Convention T and Substitutional Quantification

Convention T and Substitutional QuantificationAuthor(s): John WallaceSource: Noûs, Vol. 5, No. 2 (May, 1971), pp. 199-211Published by: WileyStable URL: http://www.jstor.org/stable/2214732 .

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Convention T and Substitutional

Quant.fication'

JOHN WALLACE

THE ROCKEFELLER UNIVERSITY

It is sometimes suggested or hinted or assumed that there are two workable approaches-which I will tag 'referential' and 'substitutional'2-to the semantics of sentences built on the pattern of first order quantification theory; i.e., that there are two adequate accounts of how the truth value of a sentence of this kind depends on semantic features of its parts. The heart of the referential approach is a recursive definition of satisfaction, a relation which connects sentences open and closed with non-linguistic things, whose characterization makes essential appeal to a range of quantification, and in terms of which truth can be explicitly defined. The substitutional approach, on the other hand, attempts a direct recursive characterization of truth. Truth conditions for closed atomic sentences are given somehow outright. And the key recursion clauses are the following ones for quantifiers:

(E) a sentence that begins with an existential quantifier is true if

1 The writing of this paper was helped along by discussions with Donald Davidson, Dana Scott, Leslie Tharp, Hao Wang, and Scott Wein- stein. Donald Davidson and I have discussed Convention T and substitutional quantification for several years; his views have contributed greatly to my understanding of both; it would not be possible to separate our contribu- tions to this paper. Some points discussed in this paper are discussed also in Donald Davidson, "True to the Facts," The Journal of Philosophy, LXIV (1969): 748-764, and in John Wallace, "On the Frame of Reference," Synthese, XXI (1970): 61-94. The reader may consult the latter paper (footnote 10, pp. 92-93) for references to the literature on substitutional quantification.

2TThese tags echo terminology of Quine's; see his Ontological Relativity and Other Essays (New York: Columbia University Press, 1969), p. 63.

199



200 NOeS

and only if at least one substitution instance of the open sentence obtained by deleting its initial quantifier is true;

(U) a sentence that begins with a universal quantifier is true if and only if every substitution instance of the open sentence obtained by deleting its initial quantifier is true.

Here open sentences are assigned no semantical role; they are merely receptacles for sentence-forming substitutions, constant singular terms being the intended substitutes. The operation of substitution is purely syntactical. Consequently, as Lesniewski pointed out,3 the substitutional approach admits of striking gen- eralizations; even punctuation marks could be substitutes and un- punctuated formulas receptacles.

Some formal differences between the referential and substitutional recursions seem philosophically significant. In making no appeal to a range of quantification the substitutional recursion seems to break the traditionally held connection between truth and extra-linguistic things. And coordinately, in making no appeal to a relation of satisfaction or denotation, it seems to undermine the importance traditionally attached to predication. The substitutional recursion travels lighter; it may go farther. It may be possible to extend it to deal with embeddings, e.g. intensional constructions, which block the satisfactional recursion. And it may then yield a natural explanation of the failure of extensionality principles for intensional contexts.

Convention T is Tarski's proposal for a necessary and sufficient condition of adequacy for accounts of truth; it has been ac- cepted by many logicians and philosophers and is regarded as necessary but too weak to be sufficient by some others.4 It provides at least a clear and precise standpoint from which to evaluate substitutional quantification. When we view substitutional quantification in this perspective an initially surprising phenomenon comes to light: the "naive" substitutional truth theories that one is initially inclined to write down do not satisfy Convention T; so if

8In a conversation with Quine; see Ontological Relativity and Other Essays, p. 106.

4 A. Tarski. "Me Concept of Truth in Formalized Languages," Logic, Semantics, Metamathematics (Oxford: Clarendon Press, 1956): 152-278; esp. pp. 187-188. Some respects in which Convention T is weak are brought out by H. Wang, "Certain Predicates Defined by Induction Schemata," The Journal of Symbolic Logic, XVIII (1953): 49-59; see especially the long footnote on p. 56.



