Intensional and higher-order modal logic: with applications to Montague semantics

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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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

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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

<

intension o r sense. Let us denote the former by Ext[Z] , the latter by Int[z] . We know what the extensions of certain sorts of English expressions should be, according to semantical conventions which we tacitly accept when we translate English sentences into the symbolism of predicate logic. For example, if < is a name (e.g., 'Jones') and we denote the universe of all individuals by D , then Ext[c] is an individual, i.e., an element of D . If t: is a common noun phrase (e.g., 'former thief') o r an intransitive verb phrase (e.g., 'run slowly'), then

Ext[<] D

set of individuals, or equivalently, an element of the set 2

is a

(the set

of former thieves, the set of individuals who run slowly).' If < is a sentence then ExtL] is simply a truth-value, o r element of the set 2 = I0,l) . Can we identify meaning with extension? For the purposes of a beginning course in symbolic logic, we often do. Thus, for example, the extension (truth-value) of the sentence 'Jones runs slowly' depends only on the extensions of 'Jones' and 'run slowly', construed as above, and Frege's

Frege [1892], English transl. in Feigl and Sellars [1949]. Carnap [1947] refers to "Frege's principles of interchangeability."

*

Here and throughout, exponentiation of sets has its usual meaning: BA denotes the set of all functions, or mappings, from A into B. As usual, we identify the number 2 with the set {O,l}.

4

INTENSIONAL LOGIC

f u n c t i o n a l i t y p r i n c i p l e i s s a t i s f i e d . P h i l o s o p h e r s have long been aware, however, of n a t u r a l examples f o r which F r e g e ' s p r i n c i p l e f a i l s when meani n g and e x t e n s i o n a r e i d e n t i f i e d , examples i n v o l v i n g " o b l i q u e " o r i n t c n siorial contexts.

For example, i n t h e s e n t e n c e ' N e c e s s a r i l y t h e morning

s t a r i s i d e n t i c a l with t h e morning s t a r ' , which should c e r t a i n l y be counted t r u e i f we u n d e r s t a n d " n e c e s s a r i l y " t o mean " i n a l l p o s s i b l e w o r l d s , " replacement o f t h e second o c c u r r e n c e o f t h e c o n s t i t u e n t ' t h e morning s t a r ' by t h e e x p r e s s i o n ' t h e evening s t a r ' , which h a s t h e same e x t e n s i o n , produces t h e f a l s e s e n t e n c e ' N e c e s s a r i l y t h e morning s t a r i s i d e n t i c a l w i t h t h e evening s t a r ' - - f a l s e b e c a u s e we can e a s i l y imagine a world i n which t h e s e s t a r s a r e n o t i d e n t i c a l . Thus t h e e x t e n s i o n o f t h e s e n t e n c e i s n o t

a f u n c t i o n of' t h e e x t e n s i o n s o f i t s p a r t s . To t a k e a n o t h e r example, l e t u s suppose t h a t J o n e s i s a t t h i s moment a member o f t h e U n i t e d S t a t e s S e n a t e ,

so t h a t t h e common noun p h r a s e ' c o l l e a g u e o f J o n e s ' h a s t h e same e x t e n s i o n as t h e p h r a s e 'United S t a t e s S e n a t o r ' . The compound p h r a s e ' f o r m e r c o l -

league of J o n e s ' has a s i t s e x t e n s i o n a c e r t a i n s e t of i n d i v i d u a l s , among whom i s J o n e s ' s o l d l a w p a r t n e r , Smith. I f we r e p l a c e t h e c o n s t i t u e n t ' c o l l e a g u e of J o n e s ' by t h e c o e x t e n s i o n a l 'U.S. S e n a t o r ' , however, we o b t a i n t h e p h r a s e ' f o r m e r U.S. S e n a t o r ' , t h e e x t e n s i o n o f which does n o t c o n t a i n Smith. T h i s shows t h a t t h e e x t e n s i o n o f ' f o r m e r c o l l e a g u e of J o n e s ' i s n o t a f u n c t i o n merely o f t h e e x t e n s i o n s o f i t s p a r t s , s o t h a t F r e g e ' s p r i n c i p l e f a i l s h e r e as w e l l . T h i s second example shows t h a t t h e d i f f i c u l t y c a n n o t be avoided by simply r e f u s i n g t o countenance t h e e x i s t e n c e o f such e n t i t i e s as " p o s s i b l e w o r l d s , " f o r even i n t h e a b s e n c e o f such n o t i o n s we e n c o u n t e r problems o f t h e same s o r t . Aware of t h e s p e c i a l problems posed by o b l i q u e c o n t e x t s , F r e g e d i d n o t abandon t h e f u n c t i o n a l i t y p r i n c i p l e f o r e x t e n s i o n s b u t r a t h e r h e l d t h e view t h a t t h e e x t e n s i o n of an e x p r e s s i o n c o n t e x t i n which i t o c c u r s . When extension IS becomes

Ext[z]

Int[<]

t:

t:

depends on t h e s y n t a c t i c

i s u s e d i n an o r d i n a r y c o n t e x t i t s

, b u t when used i n an o b l i q u e c o n t e x t i t s e x t e n s i o n

, t h e ( o r d i n a r y ) s e n s e of t:

. According

t o t h i s view, t h e

f a i l u r e of t h e functionality p r i n c i p l e f o r extensions is a t t r i b u t a b l e t o t h e ambiguity of n a t u r a l language. I n p r i n c i p l e one c o u l d e l i m i n a t e t h e d i f f i c u l t i e s b y i n t r o d u c i n g , f o r each e x p r e s s i o n

See Quine [1960] f o r d i s c u s s i o n and examples.

<

,

a new e x p r e s s i o n

NATURAL LANGUAGE AND INTENSIONAL LOGIC

5

(the concept of ) whose extension is the intension of t: . The functionality principle for extensions could then be preserved by using At:

^t:

in place of

t: when

occurs in oblique contexts. Equivalently, one

could amend the functionality principle for extensions to assert that the extension of a compound expression is a function of the extensions of those of its constituents standing within ordinary contexts, together with the intensions of those constituents standing within oblique contexts. The first serious attempt to develop an intensional logic - - i.e., a logic of intensions - - along the lines suggested by Frege was that of Church [1951], who provided an axiomatic version of the Frege theory. The task of providing a referential semantics4 for the theory remained, however, due to the problem of finding an adequate interpretation for intensions. In recent years this pnoblem has been solved by means of a device which goes back to Carnap.5 Consider the examples given earlier, in which the functionality principle for extensions failed to hold. What seems essential in both examples is that the extensions of the various expressions involved depend on the particular state of affairs. For instance, the extension of the name 'the morning star' depends on the particular world in question; the extensions of the common noun phrases 'colleague of Jones' and ' U . S . Senator' both depend on the particular moment in time. When we know the extension of such an expression for a particular state o f affairs we may still need to know its extensions for other states of affairs before we can determine the extension of a phrase of which it is a constituent. For example, to determine the set of former colleagues of Jones (at this moment), it is not enough to know the set of colleagues of Jones (at this moment); we must also know the set of colleagues of Jones at each past moment. Carnap made the proposal that the intension of an expression be identified with the function on possible states of affairs whose value, at a particular state of affairs, is the extension of the expression in that state of affairs. That is, according to Carnap "extension" is really a function of two variables: we should speak of Ext[c,i] , where is an expres-

I . e., a model theory in the sense of Tarski [1954]. See Carnap [1947], $40.

INTENSIONAL LOGIC

6

sion and

.

i

is a possible state of affairs, in place of o u r earlier with the function F

I of possible states of affairs, such that F(i) = Ext[<,i] for each i C I. This definition meets an implicit criterion of Frege's that the extension of an expression should be recapturable from its intension; in the present case the former is just the value of the latter at a particular state of affairs.6 Ext[c]

We then identify Int[E]

on the set

Kaplan [1964] used Carnap's proposal to provide a semantics for Church's intensional logic. Kaplan followed a suggestion of Carnap, however, in identifying possible states of affairs with models of the underlying language, taking

I to be a set of such models. This approach has

a number of disadvantages, and it has now been widely supplanted by the approach of Kripke [1959], which takes the notion of a possible state of affairs - - in the context of modal logic, a possible world -- as primitive.' Montague [1968] and Scott [1970] suggested that a possible state of affairs be thought of as specifying the context of use appropriate to a particular language. Expressions of the language are thought of as indexical,e i.e., their extensions depend on a particular context of use. For ~

example, to evaluate the truth o r falsity of an utterance of the English sentence 'I was here yesterday', one must know the speaker s , the time

, the world w , and the spatial position p = (x,y,z) . Thus, a possible state of affairs - - in Scott's terminology an index o r point o f reference - - can be thought of as a sequence i = (s,t,w,p, ... ) , where t

the remaining coordinates specify, in general, various other aspects of the context of use.9 Given an expression Z be able to define the extension Ext[E,i]

and an index i , we should of t: at i , and then, fol-

lowing Carnap, define the intension Int[Z] Int[<](i)

=

Ext[<,i]

for

i

C

of Z by the equation I , where I is the set of indices.

What sorts of entities serve as the intensions of the particular types of English expressions considered earlier? Suppose we let D

See Church [1956], p. 9. Cf. Bayart [1958] for an early approach along these lines. The term is due to C.S. Peirce. See Bar-llillel [1954]. See the discussion in Lewis [1970].

now

NATURAL LANGUAGE AND INTENSIONAL LOGIC

7

represent the set of all possible individuals, i.e., individuals which exist with respect to at least one index i C I Int[c]

.

If

<

is a name then

will be an individual concept, i.e., an element of D1

a common noun phrase or intransitive verb phrase, Int[r]

.

If

will be a

t:

is

prop-

erty of individuals, i.e., an element of [Z0]' . If t: is a sentence, Int[<] will be a proposi.tion, i.e., an element of 2' . l o The potential which Carnap's idea holds for the analysis of natural language only becomes fully clear when we carry the construction to higher orders, for we can then assign intensions to many parts of speech whose analysis eludes ordinary predicate logic. Montague' first suggested such an approach for verbs which take a propositional object, such as 'know' and 'believe', which play a key role in a number of problems in the philosophy of language. Montague's proposal was to assign to such verbs intensions in the set

or in other words extensions which are functions mapping an individual and a proposition to a truth-value. The intension of the sentence 'Jones believes that snow is white' could be expressed in terms o f the intensions of its constituents by the condition: Int['Jones believes that snow is white'](i)

=

Int['believe'](i)(Int['Jones'](i),Int['Sr~ow

is white'])

Although discussion of the proper treatment of belief continues among philosophers and seems to have no single satisfactory answer, Montague's higher-order intensional approach at least provides an adequate framework for the analysis of one sense of belief, according to which the object of belief is a proposition (as opposed to a sentence or some other entity).12

l o Some authors, e.g., Scott [1970] and Lewis [1970], take concepts to be partial functions on I, possibly undefined for some indices.

Talk given in 1967, reported in Montague [1970a]. l 2 See Partee [1973].

INTENSIONAL LOGIC

8

blontague's s t u d e n t Kampl

pointed out t h a t t h e intensions of a d j e c t i v e s

could be t a k c n t o h e elements o f t h e s e t

so t h a t t h e e x t e n s i o n s o f a n a d j e c t i v e map p r o p e r t i e s o f i n d i v i d u a l s t o s e t s of i n d i v i d u a l s . For t h e common noun p h r a s e ' f o r m e r c o l l e a g u e of J o n e s ' , f o r example, we would have t h e s e m a n t i c c o n d i t i o n : I n t [ ' f o r m e r c o l l e a g u e of J o n e s ' ] ( i ) = ~nt['former'](i)(Int['colleague

.

of J o n e s ' ] )

I n blontaguc [1970h] t h e method was extended t o encompass a l i m i t e d b u t comp l e t e l y f o r m a l i z e d fragment o f E n g l i s h , for which a p r e c i s e s y n t a x was provided i n terms of grammatical c a t e g o r i e s and f o r m a t i o n r u l e s , a s w e l l as

; i

r e f e r e n t i a l s e m a n t i c s . The r i c h e r fragment of hlontaguc [197Oc] accommodates s e n t e n c e s , common and p r o p e r nouns, r e l a t i v e c l a u s e s , s i n g u l a r t e r m s , adj e c t i v e s , t r a n s i t i v e and i n t r a n s i t i v e v e r b s ( i n c l u d i n g s o - c a l l e d " i n t e n s i o n a l " v e r b s ) , and v e r b s t a k i n g a p r o p o s i t i o n a l o b j e c t . The same p a p e r a l s o o u t l i n e s a g e n e r a l t h e o r y o f grammar and s e m a n t i c s . 1 4 I n blontague [1973], t h e t r e a t m e n t i s s i m p l i f i e d somewhat and extended i n v a r i o u s ways, e . g . , t o accommodate i n t e n s i o n a l p r e p o s i t i o n s and a d v c r h s . l ' a r i o u s p h i l o s o p h e r s and l i n g u i s t s i n r e c e n t y e a r s , among them Davids o n , Parsons and Lewis, have been i n t e r e s t e d In g i v i n g a p r e c i s e s e m a n t i c a l account of n a t u r a l language. The E n g l i s h fragments c o n s t r u c t e d by Montague c o n s t i t u t e an i m p o r t a n t s t e p i n t h i s d i r e c t i o n . They a r e r i c h enough t o p r o v i d e an a n a l y s i s of c e r t a i n p h i l o s o p h i c a l problems which h i n g e on i n t e n s i o n a l n o t i o n s , and t h e i r u n d e r l y i n g method has a l r e a d y been extended i n v a r i o u s ways.15 I t i s worth n o t i n g t h a t t h e E n g l i s h f r a g m e n t s o b t a i n e d i n t h i s way, as w e l l a s t h e formal l o g i c IL t o b e d i s c u s s e d i n $2, s a t i s f y Frege's functionality principle for intensions

--

t h e i n t e n s i o n of a com-

pound i s a f u n c t i o n o f t h e i n t e n s i o n s o f i t s c o n s t i t u e n t s

--

and a l s o t h e

f u n c t i o n a l i t y p r i n c i p l e € o r e x t e n s i o n s , a s amended t o t a k e i n t o a c c o u n t

l 3 A l s o , i n d e p e n d e n t l y , P a r s o n s [ 19681

.

l 4 C f . Lewis [1970] and P a r t e e [1975a].

S e e P a r t e e [1975b].

NATURAL LANGUAGE AND INTENSIONAL LOGIC

o b l i q u e c o n t e x t s . Whether o r n o t t h e i n t e n s i o n

<

f i e d w i t h t h e meaning of

Int[Z]

9

should be i d e n t i -

i s an a r g u a b l e q u e s t i o n ; as Lewis p o i n t s o u t ,

however, " i n t e n s i o n s a r e p a r t o f t h e way t o meanings

...

and t h e y a r e o f

i n t e r e s t i n t h e i r own r i g h t . " l 6 Let u s u s e t h e term E n g l i s h t o r e f e r t o one o f t h e f o r m a l i z e d f r a g m e n t s i n Montague [197Oc], [1973]. In each c a s e we can s i n g l e o u t a c e r t a i n subs e t of t h e b a s i c v o c a b u l a r y which c o n s i s t s o f e x t e n s i o n a l words: l o o s e l y s p e a k i n g , t h o s e which do n o t c r e a t e o b l i q u e c o n t e x t s . l 7 C o n s i d e r , f o r example, t h e a d j e c t i v e s ' f o r m e r ' and ' t a l l ' . Each t a s an i n t e n s i o n i n t h e set

but they d i f f e r i n t h a t t h e extension, a t

i

, of ' f o r m e r c o l l e a g u e o f

J o n e s ' depends on t h e e n t i r e i n t e n s i o n of ' c o l l e a g u e o f J o n e s ' , whereas the extension, a t extension a t

i

i

, of

' t a l l c o l l e a g u e o f J o n e s ' depends o n l y on t h e

of 'colleague of Jones'. Put d i f f e r e n t l y , t h e s e t of t a l l

colleagues of Jones, a t

i

, i s determined once we know t h e s e t o f c o l -

leagues of Jones a t

i ; t h e same i s n o t t r u e o f t h e s e t o f former c o l -

leagues of Jones a t

i

.

The a d j e c t i v e ' t a l l ' i s t h e r e f o r e e x t e n s i o n a l ,

i n c o n t r a s t t o t h e intensional a d j e c t i v e 'former'. Extensionality of an a d j e c t i v e can b e e x p r e s s e d by t h e f o l l o w i n g c o n d i t i o n on i t s i n t e n s i o n

If i

E

I , G , H 6 [ZD]I , and G ( i )

= tI(i)

then

F :

, F ( i ) (G) = F ( i ) (11)

.

S i m i l a r l y one can d i s t i n g u i s h between e x t e n s i o n a l and i n t e n s i o n a l v e r b s , p r e p o s i t i o n s , e t c . , and r e q u i r e of e v e r y model of E n g l i s h t h a t t h e i n t e n s i o n s of a l l e x t e n s i o n a l words s a t i s f y t h e a p p r o p r i a t e s e m a n t i c r e s t r i c t i o n . Those e x p r e s s i o n s which o n l y i n v o l v e e x t e n s i o n a l w o r d s l a t h e n comp r i s e a sublanguage which we s h a l l c a l l E x t e n s i o n a l E n g l i s h . We r e f e r t o t h i s language b r i e f l y i n 54.

l6 l7

Lewis [1970], p . 25. S e e Montague [197Oc], pp. 395-396. C e r t a i n o f t h e f o r m a t i o n r u l e s may have t o be r e s t r i c t e d a l s o .

10

INTENSIONAL LOGIC

The Logic IL

52.

Montague makes use, in Montague [197Dc], [1973], of an auxiliary formal logic which he calls Intensional Logic, here denoted by IL. The language of IL is broad enough to encompass a wide variety of intensional notions; Montague uses the semantics for IL, in fact, to provide his semantics for English, by means of a translatim nf English into the symbolism of 1L.l Our principal object of study in this chapter will be the logic I t . In constructing domains of entities to serve as intensions of various expressions of English, we have observed the need to pass from a given domain

A to another domain A ' , and from given domains A and B to the domain BA . This observation motivates the choice of symbolism for

IL, which is based on the theory of types as formulated in Church [1940], in which functional abstraction is taken as a primitive notion.

Types. Let e , t , s be any three objects, none of which is an ordered pair. The set of types of IL is the smallest set T satisfying:

(i)

e , t < T ,

(ii)

u , p C T imply

(iii)

a

F T implies

(a,p) C (s,a)

E T

T ,

.

As will emerge later, objects of type e will be possible entities o r individuals, objects of type t will he truth-values, objects of type (a,P) will he functions from objects of type u to objects of type p , and objects of type

(s,a)

will be functions from indices to objects of type a ,

i.e., senses appropriate to denotations of type e up

for

(a,P)

and sa

for

(s,a)

.

We frequently write

.

To accommodate a fragment of English admitting tenses, one would add tense operators to the formalism of IL, as in Montague [1973]. The semantics of IL would then be extended to take account of the tense operators. See Cocchiarella [1966].

THE LOGIC IL Primitive Symbols. For each

a F

11

T we have a denumerable list of

variables

and non-logical constants'

of type

a

, together with the improper symbols

,X ,

B

,. ,

"

, [ , ]

.

We also denote the variables of type a , in their proper order, by

and the constants by

and c i is c 4 . We use the letters ' X I , 'y', is x8 'h' (serif type), with or without superscripts or primes, as syntactical variables ranging over formal variables of IL, and similarly

so that, e.g., ha k!,

... ,

we use 'c', I d ' , with or without superscripts or primes, t o range over constants of IL. Terms. a

We characterize recursively the set Tma of terms of IL

of

, as follows:

(i)

Every variable of type a belongs to Tma

,

(ii)

Every constant of type

,

B E Tma imply

(iii) A E Tm

aP '

(iv)

A 6 Tm

(v)

A , B

(vi)

A 6 Tma

(vii)

A E TmSa implies "A

P ' E

belongs to Tma

a

x

[AB] E Tm

a variable of type a

Tma imply

[A

implies ^ A

5

B]

E

P ' imply Xx A E Tm

aP '

Tmt '

TmSa ,

F E

Tm

.

One could allow here an arbitrary set of constants, not necessarily denumerable. See comment at the end of $ 3 .

INTENSIONAL LOGIC

12

lVe b r i t e

for

Au

when

A

Tma , and adopt t h e Lisual c o n v e n t i o n s r e -

A C

g a r d i n g grouping i n terms. In p a r t i c u l a r , we sometimes u s e p a r e n t h c s e s )

in place of brackcts Semantics.

based on

Let

D

[ , and

D

1 ,

,

and o u t e r m o s t b r a c k e t s may be o m i t t e d .

be non-empty s e t s . By t h e s t a n d a r d frame

I

of s e t s ,

we u n d e r s t a n d t h e indexed f a m i l y

I

(

where

,

(i)

Me = D

(ii)

bf

(iii)

hfup=

blp

(iv)

Msa

Mu

t

= 2 = {0,1)

=

a

I

=

, {FI F : M a + M

)

P .

