Peacocke What is a Logical Constant

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What is a Logical Constant?Author(s): Christopher PeacockeSource: The Journal of Philosophy, Vol. 73, No. 9 (May 6, 1976), pp. 221-240Published by: Journal of Philosophy, Inc.Stable URL: http://www.jstor.org/stable/2025420 .Accessed: 27/09/2013 05:56

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THE JOURNAL OF PHILOSOPHY VOLUME LXXIII, NO. 9, MAY 6, i976

WHAT IS A LOGICAL CONSTANT? * No objective grounds are known to me which permit us to draw a sharp boundary between [logical and nonlogical expressions].

(Alfred Tarski, 1935) M t Adz Y aim is to give a criterion for an expression to be a logical

constant in a given language, a criterion that does not reduce to a list and that involves as small an element as

possible of arbitrary stipulation. The plan is then to explain by that criterion why other criteria for the logical constants seem plausible where they do and to account for some properties of the logical constants in terms of the proposal about what it is to be such an expression. The main criterion to be discussed applies to expressions for which the appropriate specification of meaning is an account of their contribution to truth conditions of sentences; but we shall see toward the end how this criterion is an instance of a theory of wider applicability.

I The prospects seem very poor for any theory that attempts to isolate the logical constants via properties of whole sentences or arguments in which they occur and which at the same time attempts thereby to pick out what is fundamentally distinctive of those constants. Such theories naturally start from the rough idea that a is a logical constant just in case there are sentences containing a that remain true under uniform substitutions for their parts other than

* I am indebted to Michael Dummett, Gareth Evans, Colin McGinn, and Marie McGinn for comments on earlier drafts.

For brevity I have had to omit certain generalizations and refinements of the criterion of logical-constanthood presented here which are needed in order to include such devices as substitutional quantifiers, higher-order quantifiers, and connectives linking feature-placing expressions ('cold and foggy'), and to exclude, under the objectual interpretation of first-order arithmetic, arithmetical functors and predicates. The additions required are all natural, given the intuitive motivation behind the criterion that is stated here.

My use of corner quotes essentially follows that of Quine in ?6 of Mathematical Logic (Cambridge, Mass.: Harvard, 1940).

22I


222 THE JOURNAL OF PHILOSOPHY

a and certain other expressions. The difficulty plainly lies in the phrase 'certain other expressions' and specification of the legitimate substitutions. On the one hand we have to prevent the truth of all instances of 'Everything that is F and is blue is colored', 'If a knows that p, then p', 'If a prevented it from being the case that p, then -p' from making all of 'blue', 'colored', 'knows', and 'prevents' into logical constants; on the other hand, we do not want it to be relevant that instances of the schema '((p v q) D r) D (p D r)' can be turned false by substitution of '&' for 'v'. What are the general principles according to which such unintended cases are excluded? It would be circular at this point to say that for the purposes of substitution "phrases are classified together if systematic substitution of one for the other does not affect the validity of arguments" (thus, Davidson and Harman'). We do not have a conception of the validity of arguments in advance of a selection of the logical constants; we have only a notion of truth-preservingness and of an argument's necessarily being truth-preserving. These last two would not, by themselves, yield exactly what we wanted. For instance, this argument

John drank some water .,. John drank some H20

is such that necessarily, if its premise is true, its conclusion is true. But substituting some other one-place predicate in place of occur- rences of 'water' plainly does not preserve this property. Hence, under such a proposal based on substitution, we will not obtain the traditional category of simple and complex one-place predicates we want. Let us suppose, however, that by a series of stipulations on legitimate substitutions a relatively traditional stock of constants were isolated. Then three serious questions would arise. First, what would be the interest of the concept thus defined? Is there anything special about this notion that distinguishes it from others at which we would have arrived by making the stipulations slightly differ- ently? Second, the stipulations, in allowing substitutions for 'blue', 'prevents', and the rest but not for the logical constants, must appeal, if they are to be adequately general, to features differentiat- ing those expressions from the constants. But if such features can be specified, why not use them directly in the criterion and cut out the detour through sentences or arguments? The third question is about the notion of validity immediately generated by such an account, the notion of truth of a sentence s under all uniform substitutions for simple components of s other than the logical constants.

