A new semantics for first-order logic, multivalent and mostly intensional

A New Semantics for First-Order Logic, Multivalent and Mostly Intensional Hugues Leblanc

Cited by many as distinctive of first-order logic are the

bivalence of its statements and the extensionality of its

operators, among them the three operators ' ~ ' , '&', and 'V'. It is to that bivalence and that extensionality, we are

told, that the logical entailments and, hence, logical truths

peculiar to the logic are due. But first-order logic is a far more diverse thing than its exponents usually allow. As

proof I submit here a new semantics for it, one in which

statements are susceptible of up to 2 t~~ values, the

operators '&' and 'V ' are intensional, and yet the logical entailments and logical truths that first-order logic acknowl-

edges are all preserved.

Credit should go to Karl Popper, from whom I largely

borrowed Constraints D1-D6 on p. 57; to Kent Bendall, from whom I borrowed Constraint D7 on that page; and to

Nicholas Rescher, who pointed out in 1963 that '&' - as interpreted here - is intensional in the presence of ' ~ ' .

A logic begets languages after its own image, and it is to an arbitrary one of those begotten by first-order logic that I direct my attention here. Call the language L.

L is to have as its primitive signs (i) one or more predicates, each identified as being of a certain degree d,

(ii) one or more (individual) terms, 0ii) ~o (individual) variables, (iv) the three operators ' ~ ' , '&', and 'V ' men- tioned earlier, 1 and (v) the two parentheses ' ( ' and ' ) ' ,

and the comma ' , ' . L is to have as its formulas all finite (but non-empty) sequences of primitive signs of L. And L is to have as its statements (i) all formulas of L of the sort Q(T1, T2 . . . . . Ta), where Q is a predicate of L of degree d and T1, T2, . . . , Ta are (not necessarily distinct) terms of L, (ii) all formulas of L of the sort " A , whereA is a statement of L, (iii) all formulas of L of the sort (A & B), where A and B are (not necessarily distinct) statements of L, 2 and (iv) all formulas of the sort (VX)A, where for any term T of L you please A(T/X) (= the result of putting the term T

for every occurrence of the variable X in A) is a statement ofL.

L will have as its axioms all statements of L of the following sorts:

A1. A D (.4 &A)

A2. (A & B ) D A

A3. (.4 D B ) D ( " ( B & C ) D " ( C & A ) )

A4. A D (VX)A

A5. (V J0 A D A (T/X) A6. (VX)(A ((VX)A (VX)8),

plus all those of the sort (VJO(A(X/T)), where A is an

axiom of L; and L will have as its one rule of inference Modus Ponens - the ponential o f two statements A and A D B o f L being B.

Where A is a statement and S a set of statements of L,

I shall understand by a proof in L of A from S any finite column of statements of L such that (i) each entry in the column is a member of S, an axiom of L, or the ponential

of two earlier entries in the column and (ii) the last entry in the column is A. I shall say that A is provable in L from S - S ] -A, for short - if there exists a p roo f inL of A f r o m

S; and I shall say that A is provable in L - I--A, for short -

if 0 t -A. And, with an eye to the Strong Completeness

Proof on pp. 57-58, I shall say that S is (inJconsistent in L if there is no (there is a) statement ofA of L such that both

S J-A and S ~ ~A. From there on, use will be made of the term-extensions

o f L. These are to be all languages exactly like L except for having anywhere from 1 to S0 terms besides those of L.a

I shall refer to the term-extensions of L by means of 'L § With L § an arbitrary term-extension of L, I shall assume that the terms o fL § come in some alphabetic order, refer to the alphabetically i-th ( i= 1,2, 3 . . . . ) of them by means of ' t ; ' , and write 'S ~ § for 'A is provable in L § from S' . And, with an eye again to the Strong Completeness Proof on

pp. 57-58 , I shall say that a set S of statements o fL § is (i) (in)consistent in L § if there is no (there is a) statement A

of L § such that both S P* A and S t -§ ~A, (ii) maximally

Topoi3 (1984), 55-62. 0167-7411/84/0031-0055501.20. (~ 1984 by D. Reidel Publishing Company.