= {FI F:I +Ma}

A ( s t a n d a r d ) i n o d e l o f I L based on

n

and

I i s a System

,

= (hiu,

where

i s t h e s t a n d a r d frame based on

(Ma)uET

(ii)

in ( t h e meaning f u n c t i o n ) i s a mapping which a s s i g n s t o each c o n s t a n t I c a f u n c t i o n from I i n t o Ma ; i n symbols, m(c,) C Ma .

I n t u i t i v e l y , a constant

c

of type

D

and

,

(I)

I

r e p r e s e n t s a name, l i k e ' J o n e s '

e

or ' t h e morning s t a r ' , and must t h e r e f o r e be a s s i g n e d an i n d i v i d u a l conc e p t , r a t h e r t h a n an i n d i v i d u a l , as i t s meaning o r i n t e n s i o n . S i m i l a r l y , a constant

c

o f t y p e e t might r e p r e s e n t a b a s i c common noun, l i k e et ' t h i e f ' , which must b e a s s i g n e d a p r o p e r t y o f i n d i v i d u a l s , r a t h e r t h a n a

s e t , a s i t s meaning. If

i s a model based on

M

n o t e d by

Doni(M)

d e n o t e by

As(M)

a

I , t h e domain

and

D

, and t h e i n d e x s e t

of

I

of type

a

then

a(x/X)

able

y

.

i s equal t o

i

a 6 As(M)

, xa

X

if

value

M

y

V. (A,) i,a and t h e assignment

is

a' xu

a(x,)

, by

a(y)

M

i s de-

Ind(M)

.

We

M , i . e . , a l l functions C Ma

whose v a l u e and

M

a

of

f o r every v a r i a b l e

i s a variable of type

d e n o t e s t h e assignment

We d e f i n e t h e t h e index

If

D

i s d e n o t e d by

t h e s e t of a l l assignments over

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

xa

M

a

al(y)

, and

X C Mu

for a vari-

otherwise.

o f t h e term

Aa

with respect t o

t h e f o l l o w i n g r e c u r s i o n on t h e

13

THE LOGIC IL

term

o f IL (we s u p p r e s s t h e s u p e r s c r i p t ' b f ' ) :

A

(1)

v l. , a (xu)

=

a(xa) ,

(2)

Vi,a(ca)

=

m(c,)(i)

(3)

Vi,a(A,pBa)

(4)

V. (Ax, A ) = t h e f u n c t i o n I: on Ma whose v a l u e a t l,a P e q u a l t o Vi,,,(Ap) , where a ' = a(x/X) ,

(5)

Vi,a(Aa

(6)

V.

(^Aa) =

to

V.

i,a

j

=

v i. , a (Aup I[\'. i , a ( B a l l ,

=

1 if

B ) =

V.

1 ,a

the function

,a(*u)

Vi,a("Asu)

(7)

,

(A,)

V.

=

F

(Be)

l,a

on

I

, and

is

X E bl,

0 otherwise,

whose v a l u e a t

i s equal

j E I

1

V.

=

l,a

(Asa)(i)

.

I t i s e a s i l y s e e n t h a t we always have

V.

l,a

(A,)

C Flu

.

fiere

value p l a y s

the

r o l e of extension i n our e a r l i e r discussion, although t o allow f o r f r e e v a r i a b l e s b o t h t h e e x t e n s i o n and t h e i n t e n s i o n o f a term on t h e assignment and Inta[Aa] Viaa(Aa) . 3

a

.

P r e c i s e l y , we d e f i n e

t o be t h e function

on

F

I

w i l l depend

A,

Cxta[Aa,i]

t o be

whose v a l u e a t

V.

i,a

s

,

is

i C I

C l a u s e s (6) and ( 7 ) d e s e r v e s p e c i a l a t t e n t i o n . The c a p o p e r a t o r a c t s a s a functional a b s t r a c t o r over i n d i c e s , although

(A,)

^

itself is not a

t y p e and no v a r i a b l e s r a n g i n g o v e r i n d i c e s are p r e s e n t i n I L . 4 Given a term Au

, an index

i

and an assignment

i . e . , f o r each term

A

tension (with r e s p e c t t o (with respect t o i s independent o f

t i o n on

I

a term

ASe

.

a

, we have

E x t a [ " A u , i ] = Inta[Aa] ;

o f IL we can produce a n o t h e r t e r m a

and any index

, so t h a t

The cup o p e r a t o r

"

Inta[^A,]

whose ex-

i ) i s t h e intension of

a ) . In p a r t i c u l a r , t h e extension

i E I

"A,

Exta["Aa,i]

of

Aa "Aa

i s always a c o n s t a n t func-

i s an i n v e r s e t o t h e c a p o p e r a t o r . Given

, f o r example, which d e n o t e s f o r a p a r t i c u l a r i n d e x and a s s i g n -

We adopt t h i s c o u r s e f o r r e a s o n s of convenience. There would be no d i f f i c u l t y , however, i n making t h e assignment p a r t of t h e c o n t e x t of u s e , by r e p l a c i n g o u r p r e s e n t i n d i c e s , i , by p a i r s (i,a). I n Chapter 2 we c o n s i d e r a t w o - s o r t e d t h e o r y o f t y p e s which c o n t a i n s such v a r i a b l e s .

14

INTENSIONAL L O G I C

ment a c e r t a i n i n d i v i d u a l c o n c e p t , t h e term

is o f type

"A

e

and d e n o t e s

t h e i n d i v i d u a l which i s t h e v a l u e of t h e c o n c e p t f o r t h e i n d e x i n q u e s t i o n . The terms

and

Aa

w i l l always have t h e same e x t e n s i o n s , and hence

""A,

a l s o t h e same i n t e n s i o n s . flowever, f o r a n a r b i t r a r y term n o t b e of t h e form ^ B a , t h e terms

and

A

, which may

As,

need n o t have t h e same

^'A

extensions. An o c c u r r e n c e o f a v a r i a b l e pends o n l y on t h e v a l u e s

i n a term

x

P

free. A s

hxp By , o t h e r w i s e

within a p a r t

a(xP)

for

x

P

free in

contains exactly t h e d i s t i n c t variables

A, Y € ,!b

, we

is

A,

bound i f

usual, the value A,

xP '

yY

f o r any assignment

a

can w r i t e

it occurs

Vi,a(A,)

de-

, so t h a t , e . g . , i f f r e e , and

X E M

such t h a t

a(x) = X ,

P '

'i ;X , y ( A a )

t o denote t h e value

.

a(y) = Y

V. (A,) l,a In p a r t i c u l a r , if

ables, then simply

f o r every v a r i a b l e

xa E MC

(ii)

"A,

(iii)

[A

(iv)

[Aa

(v)

Xx, AP C MC

6 MC

f o r e v e r y term

B ] C MC

whenever

a

B, ] C MC

AaP

whenever

whenever

i C I

i s independent o f

A

C MC ,

B,

,

B,

€ MC

.

A,

P

,

C MC ,

o f a modally c l o s e d t e r m

V. (A,) l,a

i s b o t h c l o s e d and modally c l o s e d we w r i t e

and

i

M V.

l,a

At

of type

t

V(A,)

. Given

a , we s a y t h a t t h e formula

and an assignment

A

.

A,

If

. a model

M

, an index

is satisfied i n

M

a , and write

M, i , a if

,

, and we can t h e r e f o r e write simply Va(Aa)

A formula o f IL i s a t e r m i

and we w r i t e

x, ,

A,

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

9

,

o f modally c l o s e d terms i s t h e s m a l l e s t c l a s s such t h a t

MC

(i)

Aa

a

.

The c l a s s

5

A,

i s independent of t h e a s s i g n m e n t

V. (A,) l,a

Vi(A,)

aP

i s c l o s e d , i . e . , c o n t a i n s no f r e e v a r i -

(A) = 1

sat

.

A

,

In t h e case that

A

i s c l o s e d , modally c l o s e d , o r b o t h ,

TtIE LOGIC IL

15

we write respectively M, i sat A , M, a sat A , or we give the obvious meaning to an expression such as

when

A

.

A is a formula containing exactly the distinct free variables and X , Y are elements of hi , bly respectively. A formula A

yY ’ true i n M

Also,

sat A ,

i; X,Y

bl;

M sat

if M, i, a sat

for every index i

A

xP

and assignment a

is

’

.

set I: of formulas is satisfied in h1 by i a , and we write M, i, a sat C , if M, i, a sat A for every A 6 Z . Z is satisfiable in IL if M, i, a sat C for some model F1 , index i and assignment a . A formula A is a semantical consequence, in IL, of a Set r A

of formulas, and we write

r

in IL,

A

if M, i, a sat A whenever bl, i, a

sat

r .

A

formula A

is

9 in

IL, and we write in IL,

/= A

if A is a semantical consequence i n IL of the empty set of formulas, or equivalently, if A is true in every model of 1 L . We introduce the sentential connectives, quantifiers and modal operators in IL by definition5: T =

[hxtxt.Xx

F =

[ Xxt xt

-

5

x t t

1 ,

Axt T ] ,

=

XX

[F

A =

Ax

Xy

-+=

AXt

AYt [ [x

=

x

v =

AXt

Xyt [ - x - t y ]

,

t

Vx A =

t

5

xt] , [ hftt [fx z y] A

Yl

[ X x A c X xa T l

=

Xf

tt [f T I ]

,

1 ,

>

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

h1

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

j

c

.

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

j u s t in case either

M, i , a

B , and

sat

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

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)

At(Ba)

(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

, if

17

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 (i)

Me

(ii)

Mt = 2 = {O,l} ,

(iii) E.1 (iv) A

aP

Msa

=

D ,

is a non-empty subset of F1

F'U

is a non-empty subset of \la

I

P

' .

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

1) '0

AS4.3

(Ax,

AS4.4

(Xx, [AP

E

(Axa Xy

A 1 Ba

AS4.5

y

[A C PY

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

Ba

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

A;

f

t

and

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

1<6.

From

Asa

R7.

From

A

R8.

From

A

A

proof

111

a

u

a

a

5

ALa

c A'

a

5

A' n

t o infer

A

,

A'

A' ha ,

t

H

F

,

A'

UP

A z Ax ^A'

B

'A

A'

,

,

z "A'

,

A ] a [B

to infer

[B

to infer

[A c B ] z [A' s

z

t

A'] ,

B ]

.

I L i s a sequence of formulas cnch of which i s e i t h e r an axiom

o r c l 5 e is o b t a i n a b l e from e a r l i e r formulas by R u l e K . A formula

A

is

r

is a

p r o v a b l c i n I L , o r a theorem of IL, and w e w r i t e

1-

i n IL,

A

i f i t i s t h e l a s t l i n e of a p r o o f i n IL. I f

A

i s a formula and

s e t of f o r m u l a s , we w r i t e

r 1-

in IL

A

and s a y t h a t A 1 1

B ,

... , Bo

+

.

H

in

1

+

r

F

i n I L , i f t h e r e e x i s t formulas

B

0

such t h a t

. ...

i s a theorem of IL. A set mula

r

i s d e r i v a b l e from

4 .

C

i < n o t d e r i v a h l e from

1

+ A

of f o r m u l a s i s c o n s i s t e n t i n IL i f t h e f o r -

Z i n IL.

,

GENERALIZED COMPLETENESS OF IL

21

TIEOREM 3.1 (Soundness Theorem for IL)

1-

(i)

kg A in implies r hg A

in 11, implies

A

IL,

(ii)

r 1-

(iii)

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

A

in IL

in IL,

Proof: The axioms of IL are easily seen to be g-valid (in the case of ___ AS4 one can either show this directly o r verify the g-validity of the equivalent AS4.1 through AS4.7), and Rule R clearly preserves g-validity. This proves (i), from which (ii) and (iii) follow immediately. The Generalized Completeness Theorem for IL (Theorem 3.3) is the converse to Theorem 3.1. Before we can prove it we need some additional information about the theory IL, which is provided in the following list of metatheorems. The reader can easily reconstruct the proofs of these theorems, in the order listed, by consulting llenkin [1963] and Andrews [1963]; we remark that later theorems in the list frequently imply earlier ones (e.g., T35 implies T5.1), but in general these later theorems depend on earlier ones for their proof. and a term Ba ,

Some preliminary terminology: Given a term A (x )

B

we s a y that

is free for x

5

P a

if no free occurrence of x

A(x)

lies within a part Xy C , where y

A(x)

tial formula type t

occurs free in

is one built up from the formulas T , F

by means of the connectives

5

,

-,

A

,

--+

B

.

in

A senten-

and variables of

, v

.

A sentential

formula is a tautology if it is valid (or equivalently, g-valid) in IL. An arbitrary formula of IL is tautologous if it comes from a tautology by uniform substitution of formulas of IL for free variables. Metatheorems of IL. T1.

I-

A

T2.

1

T

T3.

1I/-1-

T4. ~5.1 T5.2

a

E

Aa

Vxa T

0T

Aa

5

Ba

implies

Aa

E

Ba

and

/--

tBa

B~ P

C

f

a

imply

I-

Aa

=

C

INTENSIONAL LOGIC

22

T6. T7. T8.

T9.

III-

T A T

I-

TI

[A

P

A

implies

Vx, A(x)

mula

A(x)

1-

vx

A

1

implies

i s any formula.

A

, where A(B)

A(Ba)

comes from t h e f o r -

by 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

is f r e e f o r

and ( i ) B rence of

in

x

A(x)

x

in

x

by t h e term

B,

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

A(x)

-

l i e s w i t h i n t h e scope o f

, or

else ( i i ' )

i s modally c l o s e d .

B

T10.

, where

A

5

1-

A(x,)

1-

implies

, where A(x)

A(B,)

and

B

satisfy the

c o n d i t i o n s o f 1'9. T11.

T12.

I-

A

1-

Xxa A ( x ) a Xy, A (y) , where

I-

and

P

except t h a t

B

I-

Vxa A(x)

Vy,

B

A A B

A(x)

P

A(x)

and

has f r e e occurrences of

f r e e occurrences of T13.

I-

imply

y

A{y) x

arc identical

where

A(y)

has

, and v i c e - v e r s a . , where

A(y)

A(XJ

and

A(y)

a r e formulas s a t -

i s f y i n g t h e c o n d i t i o n s of T12. T14.

T1S.

b

[ Aapx 5 BUpx ] occurring f r e e i n A o r

b

Vx,

A(T)

I-

and

A(F)

come from t h e formula

x T16.

T17.

Ib

T22.

III1I-

T23.

I-

T18. T19. T20.

T21.

by

and

T

T A F

[T

A

-+

xu f

aP

A

-+

g

E

if

5

aP

I-

A(xt)

B . imply

A(x)

A f +

imply

XEf y aP UP f x u E gxu

i s a tautology

, where A(T)

and

A(F)

by 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

t

t A

, where x i s any v a r i a b l e n o t

xt

and y,

F + x

x

xt]

E

B]

F 5

B

-+

E

respectively.

F

E

TAxt

[A

E

I-

B

GENERALIZED COMPLETENESS OF I L

T24.

1-

T25.

1- vxa A(x) of T9.

T26. T27. T28. T29. T30.

I-

if A

A

A(B,)

11I-

3x

I-

Vxa

3x

t

t

+

x

23

is tautologous.

, where A ( x )

A(B,)

+

B

and

satisfy the conditions

3xa A ( x ) , under the same conditions.

t

-

x

t

3 xa [A a

x] , where

E

[ A * B]

[A

-+

-+

is any variable not free in A

x

V x B] , where

.

is any variable n o t free

x

in A . T31.

I-

=

B a

C

A (8)

A (C)

5

P

-+

a

P

, where A ( B ) , A ( C )

by replacing all free occurrences of x

come from A ( x a ) by the terms B , C re-

spectively, and (i) B and C are free for x in A ( x ) , and either (ii) no free occurrence of x in A ( x ) lies within the scope of

T36.

II11I-

T37.

1-

T32. T33. T34. T35.

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

A

B

a AaP

=B ' a = A'

A

E

ASa

eBa a

-+

"A

T39.

[ B

+

'

Ba

'A'

E

a

s C

a

-+

A e C ]

+

II-

B

a

= C

--c

a

and C

B

, if

CI A * A

, where

T43.

)-U[A-+B]

T44.

I-

are free for x

A

is a modally closed formula. is any formula.

A

A + O A - 0 A

A

+

0 - U A

implies

in A ( x )

0 [BaC]

A -+ElA

T42.

T41.

a A

B'

= Ca A (B) E A (C) , where A ( B ) , A ( C ) come from A ( x a ) a P P by replacing all free occurrences of x by the terms B , C reB

11I-

T40.

aP

B s A

spectively, and T38.

B e A

A B

-+

a B a

aP

-+

aP

ASa A

A

--t

[ O A + O B ]

+

t-

A

.

24

INTENSI ONAI, LOCIC

T48.

I- - n A 1- - O A 1- O O A I- nnA

T49. T50.

T45. 1'46. 1'47.

s

0 - A

s

n-A C)A

s

CI~A

(-0oA

z

UA

I-OOA

z

OA

T66.

I- 0 [ A h B] n A A O B 1- O [ A V B ] t O A V O B I- O A V U B n [ A v B ] I- O[A A B ] OA A O B 1- U [ A - + B ] [OA-tOB] 1- U Vxa A VX U A I- 0 3 x a A z 3x 0A I- 3xa n A n 3x A V X OA /- Ovx, A I["fsa I "gsa] I [f I g ] 1[ B z~ ca] B 5 c I- ^ A I~ fsa [A B "f] A E r implies r 1 A r I- A and r 5 A imply A 1- A t A implies r I- A I- A and r I- [A B] imply r 1-

T67.

r I-

A

T68.

r u

{A}

T69.

r I-

A

T51.

5

T52. T53.

-+

T54.

-+

T55.

-+

T56.

2

T57. T58.

-+

~ 5 9 . T60. T61. T62. T63. T64. T65.

--t

-+

-+

B

-+

implies ring free in r .

1

B

r I-

Vx,, A

if and only if

if and only if

r

U {

, where

r I-

-

A }

x

[A

-+

is any variable not occur-

B]

is inconsistent.

We can now prove a key lemma needed for the Generalized Completeness Theorem for I L :

GENERALIZED COMPLETENESS OF I L

Z be a s e t o f f o r m u l a s , c o n s i s t e n t i n IL, and suppose

Let

LEMMA 3 . 2 .

25

t h e r e a r c i n f i n i t e l y many v a r i a b l e s o f each t y p e which do n o t o c c u r i n any

.

formula i n

Then t h e r e e x i s t s a sequence

-

-

Z

= ( Zi

of s e t s o f

)iCo

formulas such t h a t :

(ii)

For each

, Z.

i C w

i s a maximal c o n s i s t e n t s e t o f formulas i n

IL,

(iii)

i C w

For each

and each formula

i f and o n l y i f x

in

only i f Proof: -

If

C

Ei

, we have 3x B(x)

B(x,)

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

i C o

f o r some

1

j C o

Cnj(A)

c o n j u n c t i o n , i n some s t a n d a r d o r d e r , o f t h e formulas of T

ri

)iCw

if

Ai

5 Ti

for all

r

that

A

i s r e l a t i v e l y i - c o n s i s t e n t with

from

r

by adding 0 (iO,A )

Let

i C w k

where quence

C

=

values of (2)

, (il,A

1

)

, which we t a k e

r

r

=

finite, the set

A.

A

and

i E w

, we s a y

i n IL i f t h e sequence o b t a i n e d

i s r e l a t i v e l y c o n s i s t e n t i n IL.

, ...

b e an enumeration o f a l l p a i r s

For each

, Zik C w ,

k C w

(i,A)

,

i

= 4 ( t h e empty s e t ) f o r a l l b u t f i n i t e l y many

k C o , t h e r e a r e i n f i n i t e l y many v a r i a b l e s o f each t y p e k Zi f o r i € w .

which do n o t occiir i n any formula i n any For

A

denote t h e

and A i s a formula o f IL. W e d e f i n e f o r each k 6 w a sek ( Zi )iCw o f sets o f formulas, s a t i s f y i n g t h e c o n d i t i o n s :

For each

(1)

ri

to

i f and

is r e l a t i v e l y consistent

, each

i C w

o f f o r m u l a s i s c o n s i s t e n t i n IL. Given a formula A

Zi

i s empty. Given an a r b i t r a r y sequence

A

o f s e t s o f f o r m u l a s , we s a y t h a t

i n IL i f whenever

C

.

i s a f i n i t e set o f f o r m u l a s , l e t

A

OB

B , we have

and each f o r m u l a

F.

B C

t o b e t h e formula (

Fi

E

which i s f r e e f o r

y

,

B(x)

For each

(iv)

B(y,)

k = 0

we put

0 Zo =

Z , Zi0

= 4

for

i z 0

.

INTENSIONAL LOGIC

26

Given Ck satisfying (1) and (Z), we define Zk+' as follows: Suppose k (ik,A ) is the pair (i,A)

.

Case 1. A k t o be Z .

B or

OB .

u

.

Case 2.1. A does not have the form ]xu be the same as C k , except put i' k+l = 2 1

.

