisfiable in IL.

Proof: -

Parts (i) and (ii) follow easily from (iii) as usual, and in

fact we show the somewhat stronger: LEMMA 3.3.1. Suppose I:

is consistent in IL. Then 2

is g-satisfi-

able in a g-model M = (Ma, m)acT of IL based on sets D and I , where I is denumerable and D , as well as each domain Ma , is at most denumerable. Proof: We may assume without l o s s of generality that there are infinitely many variables of each type not occurring in any formula of Z , since, e.g., replacing each variable :X by xin throughout 2 produces a set with this property, and clearly affects neither the consistency nor the g-satisfiability of Z . Let 2 = ( Xi )iCo be a sequence of sets of formulas satisfying (i) through (iv) of Lemma 3.2, and therefore satisfying also (v) and (vi) of Remark 3.2.9. Suppose a E T , i t w Aa

n.

Ba (mod i)

-

. The relation

if and only if

[A s B] C

Ti ,

In particular, to those extensions of the modal propositional logics K,

M (also called T), B, and 54 (see Kaplan [1966]) in which the Barcan f o r mula [Vx o A O V X A ] and the usual predicate laws are valid. Cf. the completeness proofs given in Cresswell [I9671 and Thomason [1970].

GENERALIZED COMPLETENESS OF I L

31

u , i s c l e a r l y a n e q u i v a l e n c e r e l a t i o n on

between terms of t y p e Var,

. Then by T29 and p r o p e r t y ( i i i ) of

Tma

and

i E w

x E Var

some

A

2

.

x (mod i )

=

x

...

trxo

A

Vxn-l [x

5

A E

In f a c t , prop-

e r t y ( i i i ) i m p l i e s t h a t t h e r e a r e i n f i n i t e l y many such v a r i a b l e s 0 1 n- 1 f o r any d i s t i n c t x , x , ... , x E Var, t h e formula 3x [ A

x

, since

x] ]

i s p r o v a b l e i n I L , where x i s n o t f r e e i n A and i s d i s t i n c t from . . . , x n- 1 . I f x , y E Var, t h e n by T39 t h e formula

i s p r o v a b l e i n IL, from which i t f o l l o w s t h a t t h e r e l a t i o n i s independent o f x

*

y . Let

x/=

x/=

for

-

from

,

For

(2)

For e v e r y

and

, x

j E I

:

E Var,

i E I ),

rr

X E Ma

i C I

,

B (mod i )

A

,

B E Tma :

Var,

z

of t y p e

t

x , we must .

and a map-

Ma

,

sgch t h a t

X = pa(x)(i)

i f and o n l y i f

, and p u t

pt(At) (i) = 1 i f

o t h e r w i s e . Then (1) f o l l o w s from t h e f a c t t h a t t h e f o r m u l a

,

I = o ,

.

pa(A)(i) = ku(B)(i)

i s p r o v a b l e i n I L , by T39. To v e r i f y ( Z ) , property ( i i i ) of

under t h i s

.

Mt = 2 = {O,l}

Let

I

pu(x)(i) = ka(x)(j)

(which by (1) i s i n d e p e n d e n t of

For

in

c o n s i s t i n g of a l l c l a s -

, s a t i s f y i n g t h e following t h r e e conditions:

Ma1

into

x E Var,

a = t :

y

D

there exists

A

0

i

xa

a E T , we s i m u l t a n e o u s l y d e f i n e a s e t

Tma

(1)

,

y (mod i )

i s a t most denumerable. L e t t i n g

D

M = (Ma, m)aET o f IL based on

pu

(3)

Then

' 1

0

Z t o c o n s t r u c t , i n t h e manner o f Henkin [1950], a

By r e c u r s i o n on ping

.

x E Var

x

x

, y , and we c a n w r i t e s i m p l y Var /=

b e t h e q u o t i e n t set

D

we u s e t h e sequence g-model

x

denote t h e equivalence c l a s s of

r e l a t i o n , and l e t

ses

for variables

i E o

in

u by

, t h e r e e x i s t s f o r each

f o r which

,

Tma

view o f T 1 , T35 and T36. Let u s d e n o t e t h e s e t of v a r i a b l e s o f type

Hence

z

have

y E

+ T i , and

< Ti

At

and

xt + U xt

o b s e r v e t h a t b y T27, T28 and

zi ,

-

z E

Ti

kt(y)(i) = 1

f o r some v a r i a b l e s

, kt(z)(i)

C o n d i t i o n (3) f o l l o w s e a s i l y from t h e maximal c o n s i s t e n c y o f

= 0

-

Zi

.

.

INTENSIONAL LOGIC

32

a = e :

Let Me = D = Var

/c=

, and define pe(Ae)(i)

to be x/- ,

where x

is any variable of type e for which A rr x (mod i) (this is clearly well-defined). In particular, pe(xe)(i) = x/= , so that clearly (1) and (2) hold. The verification of (3) is straightforward. have already been defined, and a = Py : Suppose MP , pP , My , pY conditions (l), (2) and (3) hold f o r p and y . We first define the map-

ping pa and X

C

from Tma

M

P '

[My

into

let x C Var

P

731 I

, as follows: Given A

€

Tma

,i

I ,

C

, and put

be chosen such that X = pp(x)(i)

~~(A)(i)(x) = py(Ax)(i) . Such a variable x exists by condition (2) for p ; to see that the value is well-defined, suppose that y C Var and

P

.

Then by condition (3) for ~ ~ ( (i) y ) = X = pP(x) (i) , x rr y (mod i ) , whence by T32 Ax = Ay (mod i) and therefore py(Ax) (i) = py(Ay) (i) by condition (3) for y

.

Before defining Ma we verify condition (1) for a : Suppose f C Vara and i , j C I ; we show that pa(f)(i) = pa(f)(j) . Suppose X C MP . By conditions (1) and (2) for p we have X = p (x)(i) = p (x)(j) for some

P

P

x C Var , and therefore pa(f)(i)(X) = wy(fx)(i) and p , ( f ) ( j ) ( X ) = P py(fx)(j) , so it suffices to show that py(fx)(i) = py(fx)(j) . Let y Var be such that fx rr y (mod i) . By T39 the formula Y

fx

f

y

+

C

0 [fx s y]

is provable, so that fx rr y (mod j) (1) and (3) for Y

also, and therefore using conditions

we have wy(fx)(i)

=

wY(y)(i)

=

pY(y)(j)

=

py(fx)(j)

.

We also observe that if A C Tma , i € I then p,(A)(i) = pa(f)(i) Indeed, suppose A rr f (mod i) ; then if X € M for some f € Var, P ' say X = p (x )(i) , we have by T33 Ax rr fx (mod i) , s o that by condition

.

P P

(3) for Y , cLa(A)(i)/X) therefore set

M~ =

pa(f)(i)

=

I

KY(Ax)(i)

=

pyy/fx)li) = @a(f)(i)(X)

. We

can

M

f c Var,

15

M~ P ,

which by condition (1) is independent of i C I , and condition (2) will be satisfied for a . To verify condition (3), suppose A , B C Tma , i C I . Then the following conditions are equivalent:

GENERALIZED COMPLETENESS OF I L

p ,

These e q u i v a l e n c e s employ c o n d i t i o n ( 2 ) f o r

F ,

p r o p e r t y (v) o f

We remark t h a t

A E Tm

which

=x

B

B E Tm

a '

,i

P

.

(mod i )

P

into

Suppose

.

blP

,

pp

chosen s o t h a t

A

5

f (mod i )

(ill

AB

=

Ax (mod i ) , from

= (J-,(A) (i) bP (B) (i)

J .

have a l r e a d y been d e f i n e d s o t h a t c o n d i -

P .

We f i r s t d e f i n e t h e mapping

, as f o l l o w s : Given

[Pfgi]'

,

y

imply t h a t

Then by T32 we have

t i o n s ( l ) , ( 2 ) and ( 3 ) h o l d f o r Tma

E I

/lY(AB) (i) = py(Ax) (i) = paLaA) ( i ) IPP (XI

a = sp :

condition (3) f o r

and ~ 1 4 .

py(AB) ( i ) = wa(A) ( i ) [kp (B) ( i ) 1 F o r , suppose

33

A 6 Tma

, and p u t

w e l l - d e f i n e d , f o r i f we a l s o have

,

i E I

, let

pa

from

f C Vara

be

pu(A)(i) = pp("f) C M I . T h i s is

P

g E Var,

and

A

= g (mod i ) , t h e n

, and t h e r e f o r e " f = "g (mod j ) f o r a l l j C I by T34. Using c o n d i t i o n ( 3 ) f o r P , t h i s i m p l i e s t h a t p ( " f ) ( j ) = p P ( " g ) ( j ) f o r a l l j E I , so t h a t pp(-f) = pp("g) i n P f

M

rr

g

, i.e.,

f

e

g (mod j )

for a l l

j E I

I

P

. We o b s e r v e t h a t f o r

which i s independent of earlier, i f Varu

, since

A C Tina A

rr

,

f C Var,

,

i E I

,

we have

k a ( f ) ( i ) = pp("f)

i E I ; t h u s c o n d i t i o n (1) h o l d s f o r i E I

f (mod i )

, t h e n wa(A)(i) implies

= pa(f)(i)

a

.

f o r some

pa(A)(i) = pp("f) = p a ( f ) ( i )

,

A l s o , as f E

.

INTENSIONAL LOGIC

34

We can therefore set

which by condition (1) is independent of satisfied for

.

a

i

, and condition

C I

-

(2) will be

Finally, we show that condition ( 3 ) holds for

.

a :

Sup-

f (mod i) , pose A , B E Tma , i € I Choose f , g E Var, with A B = g (mod i) . Then the following conditions are easily seen to be equivalent, using condition (3) for p , property (vi) of Z , and T60:

(b)

~ ~ ( =~ pLB("g) f ) ,

(c)

For all

(d)

For all j C I

(e)

F o r all

(f)

0 ["f

=

5

, "f , ["f

j C I

bP("g)(j)

,

wg (mod j ) , "g] C

5

2. , 3

1 '

x.

[f

(h)

f = g (mod i)

,

(i)

A

.

€

=

x.

"g] C

(g)

'LT

g]

-

, pp("f)(j)

j C 1

1 '

B (mod i)

This completes the definition of the frame

(Ma)aCT

for IL based on

D

and I . Obviously, by conditions (1) and ( 2 ) each domain Ma is at most denumerable. To complete the definition of the g-model M = (Ma, m ) a E T we define the meaning function m by putting m(c) = ba(c)

6 Ma

for each constant ca

I

. .

It remains to prove that there exists a value function $1 in M Suppose A 6 Tma , a E As(M) Suppose the free variables of A are among 0 1 the distinct variables x x , , xn- 1 , where xk is of type ak , and 0 n-1 0 1 n-1 ) for A Choose a sequence y , y , ,y of write A(x , . . . ,x distinct variables, yk of type ak , satisfying the conditions

.

... .

pa

k

k (y )(i)

=

k a(x )

(independent of i

...

C

I ) ,

GENERALIZED COMPLETENESS OF IL

(b)

yk

is free for xk

in A

35

.

k Such a sequence exists; for by conditions (1) and ( 2 ) for ak , a(x ) = ka (y)(i) , independent of i C I , for some variable y of type ak , and k as remarked earlier we have y CT y' (and therefore p(y') (i) = p(y)(i) = k a(x ) ) for infinitely many variables y' . Hence for each k < n there exist infinitely many variables yk satisfying (a), and it follows that there exists a sequence y0 , y1 , ... , yn-1 of distinct variables satisfying both (a) and (b). We call such a sequence a representing sequence for 0 1 the term A and assignment a , where the original sequence x , x , . . . , n1 x is fixed in advance to contain all free variables of A Let be 0 the term A(y , , ,yn-') which results from A by simultaneously rek placing all free occurrences of x by yk for k < n Given i C I we

.

..

.

define M V. (A) = i,a

pa(

z )(i)

C Ma

.

) (i) does not We show that this is well-defined, i.e., the value ka( 0 n-1 0 n-1 depend on the sequence y , . . . , y Indeed, suppose z , ... , z is k k another representing sequence for A and a ; then wa (y )(i) = a(x ) = k k 1 , (z ) (i) , so we have yk CT zk (mod i) , from which it follows by n apk 0 n-1 0 n-1 plications of T31 that A(y , ,y ) ". A(z , .. , z ) (mod i) , and 0 n-1 0 therefore pa[A(y , , , ,y ) ] (i) = ka[A(z , . ,zn-')] (i) , a s required.

.

. ..

.

..

.

e: vM is a value function in

M

.

We must verify the recursive clauses (1) through ( 7 ) on page 13. The verification of (1) and ( 2 ) is immediate. To verify (3)

Vi,a(AagBa) =

v.1,a (AaB )[Vi,a(B,)l

$

0 n-1 we let x , . . . , x be the distinct free variables of [AB] , and we 0 , yn- 1 for [AB] and a We then choose a representing sequence y ,

...

have V. (AB) = pp( l,a v.l,a (A)[Vi,a(B)l * (4)

)(i)

=

w ( x E )(i) B

.

=

w

aB

(

a )(i)[ka(

)(i)]

V. (Xx, A ) = the function F on Ma whose value at Y B l,a equal to Vi,al(AB) , where a' = a(x/Y)

.

€

=

Ma

is

INTENSIONAL LOGIC

36

For, l e t

, at

Y 6 Ma

0

1

...

x , x ,x ,

Let

,x

n- 1

,

a l l f r e e variables of

A

r e p r e s e n t i n g sequence

y ,

yk

.

x

free for 0

y ,

... ,

y

n- 1

V.

1,a

... ,

y

n- 1

y,

Let

for

hx A

and

.

pa(y)(i) = Y

forms a r e p r e s e n t i n g sequence f o r (A) = pP(

Vi,a, (XX

, where

j ( i ) (Y)

= F

aP

is t h e term

t h e r e f o r e s u f f i c e s t o show t h a t

(Xx A ( x , y

equivalently that

0

,

pLp[(

Xx A ) y

n- 1 ,y ))y

...

, y ,

Then t h e sequence A

and

(

a t , and we

GT ) ( i ) [ p a ( y j ( i ) ]

_ I _

pLp[( Xx A ) y ] ( i )

yk

0 n-1) , where A i s A(y,y , . . . , y ,

) (i)

X i 3

A)(Y) = p ( aP

, where o f c o u r s e

a

be a v a r i a h l c d i s t i n c t from each

A , and s a t i s f y i n g

in

t h e r e f o r e have and a l s o

0

xk

corresponds t o

V. (hx A)(Y) = V i , a l ( A ) . l,a b e a l i s t of d i s t i n c t v a r i a b l e s which i n c l u d e s and w r i t e A ( x , x0 , ... , x n- 1) f o r A . Choose a

= a(x/Y) ; we show t h a t

Xx A(x,y

0

, ... .Y n-1)

] ( i ) = pP( T;; )(I) 0

= A(y,y ,

...

,

=

.

It

or

n- 1 ,y ) (mod i )

.

But t h i s f o l l o w s immediately from axiom AS4 of 11,. The v e r i f i c a t i o n of c l a u s e ( 5 ) i s s t r a i g h t f o r w a r d , s i m i l a r t o t h a t f o r c l a u s e ( 3 ) . To v e r i f y V. ( A A a ) = t h e f u n c t i o n l,a t o Vj,a(Aa) ,

(6)

suppose t h a t ables of a

.

j E I

.

A

and l e t

"iT

is equal

he t h e d i s t i n c t f r e e v a r i 0 y , , y n- 1 f o r "A and

...

Vi,a("Asa)

For, l e t

x

0

2

=

V. ( A s u ) ( i ) 1,a

xc= f

V.

= pa(

"A?

"f (mod i )

i,a

("A)

.

h e t h e d i s t i n c t f r e e v a r i a b l e s o f A , and choose 0 . . . , y n- 1 f o r "A and a . Let f C Varsa

y ,

a r e p r e s e n t i n g sequence and

pa("f)(j) T62 and

.

, ... , x n- 1

b e such t h a t

. I t t h e r e f o r e s u f f i c e s t o show t h a t A - " f (mod j ) . But t h i s f o l l o w s from

(mod i )

, i.e., that

p r o p e r t y ( v i ) of

-

n- 1 x

j E I

a

^n- f

)(j)

(7)

, ... ,

whose v a l u e a t

V. (A) = pu( ) ( j ) , and i n a d d i t i o n V. (-A)(j) = I>a 1 ,a f 6 Varsa i s cho) ( i ) ( j )= psu( ^ A ) ( i ) ( j ) = p a ( " f ) ( j ) , where

sen so t h a t = pu(

0

I

on

Choose a r e p r e s e n t i n g sequence

Then we have

psa(

x

F

(mod i). Then ) ( i ) = pa(

V. ( A ) ( i ) l,a

"A ) ( i )

= psa(

) ( i ) ( i )= p a ( " f ) ( i )

, s o i t s u f f i c e s t o show t h a t

, which f o l l o w s from T34.

T h i s p r o v e s t h e c l a i m , and t h e r e f o r e completes t h e p r o o f t h a t a g-model of IL. Now l e t

a C As(M)

be d e f i n e d b y :

M

is

37

PERSISTENCE IN IL a(x ) =

pa(x)(i)

(independent of i

C I )

.

If Aa is any term, we clearly have V. (A) = pa(A)(i) 1 ,a and this implies in particular that for any formula A

for any

,

i C I

of IL, M, i, a

-

sat A sat Z

if and only if A C Ti . Since Z 5 Zo , it follows that M, i, a when i = 0 . This completes the proof of Lemma 3.3.1 and Theorem

3.3. Combining Theorems 3.1 and 3.3 we immediately obtain the following generalized compactness theorem for IL, a result which can also be proved directly using ultraproducts: COROLLARY 3.4. Let C be a set of formulas of IL. Then C is g-satisfiable in IL if and only if every finite subset 2’ of Z is g-satisfiable in IL. We conclude this section by remarking that the entire development to this point admits a natural generalization to the case of a non-denumerable language; i.e., a formulation of IL which permits, for some cardinal constants c5

$4.

for a E T and 5 c

K

K

,

.

Persistence in I L The Generalized Completeness Theorem proved in the last section relates

the axiomatic theory IL to the generalized semantics for IL. However, we can also obtain from it a useful relationship between the theory IL and the standard semantics for IL described in $ 2 . This relationship is expressed in Theorem 4.2, which shows that with respect to a suitably restricted class of formulas of IL, the theory IL is complete for the standard semantics. Suppose M = (Ma,

mIaeT

I . Let For each a E T we

is a g-model of IL based on D and

be the standard frame based on D define an element A; € MA :

and

I

.