56 HUGUES LEBLANC

consistent in L § if S is consistent in L + but, for any statement A o f L § not in S, S u {A} is inconsistent in L +, and (iii) omega-complete in L + if, no matter the quantification

(VX)A of L +, S 1-+ (VX)A if S [-+ A (t~./)O for each i from 1 on.

Proofs of the Strong Soundness and Strong Complete- ness Theorems for L that use term-extensions are easily

edited to yield:

THEOREM 1. S ~-A if, and only if, A is logically entailed by S in the standard sense,

II

The brand of standard semantics for L that most are famil-

iar with uses models, i.e., domains and interpretations relative to these domains of the predicates and terms ofL. Preference is given here to another, but equivalent, brand which uses valuation functions for L and its term-extensions and interprets quantifiers substitutionally.

With L § an arbitrary term-extension of L, 0 the (truth- value) false, and 1 the (truth-value) true, I shall understand by a valuation function for L § any function V § from the statements of L § to {0, 1} that meets the following three constraints:

B1. V+(~A) = 1 if, and only if, W(A) q~ 1 B2. V*(A &B) = 1 if, and only if, V+(A) = 1 and

V+ (B) = 1 B3. V+((VX)A) = 1 if, and only if, V+(A(t~./X)) = 1

for each i f rom/-on,

or, equivalently, the following three:

C1. V+(~'A)= 1 - V+(A) c2. V§ &B) = V§ x V§

V § (i_I11 (,4 (t~./X))) ifL § has] terms

C3. V§ / /Limit - l

V§ ( II (A (t[/X))) if L § has t 1-.o0 i=1 ~0 terms,

/ where in C3 ' II (.4 (t~. /))' is short for '(...(A (t~/X) &

i=1 A ( t ; /X)) ...) & A (t;/X)'.

I shall call a function V from the statements of L to {0, 1} a valuation function for L ff for some term-extension L § of L there exists a valuation function V § for L § of which V is the restriction to L. 4 I shall say of a statement A and a set S of statements of L that A is logically entailed by S in the standard sense if A evaluates to 1 (= A has the truth- value true) on every valuation function for L on which all members of S do. And I shall say of a statement A of L that A is logically true in the standard sense ff A is logically entailed by ~ in that sense (i.e. ifA evaluates to 1 on every valuation function V for L).

and, hence, with 0 as S:

THEOREM 2. k A if, and only if, A is logically true in standard sense, s

The logical entailments and logical truths of L in the standard sense are of course those I shall want preserved by the semantics of the next section.

(Readers more familiar with models than with valuation functions know that (i) S ~-A if, and only if, A is true in every model for L in which all members of S are, and as a result (ii) ~ A if, and only if, A is true in every model for L. So, in view of Theorems 1-2, (iii)A is logically entailed by S in what I call the standard sense if, and only if, A is true in every model for L in which all members of S are true, and in particular (iv) A is logically true in what I call the standard sense if, and only if, A is true in every model for L. So, said readers need not fret over my account of things. Our logical entailments and logical truths are the same.)

Clearly, the statements of L and its term-extensions are bivalent according to the semantics of this section: the valuation functions just defined can accord only 0 and 1 to those statements. And, equally clearly, ' ~ ' , '&', and 'V' are extensional according to that semantics. To elaborate on the second point, say that

(i) "~ ' is extensional if, no matter the telm-extension L § of L, the valuation function V § for L § and the negations "~A and ~A' of L §

V§ = V§ ') if V+(A) = V§

(ii) '&' is extensional if, no matter the term-extension L § of L, the valuation function V* forL § and the conjunc- tionsA &B and A' &B' of L §

V+(A &B) = V+(A ' &B' ) ff V§ = V+(A ') and V§ = V§

and

Oii) 'V' is extensional if, no matter the term-extension L + of L, the valuation function V + forL § and the quantifi- cations (VX)A and (VX ' )A ' of L +,

V*((VX)A) = V§ ') if V+(A (tT/X)) = V§

A NEW SEMANTICS FOR F I R S T - O R D E R LOGIC 57

for each i from 1 on.

'~-' and '&' are extensional by virtue of C1-C2, and - ii j § with i= I(A (ti/X)) equal by virtue of C2 to V§ (t;/X)) x

V§ x ... x V§ ( t T / X ) ) - 'V' is extensional by virtue of C3.