Case 2.2. A is ]xu B(x) k+l - k except put Zi - Ci U {A, B(y,)}

{A}

Then let Ck+'

Then let

be the same as

Zk ,

, where y is the first variable of

type a which does not occur in Zk o r B(x)

Then take zk+l

.

is relatively i-consistent with Zk

Case 2. A

2k+ 1

.

is not relatively i-consistent with Zk

B(x)

by replacing all free occurrences of x

, and by

B(y)

comes from

.

y

Case 2.3. A is O B . Then let Zk+' be the same as Ck , except k Zk+l = {B] , where j i s the smallest numPut i = Ci U {A} , and put Ck+' j ber different from i for which C k =

.

1

LEMMA 3.2.1. Each

Proof:

Ck

By induction on k

Bo

A

B1

A

...

.

If Co ,,n- 1

, ... ,

for some formulas go, B'

0[

is relatively consistent in IL.

A

were relatively inconsistent then in Z the formula

Bn-l ]

would be inconsistent in IL, and hence 2

itself would be inconsistent in

IL by T41, contradicting the hypothesis of Lemma 3.2. Assume Zk relativek We show that C k+ 1 ly consistent, and let (ik,A ) be the pair (i,A)

.

is relatively consistent: Case 1. A i s not relatively i-consistent with Zk and is therefore relatively consistent. Case 2.

A

is relatively i-consistent with Zk

Case 2.1. A

. Then

Zk+'

= Ck

.

does not have the form 3xa B o r O B

. Then

C

k+ 1

is obtained from Zk by adding A to 2: , and is therefore relatively consistent by the assumption under Case 2 .

GENERALIZED COMPLETENESS OF IL

A

is

3xa B(x)

.

not occurring i n

Zk

or

, and suppose t h a t z k + l were r e l a t i v e k A. C C . , j C w , t h e s e t

Case 2 . 2 . a

27

B(x)

Let

y

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

l y i n c o n s i s t e n t . Then f o r some f i n i t e s e t s { OCnj(Aj)

1

j # i

1 U I

O [Cnj(Ai)

1 -

A

A

would b e i n c o n s i s t e n t i n IL. But t h e n , l e t t i n g

j # i }

J

A

B(Y)

r

1 1

be t h e s e t

{OCnj(Aj)

I

and u s i n g T13, T56, T67, T69 and o t h e r s , we would h a v e :

t O c n j ( ~ I~ j) #

i

1 u

{

0[

cnj(lli)

A

A

1

}

would b e i n c o n s i s t e n t i n IL, c o n t r a d i c t i n g t h e assumption under Case 2 t h a t A

is r e l a t i v e l y i-consistent with Case 2 . 3 .

from

i

f o r which

A

is k

OB .

C. = 6 J

Let

IL

.

j

b e t h e s m a l l e s t number d i f f e r e n t

, and suppose t h a t

s i s t e n t . Then f o r some f i n i t e sets IOCnj(AL)

Ck

# i,j 1 U I

AL

5 Z Lk

were r e l a t i v e l y i n c o n -

Ck+’

, 1 C w , t h e set

O [Cnj(Ai)

A

A

1 1

U {

OB 1

would b e i n c o n s i s t e n t i n IL. But by T50, T54 we have

would b e i n c o n s i s t e n t i n IL, c o n t r a d i c t i n g t h e assumption under Case 2.

INTENSIONAL LOGIC

28

The sequence Z = ( y. ) .

LEMMA 3 . 2 . 2 .

is relatively consistent,

1Cw

1

where -

zi = u kEw zki

Proof:

'

5 ';2

Clearly C ki

for k , i

. The

C w

assertion now follows

in the usual way, using Lemma 3 . 2 . 1 . -

LEMMA 3 . 2 . 3 .

~

is consistent in IL for each

2.

i

C w

.

Proof: Follows from Lemma 3 . 2 . 2 .

If A is a formula, i C w , and A sistent with Z , then A C . LEMMA 3 . 2 . 4 .

-

zi

___ Proof: Choose k

so that the pair

,

is relatively i-consistent with Zk LEMMA 3 . 2 . 5 .

Proof:

If A

. Clearly

(ik,Ak

is

LFi .

so by construction A C"!Z

is a formula and

F.

[-A , then A C

A

Ti .

Straightforward, using Lemmas 3 . 2 . 2 and 3 . 2 . 4 . 2. is a maximal consistent set in IL; i.e., f o r every

LEMMA 3 . 2 . 6 .

formula A

(i,A)

is relatively i-con-

either A

C

-

z. or else

A C

r. .

Proof: Suppose neither of the formulas A ,

-

A

-

belongs to 2.

. Then

by Lemma 3 . 2 . 4 neither formula is relatively i-consistent with . It folsuch that lows that there exist, for j 6 w , finite sets A! , :'A C 1

Ijz

u

i

o [ cnj(~f)

A

I 0 Cnj (A;) I j # i 1 u

{

0[

A

I 0 cnj(~;)

i 1

A

1 1

-

A ] }

I -

c.J

and Cnj (A;)

are both inconsistent in IL. But then, letting A. = A! U :'A 1 1 1 T54, we see that

r I-

0 - [ cnj(Ai)

r I- o where

r

= {

-

[ cnj(Ai)

C>Cnj(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

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

3 x B(x) E

in

c

0

}

is inconsistent in IL,

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

zi

free for x

j

29

B(x) .

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

for some variable y

LEMMA 3 . 2 . 8 .

For each

free for x

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) .

(vi)

For each

xi

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

some

i E w

x E Var

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 E o

i s independent o f x

*

y . Let

x/=

x/=

ses

for

-

from

pu

,

For

(2)

For e v e r y

i

, x

and

:

E Var,

a = t :

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)

i E I ),

rr

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:

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

A

y

D

x E Var,

X E Ma

,

,

y (mod i )

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

D

Ma1

into

j E I

i C I

xa

Var /=

there exists

For

' 1

0

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

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)

0

Then

x

x

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

(3)

.

x E Var

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

g-model

x

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

D

we u s e t h e sequence

ping

for variables

denote t h e equivalence c l a s s of

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

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

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

,i

I ,

C

, and put

~~(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)

f c Var,

=

15

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

. We

can

M

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

i E I

, let

from

f C Vara

be

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

, 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

,

pa

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

j C I

, ["f

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

, at

Y 6 Ma

For, l e t

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 sat A sat Z

-

for any

,

i C I

of IL, M, i, a

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

E Per

t

implies ^A E Per

€

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

(x)

If At , Ba

C

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

formula 3x [ B ~

Proof:

=

x

A

A ]

(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

.

belongs to Per

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 .

ALTERNATIVE FORMULATIONS OF IL

Modal T-Logic

$5.

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

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

-

MLT.l

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. Let A

be either an atomic term or a formula of MLT. We say 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

A

is modally closed for any f o r -

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'

ALTERNATIVE FORMULATIONS OF SL

44

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]

and A

to infer B ,

R1.

From

R2.

From A

to infer vxa A ,

R3.

From A

to infer 0 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

(s(at))(sa)

, and therefore M sat

.

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 -

48

ALTERNATIVE FORMULATIONS OF I L

f

where

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

is of type

b1

sat

.

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

: 0 3!xa A

3f

-+

0 Vxa [ x

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@

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

[ 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

<

I

, a E As(M)

sat

Vxa 3 ! y

f o r which

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

P

. A

M = (Ma,

.

sat

be a

M, i , a

Then for e v e r y

M; i ; a , X , Y

a(x/X)(y/Y)

m)acT

We show t h a t

.

o f IL,

A

.

A

By Lemma 6 . 1 ,

sat

X C Ma

, where M

sat

DP,

and t h e r e f o r e

f o r some t i n c t from

F’ C M

xa

, where we can assume t h a t t h e v a r i a b l e (PtIP and does n o t o c c u r f r e e i n A . L e t t i n g

f’

is dis-

EXTENSIONS OF I L AND blLT

49

M

V. A ] ) C h1 i ; a , F ' ( Axa [ f ' Xy P aP '

F =

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

sat

M; i ; a , F

where

f

Vx,

Vy

[ y

P

P

fx

is the f i r s t variable of type

,

A ]

+

not occurring f r e e i n

ap

, and

A

therefore

M, i, a so that

M, i , a

We show t h a t

.

A

sat

RaJP'*

.

M, a

sat

RaJA

i C I

t h a t f o r every sat

3 f Vx Vy [ y z f x

sat

M

sat

Vgs(at)

[

,

A ]

: Assume t h a t

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

By Lemma 6 . 1 ,

M; F '

+

M, a

sat

0 3 ! x a A , so

such t h a t

X C Ma

, and t h e r e f o r e

sat

3 ! x a "9x1

+

1

0 "s"f'gl1

f o r some F ' C M , where f ' i s a v a r i a b l e o f t y p e ( s ( a t ) 1 (sa) n o t o c c u r r i n g f r e e i n t h e formula A . L e t t i n g F =

V

M ; i ; a,X

M ( f ' ^Axa A ) C Msa a,F'

( s ( a t ) ) (sa)

,

i t i s s t r a i g h t f o r w a r d t o check t h a t

where

0 Vxa [ x

sat

M; a,F

=

'f

-+

A ]

sa n o t o c c u r r i n g f r e e i n

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

f

,

.

A

Therefore M,a

sat

3fOVx[xE'f

+

A ] ,

which completes t h e p r o o f o f Lemma 6 . 2 . 1 . The s e m a n t i c a l argument g i v e n h e r e can, o f c o u r s e , b e e f f e c t i v e l y rep l a c e d by a d i r e c t s y n t a c t i c a l p r o o f i n t h e t h e o r y

a , and

p ,

and

A

ia'Aw i l l

.

IL+D

, f o r any g i v e n

The lemma shows t h a t , i n p a r t i c u l a r , a l l i n s t a n c e s

be provable i n

IL+D when

A

i s a formula of t h e s u b l a n -

guage MLT o f IL, s o by t h e g e n e r a l i z e d completeness o f COROLLARY 6 . 2 . 2 .

Every g-model of

IL+D

RaJpJA

IL+D

i s a g-model o f

we o b t a i n : MLT+R

.

ALTERNATIVE FORMULATIONS OF I L

50

Before p r o v i n g t h e c o n v e r s e t o C o r o l l a r y 6 . 2 . 2 , which i s i n c l u d e d i n Theorem 6 . 2 , we need an a n a l o g u e o f Lemma 6 . 2 . 1 . We f i r s t o b s e r v e t h a t t h e description principles LEMMA 6 . 2 . 3 .

provable i n

that

M

.

Da

[ 3!Xa [gx] o f MLT. Choosing M; X '

a C T , t h e formulas

T'

Ga

and

Da

are

.

M = (Ma,

Let

sat

are a l l formulas o f M L

For e v e r y t y p e

MLT+R

Proof:

ia

,

Da

Let A

XI

sat

A(gat,xa,xk)

gX ] C Ma

MLT+R ; we f i r s t show

b e a g-model of

V

[

be t h e formula [gx]

3!X

]

A X r; X'

a r b i t r a r i l y , we s e e t h a t

.

Vgat 3!xa A(g,x,x')

Using replacement and r e w r i t e of bound v a r i a b l e s , i t f o l l o w s t h a t M; F,X'

Vgat Vxa [ x

sat

:f g

A(g,x,x')

, where t h e v a r i a b l e (atla But t h e n c l e a r l y

f o r some

F C M

A(g,x,x')

.

Vg a t [ 3 ! x a [gxl

sat

M; F

.

To see t h a t

[ 0 3!xa ["gx]

A

0 ['g'h]

sat

M

f

sat

M

] does n o t occur free i n

(atla

1

g [ f ( a t ) a 91

+

Da

and hence

+

>

, let

A(gs(at),hsa,hga)

b e t h e formula

o f MLT.

Let

HI

6 Msa

] v [

N

0 3!x,

['gx]

A

h

P

h'

]

b e a r b i t r a r y , and suppose t h a t we can show t h e f o l -

lowing c o n d i t i o n : (*)

M;

HI

sat

.

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

Then by replacement and rewrite o f bound v a r i a b l e s ,

M; where

f

formula

H I

sat

3f Vg Vh [ h

7

i s a variable of type A(g,h,h')

M; F , H '

sat

.

fg

-f

] ,

A(g,h,h')

(s(at))(sa)

not occurring f r e e i n t h e

We t h e r e f o r e have Vg Vh [ h

5

fg

+

A(g,h,h')

]

EXTENSIONS OF IL AND MLT

f o r some

F E M

51

; b u t from t h i s i t i s s t r a i g h t f o r w a r d t o check

(s (at) 1(sa)

that sat

M; F

where

f

Vg

[ 0 3!x,

s (at)

"9x3

( s ( a t ) ) ( s a ) , and h e n c e t h a t

i s of t y p e

mains o n l y t o p r o v e c o n d i t i o n ( * ) . Assume t h a t M; G , H '

sat

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

.

1

0 ['g'[fgll

+

sat

M

G t M

M; G

F i r s t , in the case

(at) sat

>

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

B

,

y

A

fx ] .

5

not f r e e i n

fa

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

[B

is

-= C

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

P

‘f ]

,

not free in

.

Aa

Let

which is n o t f r e e i n

sa

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

0A

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

.

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

Case 2 .

is

Qo 1

3

.

.

0A

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

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

nA

(3)

-

+-.

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

where

f

.

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

recursively:1

va

9

2 ,a

wa

7

fa

9

Variables and constants of type

(ii)

A C Tm

(iii) A 6 Tm

298

A ,B

(iv)

, B c Tm

2 ,a

implies Xx

TmZ,,

imply

imply

A [A

C e

B]

for Ty2 based on I)

g&

a

are characterized

belong to Tm2,a ,

a

[AB] c Tm2,p ,

Tm

Generalized Semantics. Let D ~

.

ha

J

of terms of Ty2 of type

(i)

2,aP

9,

I

59

’

2,ap

Tm

C

and

2,t

.

be non-empty sets. By a

I

I we understand an indexed family

(E.fa)aET

of sets, where (i)

M

(ii)

E.1 = 2 = [O,l}

(iii)

M

(iv)

bl

= D

aP

2

,

t

=

frame

,

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

2 for each constant

I ,

(ii)

m(ca) E Ma

ca ,

(iii)

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

=

on Ma

whose value at X

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

V

(5)

(Aa : Ba)

1 if Va(Aa) = Va(Ba)

=

C Ma

,

, and 0 otherwise.

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

bg

. 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.

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

(iii) 2

satisfiable in IL if and only if 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 that m(c:) Aa

= m"(cn

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" = (M:,

m")uFT2

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

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

2 : '

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

(no,ul,...,u

imply

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

n- 1) E P

,

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'

u

o f type

we u s e

...

vxo Vxl

U

[s

s

s']

. . , u ~ - ~ and )

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

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,.

where 3!x

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 b!,

sat

a

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)

oo

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

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

0

x

)

Atu

I

n-1

is a non-empty subset of

.

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 bl, i, a

where

i

(i)

bf,

E I

sat A , 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

0A

sat

-,

[so

, Vxu ,

M, j , a

i f and o n l y i f

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

-+

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, a

PI,

X C Mo

-

i f and o n l y i f

3!x

a

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

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

e x i s t s a unique

-

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 -

. Moreover,

u

o f symbols o f t y p e

. 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

<

.

Next, we

n ; we d e f i n e a map-

k

k

ping

.

scT 2 ' s'

into

Sym

0

P(M,

...

x

u = (oo,.. .,u~-~)

where

bf,

x

n- 1

0

by p u t t i n g t h e sequence

0 (s,, 1

( Pu

0

into

IL

0

>

*..

( 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

,

-

Z a t the quantiZ when i = 0 and

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

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

sat A

A formula A

of ML P is called M-persistent if

if and only if M ' , i, 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 .

+

A]

and 3xa [ F ( x )

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

o

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

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

9

d e n o t e by

MLP+C+EC

t o t h e axioms o f

u

0

ak

...

x

for

n-1 c--f

k < n

,

A ] ]

, and fu i s

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]

i E I

such t h a t f o r

, 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

HIGIIER-ORDER MODAL LOGIC

80

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 .

Let M

be a g-model of ML

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

Proof: _ _

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. 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

I onto

o = (uo , . . . ,u

(x,) ,...,Oa

OO (iv)

be g-models of MLp based on

respectively. An indicial homomorphism from hI

9

(9

\I'

onto

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

(i)

( i i i ) For each

and

n- 1

I' ,

is a one-to-one mapping from blo )

n-1

,

Mo , i E I

F E

(k < n),

Xk C hlo

k

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

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

and

onto MA ,

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

iff

(Xo,...,Xn-l)

F(i)

I

, 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

D

i d e n t i t y mapping on

.

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

, 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

i E I 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

we c m choose

# 4

tion of Xo

i

C Y ;

but then

i E Xo , contradicting the defini-

.

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

other axioms. In 815 we prove that At

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

M; P,F

-

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

for all

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 E.1

prehension,

whcre

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

j 6 I

and

for a l l only i f

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,

C G(j)

i f and

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)

.

X E &lo

f o r every

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

for some

P(j) = 1 A

,

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 ,

93

PROPOSITIONAL OPERATORS

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

NA

sat

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]]

-+

(1)

M, i

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

<

Md ,

Suppose

Q C N(i) Q E M+

,

implies

Q(j) = I

and assume f i r s t t h a t

, i . e . , Q C N ( i ) . Then from ( 2 ) , we h a v e Q ( j ) = 1 when1 , so M s a t 0 [P + Q] . On t h e o t h e r hand, assume t h a t

-+

tain first that

Q] ; t h e n by M

sat

A2(N)

0NIP

+

,

R (N) 2

and comprehension we e a s i l y ob-

Q] and t h e n t h a t

M

sat

0 [NP

+

NQ]

.

PROPOSITIONAL OPERATORS

97

Thus, i f we can show M, i

(**)

sat

,

NP

t h e n we can conclude t h a t

M, i

sat

N-P

]

NQ

, and t h e p r o o f

w i l l b e complete.

W e f i r s t show

(t)

M, j

F o r , suppose Qo(j) = 0 But

0 [ -P

sat

M

bl, i

sat

, i.e., sat

+.

M

i.e.,

so using N

-

0 [P

,

A2(N)

.

P ]

Therefore

conclude t h a t

-

+

NQ,]

.

0 A5(N)

, M,

implies

0 [Q,

sat

Irl

Ql(if) = 1

I t follows t h a t

[

sat

bl

and comprehension we o b t a i n

sat

j

-

M, i

we conclude t h a t

be such t h a t

E N(i)

Q,

-

NQ,

sat

bl

-

-

-

N-

sat

P

.

M

0[

sat

-

P

-

NQ,.

.

NNQ,

-

pa

,

P ]

+

0[ N NQ, NQ,

-

NQ,

]

.

+

we

(t).

This completes t h e proof of

A3(N) , (t) i m p l i e s t h a t

sat

with

NQ ] . Then 0 for a l l i' C I ,

A (N) we s e e t h a t M s a t 0 [ NQo + N 6 0 [ NQ, + N P ] , and s i n c e M, j s a t

M, j

I n view o f

there exists

. llence u s i n g A3(N)

Q

6 (2), P(i') = 1

R2(N)

sat

Then by ( 2 ) ,

Q, C M

But u s i n g M

.

, s o using

, whence by sat

P

sat

hi, j

NQ,

By comprehension, l e t Q, C N ( i )

+

.

N

-

P ]

, or

equivalently,

which by Lemma 1 2 1 y i e l d s (4)

M

sat

0

NP-N-N-P],

But by (2), u s i n g have

PI, i

M, i

sat

N

A3(N)

- -

sat

- N

N

P

.

, we c l e a r l y have P ( i )

P , and t h e r e f o r e u s i n g

T h i s w i t h (4) y i e l d s

= 1

,

A (N) 6

M, i

sat

whence by ( 3 ) , we and comprehension, NP

, so t h a t

(**)

h o l d s and t h e theorem i s proved. We have s e e n t h a t v a r i o u s c l a s s e s o f modal o p e r a t o r s - - e . g . ,

those

obeying s p e c i f i e d modal axioms, o r t h o s e 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 between i n d i c e s - - can b e c h a r a c t e r i z e d i n a n a t u r a l way by means o f f o r mal c o n d i t i o n s e x p r e s s i b l e i n M L p .

I t would b e i n t e r e s t i n g t o know t o what

f u r t h e r e x t e n t t h e language o f MLp can b e used i n c l a s s i f y i n g p r o p o s i t i o n a l operators.

HIGHER-ORDER MODAL LOGIC

98

$13. Relative Strength of IL and MLp We now compare the logics IL and MLp by means of respective translations of the formulas of each language into formulas of the other. In each case we have the expected result that the translation preserves the standard semantics: A formula of IL is valid in IL if and only if its translate is valid in MLp, and vice-versa. However, these translations do not preserve the deductive theories IL and MLp, o r equivalently, the generalized semantics for these logics; in particular, there are theorems of IL whose translates are not theorems of MLp. We therefore consider as well the extended theories I L + D and MLp+C+EC , for which we prove strong relative interpretability, in the following sense: A formula is provable in one of these theories if and only if its translate is provable in the other. Interpretability of MLp

IL. For each u C P we define a corre-

sponding type a[a] C T as follows: (i)

a[e] = e ,

(ii)

] . [ a

=

To each symbol

(i)

(ii)

(s,(a[a0],( s

of ML

0

. . . (a [ ~ ~ - ~ ] , t ...))) )

when u = (uo,...,an-1)

we make correspond a symbol s

- at01

P

If s is xz then 2 is xn a[aI ' n If s is c z then 5 is c ].[a

-

F o r each formula A

of MLP we define a translate

A

0

n-1 ... s ,

...