A somewhat weaker form of this result is anticipated in Montague [1970a], pp. 88-89.

INTENSIONAL LOGIC

38

(i)

A;

E D , chosen arbitrarily,

(ii)

A:

=

(iii)

A ' (XI) = A '

(iv)

A;a(i)

0 ,

P

aP

= 11;

for every X ' E MA for every i C I

,

.

+

By recursion on a C T we define a one-to-one mapping

+e

(i)

is the identity mapping on M

: Ma +b!;

:

= D = M'

e' Qt is the identity mapping on M t = 2 = M't '

(ii) (iii)

+ap(F)

, for F E M

aP all X ' C MA , F'(X')

range of (iv)

+a

, is the function =

F ' E M'

such that for

if X'

belongs to the

+p[F(+a-l(X1))]

, and F'(X')

aP

= A'

P Otherwise,

, for F C Msa , is the function F' all i C I , F'(i) = +=[F(i)] .

+sa(F)

Define a standard model M' = (MA, m')aET

E

M&

such that for

of IL based on D

and

I by

for each constant ca . A term Aa of I L is called M-persistent if, for every i C I and a E As(M) , and for every choice of A; E D , we have letting m'(ca)(i)

where a' C As(M')

.

x i

E

= +a[m(ca)(i)]

is defined by

a'(x,)

=

+,[a(x;)]

for every variable

In particular, then, a formula A will be M-persistent if, for every I and a C As(M) , and f o r every choice of A; E D , the equivalence M, i, a sat A

holds. A erm Aa M of IL.

if and only if M', i , a'

sat A

is persistent if it is M-persistent for every g-model

It is immediate from Theorem 3.1 that any term provably equal, in I L , to a persistent term is itself persistent. Also, a term Aa is persistent if and only if the formula [ A z x] is persistent, where x is a variable of type a not occurring free in A

.

The next theorem shows a large recursive class of terms to be persis-

PERSISTENCE IN IL

39

tent, and therefore provides a partial characterization of the class o f all persistent terms.2 A truth-function& type T E T is one which is built up from the type t alone by pairing; e.g., the type (tt)(t(tt)) 0 1 cn-1 If B , C , C , . . . , are terms of IL of types a , a. , al ,

.

...

,

a respectively, where n- 1 a =

...

a 0(a1 (

then BC°C1

. ..

(an_l P I . . . ) )

,

Cn-l stands for the term

p , or B alone in the case n

= 0

0 1 [ . . . [[BC ]C 1. ..]Cn-'

of type

.

THEOREM 4.1. Let Per denote the class of all persistent terms of IL. Then : (i)

All variables and constants belong to Per ,

(ii) (iii)

A , Ba E Per imply [AB] C Per , aP Aa , Ba E Per imply [A e B] C Per ,

(iv)

A

(v)

Asa C Per implies "A

(vi)

At , B

(vii)

Aa C Per implies Xx A C Per , A E Per implies vxe A , 3x A E Per , t

implies ^A E Per

E Per

t

€

Per imply

Per ,

C

-

,

A

, [A

A

B] , 0 A

Per ,

€

(viii) Aa E Per implies XxT A E Per , A (ix)

t

E Per

implies VxT A , 3xT A C Per

( T a truth-functional type)

...

E Per and F (x ) is the formula BC°C1 x ... Cn t P t where Cm has type am for m 5 n , Ck is the term xB , and B

Suppose A

has type a (a ( 0

free in B

1

...

(an t) . . . ) )

.

Suppose also that

is not Then

, and the terms B , C0, _ ., , Cn belong to Per

the formulas Vx [F(x)

+

A]

and 3x [F(x)

A

A]

,

.

belong to Per

.

This notion of persistence is a generalization to IL o f a related notion for ordinary (non-modal) higher-order predicate logic. The question of a complete syntactic characterization for that case is discussed in Orey 119591. See also Mostowski [1947].

INTENSIONAL LOGIC

40

If At , Ba

(x)

C

Per and xa does not occur free in Ba , then the

formula 3x [ B Proof:

=

x

A

A ]

.

belongs to Per

(i) through (vi) are straightforward. For (vii) and (viii) it

~

suffices to observe that for a given g-model bZ mappings "d, when a types

a

of IL, if M'

and the

are obtained from hi as earlier, then maps Ma onto is e o r a truth-functional type T . In fact, for these

it holds that Ma =

and

is just the identity mapping.

Qa

For a = e this is obvious; for truth-functional types T it follows easily from a result of tlenkin.3 Conditions (ix) and (x) are straightforward. From the Generalized Completeness Theorem we now obtain: TIIEOREM 4 . 2 .

Let

r

and C be sets of persistent formulas, A

a

persistent formula. Then: A

(i)

in IL if and only if

1-

A

t

(ii)

r )=

(iii)

Z is consistent in IL if and only if Z

Proof:

A

in I L if and only if

r

in IL, A

in IL, is satisfiable in IL.

(iii) follows immediately from Theorem 3 . 3 (iii) and the defi-

nition of persistence. (i) and (ii) follow easily from (iii), as does the following COROLLARY 4.3.

Let Z

be a set of persistent formulas. Then C

satisfiable in IL if and only if every finite subset Z' fiable in IL.

of Z

is

is satis-

Theorem 4.2 has application to various fragments of English, as described in $1. In particular, it is possible to show that Extensional English can be translated into IL in such a way that the translate of every sentence is a persistent closed formula of IL. This implies by Theorem 4 . 2 that a sentence of Extensional English will be valid if and only if its translate is provable in IL, and since the translation is effective it follows that the valid sentences of Extensional English are recursively enumerable. Moreover, Corollary 4 . 3 yields a compactness theorem for this fragment of English. Henkin [1963], $ 4 .

CHAPTER 2 .

$5.

ALTERNATIVE FORMULATIONS OF IL

Modal T-Logic

We take up now the first of several alternative formulations o f the logic IL of Chapter 1. The most natural first step is to try to eliminate the functional and intensional abstractors X

and

*

in favor of the more

familiar quantifier V and modal operator , particularly in view of the were responsible for the impredicativity fact that the abstractors X , of the notion of a general model of IL. For the moment, however, we choose to retain the full "functional" type structure, i.e., to allow variables

.

and constants of all types a C T The resulting logic we refer to as Modal T-Logic, and denote by MLT. Since tl and 0 are already defined in IL, the language o f MLT can be described as a sublanguage of the language of IL. We shall adopt this course, since it will facilitate a later comparison of the two logics. Grammar. The atomic terms of MLT comprise the smallest set ATm

of

terms of IL such that: (i)

All variables and constants belong to ATm

(ii)

A

(iii)

ASa f ATm

aP

'

Ba C ATm

imply

implies "A

An example is the term

[AB] C

ATm

C

,

ATm ,

.

["fs(et)[cee"gse]]

of type t

.

A formula A

of

IL is an atomic formula of MLT if one of the following holds:

(i)

A

is an atomic term of MLT of type t ,

(ii)

A

is

[B

5

C] , where Ba

and Ca

are atomic terms.

The formulas of MLT are generated from the atomic formulas by the connectives , A , -+ , v , the quantifier Vxa where xa C Var, , and the necessity operator 0 Thus, every formula of MLT is a l s o a formula of IL,

-

.

ALTERNATIVE FORMULATIONS OF IL

42

but not conversely. Since 3 and and 0

0

are defined in IL in terms of V ,

, these operators are also defined in

Generalized Semantics. Let D and

-

MLT.l

I

be non-empty sets. By a general

I

we understand a system M =

l _ _ l _ _ _ _

model (g-model) of MLT ___ based on D (Ma, m)aCT such that: (i)

(MaIUCT is a frame based on

(ii)

m

I

D and

(see page 17),

is a mapping which assigns to each constant c

I into hIu

a function from

.

Thus, a g-model of h1L

T

differs from a g-model of IL only in that no value

M .is assumed to exist in the former, and indeed none will exist function V in general. However, for formulas of ML the assumption that VM exists is T

unnecessary. Specifically, let M = (Mu, m ) u C T

be a g-model of MLT based

I , a C A s ( M ) . Then defined by recursion for every atomic term Au in notion on D and

I

, and let i

M, i , a sat A

A

€

V.M

(A,) C Ma can be %,a the usual way, and the

,

is a formula of MLT, can then be defined as follows:

M, i, a sat A if and only if term of type t ,

VM

l,a

(A) = 1 , when A

M, i, a sat [B E C] if and only if VM (B) l,a and C are atomic terms, Usual satisfaction clauses for

M, i, a sat Vxa

A

-

,

A

,

--f

,

V

=

is an atomic

M V. (C) , when 1,a

B

,

if and only if M, i, a(x/X)

sat A

for every

XCMa’

M, i , a sat O A

-

if and only if M, j , a sat A

for all

j C I

We could take and + as the only connectives in MLT, defining the other connectives in terms of these two as usual. However, the formulas [A A B] and [A v B] would then have two readings, one in IL and another in MLT. This would unnecessarily complicate the later exposition.

.

MODAL T-LOGIC

43

This definition of satisfaction coincides with that given in §3 in the case when M

is a g-model of IL and A

is a formula of MLT, s o the same nota-

tion can be used without confusion. be either an atomic term or a formula of MLT. We say A

Let A

is

modally closed if it is modally closed as a term of IL. For atomic terms or atomic formulas A , this holds just in case A contains no constants and no occurrence of "

.

The formula

is modally closed for any f o r -

A

mula A , and the set of modally closed formulas of MLT is closed under the

-

.

connectives , A , + , v and the quantifier Vxa If A is a modally closed formula we can write M, a sat A instead of M, i, a sat A , as earlier. Corresponding to o u r generalized semantical notions for IL, WE have for formulas of MLT the notions r p A in MLT, A in MLT, g is g-satisfiable in MLT, obtained from the corresponding notions in and

hg

$ 3 (page 18) by replacing g-model of IL by g-model of MLT throughout. It A in IL does not imply rI= A in should be noted, however, that r g g MLT, even for formulas of ML

T'

The Theory MLT.

We can give an intrinsic axiomatization for MLT, as

follows: Axioms of MLT. AS1.

A , where A

A2.

Xt

A3.

-Xt

AS4.

Vxa

AS5.

Vx(, A ( x )

is tautologous in

[Yt+X+Y

-+

-

,

A

,+ ,

v ,

I ,

[-Yt+X.Y],

-+

[A --* B] -+ [A + Vx B] , where x free in the formula A , -+

A(B,)

, where

A(B)

is any variable not occurring

comes from the formula A ( x )

by re-

placing all free occurrences of x by the atomic term B , and (i) B is free for x in A ( x ) , and either (ii) no free occurrence of in A(x) lies within the scope o f 0 , o r else (ii') ally closed,

x

AS6

A7.

I

B

is mod-

Vx [ A x e Ba P ] ~ + A E B , where x is any variable not occura UP ring free in either of the atomic terms A , B ,

x

a

s x

a'

44

ALTERNATIVE FORMULATIONS OF SL

AS8.

Ba E Ca + mula A(xa)

+ A(C) 3 , where A(B) , A(C) come from the forby replacing all free occurrences of x by the atomic

[ A(B)

terms B , C respectively, and (i) B and C are free for x in , and either (ii) no free occurrence of x in A(x) lies

A(x)

within the scope of 0 closed, AS9.

, or else (ii') B and C are both modally

O A + A ,

AS10. D [ A - + B ] *

[ O A + O B ] ,

AS11.

A -0 A , if A

is modally closed,

A12.

0 [ I f s a D Ygsa]

+

f

5

g

.

Rules of Inference. [A * B]

to infer B ,

R1.

From

R2.

From A

to infer vxa A ,

R3.

From A

to infer 0 A

and A

.

We state without proof the following THEOREM 5.1 (Generalized Completeness Theorem for MLT) (i) (ii)

fg A

r

(iii) C

$6.

hg A

in MLT if and only if in MLT if and only if

1-

A

r I-

in MLT, A

in MLT,

is consistent in MLT if and only if Z is g-satisfiable in MLT'

Extensions of IL and MLT Since an arbitrary frame and meaning function determine a general model

of MLT, it is natural to seek a set of conditions, formulated in the language of MLT. which will be satisfied in exactly those g-models of MLT which are also g-models of IL. For this purpose the theory IL proves to be slightly too weak, however, so we now consider a natural axiomatic extension of it.

EXTENSIONS OF IL AND MLT

45

Intensional Logic with Descriptors. In both IL and MLT we introduce the abbreviations [A

B]

++

3!xa A where x‘

for

[A

+

B]

[B

A

for 3x; VxZ [ A

+

++

A] , x

=

is the first variable of type

curring free in the formula A

.

XI

],

a

We denote by

distinct from x and not ocDe the formula

which we call the axiom of description for individuals. Since De is both closed and modally closed, it will be either true or false in any g-model of IL, independent of the index and assignment. We dendte by

IL+D the theory obtained from IL by adding De as a

new axiom, and we write I- A in IL+D when the formula A is provable in this theory. By a general model (g-model) of IL+D we understand a g-model of IL in which De is true. From Theorem 3.3 it is easy to see that generalized completeness extends to the logic IL+D That

.

IL+D is a natural extension of IL is evidenced by the fact’ that

many familiar validities of type-theoretic predicate logic depend for their proof on the axiom De

.

The intuitive content of De is just the assertion that there exists a function on sets of individuals, whose value for

any singleton is its unique member. Thus, De is valid in IL, as are the additional description principles below:

The formula Da is the analogous principle of description for objects of is an intensional principle of description for such type a , while

ie

objects. In particular, asserts the existence of a function F from properties of individuals to individual concepts, such that for any property G , if it is necessarily true that G is satisfied by exactly one object, then F(G) is the concept of the unique individual satisfying G

Observed in Henkin [1963], p. 343.

.

ALTERNATIVE FORMULATIONS OF IL

46

The following result, which generalizes a similar result for tvpe the-

, ia to

o r y due to Church, shows that we need not add the formulas Da

IL+D

as

axioms.

LEMMA 6.1. For every type a 6 T

, the formulas Da and

are

provable in IL+D .

Proof: We first use generalized completeness to show that the formula [Da .+ fia] is provable in IL for each a . Let M = (Ma, be a g-model of IL such that M sat Da . By a rewrite of bound variables (for notational convenience only), we have

and therefore we have

for some F ' E M (at)a , No index i need be specified, since the formula in question is modally closed. We now let

is of type

where f

, and therefore M sat

(s(at))(sa)

.

This proves

the assertion. It therefore remains only t o show that Da is provable in IL+D for We use the generalized completeness of IL+D : Suppose M = is a g-model of IL+D ; we show that M sat Da by induction ( M a , m)acT on the type a . The case a z- e is immediate. For a = t we let

every a

.

M

F = V ( X g t t [ g ~ X Xt X t

1

M(tt)t

'

and verify that M; F

sat

btt[ 3!xt

from which it follows that M

[gxl

-+

g[f(tt)t

91

1

I

sat Dt (Cf. Henkin [1963], p. 3 2 8 ) .

EXTENSIONS OF I L AND MLT

Now assume t h a t

sat

M

Dp ; we show t h a t

47

sat

M

Dap

. First,

by a

rewrite o f bound v a r i a b l e s ,

f o r some

F ’ C bl

where

Vg

sat

M; F

i s of t y p e

f

.

CPt)P

Now l e t

[ 3!h

(ap 1t

b e t h e term

A(f’)

aB

((ap)t)(ap)

[ghl

+

g[fgl

1 ,

, and hence t h a t M sat

Dap

(Cf. Church

[1940], p . 6 2 ) .

M

F i n a l l y , assume t h a t

sat

Da ; we show t h a t

M

sat

w r i t e o f bound v a r i a b l e s , we have 3 f ’( a t ) a VgLt

sat

M

I 3!Xa

[s’xl

+

I ,

g”f’g’1

and hence

sat

M ; F’

F’ C M

f o r some

A(f’)

M

V,,(A(f’))

]

g’[f’g’]

--*

be t h e term

*

X Y a 3hsa [ gh

( ( s a )t ) (sa)

F =

Let

(at)a .

^[

Xg(sa)t of type

[ 3 ! x a [g’x]

VgLt

Y

’h

1 1

s[fgl

I

f

, and l e t C

M

( ( s a )t ) ( s a )

‘

One v e r i f i e s e a s i l y t h a t

M; F

sat

Vg

( s a )t

[ 3!hsa

[ghl

+

>

DSa

.

By a r e -

ALTERNATIVE FORMULATIONS OF I L

48

f

where

( ( s a ) t ) ( s a ) , so t h a t

is of type

sat

b1

.

DSa

This proves

Lemma 6 . 1 . In order t o characterize t h e gencral

Modal T-Logic w i t h Replacement. models o f

IL+D i t i s n e c e s s a r y t o add t o t h e t h e o r y h1L c e r t a i n n a t u r a l T p r i n c i p l e s o f comprehension f o r f u n c t i o n s , which we c a l l r e p l a c e m e n t p r i n c i p l e s s i n c e t h e y b e a r a formal resemblance t o t h e replacement schema o f a x i o m a t i c s e t t h e o r y . The schemata i n q u e s t i o n a r c t h e f o l l o w i n g : Vxa 3 ! y

Ra’P’A:

P

A

3f

+

Vxa Vy

aP

f i r s t v a r i a b l e of t y p e : 0 3!xa A

0 Vxa [ x

3f

-+

sa v a r i a b l e of t y p e sa

We d e n o t e by

P

a@

[ y

=

fx

A ]

-+

, where f

i s the

n o t o c c u r r i n g f r e e i n t h e formula 5

‘f

A ]

-+

, where

f

A

,

is t h e f i r s t

n o t o c c u r r i n g f r e e i n t h e formula

A

.

t h e t h e o r y o b t a i n e d from bfLT by adding a l l i n s t a n c e s

FILT+R

o f t h e replacement schemata as new axioms, i . e . , by a d d i n g a l l f o r m u l a s Ra’P’A

and

e l ) of

MLT+R

i a I A

where

i s any formula of MLT. A g e n e r a l model (g-mod-

A

i s a g-model o f MLT i n which a l l i n s t a n c e s o f t h e r e p l a c e -

mcnt schemata a r e t r u e , i . e . , s a t i s f i e d by e v e r y i n d e x and a s s i g n m e n t . A s e a r l i e r , i t f o l l o w s from Theorem 5 . 1 t h a t g e n e r a l i z e d completeness e x t e n d s t o t h e p r e s e n t l o g i c , s o t h a t , i n p a r t i c u l a r , a formula able in

i f and o n l y i f i t i s g - v a l i d i n

blLT+R

e v e r y g-model o f

LEMMA 6 . 2 . 1 . t h e formulas

Proof: g-model o f RaJPjA

.