III

Now the semantics advertised on p. 56. As a first step, suppose that in the account on p. 56

of a valuation function for L § you replaced C1-C3 by

D1-D7 below. The new constraints would pick out exactly the same valuation functions for L + (and, hence, exactly the same valuation functions for L) as the old ones did, a result obtained in 1981 and reported with proof in Section 5 of

[7]. Suppose, however, that as a next step you allowed a valuation function forL § to take the statements o fL § (and, hence, a valuation function for L to take the statements of L) into a 2 ~~ -membered set of reals (those from 0 through

1) rather than the 2-membered one (0, 1 ). The semantics exploiting these functions rather than the functions of p. 56 would be the semantics I promised, one in which state-

ments would be multivalent, 6 in which ' ~ ' would remain extensional but neither of '&' and 'V ' would, and yet in which the logical entailments and logical truths isolated in

Section II would all be preserved. Because Constraints

D1-D7 were originally intended for (one-argument or

unary) probability functions, I shall label this semantics a

probabilistic semantics. Details are as follows.

With L § an arbitrary term-extension of L, I shall understand by a valuation for L § any function V + from the state-

ments of L § to the reals that meets the following seven

constraints:

D1. 0~< V§ D2. V§ & "~A )) = 1 03. V§ = V§ &B) + V§ & ~B) D4. V+(A) ~< V§ &A)

D5. V* (A&B)<~V* (B&A) D6. V§ & (S & C)) <. V*((A &B) & C)

[ ] [V+(A & II (B(t~./X))) i fL§

D7. V§ & (VX)B) = t i=1 ] j terms

]Limit V*(A & rl(B(t~./X))) if

L § has So terms, l

where in D7 is short for ' ( . . . (B(t~/X)& ' n (R(t;/x))' i=1

B(t~/X)) & ... ) & B(t; /X) ' . 7 I shall call a function V from the statements of L to the

reals a valuation function for L if for some term-extension L + of L there exists a valuation function V + for L + (of the

sort just defined) of which V is the restriction toL . I shall

say of a statement A and a set S of statements of L that

A is logically entailed by S in the probabilistic sense if A evaluates to 1 on every valuation function for L (of the sort

just defined) on which all members of S do. And I shall say of a statement A of L that A is logically true in the probabilistic sense ifA is logically entailed by 0 in that sense. 8

I first prove the counterparts here of Theorems 1-2 , and

then attend to questions of extensionality. Proof that A, if provable in L from S, is logically en-

tailed by S in the probabilistic sense, calls for two lemmas. These are to the effect that, for any term-extension L + of

L and any valuation function V + for L +, (i) each axiom of L + evaluates to 1 on V + and (ii) the ponential of any two statements of L + that evaluate to 1 on V + likewise evaluates

to 1 on V +. To prove (i) you first show that each axiom o fL + of any

of the sorts A1-A6 on p. 55 evaluates to 1 on V+; and you show that if an axiom A ofL + evaluates to 1 on V +, so does any term-variant in L + of the axiom, 9 and hence by virtue

of (15)-(16) in the Appendix so does (VX)(A (X/T)). The details are in [10]. 1~ Proof of (ii) is in the Appendix (see

(10) there). This done, suppose there is a proof in L - and, hence, in

L § - of A from S, and suppose all members of S evaluate to 1 on V +. Then any entry in the proof of A from S that

belongs to S is sure of course to evaluate to 1 on V +, any

one that is an axiom o f L - and, hence, o fL § - is sure by (i) to evaluate to 1 on V +, and the ponential of any two

entries in the proof that evaluate to 1 on V* is sure by (ii)

to evaluate to 1 on V +. So, a simple induction on the number of entries in the proof of A from S will guarantee

that each of these entries - and, hence, the last one, A - evaluates to 1 on V +. Hence, by the definition of logical

entailment above,

THEOREM 3. I f S ~ A, then A is logically entailed by S in the probabilistic sense, (= The Strong Soundness Theorem forL)

and, hence,

THEOREM 4. I f I-A, then A is logically true in the probabilistic sense. ( - The Weak Soundness Theorem for L)