(i)

If A is

sas

(ii)

If A

is

[se

(iii)

If A

is

(iv)

If A

is

(v)

If A

is Vxo B

(vi)

If A

is 0 B

-

n-1 s then

A

is

then

A

B then

[B

+

C]

A

then

c sh]

then then

A

A

is

[ 5z

],

11 , is

is

is

0 is " 5 %

A

[

vx

11 .

E-+C 3 , ,

in IL:

of IL:

.

RELATIVE STRENGTH OF I L AND MLp

If

Z

lates

E

i s a s e t o f formulas o f MLp we d e n o t e by

&

. A l s o , we l e t = x ] , where -

A E Z

for 3x

t h e form

[ c

a[,]

Let

THEOREM 13.1.

t h e s e t o f a l l trans-

c o n s i s t o f a l l f o r m u l as o f I L o f

AZ

c

99

is a constant occurring i n

U

be a s e t o f f o r m u las o f MLp. Then

Z

f i a b l e i n MLp i f and only i f

.

Z

Z is satis-

is satisfiable in IL.

U Az

We prove one i m p l i c a t i o n only; t h e o t h e r is s i m i l a r . Suppose

Proof:

Z i s s a t i s f i a b l e i n MLp

; s a y M , i , a s a t Z , where M = (M mIucp i s a s t a n d a r d model o f MLp based on D and I , i C I and a E A s ( M ) . Let

(MA)aeT

b e t h e s t a n d a r d frame . f o r I L based on from

M,

i s t h e i d e n t i t y mapping on

M

n o n i c a l one-to-one mappings (i)

+e

(ii)

For

Xi,

+U

u = (uo,. .

, F E

(k < n) , we p u t

C M' a [.,I

(Xo,. . .,Xn-l)

case

E F(i)

and

D

onto

,

and d e f i n e ca-

, as f o l l o w s:

M'

a [Dl

= D = M'

e '

MU = P ( MU x

...

0 +U(F)(i)(X;)

...

, where

I

) I , and n- 1 (X' ) = 1 j u s t i n n- 1 MU

x

(k < n)

Xk = +-'(Xi,)

.

Ok

We d e f i n e a meaning f u n c t i o n

m'( 5 ) ( i ) = +,[m(c,)] MLp, and l e t t i n g

da

m'

for all

m'(da)

o v e r t h e frame i C I

be an a r b i t r a r y element o f

o f I L which a r e n o t o f t h e form

by l e t t i n g

(MA)aET

, whenever cU i s a c o n s t a n t o f

c . The

system

MA1 MI

=

f o r constants

(MA, m')aCT

is a

s t a n d a r d model o f IL, and one e a s i l y proves by i n d u c t i o n : LEMMA 1 3 . 1 . 1 .

a E As(M)

,

Let

a' € As(M')

f o r ev er y v a r i a b l e

MI, i , a ' Since

A

b e a formula o f MLp

M, i , a

we c l e a r l y have

&

U AZ

Suppose t h a t

i C I

,

xU o f MLp. Then

M, i , a

sat

A

i f and o n l y i f

& .

sat

sat

Z by assumption, i f we choose a ' C A s ( M ' )

such a way t h a t (1) h o ld s t h e n t h e lemma y i e l d s that

.

, and

M'

sat

AZ

M',

i, a'

i n view o f t h e d e f i n i t i o n o f

sat

E.

in Since

m' , we see

i s s a t i s f i a b l e i n I L , which completes t h e p r o o f o f 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 Theorem 1 3 . 1 .

100

HIGHER-ORDER MODAL LOGIC

Let

COROLLARY 1 3 . 2 .

lct

A = A

.

C

r

Then

COROLLARY 13.3.

r

C =

be a formula o f MLp and l e t

A

AiA3

j u n c t i o n of t h e formulas i n

[

]

+

I=

U A

i n MLp if and o n l y i f

)=A

Let

U {A} b e a s e t o f f o r m u l a s of MLp, and

.

Then

i n ML

(= A

i n IL. be t h e con-

6A

i f and o n l y i f

P

i n IL.

Turning now t o t h e g e n e r a l i z e d s e m a n t i c s , we have t h e f o l l o w i n g : Let

TtlEOKfiM 1 3 . 4 .

be a s e t o f formulas o f MLP.

2

COROLLARY 13.5.

let

A = A

Let

2 '

COROLLARY 1 3 . 6 .

1-

1 3 . 3 . Then

A

Let

in

Proof o f 1 3 . 4 :

r

2 =

tA

r

Then

CU

If

Z i s g - s a t i s f i a b l e i n MLp+C+EC

g - s a t i s f i a b l e i n IL, t h e n

in A

is

U {A} b e a s e t o f formulas o f MLP, and

and

u

implies

MLp+C+EC

I-A

A

i n IL.

s a t i s f y t h e hypothesis of Corollary

1[

implies

MLp+C+EC

Suppose

Az

.

M', i , a '

and

D

]

5u

satisfy

(MA, m')aFT i s a g-model o f I L b a s e d on

4

--f

I

i n IL.

Az

.

, where M'

Then

M' = satisfies

each formula (1)

3xuLuI [ g e x

1 ,

where

co

C

occurs i n

for a l l constants c

U

7,

.

= D =

assume t h a t

M'

and l e t

ae

from

F ' E M' a [a1

dition that

Xi

, let F

.

(Xo,. .,Xn-l)

.

= Q

(Xk) (k < n) k' i s one-to-one on M'

a [ul

s a t i s f i e s (1)

i f n e c e s s a r y when

Mu o n t o M '

,Q

hiu

k Given

c)

b e t h e i d e n t i t y mapping on

. . , u ~ - ~ and )

u = (uo,.

m'(

IVe s i m u l t a n e o u s l y d e f i n e , by r e c u r s i o n on

and a one-to-one mapping

hiu

M

, and i n f a c t we can assume t h a t M' o f MLp, by r e d c f i n i n g

does n o t o c c u r i n

a set put

c

E P( hlu x . 0 C F(i) iff

a [Dl D , Next, we

a r e defined f o r

Mu

k < n

.

uk

..

x

hio

)'

n-1 F'(i)(Xb)

b e d e f i n e d by t h e con-

...

(XA-l) = 1 , where

I t i s e a s i l y checked t h a t t h e mapping o f ; we l e t

u E P,

: We f i r s t

be i t s r a n g e and

*o

F'

to

F

its inverse.

d e f i n e d i n t h i s way i s a frame f o r ML (Mu)uCp P We d e f i n e a meaning f u n c t i o n m by p u t t i n g m(c,)

Clearly t h e family based on

D

and

I

.

=

+i'[m'(

5 ) (i)] ,

system

M = (Mo, m)oEp i s a g-model o f MLp, and one v e r i f i e s by i n d u c t i o n

which i s independent o f

i E I

by v i r t u e o f ( 1 ) . The

RELATIVE STRENGTH OF IL AND MLp

101

that Lemma 13.1.1 holds in the present situation in exactly the form given earlier. Thus, since M', i, a' were assumed to satisfy C , we must have also M, i, a sat Z , where a is chosen to satisfy condition (1) of Lemma 13.1.1. It therefore remains only to show that M is a g-model of MLp+C+EC , i e., that the schemata of comprehension and extensional comprehension hold in M Ca,A

.

.. .

3fo 0 VX0

:

Let t/xn-l [ f x0.. . xn-1

-

A 1

k be an instance of the comprehension schema, where u = (U~,...,U~-~) ,x is of type ak for k < n , and f, is the first variable of MLp of type

which is not free in the formula A . By Lemma 13.1.1 it suffices to show that its translate is true in M' ; in fact, we show the some-

LT

Ca,A

what stronger: LEMMA 13.4.1.

3fa

VX0

,

Let a , a.

.

(s,(ao,(...(an-l,t)...)))

...

Let A

, an- 1 E T and suppose that a = be any formula of IL. Then the formula

Vxn-l [ ' f x0* .. xn-1 * A 1

...

is provable in IL, where xk

is of type ak

for k < n

and

first variable of type a which does not occur free in A

Proof: F

=

V"(

Let M

(bfa, m)aET

=

^XxO . . . Xxn-' A ) M;

a,F

be a g-model of IL, a

.

€ As(M)

.

Putting

, it is easily checked that

E Ma

sat 0 Vxo . .. Vxn-l [ ' f xo.. . xn-'

A ]

++

.

of extensional

In a similar way we verify that every instance EC''A comprehension is true in M

fa is the

by showing that the translate

&

is true

in M' , and this follows from: LEMMA 13.4.2. Under the hypotheses of Lemma 13.4.1, the formula

0 Ifa [ VXO..

.

t/x"-l [ 0 'f xo. .. 2 - l v 0 A

Vx0

...

Vxn-'

[ 'f

x

'f

xo.

.. xn-l

1

... xn - l t + A ] ]

is a theorem of IL. _ Proof: _

and suppose

Let M = (Ma, m)acT i € I

,a

€

As(M)

be a g-model o f IL based on D

. We put

M

G = V i,a ( XxO . . .

and

I ,

Xxn-' A ) E M

B '

HIGHER-ORDER MODAL LOGIC

102

p

where

= ( a o , ( a l , (.

verifies that

vx

0

...

. . ( a n - l , t ) . . . ) ) ) , and

F1; i ; F vx

n- 1

M

F = V,(*g

let

P

s a t i s f y t h e formula

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

u- " f

xo..

b'x'...

'dxn-'

[ 'f

A

.

) C M

xn-l ]

x

0

...

a

.

-

n-1 x

One

A ] ,

which y i e l d s t h e d e s i r e d r e s u l t . T h i s completes t h e p r o o f of Theorem 1 3 . 4 . We remark t h a t C o r o l l a r y 13.6 can be g i v e n a d i r e c t s y n t a c t i c p r o o f

One shows t h a t t h e s e t of formulas

A of ML having t h e p r o p e r t y t h a t P i s a theorem o f I L c o n t a i n s t h e axioms o f ML +C+EC and i s [ 6A ] P c l o s e d under t h e i n f e r e n c e r u l e s o f t h a t t h e o r y .

+A

I n t e r p r e t a b i l i t y of I L & M L P .

We now o u t l i n e a similar i n t e r p r e t a t i o n

of IL i n MLp, o m i t t i n g d e t a i l e d p r o o f s . F i r s t , we make c o r r e s p o n d t o each a C T

a type

(i)

a[e] = e

,

(ii)

u[t] = 9

,

(iii)

a[aP1 = ( a [ a l , o [ P l ) ,

(iv)

u [ s a ] = (u[a])

For each

o[a] C P :

a C T

.

and each v a r i a b l e

i s of t y p e

T~(V) ( v

v

t

(ii)

T (p,)

(iii)

Tap (fa[aP1) Vx 0

(iv)

[x

~ ~ ( x , )i s

[a1

[ Ta(x)

Tsa(fo[sal)

If

s

is

-+

is

x:

v

0 Vx s

then

f r e e , as f o l l o w s :

1 ,

Rn(f) A Vx .[a1 3!!y alp] f x y 1

is

we d e f i n e a formula

,

[ 0 p v 0- p

is

To each p r o p e r symbol (i)

XI

1

P

a

of MLp c o n t a i n i n g e x a c t l y t h e v a r i a b l e (i)

o f ML

a[aI

a[aI

[ fx

vydPI

+

[ fxy

-+

Ta(x)

.[a1

Tp(y) ]

A

' Ta(x) ]

A

0 3!!x

atal

of IL we make c o r r e s p o n d a symbol

? i s xn

A

'

fx

s

. o f MLp:

RELATIVE STRENGTH OF IL AND MLp

(ii) Let

cz

free i n

.

A

A

v

A

.

is

v

a ’

Aa

is

c

(iii)

Aa

is

[BpaCpl

a

-

[ v

is

(ii)

(iv)

is

Aa

hx

Then

(v)

Au

y A

CO

C

[B

A

= C

B A

[fxy

PI

.

A,

is

of t y p e

^B

P

o[P

0 Vx [ f x

v

where

v ).

.

-

is

Let

,

f

ex

.

x

be the first

r e s p e c t i v e l y , which A

. Then

is

Co (y)

.

fXy ]

Tp(

+-+

, p+

o[p]

Aa

.

Let

be the first vari-

y

x

and open f o r

B

.

f

.[a1

C Co ( y )

A

0 [ p

open f o r

B c o (x) ]

.

A

Au

COB()’) ]

.

.

,y

Let

x

be t h e first

which are d i s t i n c t from

Aa

which i s open f o r

-

)

p

and

is

Co (p) A

2

open f o r

A

Then

B 3 x 3y [ Co (x) (vi)

A

open f o r

d i s t i n c t v a r i a b l e s of type open f o r

o f MLp

.[a1

is

Vx Vy

is

x

d i s t i n c t from

A

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

o[y]

Co ( f ) A

x

(o[p],a[a]) , a [ P ]

(X)

B

Rn(f)

open

Aa :

Co (x)

Aa

and open f o r

P Y ’ f+l

A

is

.

]

of types

[ COB(f)

able of type

(i.e.,

A,

open f o r

a r e d i s t i n c t from 3X

=x

, P

x

which o ccu r s

together with t h e v a r i a b l e s

a r b i t r a r y . Then

‘a[a]

v a r i a b l e s of ML

3f

x

open f o r

‘a[a]

Co (x)

v

A )

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

A

Then

*

o f IL and each v a r i a b l e

Aa

codes

( x

[a11

f o r every v a r i a b l e

whose f r e e v a r i a b l e s a r e

Au

(0

, w e d e f i n e a formula

A,

is free in

(i)

cn

a v a r i a b l e o f MLp. We say t h a t

xu

For each term

Co (x) P

is

i s d i s t i n c t from

x

if

which i s open f o r

of ML

5

then

be a term o f I L ,

Aa

for A -

is

s

If

103

+-+

x

S

Y 1 1 .

. Let x B . Then

be t h e f i r s t v a r i a b l e A

Co ( f )

is

104

HIGHER-ORDER MODAL LOGIC

(vii)

is

Aa

open f o r Aa . Let f o[aI ( u [ a ] ) which i s open f o r B Then

,

"Rsa

A Co (x)

.

of type

3 f [ COB(f) For each formula

3p4 [ Co (p)

which i s open f o r

.

A

0 p ] , where

A

-

c)

o f a l l formulas (1) TSa(

x)

.

, where c

, where x

i s t h e f i r s t v a r i a b l e of t y p e A l s o , we l e t Z

occurs i n

denote

denote t h e set

,

together with a l l

.

Z

occurs f r e e i n

-

is s a t i s -

i s s a t i s f i a b l e i n MLp.

Z U A'

f i a b l e i n IL i f and o n l y i f

c

+

A'

Z be a s e t o f f o r m u l a s o f IL. Then Z

Let

TIIEOREM 1 3 . 7 .

A C Z

for

A

p

i n llLp t o b e t h e

Z o f formulas of IL, we l e t

Given a s e t

t h e s e t of a l l formulas

formulas ( 2 ) Ta(

A

o f IL we d e f i n e i t s t r a n s l a t e

A

is

.

]

A fX

A

formula

be the first variable

x

W e omit t h e proof COROLLARY 1 3 . 8 . A =

.

'A

Then

COROLLARY 1 3 . 9 .

C =

Let

r I= A

Let

[ gA -+

]

{A]

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

u

A

I=

i n MLp.

b e a formula o f I L , and l e t

A

.

AtA'

j u n c t i o n of t h e formulas i n

I=

r

i n IL i f and o n l y i f

Then

kA

gA b e t h e con-

i n IL i f and o n l y i f

i n MLp.

For t h e g e n e r a l i z e d s e m a n t i c s , we s t a t c w i t h o u t p r o o f t h e f o l l o w i n g analogue of Theorem 13.4:

satisfiable in

ML +C+EC P

A = A

. Then

r 1-

A

COROLLARY 1 3 . 1 2 . 1 3 . 9 . Then

I-

A

, then

Let

COROLLARY 13.11. C

Z b e a s e t o f formulas o f IL. If

Let

THEOREM 13.10.

in

Let

i n IL+D

Z =

Z

r u

is g-satisfiable i n

and

implies

IL+D

A'

i s g-

.

b e a s e t o f f o r m u l a s o f I L , and l e t

{A)

IL+D i m p l i e s A

cu

-

r

U A

1-x

in

MLp+C+EC

.

gA s a t i s f y t h e h y p o t h e s i s o f C o r o l l a r y

[ gA

-+

]

i n MLp+C+EC

.

We remark t h a t C o r o l l a r y 13.12, l i k e C o r o l l a r y 1 3 . 6 , can b e proved d i r e c t l y w i t h o u t u s i n g g e n e r a l i z e d c o m p l e t e n e s s . Combining C o r o l l a r i e s 1 3 . 6 and 13.12, we s e e t h a t e a c h o f t h e t h e o r i e s

IL+D

, MLp+C+EC

ly interpretable i n the other, i n the following sense:

is relative-

RELATIVE STRENGTH OF IL AND MLP

I1-

(i) (ii)

A

in ML +C+EC implies

B

in IL+D implies

P

I-

I-

[ 6*

[ gB

+

+A ]

]

105

in IL+D ,

in MLp+C+EC

.

We state without proof: THEOREM 13.13.

Let A be a formula of MLp, and let

B be the f o r -

I-

mula [ 6A + & ] o f IL. Then [ g B -+ ] in MLp+C+EC implies /- A in MLp+C+EC . Similarly, let B be a formula of IL, and let A be the formula

[ gB

+

]

.

COROLLARY 1 3 . 1 4 .

Then

[ 6A

-+A]

in IL+D implies

1-

B

in

IL+D.

The implications (i) and (ii) above can be strength-

ened to equivalence. We say that the theories

IL+D and MLP +C+EC are strongly relatively interpretable in each other, in view of Corollary 1 3 . 1 4 . We remark here, again without proof, that the theories IL+D and Ty2+D (see $8) are also equivalent in the same sense: each is strongly relatively interpretable in the other. In fact, the interpretation of

IL+D in Ty2+D was given in

Ty2+D in IL+D , we represent quantification over indices (objects of type s ) by quantification, in IL, over atomic propositions in (approximately) the sense of $11. $8 (see Theorem 8 . 3 ) . For the interpretation of

CHAPTER 4 .

$14.

ALGEBRAIC SEMANTICS

Boolean Models of MLp

In this section we describe an alternative semantics for the logic ML P of Chapter 3 , which will enable us to answer various independence questions raised earlier. The models with which we now concern ourselves are Boolean models, in distinction to the standard and general models of $ 9 . This is an adaptation to higher-order modal logic of the notion of a Boolean model of ordinary higher-order predicate logic presented in Scott [1966]. The new feature is the presence in the language of the necessity operator 0 , which turns out to be quite useful for describing various properties of the underlying algebra. Given sets Xo , tion -if R € P(Xo X

...

...

X

, Xn-l , we say that R is an (Xo,...,Xn-l)-relaXn-l) , and given a set I we say that F is an

(I;Xo,..., Xn-l)-predicate if F E P(Xo nonical set-theoretic equivalence

X

...

X

Xn-l)l

.

In view of the ca-

we can identify the (Xo,.,.,Xn-l)-relations with mappings from the product xo x ... X Xn-l into the set 2 = {O,l} whose elements represent the respective truth-values falsity and truth. Under this identification, an (Xo,...,Xn-l)-relation R assigns to each n-tuple (ao,...,an-r) a truthvalue R(ao,...,an-l) , either 0 o r 1 . If we now replace the set 2 by an arbitrary Boolean algebra B , we obtain the set B

xo x ...

'n-1

The basic idea behind the present construction is thus due to Scott, whose earlier work motivates most of this chapter. The author is indebted to Scott, in particular, for providing the general outline of $17.

BOOLEAN MODELS OF MLp of all B-valued (Xo,...,Xn-l)-relations.

107

Following Scott, we here think of

as comprising a widened class of truth-values: The zero and unit elements 0 , 1 of B represent falsity and truth, while other elements B

P E B

represent specific "degrees of truth" somewhere between them. If

R is a 9-valued (Xo,...,Xn-l)-relation, then R assigns to each n-tuple ..., an-1) C 9, (ao,...,an-1) E Xo x . x Xn-l a Boolean truth-value R(a,,, which we regard as the degree of truth of the assertion that . a , , an-1 stand in the relation R . The ordinary (Xo,...,Xn-l)-relations can be identified with those 9-valued relations which only assume the values

..

...

0 and

1

. Now, from the equivalences

BI

(2')

%

P(1)

xo

x

...

'n-1

xo

x

...