CONVENTION T AND SUBSTITUTIONAL QUANTIFICATION 201

one regards the Convention as providing even a necessary condition of adequacy, these naive theories are not adequate. One can supplement the naive theories, adding new axioms until Convention T is satisfied. But in every known case the new axioms produce a full-blown account of satisfaction at the same time they are providing an adequate account of truth. To take a plausible view of the situation, adequate substitutional truth theories appear to be mere notational variants of referential theories.

The logical situation just sketched is well-known to logicians. This paper explores its philosophical significance.

It will be well to begin by emphasizing some essential features of Convention T. (1) Expressive power of the metalanguage. The metalanguage must have resources to name and to translate every object language sentence. Our metalanguages will contain number theory and its numerals will be regarded via Godel num- bering as names of sentences. It is sufficient for our purposes to by-pass the notion of translation and to suppose simply that the metalanguage contains the object language. (2) The truth paradigm. Given a one-argument open sentence, T(x), of the metalanguage we consider all sentences obtained from the expression 'T(x) if and only if p' by putting for 'p' a closed sentence of the object language and for 'x' the canonical name e.g. Godel numeral of that sentence. These sentences will be referred to as the target conditionals. There are infinitely many of them. (3) Logical consequence. Convention T demands that an adequate theory of truth be consistent and have all target conditionals as logical conse- quences. The notion of logical consequence employed is first order logical consequence, first defined in a syntactical way by Frege and proved equivalent to universal validity by Godel (complete- ness theorem).

As described so far, Convention T would be satisfied by a theory that took all target conditionals as axioms. To guard against this trivialization one may want to add the condition that an adequate theory of truth be finitely axiomatized. For our purposes it is sufficient to meet the difficulty by restricting attention to exten- sions of elementary number theory (with quantifiers relativized when appropriate) in which the induction axiom is extended to new vocabulary but no other axioms are added. In a terminology of Shoenfield's,5 our metatheory is always a P-theory ('P1 for Peano);

5J. R. Shoenfield. Mathematical Logic (Reading, Mass.: Addison-Wes- ley Publishing Company, 1967), p. 235.



202 Noes

indeed, a minrimal P-theory. We shall say that a sentence W of such a theory makes T(x) a truth predicate (or that W is an adequate theory of truth relative to T(x) ) if every target instance of

We-- (T(x)-- p) is valid in the theory (i.e., is a theorem of the theory) and in addition W is consistent with the theory. The idea that an adequate truth theory should be finite is captured by requiring that it be a single sentence.

The intuitive idea behind Convention T is that an adequate theory of truth should determine in a non-trivial way a truth condition for every sentence of the object language. 'Truth condition' is made precise by means of the target conditionals; 'determine' is made precise by the concept of logical consequence. It is note- worthy that as a result of these two moves Convention T does not use the concept of truth, but has rat-her a purely proof-theoretic character. Looking at the situation from the point of view of the deductive-nomological model of explanation we may say that Con- vention T requires that an adequate theory of truth explain all the semantical facts, where these facts are represented by the target conditionals.6

Our version of Convention T casts formal number theory in a key role-for which it is well-suited. It gives us via Godel num- bering a convenient system of structural-descriptive names of object language sentences. And having decided to exploit the numbers in this way, the induction schema of number theory and its P-ex- tensions is strongly motivated: a system of structural description is just a system of description that supports proof by induction of universal theses about all sentences.

It may strike the reader that Convention T is an astonishingly powerful intellectual device. It places us in a position to raise in a precise way deep questions about the concept of truth, yet the conceptual resources it uses are vanishingly meager-basically, just syntax. The success of the device rests on two things. First, a great invention: the predicate calculus. Without the ability pro- vided by the calculus to represent and manipulate in a clear and natural way infinities of sentences the Convention could not get off the ground. Second, Convention T ingeniously exploits a special

6 For the deductive-nomological model see C. G. Hempel and P. Oppen- heim, "Studies in the Logic of Explanation" in C. G. Hempel, Aspects of Sci- entific Explanation (New York: The Free Press, 1965): 245-290.




characteristic of the concept of truth. It is one of those concepts whose role is exhausted by the structure it induces on all concepts. Some other concepts, e.g. causality, have been thought by some philosophers, e.g. Kant, to have a purely structural role. What is striking about truth is that the truth paradigm together with the predicate calculus enables us to describe precisely the order in- duced. We are inclined to say: any concept that induces that structure is truth. Causality so far has not found its paradigm. But satisfaction is in this respect on a par with truth; for the paradigm

(Ax) (x satisfies 'white' if and only if x is white)

is every bit as compelling as

'snow is white' is true if and only if snow is white.