LILT+R

a,X,Y

and

a

,P E

ia’Aa r e

T

and e v e r y formula

provable i n

IL+D

W e u s e g e n e r a l i z e d completeness. Let IL+D , and l e t

Assume t h a t

t h e r e e x i s t s a unique

,

.

For a l l t y p e s

RaJPPA

blLT+R

o f MLT i s provi.e., true in

A

i

M, i , a

Y C M

P




b' .

So i t re-

; w e show t h a t

0 3 ! x a ['gx]

,

we have by r e p l a c e m e n t M; G

3hsa 0 tlx

sat

[ x

'h

5

from which i t e a s i l y f o l l o w s t h a t

,

]

'gx

-+

o t h e r hand, t h i s c o n c l u s i o n i s immediate when

.

3!h A(g,h,h')

sat

M; G,H'

sat

M; G

On t h e

- 0 3 ! x a ['gx]

T h i s completes t h e p r o o f o f Lemma 6 . 2 . 3 . We can now s t a t e t h e main r e s u l t o f t h i s s e c t i o n : THEOREM 6 . 2 .

The l o g i c s

and

IL+D

have e x a c t l y t h e same gen-

MLT+R

e r a l models. P r o o f : I n view of C o r o l l a r y 6 . 2 . 2 and Lemma 6 . 2 . 3 , i t s u f f i c e s t o ___ p r o v e t h e f o l l o w i n g : Given any g-model M = (Mu, m)acT of MLT+R , t h e r e exists in

M

a value function

VM

s a t i s f y i n g t h e r e c u r s i v e c o n d i t i o n s (1)

through ( 7 ) on page 13. We f i r s t d e f i n e , f o r each term occurring f r e e i n A Eq ( x )

A

( x

,

equals

.

x

Aa

is

v

(ii)

Aa

is

ca , xu

(iii)

Aa

is

[BPuCP]

a'

x

xa

not

t o g e t h e r w i t h t h e f r e e v a r i a b l e s of

The d e f i n i t i o n i s by r e c u r s i o n on

(i)

o f I L and each v a r i a b l e

A )

o f MLT, whose f r e e v a r i a b l e s a r e Aa

Au

a formula

a

d i s t i n c t from

An

:

vu

.

a r b i t r a r y . Then

, y,

not f r e e i n

A

Eq (x) Aa

A

Then

.

Eq (x) is

Let

f

is

[c

E

x]

Pa

' xP

[v

=

x]

.

. be the f i r s t

v a r i a b l e s o f t h e i r r e s p e c t i v e t y p e s which a r e d i s t i n c t from A n o t f r e e i n A , and l e t Eq (y) b e t h e formula

y

and

ALTERNATIVE FORMULATIONS OF IL

52

B

3 f 3 x [ Eq ( f ) Aa

Xx

is B

,

y

A

fx ] .

5

not f r e e i n

fa

B

P Y ’

.

Aa

y which i s d i s t i n c t from

able of type in

C

Eq ( x )

A

A

and l e t

Eq ( f )

B

vx 3 y [ Eq (y)

fx

A

y be t h e f i r s t v a r i Y and does n o t o c c u r f r e e

Let x

b e t h e formula

.

y ]

f

, xa n o t f r e e i n Aa . Let y be t h e P ’ zP P which a r e d i s t i n c t from x and A do n o t occur f r e e i n A , and l e t Eq (x) be t h e formula

Aa

-= C

[B

is

f i r s t d i s t i n c t v a r i a b l e s of t y p e

B

C

3 y 3 z [ E q ( y ) A E q ( z ) A [ ~ - y a z ] ] . Aa

^B

is

of type

fa

P ’

not free in

Aa

.

Let

x

P which does n o t o c c u r f r e e i n

P

be the first variable

B ; and l e t

A

Eq ( f )

be

t h e formula

n 3 x [ Eq B ( x ) (vii)

Aa

, xa

-Bsa

is

x

A

able of type

sa

P

‘f ]

,

not free in

.

Aa

Let

which is n o t f r e e i n

fsa

be the f i r s t vari-

B , and l e t

A

Eq ( x )

be the

formula

B

3 f [ Eq ( f ) LEMMA 6.2.4.

A x

‘f ]

G

For e v e r y term

occurring f r e e i n

. o f IL and e v e r y v a r i a b l e

Aa

A

, t h e formula 3!xa Eq

A

(x)

xa

i s provable i n

not

MLT+R

.

The p r o o f i s s t r a i g h t f o r w a r d , u s i n g g e n e r a l i z e d completeness and i n d u c t i o n on t h e term

Aa

an assignment that

VM

a

M; i ; a,X

in

over sat

not occurring f r e e i n function i n

M

MLT+R

M = (Ma, m)aET i s a g-model of

Now suppose t h a t value function

*

as f o l l o w s : Given a term

M

, an

We d e f i n e a

index

i

and

M

, we l e t V.

M

A

E q (x) A

Aa

.

.

,

(A,) b e t h e u n i q u e X € Ma s u c h i,a where x i s t h e f i r s t v a r i a b l e o f t y p e a

I t is r o u t i n e t o v e r i f y t h a t

VM

i s a value

, i . e . , s a t i s f i e s t h e r e c u r s i v e c l a u s e s (1) through ( 7 ) o f

page 13. We omit t h e d e t a i l s .

NORMAL FORMS

53

We can s t r e n g t h e n t h e de-

I n t e n s i o n a l Logic w i t h t h e Axiom of Choice. scription principles

Da ,

6"

of

IL+D by r e p l a c i n g t h e q u a n t i f i e r

i n t h e i r a n t e c e d e n t s by t h e weaker e x i s t e n t i a l q u a n t i f i e r

3!

3 , obtaining

t h e r e b y t h e f o l l o w i n g axioms o f c h o i c e :

These p r i n c i p l e s a r e v a l i d i n IL, and i f we add them t o IL as new axioms

we o b t a i n a t h e o r y which we d e n o t e by o n l y t h e formulas

IL+Ac

.

I n f a c t , i t s u f f i c e s t o add

, s i n c e t h e i n t e n s i o n a l axioms i c U can b e shown t o

Aca

f o l l o w i n IL, a s i n t h e proof o f Lemma 6 . 1 ; however, i t does n o t s u f f i c e h e r e t o add t h e formula

Ace

a l o n e . By a g e n e r a l model (g-model) o f

IL+Ac

we u n d e r s t a n d a g-model o f IL i n which t h e s e axioms o f c h o i c e a r e a l l t r u e .

A s b e f o r e , g e n e r a l i z e d completeness e x t e n d s t o t h i s l o g i c , and c l e a r l y t h e description principles

Da

, '6

a l l hold i n

.

IL+Ac

In a similar way w e c a n

Modal T-Logic w i t h Replacement and Choice. s t r e n g t h e n o u r e a r l i e r replacement p r i n c i p l e s

Ra'P'A

and

t o give

t h e f o l l o w i n g p r i n c i p l e s o f replacement and c h o i c c : R c a S P r A: icaSA

:

fixa 3 y

A

P

+

Vx fiy [ y

3f

s

fx

--*

A

UP

0 3xa A

+

where i n each c a s e

3 f s a 0 fixa [ x z 'f

--f

A ]

3 , ,

f is t h e f i r s t v a r i a b l e of i n d i c a t e d t y p e which d o e s

n o t o c c u r f r e e i n t h e formula

A

. The t h e o r y MLT+Rc comes from MLT by

adding a l l i n s t a n c e s o f t h e s e schemata i n MLT t o t h e axioms o f MLT, g e n e r a l model (g-model) o f

ML +Rc

T

and a

i s d e f i n e d i n t h e obvious way. By i n -

s p e c t i n g t h e p r o o f s of Lemmas 6 . 2 . 1 , 6 . 2 . 3 and Theorem 6 . 2 one can p r o v e : THEOREM 6 . 3 .

The l o g i c s

IL+Ac

and

MLT+Rc have e x a c t l y t h e same

g e n e r a l models.

$7.

Normal Forms The i d e a s o f t h e p r e v i o u s s e c t i o n can b e used t o o b t a i n v a r i o u s normal

forms f o r f o r m u l a s o f I L .

ALTERNATIVE FORMULATIONS OF IL

54

THEOREM 7.1. For every formula A of IL we can effectively find a formula A' of MLT with the same constants and free variables, such that [A s A'] is provable in IL. ____ Proof: For a term

Aa

of IL and a variable xa

not free in Aa , let

EqA (x) be the formula of ML defined in the proof of Theorem 6.2. One T easily shows by induction on Aa : LEMMA 7.1.1. If xa A Eq (x)

E

[A

h

is not free in Aa

then the formula

x]

is provable in I L . Now suppose A

is a formula of IL and let x be the first variable

of type t which does not occur free in A

.

Then we can prove in IL the

formu1a A

3x[ [ A s x ] h x ] ,

s

so by Lemma 7.1.1 we can also prove

A

A 3x[Eq(x)Ax],

B

and the right-hand side of this equality is the desired formula A' COROLLARY 7.2. Let A be a formula of IL, and let A'

.

be the corre-

sponding formula of MLT, as above. Then: (i)

(ii)

11-

A

in

IL+D if and only if

A

in IL+Ac if and only if

1-

A'

1-

A'

in MLT+R , in MLT+Rc

.

Proof: By Theorems 6 . 2 , 7.1 and generalized completeness. A formula A of ML is a prenex formula if it consists of a string of T quantifiers followed by a quantifier-free matrix; i.e., A has the form 0 1 n-1 Q,x Q,x ... QnW1x M , where each Q, is V o r 3 , and the formula M

contains no quantifiers. A

is a Skolem formula if in addition no uni-

versal quantifier precedes an existential quantifier in the prefix, S O that A has the form 3xo 3x1 ... 3xm-l Vxm ... Vxn-' M , where the formula M is quantifier-free.

55

N O R M L FORMS

For e v e r y formula

THEOREM 7 . 3 .

-

Skolem formula [A

w i t h t h e same c o n s t a n t s and f r e e v a r i a b l e s , such t h a t

A''

is p r o v a b l e i n

A*]

Proof: -

-

.

ML,+Rc

B

Given a prenex formula

o f MLT, we s a y t h a t

is provable i n

A

[A

B]

ML +Rc

interchange of equivalents holds f o r t h e l o g i c LEMMA 7 . 3 . 1 .

Vx Vy [ y

Vxa 3y

(ii)

0 3xa A

c--f

3 f s a 0 Vxa [ x

Vxa A

++

Vxa 0 A

(iii)

A

a r e provable i n

If

+-+

aP

1

'f

5

-.

fx

MLT+Rc

.

i s a prenex

and

-+

,

A ]

A ]

,

where i n ( i ) and ( i i )

A

.

MLT+Rc

does n o t o c c u r f r e e i n

Proof:

A

A l l i n s t a n c e s o f t h e schemata

(i)

P

and

B

B have t h e T same c o n s t a n t s and f r e e v a r i a b l e s . W e observe t h a t t h e usual p r i n c i p l e of

form o f

if

o f MLT we can e f f e c t i v e l y f i n d a

A

,

f

i s a v a r i a b l e which

For ( i ) and ( i i ) we o n l y need t o e s t a b l i s h t h e c o n v e r s e s o f t h e

p r i n c i p l e s o f r e p l a c e m e n t and c h o i c e ; b u t t h e s e a r e immediate. ( i i i ) i s t h e s o - c a l l e d Barcan f o r m u l a , which i s e a s i l y proved u s i n g g e n e r a l i z e d completeness. LEMMA 7.3.2.

If

-

o f t h e formulas

and

A

A , (A

A

are p r e n e x formulas o f MLT t h e n f o r e a c h

B B]

,

[A

--f

B]

, and

[A v B]

w e can e f f e c t i v e l y

f i n d a p r e n e x form. Proof:

As usual.

LEMMA 7 . 3 . 3 .

If

Proof: Suppose A

i s a p r e n e x f o r m u l a o f MLT t h e n we can e f f e c t i v e l y

A

f i n d a p r e n e x form o f

0A

.

By i n d u c t i o n on t h e number o f q u a n t i f i e r s i n t h e p r e f i x o f A . 0 n- 1 Qox . . . QnW1x M . I f n = 0 then 0 A i s q u a n t i f i e r -

is

f r e e and hence i n prenex form. Otherwise we can c l e a r l y assume t h a t t h e 0 , xn- 1 a r e a l l d i s t i n c t , and we have two c a s e s : x ,

...

variables

Case 1. (1)

OA

Q, c--f

is OVx

V 0

. B ,

Then i n

ML +Rc we can p r o v e

T

ALTERNATIVE FORMULATIONS OF IL

56

0 O A - V x O B ,

(2)

1 n- 1 Q x ... Q n - l ~ bl . But 1 B is a prenex formula w i t h fewer q u a n t i f i e r s t h a n A , s o by t h e i n d u c t i o n 0 h y p o t h e s i s 0 B has a p r e n e x form C , and by ( 2 ) t h e formula Vx C w i l l

by Lemma 7 . 3 . 1 ( i i i ) , where

i s t h e formula

B

h e t h e d e s i r e d prenex form f o r

Case 2 .

is

Qo 1

3

.

.

0A

Suppose

xo

i s of t y p e

a

, and w r i t e

for the

B

n- 1 Qn-l~ M

,.

. . By Lemma 7 . 3 . 1 ( i i ) , ( i i i ) and r e w r i t e o f 1 bound v a r i a b l e s , we can prove i n hlLT+Rc : Q x

formula

(1)

0A

(2)

OA

(3)

nA

where

f

-

+-.

0

0 3xa B , If

sa

OVx

0 3fvx n

0

[ x xo

0

z'f

= 'f

B ] ,

-+

1 ,

B

+

i s t h e f i r s t v a r i a h l e of t y p e sa which d o e s n o t o c c u r f r e e i n n- 1 0 1 B W r i t i n g M(x , x , . , . ,x ) f o r t h e m a t r i x M , we can choose new 1 n- 1 variables y , ... , y , d i f f e r e n t from xo and f , s o t h a t i n MLT+Rc

.

we can p r o v e (4)

0A

0

+-.

If Vx 0 [ xo

s

'f

QIY

-+

1

. .. .

n-1 0 1 Q n - l ~ M(x , Y ,

...

1 ,

,yn-')

By t h e i n d u c t i o n h y p o t h e s i s , t h e r e f o r e , we can f i n d a prenex f o r m u l a

C

such t h a t t h e formula OA

-

i s provable i n

3fVx

0

C

MLT+Rc , and t h i s g i v e s t h e d e s i r e d p r e n e x form f o r

By Lemmas 7 . 3 . 2 , 7 . 3 . 3 and a s t r a i g h t f o r w a r d i n d u c t i o n on LEMMA 7 . 3 . 4 .

For e v e r y formula

A

A

0A

we h a v e :

o f MLT we can e f f e c t i v e l y f i n d a

prenex form. To prove Theorem 7 . 3 i t c l e a r l y s u f f i c e s t o combine Lemma 7 . 3 . 4 w i t h t h e following r e s u l t :

.

NORMAL FORMS

57

LEMMA 7.3.5. Let A be a Skolem formula with n existential quantifiers in its prefix. Then we can effectively find a Skolem form of Vxa A having at most

0

n

existential quantifiers in its prefix.

Proof: By induction on . . . 3yn-1 R , where B

the number n

.

Suppose A

is of the form

is the formula V z o ... Vzm-' M . Clearly we 0 m- 1 can assume n > 0 and x , y , ... , z distinct, by dropping any vacuous quantifiers. Using Lemma 7.3.1 (i) and rewrite of bound variables, we

3y

can prove in PILT+Rc the following formulas: (1)

Vxa A

(2)

Vxa A

++

0 1 n- 1 Vxa 3yp 3y ... 3y R , 0 0 1 3y 3f Vx Vy [ y I fx -+

UP

...

3yn-l B ] ,

f is the first variable of type ap not occurring free in B . 0 0 1 Writing M(x,y , . .. , z , ... ) for M , we can choose new variables u , . . . , un-1, v0, . . . , vm- 1 different from x , yo and f , so that in

where

MLT+Rc we can prove Vx A where C

++

3 f Vx Vy

0

C ,

is the formula

...

3u

3un-l vvo

... vvm-1

[ y

0

= fx

+

M(x,y

0

1

,u

,...,v0,... 1 I

Since C is a Skolem formula with n-1 existential quantifiers, two applications of the induction hypothesis give a Skolem formula C' with at most

n-1 existential quantifiers in its prefix, such that Vx A

++

3f C'

is provable in MLT+Ilc

.

Thus 3f C '

is the desired Skolem form of vx A

.

COROLLARY 7.4. For every formula A of IL we can effectively find a Skolem formula A* of MLT with the same constants and free variables, such that

[A

= A':;] is provable in IL+Ac

.

Proof: Theorems 6.3, 7.1 and 7 . 3 . __I_

REMARK: Dual to this existential Skolem form we have a universal Skolem form, in which no existential quantifier precedes a universal quantifier. The corresponding theorems follow from Theorem 1 . 3 and Corollary 7 . 4

ALTERNATIVE FOKMULATIONS OF IL

58

by c o n s i d e r i n g t h e e x i s t e n t i a l Skolem form of

N

A

.

I t should be noted

a l s o t h a t t h e m a t r i x of a p r e n e x formula can b e p u t i n v a r i o u s modal normal forms1 on t h e b a s i s o f t h e 5 5 axioms of MLT.

$8, Two-Sorted Type Theory As we observed i n $ 2 , t h e cap o p e r a t o r

A

a c t s a s a f u n c t i o n a l ab-

s t r a c t o r o v e r i n d i c e s , a l t h o u g h t h e grammar o f IL l a c k s v a r i a b l e s o v e r i n a l o n e i s n o t a t y p e . T h i s omission i s r e a s o n a b l e , s i n c e IL

s

dices since

was i n t e n d e d as a formal l o g i c w i t h i n t e n s i o n a l f e a t u r e s c l o s e t o t h o s e o f n a t u r a l language, and i n n a t u r a l language we do n o t r e f e r e x p l i c i t l y t o c o n t e x t s of use; i n d e e d , i f we d i d r e f e r t o them e x p l i c i t l y t h e r e would be l i t t l e j u s t i f i c a t i o n f o r t h e Carnap a p p r o a c h . From a formal p o i n t o f view, however, i t i s n a t u r a l t o c o n s i d e r i n t e r p r e t i n g IL i n an e x t e n s i o n a l t h e o r y of t y p e s having two s o r t s of i n d i v i d u a l s . We c a l l t h i s l o g i c Two-Sorted Type Theory, and d e n o t e i t by Ty2.