'n-1

9

we see that we can, for all practical purposes, identify the set of all (I;XO,...,Xn-l)-predicates

with the set of all 9-valued (Xo,...,Xn-l)-rela-

.

tions, where B is the algebra P(1) of all subsets of I This suggests that the standard semantics for MLp, which is based on domains M (Do,.

. . ,an-l

=

P ( M,

x

0

...

)'

M-

x

"n- 1

of predicates, might be replaced by a Boolean semantics based on domains M,

of B-valued relations. Of course, if B

n-1 is the algebra P ( 1 )

for some

set I then.this only amounts to a reformulation of the standard semantics for MLp; we shall be interested, therefore, in the more general case. Suppose that B = (B,

+,

*,

-, 0, 1)

is a complete Boolean algebra2

and D is a non-empty set. The 9-valued Boolean frame for MLp based on D is the family of sets, where: For the definition and basic properties of Boolean algebras, see Sikorski [1969] or Halmos [1963]. The hypothesis of completeness of the algebra will be necessary for the definition of the Boolean value of a formula; as Theorem 15.13 shows, this restriction cannot be dropped.

ALGEBRAIC SEMANTICS

108

,

Me = D

(i)

For u = (uo ,..., un-l) , M

(ii)

,=

B

...

Mu x 0

Mu

n-1

A B-valued Boolean model (b-model) of MLp based on D (Mu,

m)oEp

is a system b1 =

such that: is the B-valued Boolean frame for MLp based on D ,

(i)

(Mu),cp

(ii)

m (the meaning function) is a mapping which assigns to each constant ca an element of Ma

,

consist of all assignments over M ; i.e., all functions a on

Let As(M)

.

the set of variables such that a(x,) E Ma for each variable xa Given extend a to the set of constants by the rule that a E As(M) , let a(c,) = m(c,) E M, . For each formula A of MLp and each a C As(M) we

a

define a Boolean value

1)

A

/ l Ma

€

script

IM')

(1)

II scs O... sn-l I I =~ a(s,)(a(s

(2)

I/ se = s ;

(3)

I/

-

(4)

11

A

(5)

/I

Vxa A

B , as follows (we suppress the super-

:

A

/I

+

B

Ila

1 if

=

- /I

=

I/a

=

[

Ila

=

n

A

/I

,

= ;(sf)

/la

X

Ma

*

11

B

0

,

otherwise,

3

=

-

11 A lla

+

/I

B

,

11 A //a,X (Boolean infimum), where a,X rep-

resents the assignment a(x/X) (6)

,...,a(sn-l))

(Boolean complement),

I/ A C

-a ( s )

0

, as usual,

I/ 0 A /la = N 11 A ]la , where N is the operator in B defined b y : NP = 1 if P = 1 , NP = 0 otherwise.

It is easily verified that 11 A Ila depends only on the values of a for involves only variables occurring free in A , so that, e.g., if A(xa,y,) the distinct free variables xo where X E Mo

and

Y E bfT

, y,

then we can write

11 A(x,y) //x,y

, and if A is a closed formula we can write

simply 11 A 11 , If A is modally closed then // A /la is either 0 o r 1. formula A is true in a b-model M if 11 A 1) = 1 for every assignment a , and A is b-valid in MLp if it is true in every b-model.

A

BOOLEAN MODELS OF MLp

109

We show now that the standard semantics for MLp can in fact be viewed

D

as a special case of the Boolean semantics, as suggested earlier. Let

I be non-empty sets, let

and

be the standard frame for blLp

(hlu)oEp

based on D and I , and let (M:)uEp be the P(1)-valued Boolean frame for ML based on D . IVe define canonical one-to-one mappings +u from Ma P onto b;l =

as follows: We first let 9e be the identity mapping on bl

bit . For

kc n

.

cr = (oO,.. .

F o r each

F F M

we assume that 9 uk

U

P ( Mu

=

Mi F :lb

+,(F)

P(1)

=

...

x

x

0 x

0

...

x

M;;

D

is already defined for

Mu )I , we define its image n-1

n-1

t

by the condition that i E 9u(F)(X0,...,Xn-l) where Xk = +-l(X;) uk

=

iff

(XO,...,Xn-l) C F

.

(k < n)

Suppose bl = (Mo, m)crcp is a standard model of ML

THEOREM 14.1.

I

based on D

and

I , M"

(Mi, m")uEp

=

is a P(1)-valued b-model of MLp

based on D , and for each constant cU we have m"'(c,) Given a E As(M)

define a" C As(b1")

each variable xu

.

/I A l a"li? Proof: _ _

E

I

I

M, i, a sat A }

.

9c [m(c r u) ]

a"(xu)

=

9.,[a(xU)]

and every

a

E

As(b1)

for :

.

Straightforward, by induction on A .

COROLLARY 14.2.

then A

Then for every formula A i

= {

by putting

=

Let A be a formula of blLp. If A

is b-valid in MLp

is valid in MLp.

Before stating the next theorem on b-validity (Theorem 14.3), we need several additional lemmas. The first is a counterpart for the Boolean semantics of Lemma 9.1.1, and is easily proved by induction: LEMMA 14.3.1. Let M be a b-model of MLp and suppose the symbol scr is free for the variable xu in the formula A(x) . Then f o r every assignment a over M

we have

11

A(s)

]la

= ]/

A(x)

Ila,X , where X = a ( s )

.

ALGEBRAIC SEMANTICS

110

We r e c a l l from $9 t h a t

u

f o r any t y p e

P

C

[so=

a b b r e v i a t e s t h e formula 0 [s

sl] U

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

,

z s1]

t h en t h i s i n t u r n abbre-

viates

0 YX0

uO

... vxE-1

[ s x

0

... xn-1

0

f-,

S I X

,.. xn-1

.

]

n- 1

From t h i s we immediately o b t a i n :

Let

LEMMA 14.3.2.

be a B-valued b-model o f MLp, where

M

for ev er y

a E As(M) ,

11

s

,

su

complete Boolean a l g e b r a . Let

= s ' /la

B u

b e any symbols o f t y p e

s;

,

1 if a ( s ) = ;(st)

=

0

is a

. Then o t h er w i se.

We can now prove: THEOREM 14.3.

Proof:

Every theorem o f

MLp+C

is b - v a l i d i n ML

P'

W e r e f e r t o t h e axioms and i n f e r e n c e r u l e s o f t h e t h e o r y

s e t o u t i n $9, i . e . ,

MLp+C

axioms AS1 through AS9 (pp. 73-74), a l l i n s t a n c e s o f

t h e comprehension schema (p. 77), and i n f e r e n c e r u l e s R1

-

R3 (p. 7 4 ) . I t

i s c l e a r t h a t t h e r u l e s p r e s e r v e b - v a l i d i t y , s o i t s u f f i c e s t o show t h a t e ve r y axiom i s b - v a l i d . For axioms AS1, AS2, A4 and AS t h i s f o l l o w s immed i a t e l y from t h e d e f i n i t i o n o f Boolean v a l u e and el em en t ar y Boolean laws. Lemma 14.3.1 shows any i n s t a n c e

o f AS3 t o be b - v a l i d , s i n c e

where

X = ;(so)

(1)

so ze :s

. Similarly, +

[ A(s)

+

g iv e n a n i n s t a n c e o f AS6. s a y

A(s') ] ,

we show t h a t i t i s t r u e i n a n y b-model

M : Suppose a C As(M) ; t h e n by i m p l i e s 11 s = s ' Ila = 0 , which g i v e s (1) a ( s ) # ;(st) t h e Boolean v a l u e 1 , On t h e o t h e r hand, by Lemma 14.3.1 we have t h a t Lemma 14.3.2.,

a(s) -

= ;(st)

a ( s ) = ;(st)

/I

implies

, so

ha s Boolean v a l u e

that

1

.

=

A(s)

11

Afs)

+

I/

A(xo)

A(s')

(la

Ila,X = 1

= 11 , and

A(s')

I l a , where

X =

t h e r e f o r e a g a i n (1)

BOOLEAN MODELS OF MLp

111

The modal axioms AS7, AS8 and AS9 are readily verified to be b-valid using the definition of Boolean value. For AS9, for instance, it suffices to show that if B = (B, +, ., - , 0, 1) is a complete Boolean algebra and N

is the operator on B which interprets necessity - - i.e., NP = 1 if

-

P = 1 , NP = 0 otherwise - - then we have This is immediate. Finally, suppose that Ifu 0 vxo

:

cusA

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

...

for all P E B

NP 5 N - NP

-

.

A1 k

is an instance of the comprehension schema, where u = ( U ~ , . . . , U ~ _,~ )x is of type uk for k i n , and f, is the first variable of type u which does not occur free in A . Let M = (Mo, m)uEp be a B-valued b-mod-

M

.

el of MLp, a E As(M)

/I so that

vxo

...

vxn-l [ f x0... xn-1

/I CajA /la

=

1

x

uO Define F E B

.

-

...

Mcr

1

la,^

A

n- 1 by the condition

= 1

3

This completes the proof.

By Corollary 14.2, Theorem 14.3 and generalized completeness far MLp+C we have : COROLLARY 14.4. Let A be a formula of MLp. Of the conditions (i)

A

is g-valid in MLp+C ,

(ii)

A

is b-valid in MLp,

(iii)

A

is valid in MLp,

we have (i) implies (ii) implies (iii). We shall see presently that these implications cannot be reversed, and also that condition (ii) is actually closer to (iii) than it is to (i). M-Formulas. As in $12, we can simplify the present semantics somewhat by using constants in place of free variables. Suppose that M = (Mu, m)uEp is a b-model of MLp. For each X E Mo we add to the vocabularly a new constant of type a

to

denote X , and as earlier we simplify the discussion

112

ALGEBRAIC SEMANTICS

by agreeing to let X

act as a name for itself, extending the meaning

function m by putting m(X)

= X

for each X

€ Mu

,

u € P

.

A formula

of this extended language is called an M-formula, as before, and an

M-sen-

tence if it has no free variables. For M-sentences the Boolean value I/ A can be defined by recursion on the length of A ; in particular, we have

for any (extended) constants c , c0,

11

, cn-1, and at the quantifier

.. .

clause:

I/ vx0

A(X)

/I

=

n

Mu

It is easy t o establish that

.

0

where A(x , . . ,x"-l) of type uk (k < n)

,

11

/I A(X) II A(x

.

,... ,xn- 11 lla

0

=

11 A(XO,. ,.,Xn-l) 11 ,

is a formula of ML P with distinct free variables xk k and a(x ) = Xk . We shall make free use of M-formu-

las in subsequent sections, concluding now with the following observations: In a B-valued Boolean model M = [M0, m)uEp identified with the algebra B

, the set M4

=

B"'

can be

itself,3 so that a proposition of M

is

just an element P € B , and considering P as an M-sentence we see that 11 P 11 = P Similarly, a propositional operator of M will be a mapping

.

M F 6 M

015.

(4 1

=

B

= BB

, hence an operator on B in the usual sense.

Modal Independence Results

We remarked in $9 that the schema EC of extensional comprehension is independent o f MLp+C , and again in $11 that the axiom At of atomic propositions is independent of MLp+C

.

Indeed, by Theorem 1l.S these in-

dependence results are equivalent.

LEMMA 15.1.1. Let B be a complete Boolean algebra, M a B-valued 11 = 1 if B is atomic, and 0 otherwise.

Boolean model. Then 11 At

I

Recall that, by o u r earlier convention (page 72) regarding the Cartesian = {+} when n = 0. product of zero sets, we have M x . . . ~ M OO 'Jn-1

MODAL INDEPENDENCE RESULTS

Proof: -

113

S i n c e t h e formula

11

i s modally c l o s e d , t h e v a l u e

/I

At

is e i t h e r

or

0

1

.

But t h e f o l l o w -

ing conditions a r e equivalent:

P 5

Q and

B

, and

o t h e r w i s e . For t h e e q u i v a l e n c e of (6) and ( 7 ) i t s u f f i c e s t o

0

observe t h a t

Q] = 1 i f

N[P

The e q u i v a l e n c e o f (3) and ( 4 ) f o l l o w s from t h e f a c t t h a t

R R E B,

R

R 5 P

O <

is equal t o

P

if

P

i s an atom o f

otherwise.

0

I f we now t a k e f o r

any complete non-atomic Boolean a l g e b r a , e . g . ,

B

t h e a l g e b r a o f r e g u l a r open s u b s e t s o f t h e r e a l l i n e , ’ and t a k e an a r b i t r a r y B-valued Boolean model, we w i l l have 1 5 . 1 . 1 , s o t h a t t h e formula

At

11

At

11

= 0

M

t o be

by Lemma

i s n o t b - v a l i d . By Theorem 1 4 . 3 we immedi-

a t e l y conclude: THEOREM 1 5 . 1 . 2

The formula

At

i s not provable i n

MLp+C

.

I n view o f Lemmas 1 1 . 5 . 2 and 9 . 4 we a l s o have

-

THEOREM 1 5 . 2 .

n (n F w)

The formula

is not provable i n

MLp+C

for

cr # e ,

.

’ S e e S i k o r s k i [1969], p .

* This

ED

5.

r e s u l t can a l s o b e f o r m u l a t e d s o as t o a s s e r t t h e c o n s i s t e n c y of t h e t h e o r y MLp+C+ -At r e l a t i v e t o t h a t o f a s u f f i c i e n t l y s t r o n g t h e o r y , e . g . , h i g h e r - o r d e r number t h e o r y . The s e t - t h e o r e t i c p r o o f o f Theorem 1 5 . 1 can be r e p l a c e d i n t h i s way by a f i n i t a r y r e l a t i v e c o n s i s t e n c y p r o o f .

114

ALGEBRAIC SEMANTICS COROLLARY 15.3. The schema EC is not derivable in MLp+C

As

remarked in $11, 'E

.

is provable in MLp+C , but Theorem 15.2 does not

-

resolve the status of the formulas En for n z 0 ; we now turn to this question. Recall that, for a type u = ( U ~ , . . . , U ~ ~,~')E is the formula abbreviates 0 V f u 3gu [Rn(g) A f z g] , where Rn(g) vxo and

... vxn-1 [

abbreviates

[ f z g]

vxo

... xn-1

0

g x

...

vxn-l [ f x0... xn-l

v 0 - g XO... xn-l ]

-

... xn-1 3 .

0

g x

We first need: LEMMA 15.4.1. Let B be a complete Boolean algebra, M = (Mu, m)ucp a B-valued Boolean model, and suppose u = (uo, ...,un-1) . Then for any F € Mu , we have (1 Rn(F) )I 1 just in case F is an ordinary two-valued ( M , , ,M )-rel,ation, and 1) Rn(F) I/ = 0 otherwise. aO n-1

...

Proof: -

Straightforward.

complete Boolean algebra B is X-distributive, where X is a given cardinal (initial ordinal), if the identity

A

n 5<x c

rl<X

P

5,rl

=

cp:X

+

holds for every doubly indexed family (P ) E S T 5,rl<X equivalently, if

'KZX

[ n 5 E K

'5

l-IE+K

P 5,(P(51

l-I 5 <

of elements of B , o r

-p51 = l

for every family (P5)5<X of elements of B . 3 Every Boolean algebra is X-distributive for X < o , and a complete algebra B is atomic if and only if it is completely distributive, i.e., X-distributive for every 1

.4

LEMMA 15.4.2. Suppose n C w , 0 < n . Let B be a complete Boolean algebra, M = (Mu, m)acp a B-valued Boolean model based on a set D of Sikorski [1969], pp. 61-62. Ibid., p. 105.

MODAL INDEPENDENCE RESULTS cardinality X

.

1) En 1)

Then

= 1

115

if B is X-distributive, 0 otherwise.

Proof: The formula in question is modally closed, hence has Boolean value 0 or 1 . Note that Mn = M (e,

x

=

D

11 En /I

n

F:X

...,e) . ..

x

= 1

D

=

. The

D"

following are equivalent:

,

II 3gfi [Rn(g)

B

~

x

,

= Bx

A

SI

F

II

= 1

,

For every F:X

+

B , 2 G:X +

11 Rn(G)

For every F:X

-+

B , CG:x+2

lIF=GIl

B , C G:X +

n

For every F:X F o r every

' G:X

F:X

-+

+

F o r every family '

K

~

K

[

~

E

z

G

11

= 1

,

= I ,

/I

Fa

++

Ga I/ = 1 ,

B , G(a)

-+ 2 [

a

F

A

=

1 F(a)

.

(P ) , where 5 E;
C

PKE;

e

n

G(aj = 0 -F(a) ] K

E;{K

= Xn =

=

1 ,

1x1 , = l ,

B is K-distributive, B is X-distributive. The equivalence of ( 3 ) and (4) uses Lemma 15.4.1. The equivalence of (8) and (9) follows from the fact that X c w implies K = An < w also, in which case both (8) and (9) hold, while X ? w implies K = Xn = X , since O t n .

-

THEOREM 15.4. The formula En is not provable in MLp+C for n > 0

.

Proof: Let B be any non-atomic complete Boolean algebra. Then for some cardinal X , B is not A-distributive. Take for D any set of cardinality X , and let M be any B-valued Boolean model based on D . Then by Lemma 15.4.2, we have )I Efi1) = 0 in M , so that Efi is not b-valid and therefore, by Theorem 14.3, not provable in MLp+C . In contrast to Lemma 11.5.2, we also have the following result:

116

ALGEBRAIC SEMANTICS

THEOREM 15.5. The formula any n

€ o

[Efi

At]

+

is not provable in MLp+C for

.

Proof: If n = 0 this follows from Theorem 15.1 and the fact that Ed is provable in MLp+C . For n > 0 it suffices to observe that )I EA 11 = 1 and

I/ At 11

=

0 in any B-valued Boolean model based on D , where

chosen to be non-atomic and D The conditions E'

B is

is finite.

of extensional comprehension can be weakened; spe-

cifically, for a predicate type a

=

(ao,...,u

n-1)

wc consider the formula

This formula, which is modally closed, asserts that every predicate is coextensional, at some index, with an ordinary relation. By generalized completeness we immediately have: LEMMA 15.6.1. The formula

[Ea

+

Ey]

.

is provable in MLp+C

is provable in MLp+C , and :E is valid in MLp (i.e., with reThus, ?l 2 u # e . We now show that the spect to the standard semantics) f o r every conditions E;

are essentially weaker than the conditions Ea

.

LEMMA 15.6.2. Let B be a complete Boolean algebra. Then B less if and only if there exists a set

Proof: If i7 K

=

First assume B is atomless. Let L = B i7

is atom-

L 5 B such that for every K 5 L

and suppose K

5L

:

.

P = 0 , then the condition holds, so we may assume

that 0 < rI K . Since B is atomless, there exist P , Q € B = L with 0 < P , 0 < Q and P + Q = Il K . But then P , Q € L-K , so that we have n P C KP = I I K = P + Q 5 Z L-K P . Conversely, assume the condition holds for some set

L

5B ,

and let Q be an atom of B

.

Define

.

K 5 L by putting P € K just in case P E L and Q 5 P Since Q 5 P for all P 6 K we have Q 5 rI P C Z L-I( P , whence Q 5 P for some P € L-K , since Q must in fact be atomless.

is an atom. This is a contradiction,

Using Lemma 15.6.2, we can show the followi.ng:

SO

B

MODAL INDEPENDENCE RESULTS

LEMMA 15.6.3.

u

# e,

.

$

Let

B

be a complete Boolean a l g e -

M = (Mu , m)uEp a B-valued Boolean model based on a s e t

bra,

ID1

Suppose

117

.

2 IBl

11

Then

ED

11

1

=

B

iff

1)

i s a t o m i c , and

11

E:

, where

D

= 1 iff

B

c o n t a i n s an atom.

Proof:

.

u = (uo,. . ,on) . Then

Say

Mu = Bx

, where X i s t h e Car-

Mu x ... x Mu . I t i s e a s i l y s e e n t h a t X 2 I B I , where 0 n A s b e f o r e , i n t h e p r o o f o f Lemma 1 5 . 4 . 2 , one v e r i f i e s t h e f a c t

t e s i a n product X =

1x1 . 11

that since

/I

Eu

X

= 1

i f and o n l y i f

i s 1 - d i s t r i b u t i v e , which i s e q u i v a l e n t

B

t o the condition t h a t

2 IBI

i s completely d i s t r i b u t i v e , i . e . ,

B

atomic. On t h e o t h e r hand, we have t h e f o l l o w i n g e q u i v a l e n c e s :

11

U

I/

E2

' F:X

= 1

,

s l l I/

= 1

11 0 [F

5

I/ Iso

[Rn(s)

For e v e r y

F:X

+

B

, 2

For every

F:X

-+

B

, t h e r e i s G:X

For e v e r y

F:X

+

B , there is

+

B

a E X, G(a)

=

For every f a m i l y K

5X

n

G:X

+

G:X

'

'

1 F(a)

(Ps)c<X

, 11

GI

= 1

1) F

+

2

with

+

2

such t h a t

a E X, G(a)

of elements of

=

,

c G

0 -F(a)

11 #

0 ,

# 0 ,

B , t h e r e e x i s t s a set

such t h a t

E E K

For e v e r y

5' L

n P c K

B

O[F

A

-'s

' E + K

'

5B

# O ,

there e x i s t s a set

'4

'PcL-K

K

5L

such t h a t

p J

c o n t a i n s an atom.