Number theory (with the induction schema) is quite a powerful and complex theory. It contains a significant portion of its own semantics; e.g., the notion of satisfaction for sentences of any given bounded complexity is expressible in number theory itself. In particular, the notion of the denotation or value of a numeral is defined by a primitive recursion and so is expressible in number theory. The formalizations of the defining equations are themselves general sentences which are provable in number theory. And in fact number theory proves

(An) (Em) (vl(m) = n);

i.e., every natural number has a name. This 'vl' will turn up again below, so it will be useful to have in mind a clear reading of 'vl(m) = n': n is the denotation of the numeral with Gbdel number m.7

We turn now to formulation of truth theories for number theory. Let us suppose number theory formulated with a constant, '0', a one-place functional expression, "', for successor, two three- place predicates, 'S' and 'P', for addition and multiplication, and the two-place predicate, '=', of identity. Numerals are formed from '0' followed by primes. We represent the Godel numeral of an expression by overlining the expression. The naive substitutional truth theory adjoins to number theory a new one-place predicate

7 For a thorough study of the logical phenomena alluded to in this re- mark see D. Hilbert and P. Bernays, Grundlagen der Mathematik II (Berlin, Springer-Verlag, 1939): 329-340.



204 NOYS

symbol 'T; the proposed theory W is the conjunction of sentences ( 1)-( 5) :

(1) (Ax)(Ay)(T(conj(x,y)) -T(x) &T(y)) (2) (Ax) (T(neg(x) )=- ~ T(x) ) (3) (Ax) (T(un(x)) (Az) (T(sub(x,z)))) (4) (Ax) (T(ex(x)) (Ez) (T(sub(x,z)))) (5) (Ax) (x is the Godel number of a closed atomic sentencee--

(T(x) t R(x)) )

In (1) and (2) it is intended that the variables 'x' and 'y' range over G6del numbers of closed sentences; in (3) and (4) 'x' ranges over G6del numbers of open sentences with exactly one free variable and 'z' ranges over G6del numbers of numerals. 'conj', 'neg', 'un', 'ex', 'sub', 'is the G6del number of a closed atomic sentence' are arithmetizations of the metalinguistic operations they suggest; e.g., 'conj(x,y)' is to be read: the G6del number of the conjunction of ffie sentence whose G6del number is x with the sentence whose Godel number is y. The symbol 'R' in (5) abbreviates an open sentence of arithmetic that defines the class of G6del numbers of true closed atomic sentences of number theory. This class is recursive and thus there is an arithmetic formula that defines it. More- over, each sentence of t-he class is provable in number theory. (5) thus entails the target conditionals for closed atomic sentences. To sketch half of the argument, suppose p is a true closed atomic sentence; then p is provable, R(p) is provable, and, having proved each side independently, R(p) = p is provable; then from (5) we have T(p) p provable.8

This theory has a ring of truth about it; it correctly determines the truth conditions of atomic sentences and it says the correct thing about each logical operation by which sentences are com- pounded. To see t-hat it does not satisfy Convention T, consider the following attempt to derive from it the sentence

T((Ex)((x= x)ve cn (Ex)(o(xlx))

Assume wit- a view to conditional proof

Tl (Ex) (~ _, =t x) ) )

8 It should be mentioned that a comprehensive treatise in English on the metamathematics of number theory is S. C. Kleene, Introduction to Meta- mathematics (Amsterdam, North-Holland, 1952).




derive by the clause for the existential quantifier

(Ez) (T(sub ( -. (x = x),z) ) ) Here the derivation stalls. The formula we have reached has the logical force of an infinite disjunction

T(,(0 = 0)) v T(~(0'-=0')) v ...

For each disjunct we have

T( ~(An = An)) =-s (An = An)

(where 'an' represents the n + Ist numeral). But we cannot use any of these in our derivation unless the disjunction terminates at a finite stage. Even if all truths of arithmetic are taken as axioms, adjunction of (1)-(5) does not yield an adequate theory of truth.

