General semantic closure

PAULVINCENTSPADE

GENERAL

SEMANTIC

CLOSURE

In recent years there has been considerable progress in our understanding of the Liar and related paradoxes. As a result, the proscription against a language’s expressing its own semantic theory has been more and more eroded. In this connection, Herzberger ([2] , p. 26) distinguishes three degrees of semantic closure for a language L : Atomic Closure: L contains the means for recording the truth-value of each of its own sentences. Molecular closure: L contains the means for expressing all singular consequences of its semantic theory. General Closure: L contains the means for expressing the whole of its semantic theory. The passage from the second to the third degree of semantic closure requires the domestication of the quantifier in semantic contexts. (See [2] , pp. 2728, and [3] .) Herzberger ([2], p. 35) suggests that Skyrms’ language in [4] reaches the second degree. In fact, however, it does not - unless the notion of “expressing all singular consequences of its semantic theory” is taken very narrowly indeed. What Skyrms’ language can express are such singular consequences of its semantic theory as “if a is identical with ‘- Ta’ (T a truth-functor), then ‘- Ta’ is not true - indeed, it is neuter”. On the other hand, Skyrms’ language cannot express such ‘singular consequences’ of its semantic theory as, for instance, that such and such is a model for the language. In this sense, Skyrms’ language has not yet reached the second degree of semantic closure. In this paper I shall construct a language L which in a certain interesting sense approximates the third degree of semantic closure - general semantic closure.’ In order to make clear just what this sense is, a few words of explanation are in order. In general, we shall say that for any sentence s, and for any set Z of sentences, of a language H, Z H-semantically entails Journal Copyright

of Philosophicai 0 1911

Logic 6 (1977) by D. ReideiPublishing

209-221. All Rights Reserved. Company, Dordrecht-Holland.

210

PAUL

VINCENT

SPADE

s (Z]& s) if and only ifs is nonfalse in each interpretation of H in which every sentence in Z is true. (Compare the definition of ‘C-validity’ in [4] , p. 155. If H is a bivalent language, the above definition is equivalent to the more standard one, e.g., in [7], p. 32.) A set Z of H-sentences will be called an H-theory if and only if there is a set Y such that Z = {s: Y IkH s}. If H is a bivalent language, this definition (call it ‘Definition 1’) amounts to that given in [7], p. 40: in effect, that 2 is an H-theory (or H-‘system’) if and only if {s: Z I/-H s>S Z. (Call this ‘Definition 2’.) (We prove this as follows: Since Z H-semantically entails its own members, Z C {s: Z IkH s). Therefore, 2 is an H-theory by Definition 2 if and only if Z = {s: Z Il--H s>.It follows at once that if Z is an H-theory by Definition 2, it is also an H-theory by Definition 1, since Z itself will serve as the required Y. Conversely, if Z is an H-theory by Definition 1, there is a Y that H-semantically entails exactly the members of Z. Now if H is a bivalent language, H-semantic entailment is transitive, so that if Z It-r, s then Y lku s too, and therefore s E Z. Z is thus an H-theory by Definition 2.) Alternatively, we might have defined H-semantic entailment as follows: Z lkH s if and only ifs is true in every H-interpretation in which every member of Z is nonfalse. This definition too is equivalent to that in [7] , p. 32, if H is a bivalent language. On this alternative definition, H-semantic entailment is transitive whether H is a bivalent language or not, so that any H-theory by Definition 1 above is also an H-theory by Definition 2. If H is a bivalent language, then any H-theory by Defmition 2 is also an H-theory by Definition 1. Either of these definitions of H-semantic entailment may be employed in what follows. They will yield different results, recorded in the Corollaries to Theorem 3. Returning now to the language L to be developed below, we shall construct a set-theoretic metalanguage S for L, with a semantics that is quite straightforward and classical - its rules of valuation will be naive ones. S will contain quotation names of all expressions of L. Now although S cannot express its own semantic notions, it is able to express all the semantic notions of L. Let X be a set of S-sentences expressing the semantic notions of L. Then the semantic theory of L as expressed in S will .be the set Y = {s: X IFS s}. We shall define an obvious translation function tr between S and L, a one-one mapping from the expressions of S onto the expressions of L. Where K is a set of S-expressions, we shall also write h(K) for the set of all tr(e) such that e E K. Where X is as immediately above, and ‘A’