We u s e h e r e Lemmas 1 5 . 4 . 1 , 1 5 . 6 . 2 , and t h e f a c t t h a t p l e t e s t h e p r o o f . Choosing f o r

B

X

2 IBI

.

T h i s com-

a n a t o m l e s s complete Boolean a l g e b r a

( e . g . , t h e a l g e b r a o f r e g u l a r open s u b s e t s of t h e r e a l l i n e ) , we o b t a i n : THEOREM 1 5 . 6 .

For

u # e, 4

,

E;

i s not provable i n

THEOREM 1 5 . 7 .

For u # e , 4

,

E'

is not derivable i n

t h e s e t of formulas

{ E;

I

u f e

3 .

MLp+C

MLp+C

. from

118

ALGEBRAIC SEMANTICS Proof: -

Assume otherwise; then some formula

is provable in MLp+C and hence b-valid in MLp. Choose for B

a complete

Boolean algebra which is neither atomic nor atomless, e.g., a direct product B1 x B2 , where B1 is atomic and B2 is atomless. Let D be any set with ID1 Z IBl , and let M be any B-valued b-model based on D By

.

Lemma 15.6.3, the formula above has Boolean value 0 in M tion.

, a contradic-

It was remarked in $12 that there exist g-models of MLp+C which contain non-indicia1 SS-operators. We can now prove this, by the following argument: The question whether such operators exist is just the question whether the formula

which asserts that every SS-operator is indicial, is g-valid lently, provable - - in MLp+C .

--

or equiva-

THEOREM 15.8. The formula (*) is not b-valid in MLp, and therefore is not provable in MLp+C

.

Proof: Let B be any non-atomic complete Boolean algebra, and let = gB is any be a B-valued Boolean model. If F 6 M M = (Mo, m),cp propositional operator of M , and A is any M-sentence, then FA abbreviates the M-sentence 3p [ O[p t--c A] A Fp ] , from which it follows that 0 11 FA 11 = F(ll A 11) . Now suppose that F is the identity operator, defined by F(P) = P for all P C B . Then we have 11 FA 11 = I/ A 11 f o r every

.

M-sentence A From this it is easily verified that each of the M-senand A6(F) has Boolean value 1 in M ; tences A2(F) , R2(F) , A3(F) i.e., F is an SS-operator of the Boolean model M the M-sentence (1)

0 l P + vq+ [ Fq

asserting that F

-

0 [P

+

41

.

On the other hand,

1

is indicial, is equivalent in M to the sentence

MODAL INDEPENDENCE RESULTS

which i s p r o v ab l y e q u i v a l e n t i n

MLp+C

119

t o t h e axiom

o f atomic propo-

At

s i t i o n s . Hence, by Lemma 15.1.1, formula (1) h a s Boolean v a l u e and t h e r e f o r e (*) a l s o h a s v a l u e

0

in

in M

0

,

.

M

The independence r e s u l t s p r e s e n t e d i n t h i s s e c t i o n were a l l proved by means o f t h e Boolean methods developed i n $14. Whether t h e same r e s u l t s c ou l d be o b t ai n ed w it h o u t t h e s e methods h a s n o t been r i g o r o u s l y s e t t l e d , a lt h o u g h t h e a u t h o r s t r o n g l y b e l i e v e s t h a t d i r e c t p r o o f s u s i n g o n l y t h e g e n e r a l i z e d s em an ti c s f o r MLp are n o t p o s s i b l e . To complete t h e d i s c u s s i o n of independence, however, we mention t h e f o l lo w i n g r e s u l t , whose proof does n o t r e q u i r e Boolean s e m a n ti c s : The comprehension schema C i s n o t d e r i v a b l e i n t h e t h e -

THEOREM 15.9.

ory

whose axioms a r e t h o s e o f ML

ML +EC P

ECa’A

o f t h e e x t e n s i o n a l comprehension schema. I n f a c t , t h e i n s t a n c e

MLp+EC

i s n o t p r o v ab l e i n Proof:

Let

D

frame

and

I

. be a r b i t r a r y s e t s , w i t h

f o r MLp based on

(i)

Me = D ,

(ii)

M

(iii)

toget her with a l l i nst ances

P

I

=

P E 2

6 For u = (a,,

D

and

111 > 1

m(c,)

€

Ma

I P c o n s t a n t I = tPo,P1) , ...,un) , n 2 0 , Mu = P( Mu

X

...

x

Ma ) I

F E M(e) = P(D)I

and

M

X E D

.

.

n

b e a r b i t r a r y . I t i s obvious t h a t ev er y i n s t a n c e

o f e x t e n s i o n a l comprehension i s t r u e i n

a t e choice of

a

I

0

Let

. D ef i n e

by:

I

ECU’*

But c l e a r l y , f o r an a p p r o p r i -

we w i l l n o t have

M; F , X

sat

c4, f x V a l i d i t y and B-Validity.

We have s e e n ( C o r o l l a r y 14.2) t h a t ev er y

b - v a l i d formula of MLp i s v a l i d . The c o n v e r s e i s n o t t r u e , s i n c e by Lemma 15.1.1, t h e axiom

At

o f a to m ic p r o p o s i t i o n s i s n o t b - v a l i d . We proceed

t o show, however, t h a t i t i s o n l y b-validity.

At

t h a t s t a n d s between v a l i d i t y and

120

ALGEBRAIC SEMANTICS

THEOREM 15.10. Let A be a formula of MLp. Then A if and only if [At + A] is b-valid in MLp.

is valid in MLp

Proof: Suppose first that [At + A] is b-valid. Then it is valid, by Corollary 14.2, and since At is valid it follows that A is valid. Conversely, suppose A is valid, and let Mft = (Mi, m*)ucp be a B-valued L .

Boolean model of MLp, a* an assignment over M* , If B is non-atomic then 11 At 11 = 0 by Lemma 15.1.1, and therefore I( At * A [Ia* = 1 . We can therefore assume that B is atomic. Since B is complete, it follows that B is isomorphic to a complete field of sets; in fact, to the algebra P(1) of all subsets of the set I of atoms of B . 5 We can therefore identify B with P ( I ) . Now let (Mu)uEp be the standard frame for MLp based on D and

I , where D

from Mu

onto :M

=

M*e ' and let (Pa be the canonical one-to-one mapping for = @,'[m*(ca)] defined in $14. If we put m(c,)

each constant cu and let a(x,) then by Theorem 14.1:

= (P,'[a*(xu)]

for each variable xu ,

1 1 A l i $ = { i C 1 ~ M , i , a sat A ) = r , since

A

is valid in MLp. Thus A has Boolean value

1 in M

t

, under

the assignment a* , and the proof is complete. COROLLARY 15.11. The b-valid formulas of MLP are not axiomatizable. In fact, the set of Glldel numbers of b-valid formulas of MLp is not definable by any formula of higher-order number theory. Proof: It is known6 that the set of valid formulas of Lp is not definable in higher-order number theory, and the same result clearly holds f o r the set of valid formulas of MLp, This fact together with Theorem 15.10 shows that the set of b-valid formulas of MLp is also undefinable, and therefore not recursively enumerable. The result can also be obtained by a direct interpretation of Lp in MLp: By relativizing quantifiers to heredi-

Sikorski [1969], p. 105. See Enderton [1972], p. 272 (The argument is based on Tarski's theorem on undefinability). L. Harrington informs us that the counterpart of Corollary 15.11 f o r ordinary (non-modal) higher-order logic is also true, although the proof lies much deeper.

MODAL INDEPENDENCE RESULTS

121

tarily constant predicates, using the formula Rn(f) find for each formula A of Lp a formula A* of ML valid in Lp just in case A*

, we can effectively P

such that A

is

is b-valid in MLp. Since the mapping is re-

cursive and hence arithmetically definable, the result follows. Expressible Boolean Properties. We have seen that the property o f atomicity of a complete Boolean algebra B can be expressed in MLp by the formula At , in the sense of Lemma 15.1.1; viz., in any B-valued Boolean model we will have

1) At )I

= 1

if B is atomic, and

0 otherwi-se.There

is, in fact, a wide class of Boolean properties expressible in MLp in the same sense, i.e., by means of a closed and modally closed formula of MLp. Specifically, we can express in MLp any Boolean property expressible in the language of higher-order Boolean algebra, by which we understand a formalism containing primitives for the Boolean operations and relations, and variables over elements of the algebra, sets of elements, sets of sets, etc. To see this, we observe that variables pQ of type 4 range over elements of B in our Boolean semantics, and the Boolean relations P and and

P

4

Q

Q can be expressed by the modally closed formulas O[p 4 + q,’ [p, z qQJ , respectively. The Boolean operations P + Q , P - Q , - P =

-

can be expressed by the formulas p V q , p A q , p respectively. The formula Vp A(p) , where A(p) is modally closed, expresses quantifica6

tion over elements of the algebra, while Vf

(Q 1

[Rn(f)

+

A(f)]

expresses

quantification over subsets of the algebra, and similarly for higher o r ders. We therefore have: THEOREM 15.12.

For any sentence A

of the language of higher-order

Boolean algebra we can effectively find a closed and modally closed formula A* of MLp such that for any complete Boolean algebra B and any B-valued Boolean model M ,

11

A*

[IM

= 1 if and only if A

is true in B

.

Incomplete Boolean Models. We may attempt to relax the requirement of completeness on the algebra B in our definition of Boolean model. However, the next theorem shows that it is in fact not possible t o do so. Let B be an arbitrary Boolean algebra, not necessarily complete, D a nonempty set. A B-valued Boolean model of MLp based on D is a system M = (Mu, m)ucp (i)

such that:

(Mu)uCp

is the B-valued Boolean frame for MLp based on D

,

122

ALGEBRAIC SEMANTICS

(ii)

m assigns to each constant cU an element of

Mu ’

(iii)

There exists a function 11 /IM which assigns, to each formula A M of MLP and a C As(M) , a Boolean value I/ A Ila E B , in such a way as to satisfy the recursive conditions (1) through (6) on page 108.

Clause (iii) should be interpreted as follows with respect to condition ( 5 ) on page 108: Given a formula Vxo A and an assignment a , the Boolean inM M fimum l7 Mu (1 A //a,X exists in B and is equal to // Vx, A /la . THEOREM 15.13. If there exists a B-valued Boolean model then B is complete. Proof: Suppose M = (Mo, m)ocp is a B-valued Boolean model, where B is an arbitrary Boolean algebra. Suppose K 5 B , and define an operator F E M(+) = BB by putting F(P) = P if P E K , F(P) = 1 otherwise. Then by clause (iii) we have

(1

n P C B F(P) B

. Since

+ f ($1p+ /IMF

Vp

//

= l7

= n P c K P , so that the infimum K was arbitrary, B is complete.

6

n

f

($1

p

4

/IMF,P

-

K of K belongs to

In the next section, we consider a somewhat different generalization of the notion of a Boolean model.

$16.

Topological Models of MLp

Since the early 1940’s, one branch of the study of propositional modal logics has centered around topological interpretations, o r more generally, interpretations in such structures as topological Boolean algebras’, which are naturally related t o topological spaces. A topological Boolean algebra is a structure (B,N) , where B is a Boolean algebra and N is an operator on B satisfying: (i)

NP 5 P ,

(ii)

N[P

*

Q] = NP

9

NQ ,

Also called closure algebras; see McKinsey [1941], McKinsey and Tarski [1944]. We here employ the dual notion, as in Rasiowa [1963].

TOPOLOGICAL MODELS OF MLp (iii)

NNP = NP

(iv)

N1 = 1

123

,

.

The motivating example is a topological field of sets

(B,N)

, where X is

a topological space with interior operator N , and B is a field of subsets of X closed under N

.

It can be shown’ that every topological Bool-

ean algebra is isomorphic to a topological field of sets. In any topological algebra notions: An element P

€

B

(B,N) we can define the usual topological is open if NP = P , o r equivalently if NQ = P

.

for some Q C B The closure of an element P is defined to be -“-PI , and P is closed if it is equal to its closure, o r equivalently if -P is

...

* Pn of finitely many open elements is open. The infimum PI P2 . P of any set K of open elements again open, and the supremum 2 is open, if it exists in B . For any P < B , NP equals the supremum z Q S P , N Q = Q Q of all open elements dominated by P

.

It is easily verified that clause (ii) of the definition of topological algebra can be replaced by (ii’) N[P = Q] 5 [NP = NQ] , so that a topological algebra

(B,N)

consists of a Booleali algebra B

to-

gether with a Boolean S4-operator on B , in the obvious sense. It is this fact that provides the connection, remarked earlier, between topological algebras and modal logic. Accordingly, we say that a topological algebra (B,N)

is an SS-algebra if it satisfies also

(V)

-NP

5

N[-NP]

for all P C B , o r equivalently, if every closed element is also open (and hence conversely). In such an algebra, the open elements form a Boolean subalgebra B* if K

5 B*

of B

, which

is a complete subalgebra, in the sense that

and the supremum C K exists in B

Consequently:

’ McKinsey and Tarski [1944]. See McKinsey and Tarski [1948], p. 8 .

, then it belongs to B*

.

ALGEBRAIC SEMANTICS

124

LEMMA 16.1. Let (B,N) be a complete SS-algebra. Then the open elements form a complete subalgebra B* , and for every P C B we have (1)

NP

C

=

QCB*,QcP

Q *

Conversely, if B is any complete Boolean algebra, B" is a complete subalgebra of B , and we define an operator N C BB by condition (l), then (B,N) is a complete S5-algebra and B" consists of its open elements. Let (B,N) be a complete S5-algebra, D a non-empty set. We define the (B,N)-valued topological frame for MLP based on D to be the family (Mu)uCp of sets, where we simultaneously define the set Mu and the Boolean value I/ X = Y 11 f o r X , Y C Mu as follows: (i) (ii)

Me = D ; I]

X L

Y 11

=

1 if X

= Y

in D , 0 otherwise.

u = (U~,...,U~-~) , Mu

consists of all mappings F from Mu into B such that for any sequences X , Y C uO n-1 Mu x x Mu , ll N 11 Xk 5 Yk 11 9 [F(X) F ( Y ) ] . 4 We 0 n-1 let 11 F 5 G 11 = n M, ... [F(x) G(X)] for any Ma 0 n-1 F , G E Mu. For

M

x

... x

...

-

Q

(B,N)-valued topological model (t-model) of MLp based on D M = (Mu, m)uCp such that: A

is a system

is the (B,N)-valued topological frame based on D ,

(i)

(Mu)uCp

(ii)

m assigns to each constant cu an element of Ma

.

Let As(M) consist of all assignments over M , in the usual sense. F o r M each formula A of MLp and each a C As(M) we define (1 A (la C B as in $14, except that condition ( 6 ) on page 108 is now replaced by: (6')

11

Ila

N )I A algebra (B,N) . 0 A

=

Ila , where

N is the interior operator of the

-

In any Boolean algebra we write [P Q] for the Boolean combination P.Q + (-P)*(-Q), where P, Q C B. Condition (ii) is necessary in order to validate the equality axioms o f MLp in the topological semantics. See the proof of Theorem 16.2.

TOPOLOGICAL MODELS OF MLp

A formula A and A

is

true in a

is t-valid in ML

THEOREM 16.2.

Proof:

P

t-model M

if

125

)I A llaM

=

1 for all a E As(M)

,

if it is true in every t - m ~ d e l . ~

Every theorem of MLp+C is t-valid in ML

Clearly the inference rules R1

-

P'

R3 of MLp preserve t-validity.

The proof that every axiom of ML is t-valid proceeds as in the proof of P Theorem 14.3, except that one must verify that axiom schema AS6 can be replaced in MLp by the following axioms: A6.1

xe r ye

+

A6.2

xe

=

ye

+

A6.3

xo

F

yo

ye

G

[ ye

xe , E

ze + xe

z e"

z

... A xn-l= yn-' 0 n-1 0 , y , ... x , ... , x A

where tinct variables such that xk and

f

... xn-1

[ f x0

+

+

f

... yn-1

y0

3 ,

, yn- 1 is the first sequence of disand yk are of type uk (k < n) ,

.

is of type u = (uo,.

These axioms are easily seen to be t-valid -- in the case of A6.3 this follows from clause (ii) in the definition of a topological frame -- so that every theorem of MLp is t-valid. It remains to verify the t-validity of each instance CoSA :

... Vxn-'

3fa 0 Vxo

[ fx

... xn-1

0

+.+

A ]

of the comprehension schema. Suppose that M is a (B,N)-valued topological model and a E As(M) As in the proof of Theorem 14.3 one defines a mapinto B by letting F(XO,...,Xn-l) = ping F from Ma x ... x Mu 0 n- 1 " A "a,X0,.. . , x ~ - ~It must then be shown that F C Mu ; but this follows

.

.

from the fact that the formula vxo

... Vxn-l Vy0 ... Vyn-' -+

n-1 n-1 yo A A x E y A(x 0,..., xn-1) A(yo ,...,yn-')

[ xo

.

is provable in MLp and hence t-valid, where A

...

++

is A(x O,..., xn-')

1

, the

The idea of interpreting first-order modal logic in complete topological algebras is apparently due to Rasiowa [1951].

126

ALGEBRAIC SEMANTICS

variables y0 , ... , yn-' are distinct from x 0, ... , x n- 1 and from each other and do not occur in A , and yk is of the same type as xk (k < n). We omit the details.

REMARKS. Suppose B is a complete Boolean algebra. The trivial operator N C BB is defined by: NP = 1 if P = 1 , NP = 0 otherwise. The pair (B,N) is an SS-algebra, and a (B,N)-valued topological model is just a B-valued Boolean model, as defined in $14 (This operator corresponds to the trivial topology). Thus, every b-model is also a t-model, and every t-valid formula is b-valid. If we define N E BB by letting NP = P f o r all P E B (the discrete operator), we obtain another SS-algebra (B,N) , corresponding to the discrete topology. In any discrete topological model M , the formula 0 A * A is true f o r every formula A , so that the necessity operator 0 is vacuous and hence M is essentially the same as one of Scott's Boolean models of ordinary higher-order logic. Thus, both Scott's Boolean models and those defined in $14 can be construed as topological models. The notion of a topological model can be further generalized by permitting a.smaller set of Boolean-valued relations at each level. Thus, if (B,N) is a complete SS-algebra, a (B,N)-valued general topological frame is defined as before except for the replacement of clause (ii) by: (ii')

For

a = (U~,...,U~-~) , Mu

from Ma E Mo

x

... x

x

0

...

X

0

MD

M,

n- 1 I7 n-1 '

is some non-empty set of mappings F into B such that for any sequences X , Y N

/I

Xk

=

Yk

11

5

[F(X)

F(VI

.

This leads to the notion of a (B,N)-valued general topological model. If N is the trivial operator on B , we speak instead of a B-valued general Boolean model. Every theorem of MLp is true in every general topological model, but in general the comprehension axioms will not hold. Nevertheless, such general models prove useful, e.g., in showing the independence of the axiom of choice in ordinary higher-order logic. Homomorphisms. homomorphism from such that f o r all -S(P) ; (iii) 8(NP)

Let (B,N) and (B',N') (B,N) (B',") is a P , Q E B : (i) S(P + Q) = = N'[8(P)] Here we use

.

be topological algebras. A mapping 8 from B into B' O(P) + S(Q) ; (ii) S(-P) = + and - ambiguously to

TOPOLOGICAL MODELS OF MLp

de n o t e t h e o p e r a t i o n s o f

127

as well as t h o s e o f

B'

.A

B

homomorphism

9

exists i n

9(P)

C

Z

exists i n

P

~

B'

.

B

9 ( I:

and i s e q u a l t o

9

B ; i.e., i f

i s complete i f i t p r e s e r v e s i n f i n i t e infima and suprema i n

whenever

P )

We s h a l l b e i n t e r e s t e d i n complete homomorphisms

on complete S 5 - a l g e b r a s

.

(B,N)

I n t h i s c a s e t h e r an g e

(B',N')

of

9

i s a l s o a complete S 5 - a l g e b r a , and a l l i n f i m a and suprema are p r e s e r v e d . 6

M

Let where

= (Mu,

m)uEp be a (B,N)-valued t o p o l o g i c a l model based on

i s a complete S 5 - a lg e b r a , and suppose t h a t

(B,N)

homomorphism from

(B,N)

onto

M under

o f t h e model

.

9

MA

onto

Mu

For a l l

(1)

,

C M,

1)

(Recal l t h a t t h e v a l u e

911 X

X c Y

//

maps

M

uk

onto

MA E

where

Xi,

B'

Xi,

x

ak

...

3

I/

Xi Xi F

x

.

I/

=

9u(X)

is defined f o r

MI

aU

u mappings

au(Y)

E

,Y

X

(1

E M,

i n any t o p o -

...,un- 1)

k < n

f. B

9

be

assume

i n such a way t h a t c o n d i t i o n

Mu F C ,M

, we

u = (uO,

For

...