Types.

T2

The s e t

(i)

e , t , s C T

(ii)

a

, p E T2 T

Thus, t h e s e t

o f t y p e s o f Ty2 i s t h e s m a l l e s t s e t such t h a t :

2 ’

(G,P)

imply

E T2

.

of t y p e s of IL i s c o n t a i n e d i n t h e s e t

P r i m i t i v e Symbols.

For each

T2

.

a C T 2 , we admit v a r i a b l e s

0

’ and non-log c a l c o n s t a n t s 0 ca

of type

]

.

1

ca

2

I

ca

,

...

, which we i d e n t i f y w i t h t h e c o r r e s p o n d i n g symbols of

a

the type and

’

belongs t o

T

.

I L when

We a l s o have t h e improper symbols

E

As b e f o r e , we d e n o t e t h e f i r s t n i n e v a r i a b l e s of t y p e

a

G

,X , [ by:

E.g., t h e modal c o n j u n c t i v e normal form d e s c r i b e d i n Hughes and C r e s s w e l l [1968], pp. 54-56.

TWO-SORTED TYPE TilEORY

,

xa

Terms. -

Y,

f

2,

9

ua

The sets Tm

va

9

2 ,a

wa

7

fa

9

9,

I

59

.

ha

J

of terms of Ty2 of type

a

are characterized

recursively:1 (i)

Variables and constants of type

(ii)

A C Tm

2,aP

(iii) A 6 Tm

, B c Tm

2 ,a

implies Xx

A ,B

(iv)

TmZ,,

imply

A

C

Tm

[A

e

B]

Generalized Semantics. Let D

g&

for Ty2 based on I) ~

[AB] c Tm2,p ,

imply

298

belong to Tm2,a ,

a

’

2,ap

Tm

C

and

2,t

.

be non-empty sets. By a

I

I we understand an indexed family

frame

(E.fa)aET

2 of sets, where

,

(i)

M

(ii)

E.1 = 2 = [O,l}

(iii)

M

(iv)

bl

= D

t

=

aP

,

I , is a non-empty subset of L1

E.ffl

P

’

The frame is standard if the inclusion in condition (iv) can be replaced

and

by equality. A general model (g-model) o f Ty2 based on D satisfying: system M = (M a , m)aCT

I

is a

2

(i)

is a frame for Ty2 based on D

(bfaIaCT

and

I ,

2 ca ,

(ii)

m(ca) E Ma

(iii)

There exists a function VM which assigns, to each .assignment a

for each constant

M

over PZ and each term Au , a value Va(Aa) that the following conditions hold:

C Ma

, in such a way

We employ freely in this section various of our notational conventions for the logic IL.

ALTERNATIVE FORMULATIONS OF IL

60

(3)

V a (Aap Ba 1 = va(Aap)[Va(Ba)I

(4)

V (Axa A )

P

,

the function F

=

whose value at X

on Ma

is equal to Val(Ap) , where a' = a(x/X)

V

(5)

(Aa : Ba)

,

, and 0 otherwise.

1 if Va(Aa) = Va(Ba)

=

C Ma

If the underlying frame is standard then condition (iii) is unnecessary, and M

is called a (standard) model of Ty2. As before, a formula is a term

. The notions M , a sat A , A in Ty2> A in g Ty2, and C is g-satisfiable in Ty2, are defined in the usual way, as are their standard semantical counterparts, e.g., the notion r I= A in T y 2 . Also, we employ in Ty2 the definitions of the logical operators T , F , , A , -t , v , V , 3 given in $2.

bg

A of type t

-

The Theory Ty2. Axioms of Ty2. gtt F

Al.

gtt T

A2.

x a ~ Y a+

143.

b'x

AS4.

a

A

[ f

aB

x

f

=

(Xx, Ap(x))

VX X

z

at

gapx Ba

t

f

1

t [gx] , y ,

at 5

[f

f

sl ,

Ap(Ba) , where A (B,)

B

comes from Ap(xa)

replacing all free occurrences of x by the term B , and free for x in A(x) .

by

B is

Rule of Inference. R.

A' and the formula B to infer the formula B ' , where comes from B by replacing one occurrence of A (not immedi-

From Aa B'

s

ately preceded by X ) by the term A'

.

We have generalized completeness for the two-sorted logic Ty2, as a trivial extension of Henkin's result for ordinary type theory. It i s worth noting, in particular, that the schemata (i)

Vxa A(x) mula A(x)

A(Ba)

-+

,

, where the term B

is free for x

in the for-

TWO-SORTED TYPE THEORY

B = C a u t h e term

(ii)

A (B)

--t

P

Ap(xU)

s

A (C)

i3

,

, where

and

B

61

C

are f r e e f o r

x

in

a r e p r o v a b l e i n Ty2 w i t h o u t f u r t h e r r e s t r i c t i o n (Cf. d i s c u s s i o n a t t h e end of § Z ) .

We d e n o t e by

t h e t h e o r y o b t a i n e d from Ty2 by adding as new ax-

Tyz+D

ioms t h e formulas [gxl

+

s[fsl

1

bst [ 3 ! x s [gxl

+

g[fgl

1

De:

3f ( e t ) e vget

DS:

3f ( s t ) s

[ 3!x,

9

*

A s b e f o r e we can prove:

LEMMA 8 . 1 . Da :

In

Ty2+D t h e formulas

bat [ 3!xu [gxl

3f(ut)a

a r e p r o v a b l e f o r each t y p e

a E T2

I n t e r p r e t a b i l i t y o f IL

& Tyz.

t h e t r a n s l a t e of

Aa

(v)

[A

=

(vi)

[^A,]*

(vii)

["Asa]" =

a

= B ] a =

i n Ty,,

. For each term

of I L we d e f i n e

A:,

as f o l l o w s :

;: [A

:*;

G B ] ,

,

Axs A*

.

[A"xs] A:

are j u s t the f r e e variables of

i n some c a s e s , w i t h t h e s i n g l e v a r i a b l e cn

Aa

L

The f r e e v a r i a b l e s o f

constants

1

s[fsl

+

such t h a t

cP

occurs i n

x A

SP

o f IL, we d e n o t e by

r*

t h e set of f o r m u l a s

. .

Aa

The c o n s t a n t s o f If A"

together, A''

are the

i s a set o f f o r m u l a s for A €

r .

62

ALTERNATIVE FORMULATIONS OF IL

*

THEOREM 8.2. The translation of A semantics. Precisely, let mula of IL. Then

b

(i)

A

and C

A

in IL if and only if

preserves the standard

be sets of formulas of IL, A

I=

in IL if and only if

r

(ii)

r

into A

A"

in Ty

r* b

A*

a for-

2' in Ty

2 '

-L

satisfiable in IL if and only if 2

(iii) 2

Proof: -

satisfiable in Ty2.

(i) and (ii) follow from (iii), which in turn follows from the

following LEMMA 8.2.1. Let D and >L

I be non-empty sets, and suppose that M

=

j,

(Ma, m)acT , Mi' = (Ma, m )aCT2 are standard models o f IL and TyZ, respectively, based on D and = m"(cn

that m(c:) Aa

sa

)

I , s o that Ma = M[;

for each constant

of IL, every assignment a over M

where a* Proof: -

c:

for a 6 T

. Suppose also

of IL. Then for every term

and index i 6 I :

is the partial assignment a(x,/i)

over M"

.

Straightforward induction on Aa

Less obvious than Theorem 8.2 is the fact that the translation of A into A* provides a relative interpretation, in a sense close to that of Tarski, Mostowski and Robinson [1953], of the theory . Precisely:

IL+D in the theory

Ty2+D

THEOREM 8.3. Let

r

and Z

be sets of formulas and let A b e a for-

mula of IL. Then:

(i)

I-

(ii)

r 1-

(iii)

2"

Proof:

1-

in IL+D implies

A A

in IL+D implies

consistent in Tyz+D

A''

r': 1-

in T Y ~ + D,

:'A

implies C

in T ~ ~ , + D consistent in IL+D

.

Again (i) and (ii) follow from (iii). By generalized complete-

ness it suffices to show:

TWO-SORTED TYPE THEORY

LEMMA 8 . 3 . 1 ,

fiable in Proof: -

IL+D

Let

Z'~ i s g - s a t i s f i a b l e i n Ty2+D , t h e n Z i s g - s a t i s -

If

. m")uFT2

M" = (M:,

is s a t i s f i a b l e , based on s e t s of I L based on

D

and

b e a g-model o f

D

I

and

by l e t t i n g

I

VM

a'

i s t h e p a r t i a l assignment

is a value function i n

(t)

M, i, a

sat

M

Z" i s s a t i s f i a b l e i n M'

sat

7",

M

i s a g-model o f

[De]*

is

D e , and

,

IL+D Milr

a 6 T

.

a(xs/i)

Mil',

a(xs/i)

.

De

2 : '

a' m)aFT

and p u t t i n g

let

at

I t i s e a s i l y checked t h a t

sat

A"

A

, where a

of I L

. F As(M)

a"

Mili,

i F I.

and

I t t h e r e f o r e remains o n l y t o show t h a t

, i.e., that M sat sat

for

M = (M

and i n f a c t we can assume t h a t

Z

sat

Mi

i n which

and c l e a r l y f o r e v e r y f o r m u l a

of t h e form

a'

But t h e n by ( t ) , M, i , a

Ty2+D

D e f i n e a g-model

i F I

i f and o n l y i f

A

But

f o r some

,

,

bla =

. For a E As(E.1) and

where

63

since

Ma

De

.

But i t i s c l e a r t h a t

i s a g-model o f

Ty2+D

.

By

(t). t h e p r o o f i s t h e r e f o r e complete. We conclude w i t h two remarks. F i r s t , i t i s p o s s i b l e t o i n t e r p r e t t h e theory

Ty2+D i n t h e t h e o r y

IL+D

i n a similar s e n s e , u s i n g n o t i o n s t o

be developed i n t h e n e x t c h a p t e r ; we s h a l l r e t u r n t o t h i s q u e s t i o n b r i e f l y i n $13. Second, each t h e o r y i s s t r o n g l y i n t e r p r e t a b l e i n t h e o t h e r , i n t h e s e n s e t h a t t h e i m p l i c a t i o n s i n Theorem 8 . 3 , f o r example, c a n a c t u a l l y b e s t r e n g t h e n e d t o e q u i v a l e n c e . We omit t h e v e r y l e n g t h y p r o o f of t h i s f a c t , a l t h o u g h t h e g e n e r a l i d e a i s d i s c u s s e d a t t h e end of $13.

This Page Intentionally Left Blank

PART 11.

HIGHER-ORDER MODAL LOGIC

This Page Intentionally Left Blank

CllAPTER 3 .

$9.

I-IIGHER-ORDER MODAL LOGIC

Modal P r e d i c a t e Logic We now c o n s i d e r a n o t h e r a l t e r n a t i v e f o r m u l a t i o n o f IL, which we c a l l

Modal P r e d i c a t e Logic and d e n o t e by MLp. Like t h e system MLT o f $5, t h i s logic takes

V

and

0 a s p r i m i t i v e s ; u n l i k e MLT, however, i t s t y p e s a r e

r e s t r i c t e d t o i n c l u d e o n l y t h o s e f o r i n d i v i d u a l s and p r e d i c a t e s a t v a r i o u s l e v e l s . Here p r e d i c a t e i s used i n a p r e c i s e s e n s e employed by Montague' t o mean r e l a t i o n - i n - i n t e n s i o n . Thus, an n - p l a c e p r e d i c a t e i s t o an n - p l a c e r e l a t i o n what a p r o p e r t y i s t o a s e t . Such a r e s t r i c t i o n o f t h e s e t o f t y p e s seems n a t u r a l t o a f o r m u l a t i o n i n which

V

and

0 a r e primitive,

and i t i s p e r h a p s n o t s u r p r i s i n g t h a t s e v e r a l a u t h o r s have proceeded a l o n g t h e s e l i n e s i n g e n e r a l i z i n g modal p r e d i c a t e l o g i c t o v a r i o u s h i g h e r o r d e r s . Bayart [1959] and C o c c h i a r e l l a [1969] g i v e g e n e r a l i z e d completeness t h e o rems f o r systems o f s e c o n d - o r d e r 55; B a y a r t ' s methods, however, do n o t seem t o g e n e r a l i z e r e a d i l y t o h i g h e r o r d e r s . Bressan [1964] h a s a p p l i e d h i g h e r o r d e r S5 t o problems a r i s i n g i n t h e f o u n d a t i o n s o f p h y s i c s , and i n h i s most r e c e n t work [1972] h e d e v e l o p s i n d e t a i l a l o g i c similar t o MLp,

allowing

u n l i m i t e d p r e d i c a t e t y p e s . Montague [1970a] i n d e p e n d e n t l y employed a s e c o n d - o r d e r modal l o g i c i n c o n n e c t i o n w i t h h i s a n a l y s i s of b e l i e f c o n t e x t s , mentioned i n $1, and remarked t h a t t h e same c o n s t r u c t i o n c o u l d b e c a r r i e d t o h i g h e r (and even t r a n s f i n i t e ) o r d e r s . The l o g i c MLp i s t h e r e f o r e a n a t u r a l and u s e f u l a l t e r n a t i v e t o IL; moreover, we s h a l l see t h a t MLp h a s some d i s t i n c t a d v a n t a g e s o v e r I L when we come t o c o n s i d e r t h e Boolean s e m a n t i c s of Chapter 4. Higher-Order P r e d i c a t e Logic.

Before d e f i n i n g t h e s y n t a x and s e m a n t i c s

o f MLp, we c o n s i d e r a f o r m u l a t i o n o f o r d i n a r y (non-modal) h i g h e r - o r d e r p r e d i c a t e l o g i c , which we d e n o t e by L p . T h i s l o g i c , which i s e s s e n t i a l l y t h e v e r s i o n p r e s e n t e d i n Orey [1959], w i l l b e u s e f u l i n i t s own r i g h t i n Montague [1970a], p . 71.

68

HIGHER-ORDER MODAL LOGIC

a l a t e r s e c t i o n , and i t s s y n t a x and s e m a n t i c s w i l l b e c l o s e l y p a r a l l e l e d by t h o s e o f t h e l o g i c ML,,. P r e d i c a t e Types. The s e t

e

(ii)

a .

e

ue any symbol which i s n o t a f i n i t e s e q u e n c e .

o f p r e d i c a t e t y p e s i s t h e s m a l l e s t s e t such t h a t :

P

(i)

Let

C P

,

, al ,

... ,

un-l 6 P

imply

n- 1) E P

(no,ul,...,u

That i s , t h e s e t o f p r e d i c a t e t y p e s c o n t a i n s

e

,

and i s c l o s e d under t h e

f o r m a t i o n of a r b i t r a r y f i n i t e sequences. O b j e c t s o f t y p e

e

w i l l be indi-

(uo,u],..., u ) w i l l be relations of n n- 1 arguments, o f which t h e f i r s t i s an o b j e c t o f t y p e uo , t h e second an ob-

v i d u a l s , and o b j e c t s of t y p e j e c t of type

al , e t c .

P r i m i t i v e Symbols.

For each

u C P

we have a denumerable l i s t of

variables

and n o n - l o g i c a l c o n s t a n t s '

u

of t y p e

,

t o g e t h e r w i t h t h e improper symbols

We a l s o d e n o t e t h e v a r i a b l e s of t y p e

and w e u s e t h e l e t t e r s

' X I ,

'y?,

u

... ,

,

5

,

-,

+

,

V

, [ ,]

.

i n t h e i r p r o p e r o r d e r , by

I r I ,

with or without s u p e r s c r i p t s

o r primes, t o r a n g e over formal v a r i a b l e s of Lp. A symbol

s0 of t y p e

u

is a v a r i a b l e o r constant of t h a t type.

Grammar.

An atomic formula of Lp is an e x p r e s s i o n of one of t h e forms

s s o s l * . . sn-l where

s

i s of t y p e

, u

=

(uO,ul,... , o n - l )

and

sk

i s a symbol of t y p e

We f i x t h e s e t o f c o n s t a n t s h e r e f o r r e a s o n s o f convenience. One c o u l d a l l o w an a r b i t r a r y s e t of c o n s t a n t s , n o t n e c e s s a r i l y denumerable.

MODAL PREDICATE L O G I C

uk

for

k < n ; or

[s

5

,

sl]

s , s'

where

a r e symbols o f t y p e

.

e

The f o r m u l a s of Lp a r e g e n e r a t e d

from t h e atomic formulas by t h e c o n n e c t i v e s Vxu

69

-,

and t h e q u a n t i f i e r

-+

, where xU i s an a r b i t r a r y v a r i a b l e . d

I t i s i m p o r t a n t t o n o t e t h a t t h e empty sequence t h a t a symbol

The s e n t e n t i a l c o n n e c t i v e s

,

A

v

, t+ and t h e q u a n t i f i e r 3xo a r e u # e

d e f i n e d a s u s u a l . For an a r b i t r a r y p r e d i c a t e t y p e

s'

o f type

we u s e

u

...

vxo Vxl where 3!x

U

[s

s

s']

. . , u ~ - ~ and )

s ,

and symbols

as an a b b r e v i a t i o n f o r t h e formula

[ s x0 x 1. . . x n-1

vx"-l

u = (oO,al,.

xk

+ . +

0 1

s'x x

is of type

ak

...

,

2-l ]

.

k c n

for

We u s e

a s a n a b b r e v i a t i o n f o r t h e formula

A

3xh Vxu [ A

+--f

=

x

,

x' ]

x' is t h e first v a r i a b l e of type u c u r r i n g f r e e i n t h e formula A .

where

Generalized Semantics.

Given a s e t

s e t , o r s e t o f a l l s u b s e t s , of

x0 x . . .

x

quences

(ao,.

Let

P , so

belongs t o

s t a n d i n g a l o n e i s an a t o m i c f o r m u l a .

sd

D

X

u

d i f f e r e n t from

X , we d e n o t e by

...

. Given s e t s Xo ,

denote t h e i r Cartesian product, i . e . ,

Xn-l

. . , a n - l ) , where

ak C Xk

for

k < n

Me = D

(ii)

F o r each t y p e x

0

t h e power

, we

let

t h e s e t of a l l s e -

. D

we under-

o f s e t s , where

,

(i)

P(Mu

(Mu)uCp

and n o t oc-

P(X)

, Xn-l

h e a non-empty s e t . By a frame f o r Lp based on

s t a n d an indexed f a m i l y

x

...