GENERAL

SEMANTIC

CLOSL’RE

211

ranges over L-sentences, we shall say that the semantic theo~ oJ’L as expressed in L is the set 2 = ‘$I: rr(X) Ikr, A}. Now we say that L would reach general semantic closure, in the sense of this paper, if and only if the semantic theory of L as expressed in L were exactly the translation of the semantic theory of L as expressed in S - that is, if and only if Z = t,(Y). While general semantic closure in this sense is an interesting notion, nevertheless L will not reach it. For tr will be defined so that where e and e’ are expressions of S, tr(ree’l) = rtr(e)tr(e’)l. h’ow where q(e) is the quotation-name in S of [tie), let us say that e’ is a diagunalization of e if and only if there is a variable y free in e and such that e’ is formed from e by replacing all free occurrences ofy in e by occurrences of s(e). Let us also say that rr(e’) is a diagonalization of tr(e) if and only if e’ is a diagonalization of e. Then since S can express all the semantic notions of L1 there is a sentence r@yl of S expressing ‘y is an expression of L no diagonalization of which is contained in L’s semantic theory as formulated in L’. It follows that for every expression e of .S,r@(e)’ E Y if and only if for every diagonalization e’ of e, tr(e’) 4 2. Hence. if L reached general semantic closure, so that 2 = P(Y), we would have trr8s

(t&l)

(5 2 if and only if r-94 @y)l E Y if and only if tr(r@q(@y)l)

$Z2.

Thus L will not reach general semantic closure. The semantic theory of L as expressed in L will not be exactly the translation of its semantic theory as expressed in S. Nevertheless, L will ‘approximate’ general semantic closure in two senses: (I) On the first definition of semantic entailment, above, [r(Y) is a proper subset of 2 - that is, L’s semantic theory as expressed in L will include the translation of its semantic theory as expressed in S, but will contain more besides; this approximation is thus an ‘outer’ one (see [ I] , pp. 16 i- 162). (2) On the second definition of seman tic entailment, above, Z is a proper subset of P(Y) - L’s semantic theory as formulated in L is a proper subset of the translation of L’s semantic theory as formulated in S; the approximation is thus an ‘inner’ one (ibid.). Note incidentally that L will have a ‘mildly global truth predicate’ (see [4], p. 160), and that quotation names will be ‘safe’ in L in the sense that atomic sentences with quotation names in ‘subject’ position will be always bivalent (see [4] , p. 155). These are attractive features of L (as they are of Skyrms’ language in [4]), but they are in no way required for the main results of this paper.

PAUL

212

VINCENT

SPADE

II

We construct the language L as follows: Primitive vocabulary of L : (1)

logical constants -, &, v, 3, --, (, ), Q (a quotation functor), T (a truth functor), E, =. (Wealso use ‘T’, ‘E’ and ‘=’ as metalinguistic signs in our semiformal discussion.)

(2)

the symbols a, x, ’ (a ‘prime’).

Formation rules of L : An expression of L will be any sequence from the primitive vocabulary ofL.Wheree,,e*,. . . , e, are from the primitive vocabulary of L, we write h,e2,. . . , e, >simply as re, e2 . . . e, 1. Where e is an expression of L, ‘-Q(e)1 is a quotation name of e in L. An individual constant of L will be any expression of L consisting of a followed by a string (possibly nuI1) of n primes. An individual variable of L will be any expression of L consisting ofx followed by a string (possibly null) of n primes. For convenience, we write simply: a,, x, , respectively. The terns t1 , t2, . . . df L will comprise exactly the quotation names, individual constants and individual variables of L. Subscripts will be dropped where convenient. Where B is formed from A by replacing every occurrence of the individual constant ai in A by an occurrence of xi, and ai does not occur within the scope of a quantifier ‘(x/)1 or a quotation functor in A, we write B = AXj/ai. Given this, we have

(2)

r(ti = is an (atomic) sentence of L. r(ti E tj)l is an (atomic) sentence of L.