X

X

Mu

0

n- 1

for

k

by:

-= n

, we

define

,...

au(F)(X;) ,X;-l) = SIF(XO,...,Xn-l)I , i s w e l l - d e f i n e d , f o r i f we a l s o have

. This

k < n , th e n u s i n g c o n d i t i o n (1) f o r

for

11 .

i s as follows: W e first let

.

for

Given

n- 1

=

be t h e (B',N')-valued

MA

0

I/

I/

Y

aU

Ok

1 =

(MA)uEp

k

= 9uk(Xk)

(Yk)

= 9

MA

Ok

(1) h o l d s f o r each 9,(F)

= D =

M

t h e i d e n t i t y mapping on 9

topo-

, which we c a l l t h e image

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

l o g i c a l frame.) The d e f i n i t i o n of that

D

s a t i s f y i n g t h e f o l lo w i n g c o n d i t i o n :

,Y

X

,

D

We d e f i n e a ( B',N ') - v al u ed

a l s o based on

We first l e t

t o p o l o g i c a l frame based on from

.

(B',N')

M' = (MA, ml),Cp

l o g i c a l model

,

D

i s a complete

9

(X,)

9 Ok

E

(Y,)

9

11

= 911 Xk

=

Yk

11

uk we see t h a t

. Since

F E Mo ,

Ok

however, t h e i n e q u a l i t y

Such a mapping 9 w i l l p r e s e r v e t h e s u b a l g e b r a o f open el em en t s. However, a complete Boolean epimorphism which p r e s e r v e s t h e s e t o f open elements may f a i l t o preserve t h e operator N.

128

ALGEBRAIC SEMANTICS

= 9[F(Y)] . In essentially the same way we show that 9,(F) that 9[F(X)] belongs to MA . For suppose XI, Y' E MA x ... x MA , and let Xi = 0 n-1 9 (Xk) , Y i = 9 (Yk) for k < n . Applying 9 to inequality ( 2 ) again,

*k

Ok

and using condition (1) for ak , we have

N' 11 X' = Y' 1) k - k

'k
[ 9[F(X)]

=

[ 9u(F)(X')

as required. Finally, we show that

Oa

and define a mapping F from Mu

...

(3)

'

F(X) =

9[F(X)1

x

0

'P

=

and Xi =

-

1 ,

9u(F)(Y')

* 9

(X,)

uk

(k < n)

9(P)

B, 9 ( P ) 5 F'(X')

C

]

9[F(Y)]

is onto MA : Suppose F' E MA , x Mu into B by: n-1

p E B, Q ( p ) 5 FI(XI)

where XI = (Xb,...,X;-l)

-

5

=

. Applying

F'(X')

9

to ( 3 ) :

,

since 9 is onto B ' , so it remains only to verify that F C Ma , i.e., . For this it sufx Ma that ( 2 ) holds for arbitrary X , Y C M x OO n- 1 fices to show that

...

N 11 Xk = Yk 11

* k < n

F(X)

*

it suffices in turn to show

and by the definition of F(Y) (4)

where Yi

N

kc

@['

= 9

uk

n k < n

[I

Xk

3

(Yk) (k < n)

Yk

11

F(X)

*

. But

11 X'k =- Y'k 11

N'

F(Y) ,

5

]

,

F'(Y')

5

(4) is equivalent to

F'(X')

,

i F'(Y')

.

(X,) (k < n) , and this clearly holds since F' C MA *k To verify condition ( 1 ) for o , suppose F , G E Mu and let X M Ok . Then: denote the product M x ... Mu n-1 aO

where Xi =

911 F

9

G II =

9[

*

xM

[F(X)

ak =

"XEXMu

[ 9[F(X)1

k

~

G(X)I

-

9[G(X)1

1 1

TOPOLOGICAL MODELS OF MLP

XI t xM;

=

[

129

9u(G)(X')

1

k =

11

aU(F)

5

9,(G)

11 ,

as required. To complete the definition of the image model M' = 9 [m(c ] ] f o r each constant c We can now state: m'(c,) u

.

u

LEMMA 16.3. Let M

be a (B,N)-valued topological model of MLp, and

let M ' be the image of M under a complete homomorphism 9 onto (B',") . Then f o r every formula A o f MLp, every a C a' E As(bl') , if a'(x ) = 9 [a(x ) ] f o r every variable x u u u U // A /IalM' = 911 A /laM . In particular, if A is a closed formula

11 A ItM'

= 811

Proof: -

A

, we define

from (B,N) As(M) and then we have of MLp then

/IM .

Straightforward, by induction on

Before applying Lemma 16.3

we

A

need some algebraic preliminaries. The

direct product (B,N) of a family of complete S5-algebras (Bi,Ni) (i is defined in the usual way: B is the ordinary direct product of the

C

I)

xi,,

Boolean algebras Bi , B = Bi , consisting of all indexed families (Pi)iEI such that Pi t Bi for all i C I , with the Boolean operations defined coordinate-wise,7 and the interior operator N

is defined by let-

It is a routine matter to verify that (B,N) 1 1 1CI * is again a complete S5-algebra, and that the projection mapping n . from I B onto B. defined by n.(Pi)itI = P. i s a complete homomorphism from I I I (B,N) onto (B.,N.) for each index j E I Let us call a complete SSting N(Pi)iCI = (N.P.).

1

.

1

algebra reducible if it is isomorphic to a direct product of trivial complete S5-algebras, i.e., algebras (Bi,Ni) in which the interior operator N. is the trivial one: N.P = 1 if P = 1 , NiP = 0 otherwise. We say that a (B,N)-valued topological model is reducible when the algebra (B,N) is reducible. It is not difficult to show that (B,N) i s reducible if and only if the complete subalgebra B*

of open elements of

(B,N)

is atomic.

We omit the proof. THEOREM 16.4. A formula A

of MLP is b-valid if and only if it is

true in every reducible topological model of MLp.

Sikorski calls this the direct union; see Sikorski [1969], p. 50.

ALGEBRAIC SEMANTICS

130

Proof: Since every Boolean model of MLp is a reducible (in fact, trivial) topological model, one implication is immediate. For the converse, assume A is b-valid. We can assume without loss of generality that A is closed, since A can be replaced by its universal closure otherwise. Let M be any (B,N)-valued topological model of MLp, where (B,N) is a reducible complete S5-algebra. We can assume that (B,N) is itself the direct (Bi,Ni) (i C I) . If A is not true in M product of trivial %-algebras then /I A /IM # 1 , so clearly nil/ A /IM # 1 in Bi for some i C I Let

.

Mi be the image of the model M under the complete homomorphism ni . Mi is a (Bi' N.)-valued topological model of MLp, hence a Boolean model, since 1 Ni is trivial. In view of the assumption that A is b-valid, we conclude that 11 A 1) = 1 in Mi . On the other hand, Lemma 16.3 implies that

which is a contradiction. Let (B,N) be any topological algebra, P E B a non-zero open element. The set Bp = Q C B I Q P P 1 becomes a Boolean algebra (Bp, + p $ *pB - p , Op, I p ) if we let + p and coincide with the corresponding operations + , * of B , define - p Q to be P.(-Q) , and let Op , l p be 0 , P , respectively. If we let Np be the operator on Bp which coincides with the operator N f o r all Q E Bp , then (Bp,Np) is a topological algebra, which is complete (resp., an S5-algebra) if (B,N) is complete (resp., an SS-algebra). The mapping 8 ( Q ) = P - Q is a complete homomorphism from (B,N) o P

onto (Bp,Np) . Conversely, if (B,N) is a complete topological algebra and 9 is a complete homomorphism from (B,N) onto (B',N') , then it is easily shown that (B',N') is isomorphic to (BprNp) , where P is the Q of B . It follows, in particular, that open element Q C B, 4(Q) = 1 a trivial complete S5-algebra has no proper complete homomorphic images. THEOREM 16.5. Let A be a modally closed formula of MLp which is not t-valid. Then for some topological model M of MLP and some a € As(M) , M we have 11 A (la = 0 .

TOPOLOGICAL MODELS OF MLp

131

Proof: Let M be a (B,N)-valued topological model, a an assignment M over M such that I/ A Ila # 1 . Since A is modally closed, the formula 0 A * A is provable in MLp and therefore true ir. M , by Theorem 16.2, M so that the element P = -11 A /la # 0 is open in (B,N) . Let 9 be the complete homomorphism from (B,N) onto (Bp,Np) defined earlier, and let M' be the image of M under 9 , a' the assignment over M' defined = 9,[a(x,)] . Then by Lemma 16.3, by a'(x,)

as required, and the proof is complete. The hypothesis that A be modally closed cannot be dropped from Theorem 16.5. For example, the formula [pQ v 0- p ] fails to satisfy the Q conclusion of the theorem, but it is not even valid in MLp, and therefore a fortiori not t-valid. We conclude with several remarks. It can be shown that there exist b-valid formulas of MLp which are not t-valid. For it follows from the results of Scott [1966] that there are sentences of ordinary higher-order logic which are valid (true in every standard model of Lp) but not true in every Boolean model, in his sense. By relativizing higher-order quantifiers to hereditarily constant predicates, we obtain a closed and modally closed formula A of MLP which is valid in MLp but fails in some discrete topological model M , in the sense that /I A )I # 1 in M . Since A is valid in MLp, Theorem 15.10 implies that the formula [At + A] is b-valid; but if this foimula were t-valid then it would be true in M , which is impossible since it is easily checked that /I At 11 = 1 in any discrete complete S5-algebra. The problem of classifying the various types of complete S5-algebras -or equivalently, the various types of pairs (B,B*) where B" is a complete subalgebra of B - - is probably not trivial, in view of a result of Kripke' which implies that any complete Boolean algebra B" is a complete subalgebra of some countably generated complete Boolean algebra B .

Kripke [1967].

ALGEBRAIC SEMANTICS

132

$17.

Cohen's Independence Results

We have seen in $15 that the Boolean semantics for MLp yields a number of independence results in modal logic; i.e., theorems to the effect that a particular modal formula is not provable in a particular modal theory. We can also use the Boolean semantics for MLp to study certain of the wellknown independence results in set theory first proved by Cohen.' It was a joint discovery of Scott and Solovay2 that Cohen's indepen-

dence proofs could be recast in the framework of Boolean-valued logic, by means of Boolean models similar to those introduced in $14. In particular, one can show in this way that the continuum hypothesis cannot be proved in set theory even when the axiom of choice is added. In the first published exposition of the Boolean m e t h ~ d ,Scott ~ actually proved a somewhat weaker result, viz., that the continuum hypothesis cannot be proved in higherorder number theory with the axiom of choice. This proof involves all the ideas needed for the stronger result and avoids some of the messier details connected with Boolean models of the full theory of sets. We examine now a proof, similar to Scott's, of the independence of the continuum hypothesis in higher-order number theory with the axiom of choice. Our proof, however, employs formulas of MLp as modal "interpolants" at various stages of the argument. It should be remarked that the present approach could be adapted to the context of set theory by replacing the higher-order modal logic MLp by a f u l l modal set theory.4 Higher-Order Number Theory. We augment the axioms of the theory Lp+C defined on pages 70-71 by several additional axioms to obtain a theory Np, Higher-Order Number Theory, which will play the role of set theory in our formulation of the independence proof. We first add, for each type u

Cohen [1963]. See Scott [ 19661, [1967a], [1967b], Rosser [1969]. Scott [1967a]. See Lemmon [1963].

, an

COHEN'S INDEPENDENCE RESULTS

133

axiom of choice: Ac'

:

where Wo'(f)

3f(u,u) WoU(f) , is the formula

Ac' asserts that a well-ordering of the objects of type u therefore valid, i.e., true in all standard models o f Lp.

exists; it is

Consider now a standard model of LP based on the set w of natural numbers. If we let < be the first constant of type (e,e) and write x < y for <xy , we can characterize the usual ordering on w by the following second-order Peano axiom:

which asserts that the usual ordering of w is a well-ordering with no greatest element in which every element except the least has an immediate predecessor. The theory Np takes for its axioms all those axioms o f Lp+C which contain no constants other than c , together with the Peano axiom Pe and the axioms Acu for u C P . The rules of inference are those of the theory Lp (Rules R1, R 2 on page 71). Clearly, any standard model of Np is isomorphic to the standard model M = (Mu, m)uCp based on w , in which m(<) is the usual ordering on o .5 We can easily define the usual number-theoretic operations as relations in Np, using higher-order quantifiers, and rational and real numbers can be defined as higher-order entities and their usual properties established in Np. Thus, the theory Np incorporates the usual body of algebra and analysis normally required in mathematical arguments. The continuum hypothesis is the assertion that every class of subsets of o is either denumerable or else equinumerous with P(w) . Since the Here we ignore the values m(c,)

for constants c, other than c.

134

ALGEBRAIC SEMANTICS

axiom of choice is present, we can formulate this as follows: Every nonempty subset G of P(o) is either the range of a function on w o r else the domain of a function onto P(o)

.

We therefore seek to establish that the formula Ch below is not provable in the theory Np:

Extensional Predicates. We introduced at the end of $1 a distinction between extensional and intensional words of English, and we indicated how the extensionality, f o r example, of an adjective L' could be expressed by a semantic condition on its intension Int[z] in a model of formalized English. This distinction can be carried over to the formal logic MLp, where it proves to be an important element in the independence proof. Let M = (Mu,

m)uEp

gle out a class Eu

be a g-model of MLp. For each type u # e

of (hereditarily) extensional predicates F E Mu

convenience, we define Eu the following form: (i)

E = D = M

(ii)

For

E

x

...

e'

5 Eu

... x

x

, Eu consists of a1

Eu

for all

n-1 and any i E I

0

Eu

n-1 and Xk = Yk when uk = e OO

. For

also for u = e , so that the definition takes

u = (ao,..., un-l)

(a) F(i)

we sin-

, if

, then X E

i € I

F

C

Mu such that

(b) For any X

Xk(i) = Yk(i) F(i)

,Y E

when uk # e

iff Y € F(i)

.

That is, the question whether a given sequence (Xo,,..,Xn-l) of extensional objects belongs to F(i) depends only on the values Xk(i) where

.

and the values Xk where uk = e We can define the classes Eo formally in MLp. Precisely, we define for each u C P and each variable x a formula U

uk # e

Extu(x)

( x

is extensional )

of MLP with exactly x

free, as follows:

U

(i)

Exte(x )

is

[x

E

x]

,

COHEN'S INDEPENDENCE RESULTS

(ii)

For u = (uo,.. . , u ~ - ~,) Ext'(f

U

...

0 Vxo

... vxn-1 Vyo ... tlyn- 1

Extu k( yk ) A x k 'y k ]

-c

is the formula

)

vxn-l [ f x 0... xn-1

OVXO

135

Cnj k
+

[ cnj k
A

A

[ f x0

... xn-1 * f y . .0. y

n-1

1 1 ,

where x0, , xn-1 , y0, , yn- 1 is the first sequence of distinct variables such that xk and yk are of type uk (k < n) . 6

...

...

The formula Ext'(x)

is modally closed, in the extended sense that the equivalence 0 Ext'(x) tf E xt'(x) is provable in MLp, and if M is a gmodel of MLp and X C Mu then clearly X C ED if and only if M satisfies

.

Ext'(X)

Ext'(x)

We shall henceforth drop the superscript on the formula

, the type u being clear from context. LEMMA 17.1.

and u =

n (n

The formulas Vx

U

Ext(x)

are provable in MLp for u = e is provable in MLp+C

, and the formula 3xu Ext(x)

C w)

.

f o r every type u

The proof is omitted. is a B-valued Boolean model of MLp based on D , where If M = (Mu, m)ocp B is a complete Boolean algebra, then for each X C Mu w e clearly have

I( Ext(X) I( = 0 or 1 , and if we let Eu = { X C M, obtain the following recursive characterization: (i)

E = D = M

(ii)

For

1 11

Ext(X)

11

= 1 }

, we

e '

, Eu consists of all F

Mu such that (a) F o r X C Mu x .. x M v , F(X) = 0 unless Xk € E (k c n); 0 n- 1 "k x Eu , F satisfies the inequality (b) F o r X , Y C Eu x 0 n-1 fl k < 11 xk B yk 11 9 [F(X) F(Y)I . (5

=

(uo,..

€

.

...

Thus, if m(c,) C Eu for each constant cu , then the extensional objects in M form a B-valued general Boolean model of MLp, in the sense of $16,

15

Here Cnj kc,, A

k

denotes the conjunction [ A

0

A

...

A

An-'].

136

ALGEBRAIC SEMANTICS

which is isomorphic to a (B,N)-valued topological model, where N discrete operator on B .

is the

Given a formula A of Lp, we let A(Ext) denote the formula of MLp obtained by relativizing all quantifiers of A to the formula Ext(x) ; i.e., replacing each subformula VxU B of A by Vxu [Ext(x) + B] . Let AA consist of all formulas Ext(sa) , where su is either a constant in A or a free variable of A . The next theorem shows that, loosely speaking, the extensional predicates in a model of higher-order modal logic form a model of ordinary (non-modal) higher-order logic. Precisely: THEOREM 17.2. If A from AA in MLp+C .

is provable in Lp+C then A(Ext)

is derivable

One shows that the class of formulas A for which A(Ext) is derivable from AA contains all the axioms of L +C and is closed under P the rules R1, R2 of Lp. As in the proof of Theorem 16.2, it is necessary to use the fact that axiom ASS of Lp (page 71) can be replaced by:

Proof:

ye

ye

xe ,

A5.1

xe

A5.2

xe'ye -

A5.3

[ f x0... xn-1 + f y .0. . y n-1 1 , where x0, ... , xn-1 , y0, ... , yn-1 is the first sequence of distinct variables such that xk and yk are of type uk (k < n) , and f is of type cr = (uo,. . . , u ~ - ~. ) xo

E

e

-+

-+

0

y

A

E

[ y e z z -+x E e e n-1 ... A xn-l e y

Z

e l *

+

In addition, the proof makes use of Lemma 17.1, and (for the verification of A5.3) the following result, which we state without proof: LEMMA 17.2.1. For any type cr the formula

is provable in MLp. Modal Number Theory. We consider now a modal analogue of Np, which we denote by MNp. Its axioms are those axioms of MLP+C containing only the constant c of type (e,e) , together with the axioms of choice:

COHEN'S INDEPENDENCE RESULTS

where

137

i s t h e formula

WO'(f) Rn(f)

vx

A

u

by

u

[ fxy

x

V

y

0 y o ) [ 3xu gx *

V

fyx ]

[ gx

3Xu

A

[ 9Y

VYu

A

+

-

fYX

3 3 1

>

and t h e Peano axiom PE :

Woe(<)

bxe 3 y e [x < y ]

A

*

Vye [ 3xe [ x < y ]

A

3xeVze[z
Z < X V Z . X I ] .

+-+

The i n f e r e n c e r u l e s o f MNp a r e t h e r u l e s R 1 , R 2 , R 3 o f MLp (page 7 4 ) . I t

i s c l e a r t h a t t h e axioms o f MNp w i l l a l l h o l d t r u e i n t h e s t a n d a r d model M = (Mu, m)ucp for

m(c)

d e r i n g on ing

t h e f u n c t i o n on

.

u

(Xs)s,x

F(i) =

of

LEMMA 1 7 . 3 . 1 .

sat

Let

such t h a t WO'(F) M, i

M

.

< r)

AC'

3f

(u,u)

y

'

sat

= P(Mu x Mu)'

by p u t t i n g

i s p r o v ab l e i n

be a g-model o f

[ Ext(f)

A

MLp+C

z

E x t( x ) y'

+

Wo'(f)(EXt) (0,u)

Ext(y)

A

3x'

U

M

. Say

i € I

sat

]

Ext(F')

D

and

and

(u,u)

M

equivalently, t h a t

. M

satisfies

[ Ext(y')

Vy,',

[ Ext(x')

F C M

, or

such t h a t A

b ased on

MLp+C

[Ac'](EXt)

F' C M

I t i s eas y t o see from (1) t h a t

M, i

(a,o)

-+

sat

-

0 Vxu Vyu [ F'xy

show t h a t

AC'

M, i

By comprehension t h e r e e x i s t s (1)

M

.

and suppose

,7

We show t h a t

sat

is t h e usual or-

i € I

.

M = (Mu, m)ucp sat

c

F

( i € I)

I

The formula

u E P

f o r each t y p e

I

E

o f i n d i c e s , i f we take

I

h o l d s , it s u f f i c e s t o t a k e any w e l l - o r d e r -

AC'

and d e f i n e

Mu

I

and any s e t

whose v a l u e a t any

I

To s e e t h a t

I (XE,X,])

Proof:

u

o f MLp based on

A

3

A

x

,

s o i t remains o n l y t o

P X'

A FX'y'

] ]

.

W O ~ ( F ' ) ( ~. ~By~ Lemma ) 1 7 . 2 . 1 and some obvious s i m -

We r eg ar d t h e formulas AC' t h a t [oACa ++ ACa] and [OPE

and PE as modally c l o s e d , i n view o f t h e f a c t PE] are theorems o f MLp.

c+

ALGEBRAIC SEMANTICS

138

p l i f i c a t i o n s , i t s u f f i c e s i n t u r n t o show t h a t

satisfy

not i n

.