X

u = (uO,

Mu

)

..., un - l )

, M,

i s a non-empty s u b s e t o f

.

n- 1

The frame i s s t a n d a r d i f t h e i n c l u s i o n i n ( i i ) can b e r e p l a c e d by e q u a l i t y . A g e n e r a l model (g-model)

such t h a t :

o f Lp based on

D

i s a system

M

= (Mu,

m)oEp

HIGHER-ORDER MODAL LOGIC

70

i s a frame f o r Lp based on

(i)

(Mo)uCp

(ii)

The mapping

b1

,

D

a s s i g n s t o each c o n s t a n t

m

c,,

a n element o f

Mu

.

i s a ( s t a n d a r d ) model o f Lp i f t h e u n d e r l y i n g frame i s s t a n d a r d . We de-

n o t e by able

t h e s e t o f a l l a s s i g n m e n t s o v e r t h e g-model

As@!)

functions

.

xo

on t h e set o f v a r i a b l e s such t h a t

a

For an assignment

, we

a

a

let

a(x,)

, i.e., all

M

f o r each v a r i -

€ Mu

be t h e extension of a a(c,) = m(c,) € Mu

s e t of a l l c o n s t a n t s , d e f i n e d by t h e r u l e t h a t

t o the

. We

can

define the notion

sat

a

b!,

A

hy r e c u r s i o n on t h e formula (i)

M, a

sat

s s

(ii)

M, a

sat

[s

...

E

s

n-1 . i f and o n l y i f i f and o n l y i f

sl]

-

(afs0 ) ,..., a ( s n - ' ) )

-

a(s) =

a(s') , where

,

C a(s)

s

and

s']

for

e ,

a r e symbols o f t y p e

s'

(iii)

0

of Lp, as f o l l o w s :

A

Usual s a t i s f a c t i o n c l a u s e s f o r

-,

+

,

Vxu

.

I t i s readily v e r i f i e d t h a t t h e defined equality r e l a t i o n

[s

5

u # e r e p r e s e n t s i d e n t i t y i n any g-model o f Lp, i n t h e M, a s a t [s sl] i f and o n l y i f a ( s ) = . From t h i s

symbols o f t y p e

a(s')

sense t h a t

M, a

it f o l l o w s t h a t

f o r which

X € Mu

sat

3!xu A

M; a,X

sat

A

i f and o n l y i f t h e r e e x i s t s a u n i q u e

.

t i c a l notions: i n Lp,

A

is t r u e i n

A

(As i n e a r l i e r sections,

a,X

i s here

a(x/X) . ) We d e f i n e as u s u a l t h e seman-

an a b b r e v i a t i o n f o r t h e assignment

M , A

i s a g - s e m a n t i c a l consequence o f

r

is g - v a l i d i n Lp, e t c .

The Theory L p . Axioms o f L p .

, where

AS1.

A

AS2.

Vxa [ A

+

A

B]

i s tautologous i n -+

[A

+

f r e e i n t h e formula AS3.

Vxu A(x) mula

+

A(x)

A(so)

,

,

Vx B] A

,

-

where

and

+

,

i s any v a r i a b l e n o t o c c u r r i n g

x

,

where t h e symbol

s

is free f o r x

i n the for-

MODAL PREDICATE LOGIC

A4.

x

ASS.

s

e 0

z x

E

71

e '

s'

[ A(s)

f

0

free for

xo

] , where t h e symbols

A(s')

+

i n t h e formula

s

and

are

s'

.

A(xu)

Rules of I n f e r e n c e . R1.

From

[A

K2.

From

A

and

B]

+

t o infer

t o infer

A

B

,

.

Vxa A

I t i s well-known3 t h a t g e n e r a l i z e d completeness h o l d s f o r t h e l o g i c L p , as does t h e corresponding r e s u l t f o r t h e l o g i c

Lp+C

, P r e d i c a t e Logic w i t h

Comprehension, o b t a i n e d by adding t o t h e axioms o f Lp a l l i n s t a n c e s , i n t h e language of L p , o f t h e f o l l o w i n g schema: :

where

3fu

0 VX

vX1

u = (oo,ul,.

.. ,

...

VXn-'

u ~ - ~, ) xk

0 1 X X

...

n- 1 X

i s of type

++

uk

A ]

for

,

k < n

, and f u

o which i s n o t f r e e i n t h e formula

t h e f i r s t v a r i a b l e of t y p e Modal P r e d i c a t e Logic. o f ML,

[ f

A

is

.

A s i n d i c a t e d e a r l i e r , t h e s y n t a x and s e m a n t i c s

c l o s e l y p a r a l l e l t h e s y n t a x and s e m a n t i c s o f L p .

I n f a c t , the set

P

o f p r e d i c a t e t y p e s i s t h e same f o r t h e two l o g i c s , t h e d i f f e r e n c e l y i n g i n t h e i r i n t e n d e d i n t e r p r e t a t i o n . In MLp, o b j e c t s of t y p e w i l l be p r e d i c a t e s (relations-in-intension)

f i r s t i s an o b j e c t o f t y p e

oo

of

n

(oo,ol,.

..

arguments, o f which t h e

, t h e second an o b j e c t o f t y p e

u1

, etc.

The v a r i a b l e s and c o n s t a n t s o f MLP a r e t h e same as t h o s e of

Grammar.

L p . The improper symbols o f MLp a r e t h o s e of Lp t o g e t h e r w i t h t h e n e c e s s i t y

.

operator

The formulas o f MLP a r e g e n e r a t e d from t h e atomic f o r m u l a s

t i a l connectives

A

,

v

operator

0

[s z s']

f o r symbols o f t y p e

a r e d e f i n e d as u s u a l . W e c a r r y o v e r from Lp t h e a b b r e v i a t i o n s u f e

we a l s o w r i t e [s

-

, + , Vxu and 0 . The s e n t e n , cf , t h e q u a n t i f i e r 3xu , and t h e p o s s i b i l i t y

g i v e n e a r l i e r by means o f t h e o p e r a t o r s

= s']

for

0 [s

z

s'] ,

By t h e method o f Henkin [1950].

, and 3!x,

A ( g i v e n e a r l i e r ) . I n MLp

HIGlIER-ORDER MODAL LOGIC

72

where

and

s

]!!xu

for l x ; 'Vxo [ A

A

where x &

are symbols of arbitrary type cr , and

s'

-

x

= x'

is the first variable of type

u different from x

and not

.

free in the formula A

Generalized Semantics. Let D blLp based on

] ,

I he non-empty sets. A frame for

is an indexed family

I

I)

and

of sets, where

(L1u)ocp

= D ,

(i)

bf

(ii

For each type u = ( O ~ , . . . , U ~ -, ~ ) blo

...

P(MD x

x

)

Atu

I

is a non-empty subset of

.

n-1

0

The frame is standard if equality holds in (ii). A general model (g-model) of F.ll.p based on D

I

is a system bl = (bin, m)ucp

is a frame for LILp based on D

(I)

(blU)ucp

(ii)

The mapping

and

I

such that:

,

m assigns to each constant co an element of

Mu .

I f n = 0 we adopt the usual set-theoretic convention identifying the Cartesian product xo x . . . x Xn-l with the set containing only the empty sequence 6 . In any g-model M of FlL,, we therefore have

so that M

is always a non-empty set of propositions. A (standard) model

4

of ML is a g-modcl whose underlying frame is standard. An assignment is P defined as before, and the notion sat A ,

bl, i, a

where

i

(i)

bf,

E I

and a E As(b1) , is defined by recursion on the formula A :

i, a

sat s

s

0

...

an element of Z(s)(i) (ii)

n-1

if and only if

(a(so),..

.,a(sn-'))

is

,

sat [s I s ' ] if and only if a ( s ) = are symbols of type e ,

bl, i, a

s'

s

a(s') , where

s and

73

MODAL PREDICATE LOGIC

(iii)

Usual s a t i s f a c t i o n c l a u s e s f o r

(iv)

h!,

i, a

The d e f i n e d e q u a l i t y r e l a t i o n

[so

-+

, Vxu ,

M, j , a

i f and o n l y i f

0A

sat

-,

sat

u # e

f o r types

s sb]

for a l l

A

now r e p r e s e n t s

c o n t i n g e n t i d e n t i t y o f p r e d i c a t e s i n any g-model o f MLp: We have

[s

sat

s;]

z

M, i , a

[sU e s;]

sat

checked t h a t some

-

i f and o n l y i f

i, a

PI,

3!x

a

u n i q u e l y . On t h e o t h e r hand, X C Mu

e x i s t s a unique

a(s') .

a(s) =

bl, i , a

M ; i ; a,X

sat

3!!x

M; i ; a,X

f o r which

sat U

u,

I t is a l s o e a s i l y sat

determines

A

for

A

X(i)

just i n case there

A

sat

M, i , a

But fqr e v e r y t y p e

j u s t i n c a s e ( i ) M; i ; a,X

A

, and ( i i ) t h e c o n d i t i o n

X C Mo

-

i f and o n l y i f sat

.

a(s)(i) = a(s')(i)

.

j C I

A

.

A i s t r u e i n M , r I= A g Z i s g - s a t i s f i a b l e i n MLp. We a l s o have t h e

A s i n $ 2 and 53 we i n t r o d u c e t h e n o t i o n s

hg A

i n MLP,

i n PILp,

and

corresponding standard semantical notions

r I=

A

i n MLp,

i n MLp,

)= A

and

Z i s s a t i s f i a b l e i n MLp. The s e t o f modally c l o s e d f o r m u l a s o f MLp i s t h e s m a l l e s t s e t c o n t a i n i n g a l l a t o m i c formulas of t h e form [ s e B s,'] , a l l , -+ and formulas o f t h e form 0 A , and c l o s e d u n d e r t h e c o n n e c t i v e s t h e q u a n t i f i e r VxU . F o r such a formula A we write M , a s a t A , a s

-

e a r l i e r , s i n c e t h e index

i s irrelevant.

i

The Theory MLp. Axioms o f MLp.

, where

AS1.

A

AS2.

VxU [A

i s tautologous i n

A

B]

-+

-+

[A

-+

f r e e i n t h e formula AS3.

Vxu A(x) mula

A4.

x

AS.

x

AS6.

s

e

E X

-+

,

i s any v a r i a b l e n o t o c c u r r i n g

x

s

is f r e e f o r

x

i n the for-

,

e '

E 5;

+

OA-+A,

x

,

x e = Ye

[ A(s)

-+

free for

6 7 .

Vx B] , where

and

A ,

A(su) , where t h e symbol

A(x)

-= Y e

U

-+

N

U

-+

A(s')

]

i n t h e formula

, where t h e symbols A(x,) ,

s

and

s'

are

74

HIGHER-ORDER MODAL LOGIC

Rules o f I n f e r e n c e . R1.

From

[A

K2.

From

A

t o infer

Vxn A

R3.

From

A

t o infer

D A

1-

We write

r 1-A

--t

and

B]

A

to infer

,

.

i n blLp, i f t h e formula

A

,

B

A

i s p r o v a b l e i n t h i s t h e o r y , and

i n NLp, i f t h e formula BO+.

R1 + .

...

- + . Bn-1 + A

i s p r o v a b l e i n MLP f o r some formulas of formulas i s c o n s i s t e n t i n ML

... ,

B0 , B1 ,

nn- 1

in

r .

Z

A set

C

i f some formula i s n o t d e r i v a b l e from

P i n MLP. The soundness of t h e t h e o r y hlL P r e l a t i v e t o t h e g e n e r a l i z e d semant i c s f o r ML i s e a s i l y e s t a b l i s h e d u s i n g t h e f o l l o w i n g s t r a i g h t f o r w a r d s e -

P

m a n t i c a l lemma:

LEMMA 9 . 1 . 1 .

M

Let

is free f o r the variable i

and assignment

M, i , a where

-

sat

X = a(s)

b e a g-model of MLp, and suppose t h e symbol

x

0

i n t h e formula

A(x)

. Then

so

f o r every index

,

a

A(s)

M; i ; a,X

i f and o n l y i f

sat

A(x)

,

.

THEOKEM 9 . 1 ( G e n e r a l i z e d Completeness Theorem f o r MLp) (i)

16 A

(ii)

r

(iii)

Z i s c o n s i s t e n t i n blLp i f and o n l y i f

g

i n ML P i f and o n l y i f A

i n blL

P i f and o n l y i f

A

i n FILp,

r I-

A

i n blL P'

Z i s g - s a t i s f i a b l e i n blLp.

1Ve s k e t c h b r i e f l y t h e p r o o f , which i s c o n s i d e r a b l y s i m p l c r t h a n t h e p r o o f of Theorem 3 . 3 . As e a r l i e r , i t s u f f i c e s t o p r o v e t h e i m p l i c a t i o n from l e f t t o r i g h t i n p a r t ( i i i ) , and a g a i n we can assume t h a t t h e c o n s i s t e n t set

Z omits i n f i n i t e l y many v a r i a b l e s of each t y p e

cr

.

Lemma 3 . 2 c a r r i e s

o v e r t o t h e t h e o r y blL P ( s e e comment on pp. 29-30), s o t h e r e i s a sequence

75

MODAL PREDICATE LOGIC

-

-

( Z i ) i L u o f s e t s o f formulas o f blLp having p r o p e r t i e s ( i ) t h r o u g h ( i v ) o f Lemma 3 . 2 ( s e e page 25) and hence a l s o p r o p e r t i e s (v) and ( v i ) o f

Z

=

s , s'

Remark 3 . 2 . 9 (page 2 9 ) . Civen symbols

s

h

s'

which i s independent o f

Sym,

l a t i o n on t h e s e t s

s

we have

t h e type

0

F

1

11

Z. 1 '

C

, i s e a s i l y shown t o b e an e q u i v a l e n c e r e -

o f symbols o f t y p e

. Moreover,

u

. x u f o r i n f i n i t e l y many v a r i a b l e s

w e define a set

G

, the relation

IJ

-

[s e s']

i f and o n l y i f

of type

and a mapping

Mu

x

is

.

from

p,

f o r each symbol By r e c u r s i o n on Sym,,

into

Mo

such t h a t : (1)

p,

is o n t o

(2)

pLa(sLa)=

We f i r s t l e t

~ ( ~ ( s ; i)f and o n l y i f = D = Sym

bl

,!I

assume t h a t

,

blu

and

from

p,

0

/.- and d e f i n e

p (S ) e e have been d e f i n e d f o r

pcT

se/-

t o be

k




*..

( s )(i) 0

1 1

(s;-l

F ,

f

n-1

0

n-1

j u s t i n case t h e formula

s s

T h i s i s w e l l - d e f i n e d , by AS6, and i f we l e t

0

...

Mo

011

and

D

I =

by l e t t i n g

CI)

belongs t o

M

m(cU) = ku(c,)

= (Mu > m)ucp

f o r every

sat

A

i f and o n l y i f

A 6

Zi

M, i, a

.

then of MLp A

cu.

that

,

-

i C I , u s i n g Lemma 9 . 1 . 1 and p r o p e r t y (v) o f

f i e r s t e p . From t h i s we conclude t h a t

Z.

f o r each constant

I t i s r e a d i l y v e r i f i e d by i n d u c t i o n on t h e l e n g t h o f t h e formula

M, i, p

-

be t h e r a n g e of

c o n d i t i o n s (1) and ( 2 ) h o l d . We d e f i n e a g-model based

sn-1

sat

Z a t the quantiZ when i = 0 and

a = p , and t h e p r o o f i s complete. The n o t i o n o f p e r s i s t e n c e , d i s c u s s e d i n 54, a l s o

P e r s i s t e n c e i n MLp.

c a r r i e s o v e r t o M L p i n a much s i m p l e r form. Suppose g-model o f MLp based on frame f o r bfLp based on every

o C P

D D

and and

, and l e t

I I

.

(blA)otP

M =

(Mo, m),tp

be t h e standard

I t is e a s i l y seen t h a t

, so t h a t t h e system M'

= (MA,

m)uFp

is a

Ma

5 MA

for

i s a s t a n d a r d model of

HIGHER-ORDER MODAL LOGIC

76

5 As(M') .

MLP and As(b1) M, i , a

A formula A

of ML P is called M-persistent if

if and only if M ' , i, a sat A

sat A

for every i E I and a E As()!) , and persistent if i t is M-persistent for every g-model M of MLP. Any formula which is provably equivalent to a persistent formula is itsclf persistent, and as earlier we can prove: TIEOREM 9.2. Let Per be the set of all persistent formulas of MLp. Then: (i)

All atomic formulas belong to Per ,

(ii)

A , B

(iii)

A E Per implies 0 A

(iv)

A E Per

(v)

Suppose A E Per and F(xo) is an atomic formula of the form s s 0 . . . x ... sn-1 in which the variable x U occurs non-initially.

C

Per imply

-

A , [A E

implies Vxe A

+

B] E Per ,

Per , C

Per ,

Then the formulas WxD [ F ( x ) Per .

+

and 3xa [ F ( x )

A]

A

A]

belong to

From generalized completeness (Theorem 9.1) and the definition of persistence, we obtain THE0RI:IIf 9.3. Let

r

and 2

be sets of persistent formulas, A

a

persistent formula of blLp. Then: (i)

I=

(ii)

r I=

(iii)

Z

is consistent in ML P if and only if Z is satisfiable in MLP,

(iv)

2

is satisfiable in blLP if and only if every finite subset Z'

Z

is satisfiable in MLP.

in M L if ~ and only if

A

A

I-

in MLp if and only if I?