(3)

r(Tti)l

(4)

IfA and B are sentences of L, then r(-A)l, ‘(4 & B)l, ‘(A v B)l, ‘(A > B)l, and ‘(A = B)l are (molecular) sentences of L.

(5)

IfA is a sentence of L and ai does not occur within the scope of a quantifier ‘(xi)1 or a quotation functor in A, then r((x&4Xj/ar)l is a (generalized) sentence of L.

(1)

tj)l

is an (atomic) sentence of L.

GENERAL

SEMANTIC

213

CLOSURE

The constituents of a sentence A are all the occurrences of sentences in A. The immediate constituents of A are its atomic and generalized constituents not within the scope of a quantifier or quotation functor in A. Orders of sentences of L : We assign to each sentence A of L a unique order as follows: (1)

A is of order one if and only if A is atomic.

(2)

A is of order 2n(n > 1) if and only if A is a molecular sentence each immediate constituent of which is of order less than 2n, and at least one of order 2n - 1.

(3)

A is of order 2n + 1 (n > 1) if and only ifA is a generalized sentence r((x)s”/a)l and B is of order 2n - 1 or 2n.

Interpretations for L: We pick three arbitrary objects r, F, and N to serve as the truth-values ‘true’, ‘false’, and ‘neuter’. Then a model for L is a couple m = (D?l, where D is a set containing at least every expression of L (and no expression of the language S to be constructed in section III below), and containing the objects T, Fz and N, and where f is a denotation-function such that (1)

for each i >, 1,f(ai)

E D;

(2)

f(a,) = T, flaz) = F, f(aJ) = N;

(3)

for every expression e of L, fCae)l)

= e.

An assignment-function with respect to a model m = (D, f > for L is a function ~1such that for each i 2 1, U(Xi) E D. Where m = CD, f> is a model for L and v is an assignment function with respect to m, we shall call the couple (m, v)(briefly: mv) an interpretation for L, and let I,, be a function such that, for each i > 1, I,,(ti) = .r(ti) if ti is a quotation name or individual constant of LI and I,,(tJ = v(ti) if ti is an individual variable of L. Admissible valuations for L: An admissible valuation with respect to an interpretation mv for L is a function V,, from the set of sentences of L into the set of truth-values (T, F, N >, such that

214

PAULVINCENTSPADE

For atomic sentences:

(1) (2)

(3) (4) (5) (6)

= tj)‘) = T if and only if Z,,(tJ = Zmv(tj), and Vma(r(ti= tj)l) = F otherwise. If Z,,(ti) is not a sentence of L, or ti is a quotation name of L, then Vmv~(tiE tj)l) = T if and only if Z,,(ti) is a member Of Z,,(tj), and Vmv(r(tiE tj)l) = F otherwise. If Z,,(ti) is a sentence of L and ti is not a quotation name of L, then V,v(T(tiE tj)l) = N. If Z,,(t) is not a sentence of L, then V,,&(Tt)l) = F. If Z,,(t) is a sentence of L and t is not a quotation name of L, then V,,&(Tt)l) = N. V,&(ti

If A is a sentence of L, then V,,~(TQ(A))l) = T if and only if V,,(A) = T, and V,J(TQ(A))l) = F otherwise.