Proof that A, if logically entailed by S in the probabil-

58 H U G U E S L E B L A N C

istic sense, is provable in L from S, is of the Henkin sort. Suppose that, contrary to the assumption, S bLA, and let L** be a term-extension of L that has ~o terms besides those of L. ll Then S u {"~A} is consistent inL and - as

shown by Henkin - canbe extended into a s e t ~ ( S u {~A}) of statements of L** that is maximally consistent and omega-complete in L**. (A set of the sort ~'~(S u {"A})is often called a Henkin set, which accounts for the ~,~' here). Now, let V** be the valuation function from the statements of L** to the reals that accords 1 to all the statements of L** in ~"(S u ("A}) and 0 to the rest. It is easily verified that V** meets all of Constraints D1-D7 (in particular,

Constraint D3 because X(S u (~'A)) is maximally consistent in L**, and Constraint D7 because ~'~(S u (~A}) is omega-complete in L**). Hence, each member of S, belong- ing by construction to ~'~(S u {~A}), evaluates to 1 on

V**. But, for the same reason, so does "~A. Hence, by (7) in the Appendix, A evaluates to 0 on V**. There is thus a term- extension of L (i.e. L**) and a valuation function for that language (i.e. V**) such that all members of S evaluate to 1 on the function but A does not. Hence, by Contraposition,

THEOREM 5. I f A is logically entailed by S in the probabilistic sense, then S ~-A, (= The Strong Completeness Theorem for L) 12

and, hence,

THEOREM 6. I f A is logically true in the probabilistic sense, then [- A. (= The Weak Completeness Theorem for L)

Hence, by Theorems 1,3, and 5,

THEOREM 7. A is logically entailed by S in the probabilistic sense if, and only if, logically entailed by S in the standard sense,

and by Theorems 2, 4, and 6,

THEOREM 8. A is logically true in the probabilistic sense if, and only if, logically true in the standard sense.

So, the logical entailments and logical truths of standard semantics are all preserved by the semantics of this paper, this despite the fact that statements become multivalent...

IV

... and logical operators mostly intensional. Call ' ~ ' , '&', and 'V' extensional if they meet conditions

(i)-(iii) on pp. 56-57 (with V+ there a valuation function in the sense ofp . 57), and intensional if they do not.

Because of (7) in the Appendix, ' ~ ' is extensional here as it was in Section II. However, '&' - extensional in that section - is intensional here, this in the presence o f ' " , as was shown in [13], but also in the absence of ' " , as was shown in [10]. (In what follows, (2)-(4) , (7), (9), (12)- (14), and (16) are all consequences of D1-D7 demonstrated in the Appendix.)

In the first case, let L" be an arbitrary term-extension of L § A be an arbitrary statement of L § and V § be any valuation function for L § such that V+(A) - and, henCe, by virtue of (7), V§ - equals 1/2. Then by virtue of (2) V§ &A) equals 1/2, whereas byvirtue of (3) V*(A & ~A) equals 0.

V+(A,) = 1/2~ V+(AI & ~A2)

TABLE I

V+(AI & A~_) = 1/8 ~ V+(AI & A 2 & A 3 )

V+(AI & A 2 & ~A3)

/ V + ( A , & ~A2 &A3)

3/8

V+(AI & ~ A 2 & ~A3)

= 1/16

= 1/16

= 3/16

= 3/16

/ V + ( ~ A I & A2 & A3) = 3/16

V+(~At & A2) = 3/8

= 1 / 2 ( / V+(~AI & A2 & - A 3 ) = 3/16 V+(~AI)

\ ~.~V+(~A: & ~A~ & A3) = 1/16

V+(~AI & ~A2) 1/8

V+(--AI & --A~ & ~A3) = 1/16

A NEW SEMANTICS FOR FIRST-ORDER LOGIC

In the second case, let L § again be an arbitrary term- extension of L, and let A1, A2, and A3 be in alphabetic order the first three atomic statements of L+. 13 As results in [8] bear out, values may be accorded to the state-descriptions of L § as in Table I; and, given that valuation function V § for L § all three of V*(A x), W(A~), and V§ equal 1/2 (W(A2) by virtue of (4), and V§ by virtue of (13)), V§ &A2) equals 1/8, but by virtue of (12) V*(A~ &A3) equals 1/4.