A

1-

in M L ~ , A

in blLp,

of

Modal Predicate Logic with Comprehension. Among the various axiomatic extensions of blLP it is most natural to consider the deductive theory we denote by MLP+C , obtained by adding to the axioms of MLP all instances of the following comprehension schema:

MODAL PREDICATE LOGIC

C O ' ~:

Ifo

where

u

=

...

n vxO v X 1

0 1 n- 1 [ f x x ... x

vXn-l

( o O , u l , . . . , u ~ - ~,) xk

the f i r s t variable of type

77

i s of t y p e

uk

for

A ] ,

k < n , and

which i s n o t f r e e i n t h e formula

o

f

0

is

A .4 T h i s

schema e x p r e s s e s t h e p r i n c i p l e , v a l i d i n blLp, t h a t e v e r y formula w i t h f r e e v a r i a b l e s d e t e r m i n e s a p r e d i c a t e , i . e . , a r e l a t i o n - i n - i n t e n s i o n . A g-model of blL

i n which a l l i n s t a n c e s a r e t r u e ( i . e . , s a t i s f i e d by e v e r y i n P dex and assignment) i s c a l l e d a g e n e r a l model (g-model) o f ML + C . I t i s P e v i d e n t t h a t g e n e r a l i z e d completeness c a r r i e s o v e r t o t h e l o g i c blLP+C . I t i s r e a s o n a b l e t o ask whether t h e o r d i -

E x t e n s i o n a l Comprehension.

n a r y comprehension p r i n c i p l e , t h a t e v e r y f o r m u l a w i t h f r e e v a r i a b l e s d e t e r mines a r e l a t i o n , can a l s o be e x p r e s s e d i n t h e language o f blLP. Although t h e models of blLp admit o n l y p r e d i c a t e s a t t h e a t h t y p e l e v e l f o r each

u # e

, we can i d e n t i f y o r d i n a r y r e l a t i o n s w i t h c o n s t a n t p r e d i c a t e s , so

t h a t , e.g., a r e l a t i o n R

5

blo

x

.. .

x

blu n- 1

0

F C blu , a = (oo,...,u ) , s a t i s n- 1 i C I . That t h e v a r i a b l e f u d e n o t e s such a

would b e r e p r e s e n t e d by t h e p r e d i c a t e fying

F(i) = R

for a l l

c o n s t a n t p r e d i c a t e i s e x p r e s s i b l e i n MLP by t h e formula Rn(f) : where

VxO xk

...

Vxn-l [ D f x

i s of type

for

uk

0

...

x

k

n

c(

n- 1

.

V

0 - f x 0 . . . x n- 1 ] ,

The p r i n c i p l e o f e x t e n s i o n a l com-

p r e h e n s i o n i s t h e n e x p r e s s e d by t h e schema: ECu'A

:

0 3fo [ Rn(f)

VxO

A

u = ( o o ,. . . ,u n-1) the f i r s t variable of type

where

xk

...

Vxn-l [ f x

i s of t y p e

0

ak

...

x

for

n-1 c--f

k < n

,

A ] ]

, and fu i s

9

d e n o t e by

MLP+C+EC

t o t h e axioms o f

u

which i s n o t f r e e i n t h e f o r m u l a

A

t h e t h e o r y o b t a i n e d by adding a l l i n s t a n c e s

MLP+C

, and d e f i n e

a g e n e r a l model (g-model) o f

i n t h e obvious way. Note t h a t a g-model

M

o f PILp i s a g-model of

.

IVe

KOJA

blLP+C+EC

blLp+C

The n o t a t i o n C''A was g i v e n a d i f f e r e n t meaning on page 71, when A i s a formula o f L p . We s h a l l r e f e r t o an i n s t a n c e o f t h e comprehension schema i n L P , when i t i s n e c e s s a r y t o d i s t i n g u i s h t h e e a r l i e r formula from t h e p r e s e n t one.

HIGHER-ORDER MODAL LOGIC

78

j u s t i n c a s e t h e f o l l o w i n g c o n d i t i o n h o l d s : For e v e r y every formula a

nient

A(x

over

f o r each

,

M

i C I

0

.

,..., x n-1)

.

LEMMA 9 . 4 .

always bclong t o

The t h e o r y

fl b f 0 3gu [ Rn(g)

f o r every

has t y p e

uk

( U ~ , . . . , U ~ - ~ )

, and e v e r y a s s i g n -

Ma , where

Mu , t h e n M

Gi

d e f i n e d by

i s a g-model of

flence: MLp+C+EC

by adding t o t h e axioms of

tiu :

xk

belongs t o

F

If in addition the constant predicates

= F(i) ( j C I )

G.(J)

MLp+C+LC

where

the predicate

,

u =

n

# e

A

P

f

3

i s equivalent t o t h e theory obtained

t h e formulas

ML +C

g ]

.

Some remarks about t h e schema EC a r e i n o r d e r . I t was d i s c o v e r e d i n t h e c o u r s e of p r o v i n g t h a t t h e t h e o r i e s IL and ML have m u t u a l l y i n t e r p r e t a b l e P e x t e n s i o n s ( C o r o l l a r i e s 13.6 and 1 3 . 1 2 ) . I n i t i a l l y i t seemed t o t h e a u t h o r t h a t IL+D and ML +C would b e e q u i v a l e n t t h e o r i e s i n t h i s s e n s e , b u t i t P proved n e c e s s a r y t o add t h e schema LC f o r t h e argument t o go t h r o u g h . A l though FC seems weaker t h a n t h e more n a t u r a l schema C o f comprehension, we s h a l l s e e i n $15 t h a t n e i t h e r schema i s s t r o n g e r t h a n t h c o t h e r , and i n p a r t i c u l a r EC i s independent of BILp+C

MLp+C ; i . e . ,

t h e r e e x i s t g-models of

i n which EC f a i l s . The d i s c o v e r y t h a t EC i s i n f a c t a s t r o n g e r p r i n -

c i p l e t h a n o r i g i n a l l y s u s p e c t e d a p p a r e n t l y confirms a c o n j e c t u r e o f Breswho f i r s t made mention of an e q u i v a l e n t schema i n h i s p a p e r Bressan [1964]. W e s h a l l r e t u r n t o t h e schema EC i n $11, where we i n t r o d u c e c e r t a i n axioms o f a r a t h e r d i f f e r e n t c h a r a c t e r which n e v e r t h e l e s s prove t o b e e q u i v a l e n t t o EC.

Bressan [1972].

PROPOSITIONS I N MLP

$10.

P r o p o s i t i o n s i n ML

P

Given an a r b i t r a r y g-model f i n e , f o r each formula A

79

with respect t o

in the set

2'

of ML

bl

P with index s e t

and assignment

A

I

, we can d e -

, the intension

a

Inta[A]

such t h a t f o r

i E I

, P(i)

M, i , a

if

1

=

P ( i ) = 0 o t h e r w i s e . IVe have s e e n t h a t t h e domain general, the proposition

sat

then i n p a r t i c u l a r

A

, and from t h i s

with index s e t

I

, and l e t

Let

of

X

I

B(M)

M

Let

I

b e a g-model o f

if

i s non-empty, s i n c e

B(M)

0 [q P(i)

P , Q C

R E M

+

R(bl)

with index set

MLp+C

I

i s a s u b a l g e b r a o f t h e Boolean a l g e b r a o f a l l s u b s e t s of

Proof: 3q+

i f and

i E X

i s p u t i n one-to-one c o r -

hi4

M+

with

M+

0

.

Hence

then R(i)

=

. I

i s non-empty. I f

B(M)

bZ; P , Q 1

P C b14

then

sat

Q E M w i t h Q ( i ) = 1 i f and 4 i s c l o s e d u n d e r complements. S i m i l a r l y , 3r+ 0 [ r

i f and o n l y i f

c l o s e d under i n t e r s e c t i o n s .

P(i)

=

,

s a t i s f y the for-

M; P

, so t h e r e e x i s t s

p]

++--

=

.

.

M

by comprehension (and rewrite o f bound v a r i a b l e s ) , only if

blLp+C

We can i d e n t i f y

, which we d e n o t e by

which we c a l l t h e a l g e b r a of p r o p o s i t i o n s o f

mula

.

M

such t h a t

P ( i ) = 1. Under t h i s i d e n t i f i c a t i o n

THEOREM 1 0 . 1 .

.

b e a g-model of

b1

be a proposition of

P

respondence w i t h a c l a s s o f s u b s e t s o f

Then

which does n o t o c c u r f r e e i n

I n t a [ A ] E bid

of Propositions.

i n t h e u s u a l way w i t h t h e s u b s e t

only i f

In

s a t i s f i e s comprehension

bl

+

i t follows e a s i l y t h a t

B(M)

.

bI

satisfy

i s t h e f i r s t v a r i a b l e of t y p e

The Algebra

P

M, a

; however, i f

b16

, and

A

d e t e r m i n e d by a formula and an a s s i g n -

I n t [A]

ment may f a i l t o belong t o

p+

P

i s always a non-empty

bl+ s e t o f p r o p o s i t i o n s , which we c a l l t h e p r o p o s i t i o n s o f t h e g-model

where

of

a ; v i z . , we t a k e i t t o b e t h e u n i q u e p r o p o s i t i o n

p

A

Q(i)

q] =

, so 1

there exists

, and B(M)

is

80

HIGIIER-ORDER MODAL LOGIC

A subset X

I is called M-definable if there exist a formula A and an assignment a such that X consists of those i C I for which M, i, a sat A . Using Lemma 9.1.1 it is easily shown that: of

THEOREM 10.2.

with index set I . Then P I form a Boolean algebra, and this algebra when M is a g-model of MLp+C . be a g-model of ML

Let M

the M-definable subsets of incides with

B(M)

Indicia1 Equivalence.

Let M

be a g-model of MLp, and let

co-

i , j E I.

lVe say that the index i is equivalent to the index j , and write i = j , if for every formula A and assignment a , M, i, a sat A if and only if M, j, a sat A

.

Equivalently, i

2

j

if and only if i and

j be-

.

long to exactly the same M-definable subsets of I The relation = is an equivalence relation on I , whose equivalence classes play a role analogous to that of the "sets of indiscernibles" of model theory. TIEOREM 10.3. Let M

be a g-model of MLp+C

.

Then for all

i , j E I

the following conditions are equivalent: j ,

(i)

i

(ii)

For every u # e

(iii)

For some u # e and every F E ,I)

(iv)

For every proposition P of M , P(i) = P ( j )

(v)

For every X E R(b1)

rr

and every F

, i

C

X

€

Mu , F(i)

=

F

, F(i) = F(

if and only if j t X

By Theorem 10.2, (i) and (v) both assert that i and j belong to the same M-definable subsets of I , and are therefore equivalent. Proof: _ _

Clearly (iv) and (v) are equivalent, and (ii) implies (iv) implies (iii). We show that (iii) implies (iv) implies ( i i ) . Assume F ( i ) = F ( j ) for all P C M ; by comprehension, F E M, , where u = ( U ~ , . . . , U ~ -. ~Suppose ) d xn-l M; P sat 3 f u vx0 . . . vxn-' [ f x0 P,L

...

-

0 n- 1 is distinct from f , x , ... , x , so there exists F E Mu where p d arbitrarily for k < n , we have for every such that choosing Xk C Muk

i t € I : (Xo ,..., Xn-l) C F(i') if and only if P(i') = 1 . Since F(i) = F ( j ) , th s gives immediately P(i) = P(j) . Now assume (iv), and suppose u = ( u o , . . ,u*-~) , F C Ma , and Xk E Muk for k < n By comprehension,

.

PROPOSITIONS IN MLP

M; F,XO, . . . 9

'n- 1

rvh ere

is not among

such that

P(i') = 1

sat 3p4 f

, x 0,

[ p

.. . ,

if and only if

++

f U

x

81

x

0... x n-1

1 ,

n-1 , so that there exists

(Xo

,..., Xn-l)

E F(i')

P E bZ

, for i'

4 E I .

In particular, the sequence (Xo,. . . ,X ) belongs to F(i) just in case n-1 it belongs to F(j) , and since xo , . . . , Xn-l were arbitrary we conclude that F(i) = F ( j ) , proving (ii). It should he observed that if M is a standard model then b l = 2 I , 4 so that B(E.l) is the algebra P(1) of all subsets of I . In this case the relation = is just the identity relation on I . In an arbitrary g-model of blLI,, or even FILp+C , the relation = may not be the identity relation on I ; a g-model M of ML is said t o be simple if we have, for P every i , j E I : i ? j if and only if i = j . Equivalently, E.1 is simple if whenever i # j in I there exist a formula a such that E.1, i, a sat A but not l i f , j , a sat A

A

.

and assignment \\renow show that,

in a precise sense, every g-model of FILp can be replaced by a simple one. Indicia1 flomomorphisms. Let E.1 D , I bl'

and D' , I '

is a family

is a mapping from

(ii)

For each o E P , aU

onto

(x,) ,...,Oa

I' ,

I onto

o = (uo , . . . ,u

OO (iv)

be g-models of MLp based on

respectively. An indicial homomorphism from hI

9

(9

\I'

6 = (4, 9a)oEp of mappings such that:

(i)

( i i i ) For each

and

is a one-to-one mapping from blo )

n-1

,

Mo , i E I

F E

and

onto MA , (k < n),

Xk C hlo

k

(Xn-l)) C 4U(F)[9(i)l

iff

(Xo,...,Xn-l)

F(i)

I

n- 1

For every constant cu ' m'(cg) are the meaning functions o f M

9,[m(ca)] and M' =

, where m and m 1 respectively.

If there exists an indicial homomorphism from M onto M I we say that M is homomorphic to M' and that M' is a homomorphic image of M . If the mapping 9 is one-to-one, we say that 8 is an indicial isomorphism, and are isomorphic. Note that an indicial homomorphism 6 is that I4 and b!' completely determined by 4 and ae . The composition of two homomorphisms is again a homomorphism, and isomorphism is as usual an equivalence relation between g-models.

11 I GHER-ORDER MODAL LOG I C

82

THtORLM 1 0 . 4 .

Let

M

i n d i c i a l homomorphism from every

klL,,,

i, a

bf,

___ Proof:

sat

h1

onto =

a C As(W)

and

i C I

)I'

e[a](x,)

b e d e f i n e d by

As(E.I'j

be g-models of MLP, and l e t

and

hf'

For each

.

hll, 9 ( i ) , e[a]

COROLLARY 1 0 . 5 .

=

and

If

0

Proof: I

,

of

A

.

A

e[a](s,)

j 6 I :

i

".

j

.

= 19,[a(s,)]

. M

i s an i n d i c i a l homomorphism from i

@[a] C

iff

9(i)

2'

onto

M I

and

M'

M

.

O(j)

By Theorem 1 0 . 4 and t h e d e f i n i t i o n of i n d i c i a l e q u i v a l e n c e .

Q u o t i e n t G-Models. and

let

a r e the r e l a t i o n s of i n d i c i a l equivalence i n

respectively, then f o r a l l

I)

A

sat

-

s 0 we have

C l e a r l y f o r e v e r y symbol

The p r o o f p r o c e e d s by a r o u t i n e i n d u c t i o n on

and

a C As(M)

Then f o r e v e r y formula

, we have

i f and o n l y i f

A

.

t9,[a(xa)]

b e an

8

, and l e t

Let

h1 = (blcr,

m)nCp be a g-model o f bILp b a s e d on

b e t h e c a n o n i c a l mapping from

I9

o f e q u i v a l e n c e c l a s s e s of i n d i c e s under t h e r e l a t i o n

I

=

onto t h e s e t in

.

bl

I/-

W e define

n q u o t i e n t g-model =

b1/=

based on onto

0

and

D

Ma/=

(hf /=, m/=)

,

a€P

, and c a n o n i c a l one-to-one mappings I4 /= = D = M and

I/=

as f o l l o w s : W e f i r s t put

i d e n t i t y mapping on

D

.

For a =

( U ~ , . . . , U ~ - ~ ,)

from

19a

let

Mu

be t h e

Oe

,

Ma /=

we assume t h a t

k have a l r e a d y been d e f i n e d f o r k < n , w i t h k t o - o n e o n t o E.1 /= . For each F E blu we d e f i n e

Oa

aa k 4,(F)

mapping

one-

Mu

k i n the s e t

Ok

by:

(Qu

0

f o r any

( X O ) , . . . '9" (Xn-l)) c 9 u ( F j ti/-] i f f (Xo,. . . ,Xn-l) E F ( i ) , n- 1 X o , . . . , Xn-l . T h i s i s w e l l - d e f i n e d , s i n c e i n Theorem 1 0 . 3 i t

i s e a s i l y checked t h a t ( i ) i m p l i e s ( i i ) i n any g-model of MLp. C l e a r l y i s one-to-one on

Ma

. We can t h e r e f o r e l e t Ma/-

F i n a l l y , f o r each c o n s t a n t TIIEOREM 1 0 . 6 . d e f i n e d above. Then

Let

M

cu

we l e t

(m/>)(c,)

b e a g-model o f MLp,

8 = ( 9 , 19,)aEp

be t h e range of = 8,[m(c,)]

\I/-

tYU

aU

.

.

t h e q u o t i e n t g-model

i s a n i n d i c i a l homomorphism from

M

,

PROPOSITIONS IN MLP

onto

.

M/rr

Proof: -

Moreover, h1/=

83

is simple

By tne construction and Corollary 10.5.

COROLLARY 10.7. Every g-model is homomorphic to a simple g-model. Combining Theorems 10.4 and 10.6, we see that if is a g-model o f blLp+C then M/= will a l s o be a g-model of blLp+C . Therefore: COROLLARY 10.8. If Z is a set of formulas of b1L P and Z is g-satisfiable in MLP (respectively, FILp+C ) , then Z is g-satisfiable in a simple g-model of MLp (respectively, MLp+C ) . We a l s o have: COROLLARY 10.9. Let M

be a g-model of MLP. Then E.1

only if every indicial homomorphism on hl

is simple if and

is an isomorphism.

Proof: Theorem 10.6 and Corollary 10.5. ___ It should be remarked that the notion of a quotient g-model can be gen-

is

eralized. If bl

a

g-model of blLp based on

D and

I ,

rr

is the rela-

tion of indicial equivalence in M , and % is an equivalence relation on I for which i % j implies i "1 j , then the quotient g-model b l / Z can be defined exactly as above. For this more general notion of quotient, analogues of the usual homomorphism theorems can be proved. Moreover, one can define similar notions of indicial equivalence, homomorphism and quotient for g-models of IL. THEOREM 10.10. Let M '

be a g-model of MLp based on D'

and

I' ,

and let 9 be an arbitrary mapping from a set I onto I' . Then there exists a g-model M of blLp based on D ' and I , and an indicial homomorphism

6

from M

Proof:

onto bl'

extending 9

Suppose Pi' = (MA, r n ' ) o C p

to-one mappings

~9~

from MG

.

. We define M

= (Mu,

m)ucp

and one-

onto MA , as follows: We first put Me = D'

.

= M k and let ae be the identity mapping on D ' For u = (o0 ' . . . ,on-l) we assume that bl and are already defined for k < n , such that ak k .a maps b1 one-to-one onto MAk . For each F' C MA there exists a k Ok

liI GI IEK- OKDEII MOUAL LOG I C

84

unique c o r r e s p o n d i n g (Xo,..