A dual difficulty arises when the initial quantifier is universal; consider e.g. '(Ax) (x & 0)'. It should be clear that the difficulties here illustrated by examples are perfectly general. Reflect on how one proves that a truth theory satisfies Convention T. One argues inductively on the basis of the complexity of replacements for 'p' in 'T(x) if and only if p'. What the examples bring out is a difficulty in carrying the induction over quantifiers.

It should be clear also that the difficulties we are encounter- ing are not special to number theory. All that is essential is that the nominal substitutes called on in the clauses for quantifiers-e.g. in the case of number theory, the numerals-form a denumerable list no finite segment of which referentially covers the whole-e.g. in number theory we do not have

(Ax)(x = O V X = Ot V . . . v x = O(n))

To some experts on truth theory clause (5) of the proposed substitutional truth theory will seem a bizarre way to deal with truth conditions of atomic sentences. They will consider it more natural to use the notion of denotation already available in number

8a The facts that adjunction of the naive substitutional axioms to Peano arithmetic and even to all truths of arithmetic fails to give an adequate theory of truth, which I have suspected since 1968, so long eluded my attempts at proofs that when this paper was submitted for publication the key clause in the sentence to which this footnote is attached read 'appears not to yield' instead of the present happier 'does not yield'. Recently Leslie Tharp found elegant proofs of the facts based in the case of the weaker fact on Godel's first incompleteness theorem and in the case of the stronger on the existence of non-standard strong models of arithmetic.



206 NOeS

theory to form clauses like the following for each atomic predicate

(6) (Az,)(Az2)(T(sub(x = y, < zi,z2 >)) Vl(Zl) = VI(Z2))

Here 'z1' and 'Z2' range over Godel numbers of numerals; sub is being regarded as an operation on pairs of numerals and two-place open sentences. If we proceed in this way, since we have provable in number theory

(An) (Ez) (vl(z) = n),

we can derive the previously independent

T((Ex)((x=x))) ->(Ex)((x = x)).

Indeed, the sophisticated theory just sketched using the notion of denotation is an adequate theory of truth for number theory.

Some advocates of substitutional quantification have a motive for preferring the devious (5) to the direct (6), namely, the doctrine of the semantical primacy of closed quantifier-free sentences. This doctrine is shared by diverse philosophical viewpoints. It goes hand in hand with an empiricist or operationalist epistemology. Thus it stands out clearly in Quine's theory of radical translation.9 On that theory, evidence that bears on interpretation of an alien language consists of aliens' assents to and dissents from sentences. From this the interpreter extrapolates to alien dispositions to assent and dissent. These dispositions in turn are grist for criteria for recognizing alien truth functions; e.g., an alien is disposed to assent to a conjunction if and only if disposed to assent to each of its parts. Dispositions to assent and dissent yield also criteria for recognizing substitutional quantification; e.g., an alien is disposed to assent to a universal substitutional quantification if and only if disposed to assent to each substitution instance of its embedded open sentence. There are no analogous criteria for recognizing denotation or satisfaction. Thus when building a truth theory to conform to evidence it is natural, from this point of view, to try to match up the atomic sentences without using denotation, i.e., without a clause like (6).

A picture-theory of meaning like Wittgenstein's in the Trac- tatus provides somewhat different motivation for affiirming the semantical primacy of closed quantifier-free sentences and for pre- I - E-

9 See W. V. Quine. Word and Object (Cambridge, Mass., MIT Press, 1960), chapter two.



CONVENTION T AND SUBSTIT'UTIONAL QUANTIFICATION 207

ferring (5) to (6). In Ramsey's essay 'Facts and Propositions" the picture-theory motivation and the pragmatist or operationalist motivation combine to make a substitutional account of the meaning of general sentences doubly desirable.'0

Motivation aside, the hard logical fact is that the naive substitutional truth theory for number theory is not adequate, in Con- vention T's sense. Fuller appreciation of why it fails may be obtained by considering a passage in which Ramsey replies to an objection:

. . . it will be objected that even if I could have a list of everything in the world 'a', 'b', 'z' 'For all x, fx' would still not be equivalent to 'fa. fb . . . fz', but rather to 'fa.fb . . . fz and a, b, ... z are everything'. . . The objection would evidently, have no force if 'a, b . . . z are everything' were, as with suitable defini- tions I think it can be made to be, a tautology; for then it could be left out without altering the meaning.""1

The analogue in our theories of Ramsey's 'a,b . . . z are everything' is the sentence '(Ax) (Ey) (vl(y) = x)', i.e., 'corresponding to each natural number there is a Godel number of a numeral which de- notes it'. And this sentence is tautologous in Peano arithmetic, in the sense of being provable (as we have remarked more than once before). Ramsey's remarks suggest that under these circumstances the substitutional account of the meaning of general sentences is acceptable. But viewed from the standpoint of Convention T Ram- sey's remarks overlook the role of the clauses of the truth theory that deal with atomic sentences. 'Everything has a name' is tautologous, if you like; at least, in number theory it is asserted. But for it to contribute to the derivation of target conditionals that invoke quantifiers the notion of denotation must be used in other parts of the truth theory. It will be seen from this that Convention T has a certain holistic character; the adequacy of the substitutional approach to truth theory cannot be judged from the quantifier clauses alone.

What we are calling the sophisticated substitutional truth theory is close to a truth theory for number theory given by Hilbert

10 F. P. Ramsey. "Facts and Propositions," in The Foundations of Mathematics and Other Logical Essays (London: Kegan Paul, Trench, Trub- ner & Co., 1931): 138-155.

11 Ibid., p. 154. Russell perhaps has in mind a similar objection in his introduction to the Tractatus; see L. Wittgenstein, Tractatus Logico-Philo- sophicus, trans. D. F. Pears and B. F. McGuinness (New York: Humanities Press, 1961), p. xxi.



208 NODS

and Bernays.12 Their theory incorporates additional analysis sufficient to show that Peano number theory contains each finite ap- proximation (measured by length of prenex quantifier prefix) to its own truth predicate. Both theories make essential use of the notion of denotation of variable-free singular terms, which is already contained in number theory. In the context of a number theory weaker than Peano arithmetic, our theory and the Hilbert-Bernays theory might fail to satisfy Convention T. Indeed, if the number theory is R. Robinson's system Q the failure just contemplated occurs.13 Each correct numerical instance of

vl(m) = n

is provable in Q, but not the general statement, needed in deriva- tions of target conditionals, that every number has a name.

The sophisticated substitutional truth theory and the Hilbert- Bemays truth theory work; the naive theory does not. Do the good theories have the advantages originally foreseen for substitutional truth theories? I think not. Here the key point is that the sophisticated theory of truth, which satisfies Convention T, is also a theory of satisfaction. That is, the theory throws up a predicate-defined in terms of ', 'vl', and 'sub'-for which all instances of the satisfaction paradigm are provable. Moreover, the logical operations prov- ably perform in the expected way with respect to the predicate; e.g. any conjunction of open sentences is satisfied by a sequence of objects if and only if both of its parts are satisfied by the sequence. Remember we said that any concept that conforms to the satisfaction paradigm is satisfaction; thus the sophisticated substitutional truth theory is supporting a recursion on the concept of satisfaction. Advantages that were supposed to attach to substitutional quantification by virtue of its "lighter," less predication-committed recursion are all lost. The range for quantified variables and the bias toward extensionality are present in the derived recursion on satisfaction. Nor can I see that the fact that the recursion is derived and not axiomatic makes a significant difference. The point is that informal arguments involving the notion of satisfaction can be carried out in the formal semantics; if some of these arguments show that quantifiers have extra-linguistic ontological commitments or that what appear in surface grammar to be intensional constructions

12 D. Hilbert and P. Bernays. Grundlagen der Mathematik II: 329-340. 13 For presentation and discussion of Robinson's arithmetic see Kleene,

Introduction to Metamathematics, pp. 197, 241-245.




either block or are rendered extensional by the recursion on satisfaction-and such arguments motivated the move to substitutional quantification in the first place-these arguments will be crystallized in the semantical theory.