And 'V', extensional in Section II, is intensional here. For proof, let L be a language that has just two predicates, both of them of degree 1, referred to by means of 'QI' and 'Q2'; let L § be an arbitrary term-extension of L that has ~o terms, referred to as usual by means of 't~', 't~', 't3 § etc.," and, for each i from 1 on, let ~ i' and 'B i' be short for 'Ql(t~)' and 'Q2(ti), respectively. As results in [8] bear out, values may be accorded to the state-descriptions ofL § in (A1) , G41 ,B1), (Aa, B1, A2), and (Ax ,B1 ,A~ ,B2) as in Table II below; and, given that valuation function V § for L § all four of V+(A1), V§ V§ and V§ equal 1/2 (V§ by virtue of (13), V§ by virtue of (4), and V§ by virtue of (13)), V§ & A2) equals 1/4 by virtue of(12), but V§ &B2) equals 1/8 by virtue of(14). Now, let V§ and V*(Bi) both equal 1 for each i from 3 on. Then by virtue of (9)

59

/ W( n A~)

i=1

equals 1/4 for each] from 2 on, and hence by virtue of (16) so does V+((VX)Qa (X)). However, by the same reasoning and reckoning,

/ V*( rI B~)

i=1

equals 1/8 for each ] from 2 on, and hence so does W((VX)Q2 (X)). Hence, for each i from 1 on V+(Q1 (t~.))= V+(Q2 (t~.)), and yet V+((VX)Q1 (X)) 4:V*((VX)Q2 (X)).

So, two of the three operators that were extensional in Section II are intensional here.

V

The valuation functions of p. 56 are what Stalnaker calls in [15] truth-value functions, but I call - for short - truth functions; and those of p. 57 are of course (unary) probability functions. In these closing pages I shall refer to probability functions by means of 'P'; and, given any such function P, I shall think of P(A1), P(A2), P(Aa), etc. as degrees to which a rational agent might simultaneously believe the (alphabetically ordered) statements A1,A~,Aa, etc. of L. 14

TABLE II

V+(Ai)

V+(AI & BI)

= 1/2~X~V+(AI & ~B1)

/ V + ( ~ A ,

V+(~AI) = 1/2~N~V+(~A I

V+(A1 & B I & A2)

= 1 / 4 ~

V+(At & B t & --A~)

......--V+(A1 & BI & A2 & B2) = 1/32

= 3 / 1 6 ~ V+(A 1 & Bl & A2 & ~B2) = 5/32

. . . . ~ - V + ( A I & B~ & ~A z & B2) = 1/32

= I / I ~ & B 1 & ~A 2 & ~B2) = 1/32

V+(AI & ~BI & A2)

= 1 1 4 ~

V+(AI & ~BI & ~A2)

. . ~ V + ( A I & --Bi & A2 & B2) = 1/32

1 / I O ~ V + ( A I & ~BI & Az & ~B2) = 1/32

. . / V + ( A 1 & ~ B t & ~ A 2 & B2) = 5/32 3 / I O ~ v + ( A I & ~BI & ~A2 & ~B2) = 1/32

& BI)

. V + ( ~ A I & B1 & A2)

= 1 / 4 ~

V+(-AI & Bl & --A2)

. . ~ - V + ( ~ A I & Bt & A2 & B2) = 1/32

3 / I O % v + ( ~ A t & B~ & Az & --B2) = 5/32

. . . . ~ V + ( ~ A I & B I & ~ A 2 & B2) = 1/32

= I/IO~"'~V+(~AI & Bi & ~A2 & ~B2) = 1/32

. . . . _...~-V + (-A V+(~A~ & ~B1 & A2) = t : t O ~ V + , A I ( /

& ~ B i ) = 1 / 4 ~

. . _ _ . - . - V + ( - - A I

V+(~AI & --BI & --A2) = 3 / I O ~ V + ( ~ A I

& --B I & A 2 & B2) = 1/32

& ~BI & A2 & ~B2) = 1/32

& ~ B I & ~A2 & B2) = 5/32

& ~B1 & ~Az & --B2) = 1/32

60 HUGUES LEBLANC

It follows from Theorem 4 that, for any two statements A and B of L and any probability function P for L,

If }--A = B, then P(A) =P(B).