C F(i)

.,Xn-l)

F C P ( hlo

F'

Moreover, t h e mapping o f be i t s r a n g e and

bl,

we l e t

m(c,)

to

F

)I d e f i n e d by t h e c o n d i t i o n n- 1 ( 1 7 ~(X,) ,...,,Yo F'[a(i)l. 0 n- 1 i s c l e a r l y one-to-one, s o we can l e t x

Mn

M ,

i t s i n v e r s e . To complete t h e d e f i n i t i o n o f

fin

be chosen s o t h a t

C blo

i l y verified that

.. .

x

0 i f and o n l y i f

0 = (8, 1

9

fi,Jm(c,)]

=

C )I(:. I t i s e a s -

m'(c,)

i) s ~t h e~ d e~s i r e d homomorphism.

~

As remarked e a r l i e r , a l l s t a n d a r d models of hlLp a r e s i m p l e , a l t h o u g h

g e n e r a l models may n o t b e . I t f o l l o w s from Theorems 1 0 . 4 and 10.10 t h a t i t i s i m p o s s i b l e t o c h a r a c t e r i z e t h e s i m p l e g-models o f blL

or

MLp+C

by

means of a new axiom o r axioms. We a r e compensated, however, by t h e f a c t ( C o r o l l a r y 1 0 . 7 ) t h a t we can always p a s s from a g i v e n g-model t o i t s quot i e n t , which i s s i m p l e and s a t i s f i e s e x a c t l y t h e same f o r m u l a s . lVe can c h a r a c t e r i z e t h e s i m p l e g-models o f

M

Suppose

i s a g-model o f

P1

ositions of if

B(M)

. x c

Suppose t h a t i

c x

i

and

j

b e t h e a l g e b r a o f prop-

{

M

i s s i m p l e i f and o n l y

i # j

M

, there

# j

in

I

then

based on D and I . Then P separates points i n I ; in f a c t , i f

ti)

B(M)

i

separates

contains every subset of

.

I

and

j

and b e l o n g s t o

p r o p o s i t i o n which i s t r u e a t i

, viz., the proposition

i

j # i

and f a l s e a t e v e r y

, then

P

s t r i c t l y implies

whenever

P

is true a t

j

.

Q

.

B(M)

, since

T h i s h a s t h e i n t e r e s t i n g consequence

t h a t , i n a s t a n d a r d model, t h e r e e x i s t s f o r each i n d e x

i

I

i s a s t a n d a r d model of EilL

i s s i m p l e , and t h e r e f o r e

B(b1)

at

in

x .

Atomic I'roposi t i o n s and EC

$11.

M

I ; i . e . , whenever

R(M) with

i n a n o t h e r way:

blLp+C B(M)

Then by Theorem 1 0 . 3 we see t h a t

separates points in

exists a set

, and l e t

bfLp+C

If

Q

a strongest

i P

which i s t r u e

i s any o t h e r p r o p o s i t i o n t r u e a t

, i n t h e sense t h a t Q i s t r u e a t

j

Consequently, t h e formula

which e x p r e s s e s t h e p r i n c i p l e t h a t t h e r e n e c e s s a r i l y e x i s t s a s t r o n g e s t t r u e p r o p o s i t i o n , i s v a l i d i n MLp,

i.e., t r u e i n a l l s t a n d a r d models.

ATOMIC PROPOSITIONS AND EC

85

There are closely related conditions which we might also consider. Let us call a proposition P

atomic if (i) P

every proposition Q , P

is possibly true, and (ii) for

strictly implies either Q

o r its negation.

This can be expressed by the formula:

which we abbreviate by Atom(p,)

.

The formulas

then express the respective principles that (1) there necessarily exists a true atomic proposition, and (2) every possibly true proposition is strictly implied by an atomic proposijtion. Both Atl and At2 are valid in MLp, and clearly we have: LEMMA 11.1. Let M

be the algebra of propositions of M

let B(M)

M

(i)

be a g-model of MLp+C with index set

sat At

set X

C

for which

i E X ,

(ii)

M sat Atl if and only if every i Boolean algebra B(M) ,

(iii)

M

sat At2

Then:

i E I there is a smallest

if and only if for every

B(M)

.

I , and

if and only if B(M)

THEOREM 11.2. The formulas At

C

I belongs to an atom in the

is atomic

, Atl , A t 2 are provably equivalent

in MLp+C . Proof: a set

It is easily verified that for any algebra

I , the conditions (i) F o r every

i E I

, and (ii) Every i

B

of subsets of

there is a smallest set X

I belongs to an atom of B , are equivalent, and both imply the condition (iii) B is atomic. By Lemma in B

f o r which

i E X

C

11.1 and generalized completeness, the formulas [At ++ Atl] and [Atl -+ AtZ] are therefore provable in MLp+C . Although (iii) does not imply (i) for an arbitrary field cation

[AtZ + At]

B of sets, we can still prove the impliFor, suppose M is a g-model

in the theory MLp+C

' Cf. Fine [1970], p. 341

.'

HIGHER-ORDER MODAL LOGIC

86

of blL,+C with index set I , and M sat At2 . Then B(M) is atomic, b y I Lenmia 11.1, and it suffices to show that every index i C I belongs to a smallest set

X t B(bl)

atom i n

.

B(li1)

, or equivalently that every

-

sat 3p 0 [ p +-,

bl

i

C

I belongs to an

By comprehension, 4

3q

4

[Atom(q)

A

q] ]

from which it follows that there exists a set Xo

,

C R(M)

such that

i E Xo

just in case i belongs to no atom of B(M) . Thus, if some i belongs to no atom then X o # 6 , and therefore Xo dominates some atom Y Since

.

Y

# 4

we c m choose

tion of Xo

i

C Y ;

i E Xo , contradicting the defini-

but then

.

ii'e refer to the formula

At

as the axiom of atomic propositions, and

we denote by

bll.p+C+At the theory obtained by adding At to the axioms of E.11, +C . A general model (g-model) o f ML +C+At is defined accordingIy. I' P Axiom At originates with Kaplan [1970], who considers an extension SSQ of the usual propositional modal logic SS in which quantifiers over propositional variables are permitted, and gives an axiomatization which is complete for the (standard) possible world semantics. The formula At appears as

Axiom 8 in his formulation, and he remarks that it is independent of the

is also independent of MLp+C , a considerably stronger theory than S5Q.2 Axiom At also appears in the log-

other axioms. In 815 we prove that At

ic S5n+ of Fine [1970], which is almost identical with Kaplan's SSQ. Before proving the main result of the present section, we have the following Let bf be a g-model o f ML +C with index set I , and P be the relation of indicia1 equivalence in M . Then for each index

LLMMA 11.3.

let

c-

i t I , the following conditions are equivalent: (11

The equivalence class i/=

belongs to B(M) ,

[ii)

i/-

B(M)

(iii)

i belongs to an atom of B(M)

is the unique atom of

containing

i ,

I

Kaplan's independence proof, which is based on a normal form theorem for SSQ, does not seem to generalize to MLp+C. The Boolean methods employed in $15, however, apply equally well to SSQ.

ATOMIC PROPOSITIONS AND EC

Proof: ___ else

87

Assume ( i ) . Then by Theorem 1 0 . 3 we have e i t h e r fl X = 4

[i/.-]

i s a n atom of

f o r every containing

R(b1)

5X

[i/-]

, from which i t f o l l o w s t h a t

X € B(M)

or i/e

, and c l e a r l y such an atom must b e

i

u n i q u e . T h e r e f o r e ( i i ) h o l d s . T r i v i a l l y ( i i ) i m p l i e s ( i i i ) . Assume ( i i i ) ; say

belongs t o t h e atom

i

so i t s u f f i c e s t o show

of

Xo

i - j

X

Then

sat

hf

Hence, i f

j

Let

ti}

, as desired.

ML +C w i t h i n d e x s e t I . P i/= b e l o n g s t o B(M) f o r a l l i € I .

sat

M

if and o n l y i f

At

B(M)

contains

.

i C I

for

i ,j , whence by Theorem 1 0 . 3

B(M)

b e a g-model o f

M

i s simple then

a l l singletons

€ [i/=]

i f and o n l y i f

At

M

of

CXo ,

[i/&]

j € Xo ; t h e n c l e a r l y

0-

and t h e r e f o r e

COROLLARY 1 1 . 4 .

By Theorem 1 0 . 3 ,

c [i/e]. Suppose

X

belong t o e x a c t l y t h e same e l e m e n t s again,

.

B(M)

We can now p r o v e : THEOREM 1 1 . 5 . Proof: -

The t h e o r i e s

and

MLp+C+EC

MLp+C+At

are e q u i v a l e n t ,

I n view of Lemma 9 . 4 i t i s s u f f i c i e n t t o show t h a t t h e t h e o r y

ML + C + A t i s e q u i v a l e n t t o t h e t h e o r y o b t a i n e d by adding t o t h e axioms o f P MLp+C a l l t h e formulas E' f o r u # e . The n e x t two lemmas a c t u a l l y show

somewhat more. For each P

-

(e,e,.

Ea]

i s provable i n

MLp+C

for

u # e

LEMMA 1 1 . 5 . 2 .

[ED

-+

At]

i s provable i n

MLp+C

for

u

We u s e g e n e r a l i z e d c o m p l e t e n e s s . Let

ML +C w i t h i n d e x set P satisfies

M

0 v f o 3g0 [ Rn(g)

:

u = (uo, ..., u ) n- 1

where

...

vxo

Suppose

i C I

[Rn(g)

f

A

P(j)

P

=

g] 1

v2-l

,

.

f

A

[ f x

3

and 0

...

g ]

I

Since M

sat

,

and assume t h a t

# e,

. n (n

E w).

M = (Mo, rn)ucp

M

sat

At.

n-1

a b b r e v i a t e s t h e formula

+-+

F € Mu ; we s h a l l f i n d

is an atom o f

€

,

[f z g] x

. . ,e)

.

4

--f

We show t h a t

1

denote t h e n-tuple

[At

Proof o f 1 1 . 5 . 1 :

{ j

n

let

i s t h e type

0

LEMMA 1 1 . 5 . 1 .

be a g-model o f

E'

,

n € o

, so that in particular

At

B(M)

g x

0

...

G € Ma

, there

x

for which

exists

containing

n-1

P C FI+

M; i ; F,G such t h a t .

i . By Lemma 1 1 . 3 ,

sat

HIGHER-ORDER MODAL LOGIC

88

we have

P ( j ) = 1 i f and o n l y i f

hension i n

..

3ga 0 vxo where

xk

vxn-l [ 9 x

*

i s of t y p e

ak

Xk C Mu

and

i' E I

(k

j u s t i n case belongs t o

i

'u

,

j

%

0

. . . ,n-l

for all

-

f o r somc

We remark t h a t

such t h a t

However, t h e formula

.

F(i)

i f and o n l y

But

P(j) = 1

(Xo,...,Xn-l) From t h i s we imme-

= g ] , and t h e proof i s c o m p l e t e .

,

MLp+C

i s not provable i n

EO

.

P(j) = 1

it follows t h a t

f

A

1 1 ,

such t h a t f o r a l l

G E Mu

is i t s e l f provablc i n

E'

n-1 x

0 A f a X

(Xo ,..., Xn-l) C G ( i ' )

j

[Rn(g)

sat

Now by compre-

s a t i s f y t h e formula

O [P'

i f and o n l y i f i t b e l o n g s t o

bl; i ; F,G

.

j C I

M; P,F

Hence t h e r e e x i s t s

n) we have

, s o by Theorem 1 0 . 3

j

C ( i l )

d i a t e l y have

.

i

k ( X o , . . . , Xn-l) C F ( j )

if

i

M ( r e w r i t i n g bound v a r i a b l e s ) ,

MLp+C

as i s e a s i l y s e e n .

, 4 , as we

u # e

for

show i n $15. Proof o f L1.5.2: n E o

.

Let

a

b e a t y p e d i f f e r e n t from

u = ( u O ,..., a4,. . . ,a n )

Then

.

MLp+C

M

Let

which s a t i s f i e s prehension,

whcre

E.1

m)a(p

= (Ma,

.

Ea

M

sat

s a t i s f i e s t h e formula

-

7fo

n vxo . . . vxn [ f

x'

i s of t y p e

have

(X o , . . . , X n ) € F ( j ) G C

for a l l

and

only i f

j 6 I Xk(i) #

3p4 0 [ p

-

$C

vx

xn

E 5 n

'c0

,

Mu

Clearly

0

0 y ,

and

...

vxn [ g,x

O

...

i f and o n l y i f f o r every

n x

-+

0 3y

P E M6

X C Mu

(X,,

M; G

,

.

i E I

I

By com-

.

Since

Therefore

(8 5 n ) M

j

..., Xn)

1 ,

a r e t h e first

XE E Mue

f o r every

we have

0

with index s e t

... , ym- 1

k

Now by comprehension,

is provable

m-1 k 0 m-1 3y x y ... y ]

X (j) # 9

.

At]

,- r e~s p e c t i v e l y .

~

('

+

Suppose

...

G ( j ) = F(i)

XE

from which it f o l l o w s t h a t t h e r e e x i s t s P(j) = 1

3y

[ED

MLp+C

.

j E I

i f and o n l y i f f o r which

At

and

, ... , T

such t h a t for a l l

F € bIu

Ea , we o b t a i n

XO...

for

a'

d i s t i n c t variables of types t h e r e exists

b e a g-model o f

We show t h a t

for all

uk = ( T ~ , . . . , T " , - ~ ). We u s e

where

g e n e r a l i z e d completeness t o show t h a t t h e formula in

n

and

e

(

we

satisfies I

.

Thus, i f and

C G(j)

s a t i s f y t h e formula a * .

m-1 k 0 3y x y

...

such t h a t f o r a l l X(i) # 6 i m p l i e s

k P ( i ) = 1 , s o i t remains o n l y t o show t h a t

M; i ; P

y

m-1

I I,

j C I , X ( j ) # 9.

satisfy

PROPOSITIONAL OPERATORS

Vq4

[q

--+

[p4

+

M; Q

.

q]]

Q ( j ) = 1 whenever

3x0

sat

Q

Suppose

P(j) = 1 yy0

.

M6 , Q ( i )

C

q4

.

ym-'

...

~y

m- 1

[ x y

0

Q ( i ) = 1 , so

ever

We must show t h a t

...

y

m-1

f--f

q4

d i s t i n c t from

x

I

>

, y0 ,

...

,

X 6 hlo

(8 c m) we have TP, t h e r e f o r e f o r a l l j E I , X(j) # 4

plies

.

such t h a t f o r a l l j C I and Ye E k ( Y o , . . . , Ym-l) C X ( j ) i f and o n l y i f Q ( j ) = 1 , and

M

have

4

is t h e f i r s t variable of type

Thus, t h e r e e x i s t s

1

By comprehension,

k where

=

89

X(j)

X(i) # 4

# 6 , which i m p l i e s

P(j) = 1

,

i f and o n l y i f

, and hence f o r a l l j Q(j) = 1

. Thus

Q(j)

=

,

6 I

1

.

But we

P(j) = 1

im-

we have

Q ( j ) = 1 when-

MLp+C

t h e axiom schema

and t h e p r o o f i s complete.

Theorem 1 1 . 5 shows t h a t i n s t e a d o f a d d i n g t o

EC o f e x t e n s i o n a l comprehension, we can e q u i v a l e n t l y add t h e s i n g l e axiom At

of atomic p r o p o s i t i o n s . We r e t u r n t o c o n s i d e r v a r i o u s independence

questions r e l a t e d t o these t h e o r i e s i n Chapter 4 .

Propositional Operators

812.

Montague [1970a] o u t l i n e s a g e n e r a l t r e a t m e n t o f o n e - p l a c e p r o p o s i t i o n a l o p e r a t o r s w i t h i n h i s f o r m a l i z e d P r a g m a t i c s , and shows how such o p e r a t o r s can b e i n t e r p r e t e d as p r o p e r t i e s o f p r o p o s i t i o n s . I n t h i s s e c t i o n we d e v e l -

op t h i s i d e a , u s i n g t h e f a c t t h a t w e can e x p r e s s i n MLp v a r i o u s p r o p e r t i e s o f t h e s e o p e r a t o r s . I n p a r t i c u l a r , we s h a l l s e e t h a t we c a n accommodate within

MLp+C

modal o p e r a t o r s s a t i s f y i n g v a r i o u s o f t h e Lewis axiom s y s -

tems, even though M-Formulas.

MLp+C

i t s e l f i s b a s e d on an S5 m o d a l i t y .

F o r t h e p u r p o s e s of t h i s s e c t i o n (and a g a i n i n C h a p t e r 4)

we f i n d i t n o t a t i o n a l l y c o n v e n i e n t t o e x t e n d t h e s e m a n t i c s of MLp i n t h e M = (M ~,m)aEp

f o l l o w i n g way: Let

b e a g-model o f MLp b a s e d on

D

and

I ; w e wish t o add t o t h e v o c a b u l a r y o f MLp new c o n s t a n t symbols t o a c t a s names of t h e v a r i o u s e l e m e n t s take the object constant of type

X F Mu

for

0

C P

.

For s i m p l i c i t y , l e t us

as a name f o r i t s e l f ; i . e . , we add

X

u

whenever

X C Mu

,

X

i t s e l f a s a new

and we e x t e n d t h e meaning f u n c t i o n

90

HIGHER-ORDER MODAL LOGIC

m

of

M

m(X) = X . l A formula o f t h i s e x t e n d e d language

by l e t t i n g

(which w i l l i n g e n e r a l havc a non-denumerable v o c a b u l a r y ) w i l l b e c a l l e d an hI-formula, and an M-sentence i f i t h a s no f r e e v a r i a b l e s . For an bl-formula

A

, an i n d e x M , i, a

,

i

sat

and an assignment

a

M , the notion

over

A

i s d e f i n e d e x a c t l y a s i n $ 9 , b u t t a k i n g i n t o a c c o u n t t h e new c o n s t a n t s . I f

A(x )

i s an F1-formula c o n t a i n i n g t h e v a r i a b l e

xu

f r e e , and

c o n s t a n t of t h e extended language, i t i s e a s i l y shown' (*)

M, i, a

where

X = m(c)

sat ~

A(c)

i f and o n l y i f

sat A(x)

M; i; a,X

I t follows t h a t t h e notion

sat

M, i

i s any

cu

that

A

,

, where

A

an M-sentence, can b e d e f i n e d d i r e c t l y by r e c u r s i o n on t h e l e n g t h o f a t t h e q u a n t i f i e r c l a u s e we simply s t i p u l a t e t h a t and only i f

M, i

sat

A(X)

f o r every

.