For molecular sentences A : A classical valuation for a molecular sentence A with respect to an interpretation mv of L is a function C assigning T or F to all occurrences of sentences Br , . . . , B, as immediate constituents ofA, such that, for all i < n, if Bi is true (false) in mv, then C assignsT (respectively, F) to all occurrences Of Bi as immediate constituents of A. Then (7)

V,,(A) = T if and only if A is classically true - i.e., by a classicalbivalent truth-table analysis - under all classical valuations for A with respect to.mv. Vm,(A) = F if and only if A is classically false under all such classicalvaluations. V,,(A) = N otherwise.

For generalized sentences A = r(x)B”/a? (8)

V,,,,(A) = T if and only if, for all mv’ such that v’ is like v except perhaps for V’(X) (hereafter: mv’, v’ = x~), I/mal(BX/a) = T. VmV(A) = F if and only if, for some such mv’, Vmvp(Bx/a) = F. I/,,(A) = N otherwise.

This account of generalized sentences preserves the heuristic view of them as amounting to (infinite) conjunctions. Conjunctions are true on our

GENERAL

SEMASTIC

CLOSURE

215

semantics if and only if all conjuncts are true, and false if and only if at least one conjunct is false - unless the conjunction is a classicaltruth-table contradiction, in which case it is false even if all conjuncts are neuter. But the conjunctions that generalized sentences may heuristically be thought to be are not of that kind. THEOREM 1. There is a unique admissible valuation with respect to each interpretation for L. Proof: Clearly the valuation-rules are such that the worrisome casesarise only with sentences that head an infinite referential chain, whether cyclic or not. (For a discussion of similar chains in a somewhat different context, see [5] .) It is difficult to define a notion of ‘infinite referential chain’ that captures exactly the dangerous cases. But the following notion is broad enough to include all the dangerous cases.Let us define a relation R such that R(a, /3) if and only if for some sentences A and B and some interpretations mu and mu’ of L, (Y= (A, mu), ~3= (B, mv’>,A is of the form r(ti E ti)l or r(Tri)l, r,,(ti) has B as an atomic constituent, and v’ is like ZJexcept perhaps for the variables Xj such that B occurs within the scope of a quantifier ‘(xi)1 in Imv(fi). Then we shall say that A ‘heads an infinite referential chain’ in mu if and only if (A, mv) is R-ungrounded. (See [ I] , [5]. and [6] .) Such infinite referential chains will be problematic only if (but not in general if) for each sentence A and interpretation mu such that c4, mu) is a member of the R-chain, there are proper subsets K and K’ of the set of truth-values {T, F, N) such that it follows ‘from the valuation-rules that the value of A in mu is in K only if the value of B in md is in K’, where (B, mu’) is a subsequent member of the R-chain, but it does not follow from the valuation-rules that for every interpretation m*v* the value ofA in m*v* is in K only if for some interpretation m*v**, where v** is like v* except perhaps for places where P’ may differ from v, the value of B in mt* *** is in K’. In such a case, we shall say that the value of A in mu is ‘interpretation dependent’ on the value of B in mu’. That is, the ‘dependence’ holds only in view of the values assigned to the terms of L by mu. Thus the only problematic infinite R-chains will be such that for any sentence ,4 and interpretation mu such that cli, mu> is a member of the R-chain, the value of A in mv is interpretation dependent on the value of some sentence B in an interpretation mu’ where (B, mv’) is a subsequent member of the R-chain. But such R-chains cannot arise in L. For infinite