Popper, who proved in [12] a binary counterpart of the result (for quantifierless A's and B's), claimed on the strength of it that "in its logical interpretation, the probability calculus is a genuine generalization of the logic of derivation." i s

True enough if you think of Boolean Algebra as the logic of derivation. But stronger support for Popper's claim can be offered. Since C1-C3 on p. 56 and D1-D7 on p. 57

pick out exactly the same 2-valued functions, the truth- functions for L are 2-valued probability functions for L, and vice-versa. So, probability theory is a generalization of truth(-function) theory in a stronger sense than Popper may have envisioned: allow your truth-functions, charac- terized for the occasion by means of Constraints D1-D7, to map the statements of L into the interval [0, 1] rather than just 0 and 1, and truth(-function) theory expands into probability theory. (By the same token, restrict your probability functions to have but the end-points of the interval [0, 1 ] as their values, and probability theory con- tracts into truth(-function) theory.)

Probability theory also constitutes, we now know, a semantics - and, hence, a rationale - for first-order logic.

Truth(-value) functions being 2-valued probability functions and vice-versa, it followed from Theorems 1-2 that

a first-order statement A is provable (from a set S offirst- order statements) if, and only if, A evaluates to 1 on all 2-valued probability functions (on which the members orS do). But a more telling result was obtained, to wit: A is provable (from S) if, and only if, A evaluates to 1 on all probability functions (on which the members o f S do) (= Theorems 3-6) . As Hume once taught us and Popper recently reminded us, there can be no deductive rationale for induction - more generally, no provabilistic rationale for probability theory. But thanks to Theorems 3 - 6 there is a probabilistic rationale for provability theory.

That in probabilistic semantics a first-order statement, when logically indeterminate, may have any one of 2 s* values is appropriate: the degree to which such a statement is rationally credible may indeed be any one of the reals in the interval [0, 1 ]. And that '&' and 'V' are intensional is likewise appropriate: rational credibility has always been held an intensional matter. So, the degree to which a first- order conjunction A & B is rationally credible need not depend just on the degrees to which A and B respectively are, nor need the degree to which a first-order quantifica-

tion (VX)A is rationally credible depend just on the degrees to which A (t 1/X), A (t2/X), A(ta/X), etc., respectively are. Both multivalence a n d - as regards '& ' and 'V' - intensionality are thus befitting here. 1~

Appendix: Sixteen Consequences of D1-D7

(I) V+(A &B)<~ V+(A). Proof" V*(A) = V+(A & B) + V+(A & ~B) by D3. But

V+(A & "~B) 1> 0 by D1. Hence (1).

(2) V+(A &A) = If(A). Proof by (1) and D4.

O) V+(A a ~A) = O. Proof." V+(A) = V+(A &A) + V+(A & " A ) by D3.

Hence (3) by (2).

(4) V§ = V§ & A) + v§ & A). Proof by D3 and D5.

(s) V§ & ~A) &B) = O. Proof." V+( (A & ".4) & B) <~ W (A & "A ) by (I). Hence

V+((A & ~A) &B) <~ 0 by (3). Hence (5) by D1.

(6) V+(A) = V§ Proof by (4) and (5).

(7) V+(~A)= 1 - V§ Proof." V*(~A) = V+(~(A & "~A) & "~A) by (6), hence

V§ = W(~(A & ~A)) - V+(~(A & "A) &A) by D3,

hence V+ ('~A ) = 1 - V+ ( ~ (A &'~A ) & A) by D2, and hence V§ = 1 - V§ by (6).

(8) V+(A)<. 1. Proof" V§ = 1 - V§ by (7). Hence (8) by D1.

(9) I f V+(B) = 1, then V+(A) = V+(A & B). Proof." Suppose V+(B) = 1. Then V+('~B) = 0 by (7),

hence V+"("B & A) <, 0 by (1), hence W(~B & A) = 0 by D1, and hence V+(A & ~B) = 0 by D5. Hence V+(A)= V§ &B)by D3. Hence (9).

(10) I f V+(A)= 1 and I/'(A D B) = 1, then V+(B)= 1. Proof: Suppose V+(A D B) = 1. Then V+(A & ~B) = 0

by the definition of 'D' and (7). Suppose further that V+(A) = 1. Then V+(A & B) = 1 by D3, hence V+(B &A)= 1 by D5, and hence V+(B) = 1 by (9). Hence (10).