X E &lo

M, i

sat

is A ;

vxu A(x)

if

We can t h e r e f o r e elimi-

n a t e any r e f e r e n c e t o assignments by working w i t h M-sentences i n s t e a d of formulas o f MLp. Note t h a t e v e r y M-formula has t h e form where

A(x

0

,..., x n - l )

i s a n o r d i n a r y formula o f MLp,

d i s t i n c t variables of types

k < n

for

.

. . . , u n- 1

,

uo

.

A(XO,. . ,Xn-l) , n- 1 x , ... , x are 0

r e s p e c t i v e l y , and

Xk E Ma k

By ( * ) , t h e r e f o r e , we may t h i n k of

M, i , a

sat

A(XO,...,Xn-l)

as abbreviating the equivalent condition

M

P r o p o s i t i o n a l O p e r a t o r s of e l of

MLp+C

, with

index s e t

p r o p o s i t i o n a l o p e r a t o r of

M

I

.

. Let M = . An element

Since M

C

(6) -

(Mo, m)oCp b e a f i x e d g-modF

of

P(M6)'

M

(61

is called a

, we see t h a t s u c h op-

S t r i c t l y speaking we s h o u l d choose, f o r each u C P and X C M,,

some new

o b j e c t c z which i s n o t a l r e a d y a symbol o f MLp, i n such a way t h a t t h e mapp i n g o f (u,X) t o c i i s one-to-one. We i g n o r e t h e s e d i f f i c u l t i e s .

*

C f . Lemma 9 . 1 . 1 .

PROPOSITIONAL OPERATORS

91

erators are always properties of propositions of M . 3 Every M-formula A(p+) , with at most the variable free, determines a unique operator P4 of M ; for by comprehension, M satisfies the M-sentence

for some F C M(+) ; i.e., and consequently M sat 0 V p [Fp t+ A(p)] 4 if and only if M, i sat A(P) , for every P F M+ , we have P E F(i)

.

i C I In particular, we always have the necessity and possibility operators of M , defined by:

If s

is any symbol of type (@) and A is any M-formula which is not (61 a symbol of type 0 standing alone, we introduce the abbreviation sA for Ip,

[ 0 [p

-

A]

A

sp ] ,

where p@ is the first variable of type 4 which does not occur free in A . Using generalized completeness it is easily shown that: LEMMA 12.1. For any formulas A , B type (4) , the formula 0 [A-B]

+

[ fA-fB

is provable in MLp+C

of ML

P

and any variable

f

(4 1

of

]

.

In a g-model M of MLp+C , therefore, it follows that f o r any index i , M, i sat [ 0 [A +-+ B] + [FA t+FBI 1 , whenever A and B are M-sentences and F is a propositional operator o f M . In fact, by comprehension we can define, as in $10, the intension Int[A] of an M-sentence A as the unique P C M for which M sat 0 [P +-+ A] ; i.e., for which we 4 have, for all i C I : P(i) = 1 if and only if M, i sat A . It then follows that:

Here we identify M6 with the Cartesian product M x...x M 00 %-I’ n = 1 and uo = 6 , although these sets are slightly different.

where

92

HIGHER-CRDER MODAL LOGIC

LEMMA 1 2 . 2 .

b e a g-model o f

M

Let

propositional operator of

, i

M

E I

.

blLp+C

Then

,

a n M-sentence,

A

sat

M, i

FA

F

a

i f and o n l y i f

.

Int[A] E F ( i )

M

I n a p a r t i c u l a r g-model

t h e r e may b e v a r i o u s i n t e r e s t i n g p r o p o s i -

t i o n a l o p e r a t o r s i n a d d i t i o n t o t h e modal o p e r a t o r s d e f i n e d e a r l i e r . Tenses

M

p r o v i d e a n a t u r a l example: Let set

i s t h e s e t o f r e a l numbers, t h o u g h t o f as moments i n t i m e . For an

I

M-sentence is true i n F E M

j < i

(+1

.

M, i

A

let

M

a t time

P(E1,)'

=

sat

i

,

express t h e i n t u i t i v e condition t h a t

A

M, i

P E F(i)

j u s t in case

sat

i f and o n l y i f

FA

M, j

sat

A

P(j) = 1 A

,

for some

any

f o r some

may b e given t h e r e a d i n g " I t h a s been t h e c a s e t h a t

FA

A

Then we can d e f i n e t h e p a s t t e n s e o p e r a t o r

by l e t t i n g

From Lemma 1 2 . 2 we see t h a t f o r any M-sentence

we have Thus,

b e a s t a n d a r d model o f MLp whose i n d e x

i E I

,

. ." O t h e r

j < i

A

t e n s e s can b e t r e a t e d s i m i l a r l y as p r o p o s i t i o n a l o p e r a t o r s . Other M o d a l i t i e s .

We s h a l l b e i n t e r e s t e d i n v a r i o u s systems o f prop-

o s i t i o n a l modal l o g i c , well-known from t h e l i t e r a t u r e .

C o n s i d e r a language

a p p r o p r i a t e t o p r o p o s i t i o n a l modal l o g i c , i n which f o r m u l a s a r e b u i l t up from p r o p o s i t i o n a l v a r i a b l e s

-,

connectives

p

,

q

,

r

...

by means o f t h e s e n t e n t i a l

and t h e formal p r o p o s i t i o n a l o p e r a t o r

+

N

.

Each o f t h e

modal c a l c u l i we c o n s i d e r t a k e s i t s axiom schemata from among t h e f o l l o w ing :

,

AS1.

A

AS2.

N[A

AS3.

NA

+

AS4.

A

-+

ASS.

NA

AS6.

-

if -+

-+

NA

is tautologous i n

A

B]

-+

[NA

+

NB]

,

-+

,

,

,

A

N - N - A , NNA

,

-+

N

-

NA

,

and has as i t s i n f e r e n c e r u l e s : R1.

From

A

and

[A

R2.

From

A

to infer

-+

to infer

B] NA

.

See Hughes and C r e s s w e l l [1968].

B ,

PROPOSITIONAL OPERATORS

93

The systems we c o n s i d e r a r e K r i p k e ' s system5 K , t h e Godel-Feys-von Wright system T , t h e Brouwersche system B , and t h e Lewis systems S4 and 55. K c o n t a i n s t h e axiom schemata AS1 and AS2 a l o n e , and i s c o n t a i n e d i n t h e

o t h e r s y s t e m s . I n a d d i t i o n , T c o n t a i n s AS3, B c o n t a i n s AS3 and AS4, 54 cont a i n s AS3 and ASS, and 55 c o n t a i n s AS3 and AS6 ( o r e q u i v a l e n t l y , AS3, AS4 and ASS). For each o f t h e s e systems a n a t u r a l s e m a n t i c s h3s been p r o v i d e d by Kripke, based on s o - c a l l e d " r e l e v a n c e r e l a t i o n s " between i n d i c e s . ' c i f i c a l l y , we t a k e a model o f K t o b e a p a i r non-empty s e t and o v e r bl -

i s a b i n a r y r e l a t i o n on

R

t o be a function

f o r each v a r i a b l e

bl = ( I , R)

.

p

a

bl, i , a

We t h e n d e f i n e

M, j , a

M, i , a

sat

A

sat

A

whenever

f o r every

a model o f T i f t h e r e l a t i o n

S4) i f i n a d d i t i o n if

K

A

a

.

.

sat

Spe-

is a

a(p) 6 2

M,

A model

M

I

i n t h e u s u a l way,

A

A formula

i s r e f l e x i v e on

i, a

A

sat

NA

is true in

= ( I , R)

i f and Fl

if

o f K is c a l l e d

I , a model o f B ( r e s p . ,

i s symmetric ( r e s p . , t r a n s i t i v e ) , and a model o f 55

i s an e q u i v a l e n c e r e l a t i o n on

R

mula

R

and

i

i R j

I

I , and d e f i n e an a s s i g n m e n t

on t h e s e t o f v a r i a b l e s such t h a t

w i t h t h e f o l l o w i n g c l a u s e f o r t h e modal o p e r a t o r : only i f

, where

I

.

Kripke [1963a] proved t h a t a f o r -

i s a theorem o f K ( r e s p . , T , B, S4, S5) j u s t i n c a s e

A

is true

i n e v e r y model o f K ( r e s p . , T , B, S4, SS). Corresponding t o t h e axiom schemata A S 2 through AS6 and t h e i n f e r e n c e r u l e R 2 , we i n t r o d u c e t h e f o l l o w i n g formulas o f MLp, i n which t h e v a r i a b l e

So d e s i g n a t e d i n Kaplan [1966], p . 1 2 1 . See Kripke [1963a], p . 95.

Kripke [1963a]. The i d e a o f u s i n g r e l e v a n c e r e l a t i o n s was s u g g e s t e d e a r l i e r by blontague [1960], Kanger [1957], and H i n t i k k a [1961]. These a u t h o r s had i n mind r e l a t i o n s between models, however, i n c o n t r a s t t o t h e i n d i c i a 1 approach o f Kripke.

94

HIGHER-ORDER MODAL LOGIC

M = (Ma,

Suppose t h a t and l e t if

i s a g-model o f

s a t i s f i e s t h e M-sentences

M

MLp+C

.

M

be a propositional operator of

N

and

AZ(N)

and

A3(N)

,

A4(N)

and

A3(N)

,

A5(N)

,

I

; a T-operator ( r e s p . ,

R2(N)

B-operator, S4-operator, S5-operator) i f i n addition (resp.,

with index s e t

i s c a l l e d a K-operator

N

E.1

Aj(N)

satisfies and

A3(N) ) . To

A6(N)

s e e t h e r e l a t i o n s h i p between t h e s e o p e r a t o r s and t h e c o r r e s p o n d i n g modal c a l c u l i , suppose t h a t , e . g . , N Then f o r any M-formulas

, if

(1)

A

(2)

N[A

B]

+

i s tautologous i n

A

[NA

+

+

-,

,

+

NB]

M ( i . e . , s a t i s f i e d by e v e r y

w i l l be t r u e i n

(1')

If

A

and

(2')

If

A

is true i n

[A

MLp+C

B , t h e M-formulas

and

A

M of

i s a K-operator o f a g-model

+

M

are true in

B]

M

then

and

i

then

B

is t r u e i n

NA

M

a ) , and i n a d d i t i o n is true in

M ,

.

Thus, any M-formula which i s a n i n s t a n c e ( i n t h e language of t h e g-model

M ) o f a theorem of K w i l l b e t r u e i n

.

M

S i m i l a r remarks a p p l y t o T-oper-

a t o r s , B-operators, e t c . 'The p r o p o s i t i o n a l o p e r a t o r s a r i s i n g from r e l e v a n c e r e l a t i o n s on t h e s e t I

a r e o f c o u r s e of a s p e c i a l t y p e . We can f o r m a l l y c h a r a c t e r i z e such o p e r -

a t o r s i n MLp; s p e c i f i c a l l y , an o p e r a t o r

o f a g-model

N

of

M

is

MLp+C

indicial i f sat

M

0 3p

9

Vq

[ Nq - 0 [p

9

Suppose t h i s c o n d i t i o n h o l d s . Then t h e index s e t unique

I

N

+

determines a binary r e l a t i o n

M , as f o l l o w s : For each i C I , l e t f o r which M , i s a t Vq+ [Nq - 0 [P q]] of

P E bI 9 unique f o l l o w s from t h e o b s e r v a t i o n t h a t

by l e t t i n g

i RN j

i f and o n l y i f

relevance r e l a t i o n f o r

LEMMA 1 2 . 3 . let every

Let

N

M

i C I :

i R N j .

M, i

sat

+

M, i

Pi(j)

=

1

sat

.

NPi

Pi

.

on

RN

be the

(That

is

Pi

.) W e define

RN

This r e l a t i o n is c a l l e d t h e

, i n view of t h e f o l l o w i n g s t r a i g h t f o r w a r d b e a g-model o f

b e an i n d i c i a l o p e r a t o r of

N

.

q] ]

NA

M

.

MLp+C

with index s e t

Then f o r e v e r y M-sentence

i f and o n l y i f

M, j

sat

A

I

, and A

whenever

and

PROPOSITIONAL OPERATORS

COROLLARY 1 2 . 4 .

operator of

Let

b e a g-model o f

M

95

.

MLp+C

Then e v e r y i n d i c i a l

i s a K-operator.

bl

For i n d i c i a l o p e r a t o r s we can show t h a t t h e axioms o f t h e v a r i o u s modal c a l c u l i c h a r a c t e r i z e e x a c t l y t h e c o r r e s p o n d i n g p r o p e r t i e s of t h e r e l e v a n c e relation. Precisely: TIIEOREM 1 2 . 5 .

let

Let

be a g-model of

hl

b e an i n d i c i a l o p e r a t o r o f

N

.

bl

with index s e t

blLp+C

Then:

,

(i)

N

i s a T-operator i f f

RN

i s r e f l e x i v e on

(ii)

N

i s a B-operator i f f

RN

i s r e f l e x i v e and symmetric,

I

(iii) N

i s an S 4 - o p e r a t o r i f f

RN

i s r e f l e x i v e and t r a n s i t i v e ,

(iv)

i s an S 5 - o p e r a t o r i f f

RN

i s an e q u i v a l e n c e r e l a t i o n on

N

Proof: First, i f

, and

I

I .

We p r o v e ( i i ) ; t h e p r o o f s o f ( i ) , ( i i i ) and ( i v ) a r e s i m i l a r .

i s r e f l e x i v e and symmetric we must v e r i f y t h a t

RN

i s a B-

N

o p e r a t o r . But t h i s j u s t f o l l o w s K r i p k e ' s argument t h a t e v e r y theorem o f B i s t r u e i n e v e r y model of B , i n view o f Lemma 12.3. For t h e c o n v e r s e , we assume t h a t

i s an i n d i c i a l B - o p e r a t o r , so t h a t

N

R2(N) , A 3 ( N )

and

N

using

A3(N)

we o b t a i n

i RN i . To s e e t h a t

j RN i . Then Q C M

exists Using

A4(N)

Lemma N

-

Q

2.3

.

implies

.

A4(N)

To s e e t h a t

i s i n d i c i a l , we have

Since

6

,

hl,

i

Pi C M sat

and c l e a r l y

d '

-

P.

-

N-

-..P . ]

it follows t h a t

sat

- N

Q

sat

M, i

, i.e.,

I

1

N

.

, which

implies

.

NPi

,

= 1

so

, i.e.,

By comprehension t h e r e

, so t h a t Q . But

M, i i RN j

it i s n o t t h e c a s e t h a t

Q(i') = 0

i C I

i RN j but n o t

But t h i s c o n t r a d i c t s Lemma 1 2 . 3 , s i n c e f o r a l l P.(i') = 1 1

,

A2(N)

sat

M, i

, which i m p l i e s P i ( i )

P.

i s symmetric, suppose t h a t

RN

satisfies

is reflexive, l e t

RN

P . ( i ) = 0 , i . e . , M, i s a t J such t h a t M s a t 0 [Q ++

M, j

M

, i.e.,

sat

M, j

if C I

,

M, i'

sat

-

.

Q

, so

by

sat

j RN i f Q

.

I t i s n a t u r a l t o a s k whether t h e c o n v e r s e t o C o r o l l a r y 1 2 . 4 h o l d s ; i . e . , whether e v e r y K-operator i s i n d i c i a l . I t i s e a s y t o see, however, t h a t t h i s i s n o t t h e c a s e . I n f a c t , we can g i v e a n example o f an S4-opera t o r i n a s t a n d a r d model of MLp which i s n o t i n d i c i a l . The example i s t h e p r e s e n t p r o g r e s s i v e t e n s e of S c o t t : Let and l e t

M

I

be t h e s e t of r e a l numbers,

b e a s t a n d a r d mode1 of MLp w i t h i n d e x s e t

ositionaI operator

N C M

(9 1

= P(M+)'

by p u t t i n g

I

. Define

P E N(i)

t h e prop-

j u s t i n case

HIGlIER-ORDER MODAL LOGIC

96

P(j) = 1 for all

i n some open i n t e r v a l around

j

t h e reading "It i s being t h e case t h a t Lemma 1 2 . 2 t h a t

."

A

I f we t h i n k o f t h e

,

A

can b e g i v e n

NA

I t i s e a s i l y checked u s i n g

i s an S 4 - o p e r a t o r , b u t c l e a r l y

N

s h a l l s e e i n $15 t h a t some g-models of

.

i

i n d i c e s as moments i n time, t h e n f o r any M-sentence

is not indicial. W e

N

even c o n t a i n n o n - i n d i c i a 1

MLp+C

S 5 - o p e r a t o r s . Ilowever: TtIEOREM 12.6.

I n any g-model o f

, every S5-operator i s in-

MLp+C+EC

dicial. I n any s t a n d a r d model of MLp, e v e r y S 5 - o p e r a t o r i s i n -

COROLLARY 1 2 . 7 .

dicial. Proof of 1 2 . 6 : and l e t

3p

sat

By Theorem 1 1 . 5 , P'

0 [PI

A

Vq+ [ q

.

q]]

-+

M, i

(1)

0 [P'

-

sat

[ A

0 [P

ever

P(j) =

sat

NQ

0 [P

+

6

.

A5(N)

A2(N)

We show t h a t

.

q] ]

f o r which

.

0 [P .+ q ] ]

, or

q]]

-

0 [PI

t+

A

.

+

equivalently,

0 M, i

f o r which sat

M, i

Vq+ [q

f--f

A] ]

A l s o by comprehension, t h e r e e x i s t s

Vq+ [ 0 [P'

-+

Nq]

+

q]]

. This

P C M

d

such

t o g e t h e r w i t h (1) i m -

j C' I :

We show t h a t ( * ) h o l d s f o r M, i

0 [p

P C M

P(j) = 1 i f f f o r every

(2)

and

, so t h e r e e x i s t s P' E M

At --t

satisfies

M

I , , R2(N),

w i t h index s e t

MLp+C+EC

so t h a t

By comprehension we t h e r e f o r e have sat

sat

M

sat

M +

p l i e s t h a t f o r every

M

-

[ Nq

Vq+ [ Nq

f o r e v e r y M-sentence that

A4(N)

i C I ; we must f i n d M, i

sat

+ V q+

,

M

u s u a l p r o o f t h a t S5 e x t e n d s S4 and B , we con-

also satisfies

M

sat

Suppose (*)

b e a g-model of

M

. From t h e

A6(N)

A3(N) and clude t h a t M

Let

b e an S 5 - o p e r a t o r o f

N

Q

.

P