216

PAULVINCENTSPADE

R-chains cannot be formed in such a way that each sentence in the chain is of the form r@(A) E tj)l or r(TQJA))l, where A is a sentence of L. There must be a sentence in the chain of the form r(ri E tj)l or ‘(7’tJl where Ti is an individual constant or individual variable of L. But such a sentence will be automatically neuter in the relevant interpretation. The value of that sentence there is in no way interpretation dependent on the value of any sentence B in an interpretation mu’ such that (B, mu’) is a subsequent member of the R-chain. This completes the proof. Note that modus ponens is truth-preserving in L. For suppose V,&(A 3 B)l) = T. Then ‘(A 3 B)l is classically true under all classical valuations for ‘(A 3 B)l with respect to mu. If, in addition, V&A) = T, then B will be true under all classicalvaluations for ‘(A 3 B)l with respect to mv, so that Vm,(B) = T. On the other hand, modus ponens is not always nonfalsehood-preserving in L. For there is an interpretation mu for L such that I,,(a) = ‘(TQ)~. Then I/,,(r(Ta)l) = N, I/,,(~((TQ) 3 (Q + Q(Ta)))l) = N, but V,,(~(Q f Q(Ta))l) = F. Substitution of identicals is not always truth-preserving, or even nonfalsehood-preserving in L. For there is an interpretation mu for L such that I,,(Q) = A and V,,(A) = T. Then V,,r(a = Q(A))l) = T and V,,(r(TQ(A))l) = T, but V,,(J-(TQ)~) = N. Again, there is an interpretation mu for L such that I,,(Q) = A and V,,,,(A) # T. Then V,,&-(Q = Q(A))9 = T and V,,&(Ta)l) = N, but V&J-(TQ(A))l) = F. On the other hand. THEOREM 2. Substitution of identicals in L never leads from a truth to a falsehood. (See the definition of ‘C-validity’ in [4] , p. 155.) Proof Where B is formed from a sentence A of L by replacing zero or more occurrences of lip not within the scope of a quantifier r(ti)l or r(tj)l in A - in case tf or tj is an individual variable of L - or within the scope of a quotation functor in A, by occurrences of tj, we write: B = A’jll ti. Then we prove the Theorem by induction on the order of A. As hypothesis of the induction we assume the Theorem for all sentences of lower order than A. Case 1. A is of order one. If L’&J-(fi = fj)l) = T, then V,,(A) # V&A 9 11ti) only if i/,,(A) = N or V,,(A’j 11ti) = N. But then the Theorem holds trivially.

GENERAL

SEMANTIC

CLOSURE

217

Case 2: A is of order 2n (n > 1). Let V,,&(li = rj)l) = T and I/,,(A) = T, and suppose F’,,&I ‘41 ti) # T. Then one or more sentences Bfi (1tj which occur as an immediate constituent of A’jllti has a truth-value in mu different from the truth-value of B, which occurs as an immediate constituent of A, in mv. By the hypothesis of the induction, it is not the case that one of B and B’i 11ti is true in mv and the other false in mv. Hence one or the other will be neuter in mu. But then there is a classical valuation C for A with respect to mv exactly like some classical vaIuation C’ for A ‘i 11t, with respect to mv except that, for all B’l IICioccurring in A ‘j II ti where B occurs as an immediate constituent ofA, C(B) = C’(B’i 11ti). Since ‘v,,(A) = T, A is classically true under all classical valuations for A with respect to mv, and in particular under C. Thus A’i 11fi is classically true under C’. Hence Vmv(A 9 11ti) # F. Case 3: A = r((x>B”/a)l is of order 2n + 1 (n > 1). ThenA’jIItf = r((~)F/a’rllt~)l. Let v&J-(fi = tj)l) = T/,,(A) = T. Then for all mu’, v’ = xv, v ,,QP,fa) = T. By the hypothesis of the induction, V,,,,~(sX/a’i [Iti) # F. Hence V,,(A ‘1II rJ f F. III

The semantics of L has been described above in semi-formal terms: English supplemented by some notation. But the semantics of L can also be precisely formulated in an appropriate set-theoretic metalanguage. The language S below is such a set-theoretic metalanguage.

Primitive vocabulary of S: (1)

logical constants 1, A, v, +, *, [, ] , q(a quotation functor), W (a truth functor), E (a set-membership sign), Id (an identity sign).

(2)

the symbols b, y, *

Let tr be a translation-function between S and L, a one-one mapping of the expressions of S onto the expressions of L such that (1)

tr(b) = (I, ffi)

(2)

the values of rr for arguments 1, A, U, -+, *, [, ] ,q, W, E, and Id are -, &, V, 3, E, (, ), Q, T, E, and =, respectively.