A NEW SEMANTICS FOR F I R S T - O R D E R LOGIC 61

(11) v§ &(B&C))= V*((A &B)&C). proof." V+((A & B) & C) <~ V+(C & (,4 & B)) by D5,

hence V+((A & B) & C) <~ V*((C & A) & B) by D6, hence

W((A & B) & C) <~ V§ & (C & A )) by D5, hence V+((A & B) & C) <-< V§ & C) & A ) by D6, and hence V§ (A & B ) & C) <<. V§ & (B & C)) by D5. Hence (11) by D6.

(12) V+((A & B ) & C ) + V+((A &"B)&C) = V+(A &C). proof." V+((A & B) & C) equals V+(A & (B & C)) by

(11), hence V+((B & C) & A ) by D5, and hence V§ & (C & A ) ) by (11). But by the very same reasoning, V+((A & '~B) & C) equals V+(~B & (C & A)). Hence V+((A &B)& C) + V§ & ~B) & C) equals V§ &A) by (4), and hence

V+(A & C) by D5.

(13) V§ ,~B)aC>+ V+((A a ~ B ) a C ) + V~(("A &B) & C) + V§ & ".B) ~ 0 = V§

proof." Both V§ & B) & C) + V+(A & ""B) & C) = V+(A & C) and V+(("A & B) & C) + V+(("A & ~B)a C) = V§ & C) by (12). Hence (13) by (4).

(14) V+(((A & B) & C) & D) + V§ & B) & "-.C) & D) + W(((.,.A a B) & 63 ~ D) + V§ e,. B) &. "C) & D) = V§ ~ D).

Proof: V'(((A & B) & C) & D) + V+(((A & B) & -C) & D) equals V+((A & B) & D) by (12), and hence equals

V+(A & (B & D)) by (11). But , by the same reasoning,

V§ . & B) & C) & D) + V§ & B) & -C) & Z)) equals W("A & (B &D)). Hence (14) by (4).

i (15) V§ = V§ II (A (t~./X))) if L § has] terms.

i=1 Proof. Let B be an arbitrary s ta tement o f L § Then

V§ equals V§ & ~'B) & (VX)A) by (6), hence !

V*(~(B & -a) & II (.4 (tT/x))) by D7, and hence i=1

V§ H (A(t~./X))) by (6) again. Hence (14). i=1

(16) W((VX)A) = Limi t V+(II (A(t;/X))) ilL§ t% terms. / ~ i=1

Proof by (15) and the def ini t ion o f a l imit .

Notes

i I shall presume here the additional connectives 'D ' , 'V', and '-- ' to be defined in the customary fashion. 2 I shall drop outer parentheses, and on pp. 58 and 59 I shall asso- ciate to the left. So 'A & B & C', for example, will be short for

'((A & B) & C-3'.

3 A language is often counted one of its term-extensions. Deviating

here from that practice will make on pp. 56 and 57 for nearer accounts of a valuation function for L. See note 4 on this matter. 4 When defining a valuation function for L as the restriction to L of some valuation function for some term-extension of L, I essen- tially follow Field's practice in [3]. Note, incidentally, that if L counted one of its term-extensions, L would be accorded valuation functions twice: at Round One, when every term-extension of L is accorded valuation functions, and hence L would be, and at Round Two, when L is accorded the restrictions to L of the valuation functions for the various term-extensions of L. Of course, the valuation functions accorded to L at Round One would be among those accorded to it at Round Two, but the redundancy is inelegant. 5 Proofs of Theorems 1-2 that use term-extensions will be found in [2], in Sections 2 - 3 of [7], etc. 6 More specifically, multivalent in that, given any statement of L that is logically indeterminate (i.e. such that neither it nor its nega- tion is logically true) and any of the 2 ~0 reals in the interval [0, 1], there exists a valuation function V for L on which the statement evaluates to that real. For proof of the point, see [8]. The results there concern only quantffierless statements, but they are easily extended by means of (15) and (16) in the Appendix to cover all the statements ofL. 7 D3-D6 come from Popper's [11] and [12], and D1-D2 are simplifications of constraints there. D7 is a special case of a constraint in Bendall's [1], one that stems from Gaffman's [4]. Proof is given in [6] that, as claimed in p. 319 of [12], D1-D6 are equivalent mutatis mutandis to Kolmogorov's axioms for probability fields in [5].