F(i)

M

G(i) # 0

F'XY

.

.

WO'(F) B

X I

M, i

do n o t

sat

F'XY

. S i n ce

E Eu

X'

M, i

and

such t h a t

M, i

and

sat

T

Y'

Y

5

Y'

XI

-

=

M

sat

Y € M

,

WO'(F) Y C G(i)

u '

e x t e n s i o n a l i t y and ( I ) ,

implies

Y € G(i)

(Y,X)

implies

(Y,X)

is

(Y',X')

t h en X

C

(X',Y')

Y , there-

E

Y 1 , T h i s shows t h a t

X C Ma

there exists

sat

M, i

then

do n o t s a t i s f y

or, a fortiori,

Y1

M, i

For t h e second c o n ju n c t o f ( 2 ) , assume

S i n ce

that for a l l

Y ' € Eu

satisfy

and suppose t h a t

, i f M. i sat Y

Y ' E Eu

sat

f o r e do n o t s a t i s f y sat

6 Eu

We show u s i n g (1) t h a t

F'YX , t h e r e e x i s t s

Hence i f

F ( i ) , because M, i

.

F'YX

and f o r every

X'

T

or

Y

do n o t s a t i s f y

M, i X

=

X

,Y

X

For t h e f i r s t co n j u n c t , assume

M, i

G C E

,X

(0)

C G(i)

and

, such

f F ( i ) . But t h e n , u s i n g f F ' ( i ) . Th i s completes

t h e proof. LEMMA 1 7 . 3 . 2 .

Proof:

The formula

PE

-f

i s provable i n

P e (Ext)

Using Lemma 17.1 i t i s e a s i l y shown t h a t

a l e n t i n MLp t o

,

Pe

THEOREM 1 7 . 3 .

and c l e a r l y

PE

I f a c l o s e d formula

implies

Pe

MLp+C

i s equiv-

Pe(EXt) i n MLp.

i s a theorem of Np,

A

.

then

A Wt)

i s a theorem of MNp.

Proof:

Suppose

A

i s a theorem o f N p . Then f o r some t y p es

uo

,

...

o t h e formula n- 1

Acuo

-+

.

i s p r o v ab l e i n

U

Ac

-f

Lp+C

. ...

-+

.

U

Ac

-+

.

Pe+A

, w i th o u t u s i n g c o n s t a n t s o t h e r t h a n c

.

By Theorem

17.2 , t h e r e f o r e , t h e formula U

[Ac O](Ext)

+

. ... . -+

U

[Ac n-1 ] (Ext)

+

. p e( Ex t )

~

A ( Ex t )

i s d e r i v a b l e ( i n f a c t , w i th o u t u s i n g c o n s t a n t s o t h e r t h a n c ) from t h e formula

Ex t ( c)

in

MLp+C

.

But by Lemma 17.1,

Ex t ( c)

MLp. The r e s u l t now f o ll o w s by Lemmas 1 7 . 3 . 1 and 1 7 . 3 . 2 .

i s a theorem o f

,

COHEN’S INDEPENDENCE RESULTS

139

We now introduce two closed and modally closed formulas of MLp which will act as the modal interpolants mentioned earlier. The first is the formula Rn(y’)

A

R~(Y)

A

fxy

A

0 vxe

-

fxy

1 1 ,

which can be given the following reading: If a predicate of type (e,(e)) is necessarily functional from individuals to sets (of individuals), then some set is necessarily out of its range. The second interpolant is the formula

which can be rendered (somewhat less satisfactorily) as follows: If a predicate of type

((e),(e))

is necessarily functional from sets to properties

then some property is necessarily out of its range. It is not difficult to show that for a standard model M on sets D

and

particular if D

I , M

sat C1

just in case

is infinite then M

sat

c1

of MLp based

III.ID/ < 21D’ just in case

,

111

so that in < 2ID1

.

On the other hand, a standard model never satisfies C2 ; i.e., the formula * C2

is valid in MLp.

LEMMA 1 7 . 4 . MLp+C

The formula

-

Ch(EXt)

is derivable from C1 and C2

.

Proof:

Ch(EXt)

is equivalent in MLp to the following formula:’

b((e))[ Ext(g1 3f(e,(el) v 3f

A

3Y(e) 9Y

[ Ext(f)

((el, ( e l )

A

+

:

Vxe 3!y

(el [gy

*

[ Ext(f)

A Vy

(el

fxy]

vY(e)

[ gy

+

A V Z (e)

’ Note the meaning of 3!x

A

9Y 3!z

-+

3xe fXY

(el 3qe) [SY

A here; see pp. 69, 71 in $9.

11

fyz ] A

fyzl

1 I .

in

ALGEBRAIC SEMANTICS

140

Suppose

i s a g-model of

M

and f o r some G E M

exists

such t h a t

((el1

The following p r o p e r t i e s o f

M sat

0 V Y ( ~ )[Rn(y)

(2)

M

Ext(G) ; i . e . ,

(3)

M, i

But

sat

sat

M, i

Case 1.

3y

(el

,

C1

sat

M

By comprehension i n

M

C2

,

,

there

satisfies

are easily verified: +

,

Gyl

G E E

'

((el) (using (1))

.

Gy

sat

Ch(Ext) ; we t h e r e f o r e have two c a s e s :

For some

(4)

M, i

sat

(5)

M, i

sat

Case 2 .

M

G

(1)

sat

.

, M , i s a t Ch(EXt)

i E I

M

such t h a t

MLp+C

F E E

'

(e,(e))

Vx

3!y [Gy A Fxy] e (el V Y ( ~ )[ Gy --t 3xe Fxy ]

' E((e),(e))

For some

9

3!z (el Fyz

(6)

M, i

sat

Vy

(7)

M, i

sat

Vz (e) 3Y(e) [GY A F Y Z I

(el

[ Gy

+

1

.

First suppose Case 1 h o l d s . By comprehension, t h e r e e x i s t s

F' E M

(e, ( e l )

such t h a t

(8)

M

0 Vxe

sat

0 Vxe

Then by ( l ) , Y s

YI

exists

,

sat

M, j

Vxe 3 ! y

Vyie) [ Rn(y)

Vy

j E I

For, l e t

Fxy

cf

Vxe 3 ! y

A

(el

[Gy

.

Fxy] ]

A

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

M

We claim t h a t

[ F'xy

X E Me Rn(Y)

M, j

A

,

Y

GY

Rn(y')

, Y'

Rn(Y')

satisfy

A

A

C M

F'XY

and

A

GY'

A

F'xy'

y

+

=

y'

]

.

such t h a t

(el A

F'Xy

F'XY'

,

.

and by (8),

M, j

satisfy

M, j s a t FXY , FXY' , s o M , j s a t (el This proves t h e claim. Since M s a t C1 , we conclude t h a t t h e r e [Gy

A

Fxy]

and a l s o

.

Y E M

(el

such t h a t

M

sat

[Rn(Y)

A

I 3 Vxe

-

F'xY]

.

But by ( l ) ,

COHEN'S INDEPENDENCE RESULTS

M, i

sat

GY

, so

conclude t h a t

by (S),

M, i

sat

M, i

3x

e

,

FxY

141

whence by ( 4 ) and (8) we

3xe F'xY , a c o n t r a d i c t i o n . Suppose, on t h e o t h e r

sat

F' E M

hand, t h a t Case 2 h o l d s . By comprehension, t h e r e e x i s t s such t h a t M

(9)

0 V Y ( ~ )V Z ( ~ )[ F'yz

sat

We now cl ai m t h a t

For, l e t

, Y ,

j E I

sat

M, j By ( l ) , M , j and a l s o

sat

FYZ C2

A

F'yz

2 ' E M(e)

F'YZ

A

,

GY

sat M

,

Z

Rn(Y)

sat

M, j

c la i m . S i n ce

Vz' [ Rn(y) (el

(el

Fyz

A

Vy(e) [Gy

.

3 ! ~ ( ~Fyz] ) ]

-+

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

M

Vy(e) Vz

+-+

((el, (el)

and by ( 9 ) ,

F'yz'

z

-+

z'

G

.

]

such t h a t

F'YZ'

A

A

,

M, j

sat

,

Fyz] , (el Z z 2 ' , which p r o v es t h e

V Y ( ~ )[Gy

FYZ' , so M , j s a t , we conclude t h a t t h e r e e x i s t s

-+

3!z

Z E M

(el

su ch

that

By ( 7 ) , t h e r e e x i s t s choice of Rn(Y')

A

G

Y E M such t h a t M, i (el implies t h a t t h e r e e x i s t s Y ' E M

Y 5.Y'

.

by (6) and ( 9 ) ,

Since

M, i

F sat

is extensional,

sat

GY

FYZ

A

such t h a t

(el M, i s a t

FY'Z

, and t h e M, i

,

sat

and hence

, c o n t r a d i c t i n g (10). T h i s completes t h e

F'Y'Z

pr oo f o f Lemma 1 7 . 4 . I t should b e remarked t h a t Theorem 1 7 . 3 and Lemma 1 7 . 4 a r e s y n t a c t i c r e s u l t s , f o r which d i r e c t , c o n s t r u c t i v e p r o o f s could be p r o v i d ed . LEMMA 1 7 . 5 . 1 .

Proof:

Let

The formula

ACa

F(X ,X ) = 1 i f

, and

gc q

5r)

Mu , and d e f i n e

11

G E M(a) =

3xa Gx

.

Let by l e t t i n g

11

5

B

11

M,

M(a,u)

(1

o th e r w is e . We show t h a t

0

C l e a r l y t h e f i r s t two c o n j u h c t s o f

(1)

u

M = (Mu, m)uEp b e a B-valued Boolean model o f MLp.

(XE;)5<x b e a wel l- o r d e r in g o f

t o assume

i s b - v a l i d i n MLp for ev er y t y p e

WOa(F)

have v a l u e

1

A

VyU [ G Y

+

-

FYX I

11

= 1

, so it s u f f i c e s

and show t h a t

3xu [ Gx

WOa(F)

I 11 .

.

142

ALGEBRAIC SEMANTICS

But if we let P = G ( X ) 5 5

for

5 < X , (1) is clearly equivalent

to

and ( 2 ) holds in any complete Boolean algebra. LEMMA 17.5.2. Let M

=

(Mo, m)uEp

based on w , and let m(<) ordering on u

Proof:

. Then

1)

E M (e,e) M PE 1) = 1

be a B-valued Boolean model of MLp B o X w be the usual (two-valued)

-

.

Straightforward.

THEOREM 17.5. The theory MNp+C1+C2 is consistent. Proof: ___

If not then some formula

U

(1)

AC

U

A

AC

A

... A

'-' PE

U

AC

A

A

C1

A

C2

is inconsistent in ML +C . We c0nstruct.a Boolean model of MLp in which P (1) has Boolean value 1 , contradicting Theorem 14.3. Let I be any set with 1 1 ) > 2u , and let B be the complete algebra of all regular open I , where 2 = {0,1) has the discrete sets in the product space 2 topology. It is well-knowng that B satisfies the countable chain condition; i.e., every disjoint set of non-zero elements of B is denumerable. Let M = (Mu, m)uEp be the B-valued Boolean model based on o in which m(<) is the usual ordering on o By Lemmas 17.5.1 and 17.5.2, we have 11 ACak 11 = 11 PE I] = 1 in M , for all k < n , so it remains only to verify that 11 C1 /I = 11 C2 11 = 1 .

.

If I/ C1 11 = 0 (note that C1 is modally closed), then there exists F E M f o r which (e,(e)) (1)

11

(2)

11 3 ~ ( ~[Rn(Y) )

0 Vxe

hie) A 0

[Rn(y) Vxe

-

A

Rn(y')

Fxyl 11 = 0

A

Fxy

A

Fxy'

+

Y

E

Y']

11

=

1 ,

.

From (2) and Lemma 15.4.1, it follows that there exists for every Y E 2@ a natural number ny C o with F(ny,Y) # 0 . For some n E w , the set

Rosser [1969], p . 32.

COHEN'S INDEPENDENCE RESULTS

143

K = { Y C 2" 1 ny = n } must be non-denumerable. But by (l), Y # Y ' in K implies F(n,Y) F(n,Y') = 0 , so that { F(n,Y) I Y C K } is a nondenumerable disjoint set o f non-zero elements of B , contradicting the countable chain condition. 9

define Zi C M = B" by letting Zi(n) be the clopen (el a x I subset { p C 2 I p(n,i) = 1 1 of the product space 2 w x I . It is easily verified that

For each i € I

I1 z.1 z. I/ 3

=

S

for i # j in I

.

n

[zi(n) * Zj(n)l

= 0

By (4), there exists for each i C I some Yi C ZW

with F(Yi,Zi) # 0 . Since 111 > Z0 , there exists Y C 2O such that the set J = { i C I 1 Y. = Y 1 is non-denumerable. By ( 3 ) , i # j in J implies F(Y,Zi)

F(Y,Z.) 1

9

I/ Z. E Z. 11 1

1

= 0

,

that { F(Y,Zi) I i E J } is again a non-denumerable set of pairwise disjoint elements of B , contradicting the countable chain condition. so

COROLLARY 17.6.

-

The continuum hypothesis Ch is not provable in Np.

Proof: If Ch were provable in N then Ch(EXt) would be provable in P MNp, by Theorem 17.3. It follows from Lemma 17.4 that the theory MNp+C1+C2 would then be inconsistent, contradicting Theorem 17.5.

144

BIBLIOGRAPHY

Peter Andrews [1963] A reduction of the axioms for the theory of propositional types. Fund. Math., 52 (1963), 345-350. Yehoshua Bar-Hillel [1954] Indexical expressions. Mind, 63 (1954), 359-379. Arnould Bayart [1958] Correction de la logique modale du premier et du second ordre S 5 . Logique et analyse, 1 (1958). 28-44. [1959] Quasi-ad6quation de la logique modale du second ordre S5 et adequation de la 1.ogique modale du premier ordre S S . Logique et analyse, 2 (1959), 99-121. Aldo Bressan [1964] A general many sorted modal language ML (Contributed paper, reported in Logic, methodology and philosophy of science: proceedings of the 1964 international congress (Amsterdam, 1965)). [1972] A general interpreted modal calculus (New Haven, 1972). Rudolf Carnap 119471 Meaning and necessity (Chicago, 1947). Alonzo Church [1940] A formulation of the simple theory of types. J. Symb. Logic, 5 (1940), 56-68. [1951] A formulation o f the logic of sense and denotation. Structure, method and meaning, ed. Paul Henle et al. (New York, 1951). [1956] Introduction to mathematical logic, vol. I (Princeton, 1956). Nino Cocchiarella [1966] Tense logic: a study of temporal reference (Doctoral dissertation, University of California at Los Angeles, 1966).

[1969] A completeness theorem in second order modal logic. Theoria, 2 (1969), 81-103.

145 Paul J. Cohen [1963] The independence of the continuum hypothesis. Proc. Natl. Acad. Sci. U.S.A., 50 (1963), 1143-1148; 51 (1964), 105-110. M. J. Cresswell [1967] A Henkin completeness theorem f o r T. Notre Dame J. Form. Logic, 8 (1967), 186-190. Donald Davidson and Gilbert Harman, eds. [I9721 Semantics of natural language (Dordrecht, 1972). Herbert B. Enderton [1972] A mathematical introduction to logic (New York and London, 1972). Herbert Feigl and Wilfrid Sellars, eds. [1949] Readings in philosophical analysis (New York, 1949). Kit Fine

[1970] Propositional quantifiers in modal logic. Theoria, 36 (1970), 336-346. Gottlob Frege [1892] Uber Sinn und Bedeutung. Zeitschrift fur Philosophie und philosophische Kritik, 100 (1892), 25-50. Paul R. Halmos [ 19631

Lectures on Boolean algebras (Princeton, 1963).

Leon Henkin [ 19501 Completeness in the theory of types. J. Symb. Logic, 15 (1950),

81-91.

[ 19631 A theory of propositional types. Fund. Math., 52 (1963), 323344.

K. J. J. Hintikka [1961] Modality and quantification. Theoria, 27 (1961), 119-128. G.

E. Hughes and M. J. Cresswell [1968] An introduction to modal logic (London, 1968).

Stig Kanger [1957] Provability in logic (Stockholm, 1957).

146 David Kaplan [1964] Foundations of intensional logic (Doctoral dissertation, University of California at Los Angeles, 1964). [I9661 Review of Kripke [1963a]. J. Symb. Logic, 31 (1966), 120-122. [1970] S 5 with quantifiable propositional variables (Abstract), J . Symb. Logic, 35 (1970), 355. Saul A. Kripke [ 19591 A completeness theorem in modal logic. J. Symb. Logic, 24 (1959), 1-14.

[1963a] Semantical analysis of modal logic I. Z. Math. Logik Grundlagen Math., 9 (1963), 67-96. [ 1963bl Semantical considerations on modal logic. Acta Philosophica Fennica, 16 (1963), 83-94.

[1967] An extension of a theorem of Gaifman-Hales-Solovay.Fund. Math. 61 (1967), 29-32.

E. J. Lemmon [ 19631 A theory of attributes based on modal logic. Acta Philosophica Fennica, 16 (1963), 95-121.

David Lewis [ 19701 General semantics. S nthese, 22 (1970), 18-67. Reprinted in

Davidson and H a r m a n k

J. C.

C.

McKinsey

[1941] A solution of the decision problems for the Lewis systems S2 and S4, with an application to topology. J. Symb. Logic, 6 (1941), 117-134.

J. C. C. McKinsey and Alfred Tarski I19441 The algebra of topology. Ann. of Math., 45 (1944), 141-191. [1948] Some theorems about the sentential calculi of Lewis and Heyting. J. Symb. Logic, 13 (1948), 1-15. Richard Montague [1960] Logical necessity, physical necessity, ethics, and quantifiers. Inquiry, 4 (1960), 259-269. Reprinted in Montague [1974]. [1968] Pragmatics. Contemporary philosophy - la philosophie contemporaine, ed. Raymond Klibansky (Florence, 1968). Reprinted in Montague [1974].

-

147 [1969] On the nature of certain philosophical entities. The Monist, 53 (1969), 159-194. Reprinted in Montague [1974]. [1970a] Pragmatics and intensional logic. Synthese, 22 (1970), 68-94. Reprinted in Davidson and Harman [1972] and Montague 119741. [1970b] English as a formal language I. Linguaggi nella societa e nella tecnica, ed. Bruno Visentini et al. (Milan, 1970). Reprinted in Montague [ 19741. [197Oc] Universal grammar. Theoria, 36 (1970), 373-398. Reprinted in Montague [ 19741. [1973] The proper treatment of quantification in ordinary English. Approaches to natural language, ed. K. J. J. Hintikka et al. (Dordrecht, 1973), 221-242. Reprinted in Montague [1974]. [1974] Formal philosophy: selected papers of Richard Montague, ed. Richmond Thomason (New Haven, 1974). Andrzej Mostowski [1947] On absolute properties of relations. J. Symb. Logic, 12 (1947), 33-42. Steven Orey [1959] Model theory for the higher order predicate calculus. Trans. her. Math. SOC., 92 (1959), 72-84. Terence Parsons [1968] A semantics for English (Duplicated, 1968). An abridged version appears in Davidson and Harman [1972]. Barbara H. Partee [1973] The semantics of belief sentences. Approaches to Natural Language, ed. K. J. J. Hintikka et al. (Dordrecht, 1973), 309-336. [1975a] Montague grammar and transformational grammar. To appear in Linguistic Inquiry, 6 (1975). Barbara H. Partee, ed. [1975b] Montague grammar (New York and London: Academic Press, to appear). W. V. Quine [1960] Word and object (Cambridge, Mass., 1960). Helena Rasiowa [1951] Algebraic treatment of the functional calculi of Lewis and Langford. Fund. Math., 38 (1951), 99-126.

148 [I9631 On modal theories. Acta Philosophica Fennica, 16 (1963), 201214. J. Barkley Rosser [1969] Simplified independence proofs: Boolean valued models of set theory (New York and London, 1969).

Dana Scott [1966]

Boolean-valued models for higher-order logic (Duplicated notes, Stanford University, January 1966).

[1967a] A proof of the independence of the continuum hypothesis. Systems Theory, 1 (1967), 89-111.

Math.

[1967b] Boolean-valued models for set theory (Lecture notes for the Amer. Math. S O C . summer institute on axiomatic set theory, University of California at Los Angeles, 1967). [1970] Advice on modal logic. Philosophical problems in logic: some recent developments, ed. Karel Lambert (Dordrecht, 1970), 143173. Roman Sikorski [1969]

Boolean algebras, 3rd ed. (Berlin, 1969)

Alfred Tarski [1954] Contributions to the theory of models 1. Indag. Math., 16 (1954), 572-581. A. Tarski, A. Mostowski and R. M. Robinson

[I9531 Undecidable Theories (Amsterdam, 1953). Richmond Thomason

z-

[I9701 Some completeness results f o r modal predicate calculi. sophical problems in logic: some recent developments, ed. Karel Lambert (Dordrecht, 1970), 56-76.