= x, and w(*) = ‘.

218 (3)

PAUL

VINCENT

SPADE

tr(r ee’l) = r t(e) tr(e’)l .

The formation rules for S are exactly like those of L, i.e., such that an expression e of S is a sentence of S if and only if tr(e) is a sentence of L. Likewise for the other syntactical notions of L defined in section II above. be the terns of S. Letz1,z*,... A model for S is a couple M = (D, g) where D is a set containing at least every expression of L (but no expression of S), and containing the objects T, F, and N, and where g is a denotation-function such that

(1)

for each i> l,g(br)ED;

(2)

g(b,) = T,g@z) = F,g(b,) =N;

(3)

for every expression e of S, grq [e] 1) = P(e).

(Note that the ‘quotation’ name ‘4 [e] 1 of S does not denote the Sexpression e bound by the quotation functor 4, but rather the corresponding L-expression trfe).) An assignment-Jiuzction with respect to a model M = (D, g) for S is a function d such that for each i 2 1, d(yJ ED. Where M = CD,g) is a model for S and d is an assignment-function with respect to M, we shall call the couple CM, d) (briefly: Md) an interpretation for S, and let I,, be a function such that for each i > 1, L,,(zi) = g(zi) if Zi is a quotation name or individual constant of S, and Z,,(Zi) = d(zi) if Zi is an individual variable of S. Clearly, for every S-interpretation Md there is an L-interpretation mv, and vice versa, such that for each i > 1, fM,(Zi) = f,,(tr(Zi))in such a case we shall say that Md and mv are coordinate interpretations. An admissible valuation for S with respect to an interpretation Md is a function VMd from the set of sentences of S into the set of truth-values (T, F}, subject to the following naive or classical rules of valuation: (1)

I’,+&- [Zi Id Zj] 7) = T if and only if fMd(Zi) = IMd(Zj), and V~,(r[~iIdzj]l)= Fotherwise.

(2)

V’,,(r [zi E zj] 1) = T if and only if lMd(Zi) is a member of IMd(Zj), and k’M,(r [Zi E Zj] 1) = F otherwise. (The axiomatization of the notion of set-membership used on the right side of this rule is of course subject to incompleteness results.)

GENERAL

SEMANTIC

CLOSURE

219

(3)

V,,(r [ WZi]l) = T if and only if IMd(Zi) is a sentence of L and, where mv is the L-coordinate of Md, V,,(I,,(zJ) = T, and VM,(r [ WZi] 1) = F otherwise.

(4)

Ifs is a molecular sentence of S, then V,Md(s) = T if and only ifs is true under a classical bivalent truth-table analysis in Md, and V,,,,(s) = F if and only ifs is false under such an analysis.

(5)

Ifs is a generalized sentence r [Iv] ~‘~/b ] 1 of S, then V&s) = T if and only if sly/b is true with respect to every VMd*, d’ = Y d, and V,+,,(s) = Fif and only if sly/b is false with respect to at least one such T/Mdl.

IV

After these preliminaries, we come to the main results of this paper. THEOREM 3. If an S-sentence s is true (false) in an S-interpretation Md then [l(s) is nonfalse (nontrue) in the coordinate L-interpretation mv. Proof. By induction on the order of s. As hypothesis of the induction, we assume that the Theorem holds for all sentences of lower order than s. Then Case 1: s is of order one. Then VIWd(s)= V,,(tr(s)) unless V&W(S)) = N. But in that case P(s) is both nonfalse and nontrue in mv. Case 2: s is of order 2n (n > 1). Let V,,,(s) = T (respectively, V,,(s) = F), and let sl,. . . , si be the immediate constituents of s. By the hypothesis of the induction, for all sh( 1 Sk < i), VrMd(sh) = V&D(S~)) unless V&tr(s,)) = N. Hence there is a classicalvahrarion C for fp(s) with respect t0 T?lV such that for all Sk(1