It follows from D1-D5 that V"(A) ~< 1. So, as disclosed on p. 57, the reals into which V + maps the statements of L § are those in the interval [0,1 ]. D1 could of course be dispensed with ff the reals into which V* maps the statements of L § were identified there as non- negative reals.

Constraints very much like D1-D6 were utilized in [14] to obtain results different from, but not unrelated to, the results in this paper.

The definitions are adaptations of definitions in [6] and Section 4 of [7]. Theorems 3 -8 below, which legitimize them, were obtained in the summer of 1980. 9 A term-variant in L § of a statement A of L § is any result of replacing every occurrence of a term of L § inA by another term of L + . lo Proofs that an axiom of L of any of the five sorts A1-A2 and A4-A6 invariably evaluate to 1 are easy. Not so the proof that one of the sort A3 does. 11 L*~ is the only term-extension of L that is indispensable in the paper. But the other term-extensions of L acknowledged here make for more natural accounts of logical entailment and logical truth. See [9 ] for further uses to which they can be put. 12 See Section 4 of [7] for a more detailed proof of Theorem 5, and [8] for one in which anywhere from 2 to ~0 reals play a role. la In this counterexample and the next I presume the statements (hence, the atomic statements in particular) of any term-extension of L to be arranged in some alphabetic order. 14 The italicized passage is from a letter of Kent Bendall's.

62 HUGUES LEBLANC

as The counterpart runs:

If t--A ~ B, then P (A/C) = P (B/C) for any binary probability function P for L and any statement C of L,

with a binary probability function P for L understood to meet the constraints on p. 349 of [12]. The claim is on p. 356 of [12]. 1~ The paper was written while I was on a partial research leave from Temple University. Thanks are due to T. McGinness, J. Serem- bus, and H. J. Wilk for reading an early draft of it.

References

[1] BendaU, K.: 1979, 'Belief-Theoretic Formal Semantics for First-Order Logic and Probability,' Journal o f Philosophical Logic 8,375-394.

[2 ] Bergmann, M., J. Moor, and J. Nelson: 1980, The Logic Book, Random House, New York.

[3] Field, H.H.: 1977, 'Logic, Meaning, and Conceptual Role,' The Journal of Philosophy 74, 379-409.

[4] Gaifman, H.: 1964, 'Concerning Measures on First-Order Calculi,' Israel Journal of Mathematics 2, 1-18.

[5] Kolmogorov, A.N.: 1950, Foundations of Probability, Chelsea Publ. Co., New York (German original, Berlin, 1933).

[6] Leblanc, H.: 1982, 'Popper's 1955 Axiomatization of Ab- solute Probability,' Pacific Philosophical Quarterly 63, 133- 145.

[7] Leblanc, H.: 1983, 'Alternatives to Standard First-Order Semantics,' in D. M. Gabbay and F. Guenthner (eds.), Hand- book of Philosophical Logic, D. Reidel, Dordrecht.

[8] Leblanc, H.: 1983, 'Probability Functions and their Assump- tion Sets: The Singulary Case,' Journal of Philosophical Logic 12,379-402.

[9] Leblanc, H.: forthcoming, 'Quantification in Deductive Logic and Probability Theory,' in M. W. Wartofsky (ed.), to appear in Boston Studies in the Philosophy of Science, D. Reidel, Dordrecht.

[10] Leblanc, H.: forthcoming, 'On Characterizing Unary Prob- ability Functions and Truth-Value Functions,' Canadian Journal o f Philosophy.

[11] Popper, K. R.: 1955, 'Two Autonomous Axiom Systems for the Calculus of Probabilities,' The British Journal for the Philosophy of Science 6, p. 51-57,176,351.

[12] Popper, K.R.: 1959, The Logic o f Scientific Discovery, Basic Books, Inc., New York.

[13] Rescher, N.: 1963, 'A Probabilistic Approach to Modal Logic,' Modal and Many-Valued Logics, Acta Philosophica Fennica 16, pp. 215-226.

[14] Stalnaker, R.: 1970, 'Probability and Conditionals,' Philos- ophy of Science 37, 64-80.

Dept. of Philosophy Temple University Philadelphia, PA 19122 U.S.A.

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A new semantics for first-order logic, multivalent and mostly intensional