For example, if the sentence 'Jones believes that a* is an airedale' is analysed as containing the open sentence 'Jones believes that x is an airedale' we can use the notion of satisfaction to make the following informal argument for the referential transparency of the embedded singular term position. A general principle about satisfaction is:

(Ay) (y satisfies 'Jones believes that x is an airedale' if and only if Jones believes that y is an airedale).

In particular,

a* satisfies 'Jones believes that x is an airedale' if and only if Jones believes that a* is an airedale.

But 'satisfies' is an extensional predicate, so if

a* = b*

we have:

a* satisfies 'Jones believes that x is an airedale' if and only if b* satisfies 'Jones believes that x is an airedale'.

Whence by one more application of the general principle,

Jones believes that a* is an airedale if and only if Jones believes that b* is an airedale.

In a word, the extensionality of satisfaction is contagious. It is this sort of argument that will be crystallized in the formal semantics. Of course, the particular argument sketched applies only if 'believes' is treated as a sentential connective in the metalanguage.14

The possibility remains, of course, that Convention T demands too much of truth theories. On further study there may emerge a good test for truth theories that is less stringent than Convention Ts and that is passed by the naive substitutional truth theory. A general justification or "deduction" of Convention T is beyond the

14 The argument is closely related to ones given by Quine; the formulation that makes explicit use of the notion of satisfaction was called to my attention by Sandra Lynne Peterson, The Masker Paradox (unpublished disser- tation, Princeton, 1969): 61-71.



210 NOi)S

scope of the paper; it would require elucidation of a philosophical point of view which leads to questions about truth that can be illuminated by nothing weaker than Convention T. Still some affec- tion for Convention T may be generated by considering a familiar controversy about substitutional quantification. Everything is made to turn on whether the naive substitutional truth theory is true; i.e., from a referential point of view, on whether every object in the range of the variables has in fact a name. Thus it may be claimed on the one side that since the real numbers are non-denumerable, the substitutional truth theory for analysis is false. To which a reply is that the Lowenheim-Skolem theorem shows all but a denumerable subclass of real numbers to be semantically superfluous for truth conditions of sentences of analysis and thus shows that a naive substitutional truth theory may be true even there.15

What makes this controversy resist resolution, I think, is that questions of truth and questions of ontology are not ultimately separable.'6 But the decision whether a naive substitutional truth theory is true e.g. for analysis or for the ordinary physical object language seems to require determination, independent of an account of truth, of the ontology of the object language. And this seems difficult; really, it is impossible.

From this impasse we may turn with relief to Convention T and see new merit in its use of the concept of logical consequence. As is made vivid by the definition of validity in terms of models (or representations), the concept of logical consequence ties together, without tying down, the truth-idea and the ontology-idea. It gives us a point at infinity where truth and ontology have their characteristic interdependence and where the effects of this interdependence can be determined without determination of either truth or ontology alone.

In conclusion, Convention T provides a vantage point from which to distinguish the questions: (1) Is the substitutional approach to the semantics of quantificational languages adequate? (2) Is it significantly different from the referential approach? (3) Are the clauses for quantifiers in the naive substitutional theories,

15 This sketch of a controversy is suggested by Quine's discussions of substitutional quantification in the second half of "Ontological Relativity;" see Ontological Relativity and Other Essays: 58-68. I have also been helped in this and the following two paragraphs by reading an unpublished paper of Quine's ("Truth and Disquotation") and by correspondence with Quine.

16 As Quine has said: "We do not learn first what to talk about and then what to say about it." Word and Object, p. 16.




e.g. for large portions of ordinary language, true? i.e., from a referential point of view, does everything have a name? To (1) and (2) decisive answers are forthcoming. Naive substitutional truth theories, which preserve aspects of the substitutional approach that have excited philosophers, are not adequate in Convention T's sense. Theories of a type first given by Hilbert and Bernays retain some superficial features of the naive substitutional theories but make essential use of the notion of denotation and in fact are adequate theories not only of truth but of satisfaction. On the other hand, the question whether the naive substitutional truth theories for ordinary language are true remains open, pending clarification of what it means and of what kind of argument could settle it.



Convention T and Substitutional Quantification

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