1 Donald Davidson and Gilbert Harman, The Logic of Grammar (Encino, Calif.: Dickenson, 1975), p. 2.


WHAT IS A LOGICAL CONSTANT? 223

It is a familiar point-one which can be traced back at least to Tarski in 19352-that, in a rather weak language, in particular one that cannot express elementary number theory, certain sentences may be counted as valid on this definition which are not instances of valid schemata of quantification theory; for it may happen that there are not, in the language, open or closed sentences available which when appropriately substituted will yield a falsehood. Naturally such sentences are not model-theoretically valid; but, on these accounts, that notion of validity, unlike the substitutional definition, stands in no very intimate connection with the theory of what it is to be a logical constant.

II Instead of pursuing criticisms of this and other criteria that have been suggested, I shall move straight to an initial statement of the main criterion I have to propose. This criterion is relative to a truth theory and syntax for the object language (OL) in question; but this relativity will be suppressed for brevity of statement. The proposal is then roughly this:

a is a logical constant if a is noncomplex and, for any expressions B1, ..., On on which a operates to form expression a(#,, . ..,n), given knowledge of which sequences satisfy each of #1, ..., f3n and of the satisfaction condition of expressions of the form a(,yi . .., Yn), one can know a priori which sequences satisfy aQ(31, . /3On), in particular without knowing the properties and relations of the objects in the sequences.

I assume the truth theories are formulated using finite sequences of objects in the satisfaction relation. Thus, given a knowledge of which sequences satisfy A and knowledge that any sequence satisfies r-A1 if it does not satisfy A, one can know a priori which sequences satisfy rFA 1, viz., just those which fail to satisfy A; so '-' is a logical constant by the criterion. So too is existential quantification; if one knows which sequences satisfy A and knows the satisfaction condition of existentially quantified sentences, one can know without further information which satisfy

_3xiA 1, viz., all those differing from a satisfier of A in at most the ith place. There is of course a radical difference between sentential and quantificational constants here; for a sentential connective a, one can know whether s satisfies a

.(.., An) given only knowledge of the satisfaction clause for a and of whether s itself satisfies 31, . . ., An One does not need to consider sequences other than s. In the case of quantificational constants, this is not so; whether s satisfies

2 "On the Concept of Logical Consequence," in Logic, Semantics and Meta- mathematics (New York: Oxford, 1956).



r3xiA 1 depends in part on whether sequences other than s satisfy A. (This may be taken as an argument that the sentential connectives are logical constants in a stronger sense than are the quantifiers.)

It may not at first be clear how the criterion is to be extended to expressions the truth theories for whose languages are not of standard first-order form; temporal operators disquotationally treated, as, for instance, by Gareth Evans,3 provide an example. But in fact the extension is forced in a particular way by the admission of the existential and universal quantifiers as logical constants. In the move from truth to truth-relative-to-a-sequence (satisfaction) we allowed as antecedent knowledge in the application of the criterion to a (s1, ..., ,B3) not simply knowledge of whether a given sequence satisfied the f3i, but also knowledge for every sequence s of whether s satisfied hi. Now similarly, if we are to avoid arbitrariness, in testing whether 'in the past' ('iP') is a logical constant, we must allow antecedently given knowledge not only of whether s now satisfies A, but of whether it did so in the past and will do so in the future. It is plain that '(p' is then a logical constant by the criterion: given knowledge that it is now the case that s satisfies rFGA l if in the past s satisfied A and that it is (is not) the case that in the past s satisfied A, one can infer a priori that s satisfies (does not satisfy) FWPA -. If we allowed only knowledge of whether s now satisfies A, 'P' would not be counted a logical constant; but there would then be an un- motivated asymmetry of treatment between the quantifiers and the temporal operators. It would be an error to try to motivate such asymmetry by the thought that, given only that s satisfies A (or given that it does not), it is an a posteriori matter whether s satisfies ryA 1; for equally, given only that s does not satisfy A and no information about other sequences, it is (in general) an a posteriori matter whether s satisfies r3xiA. (The parallel can be made completely precise by comparing the existential quantifier with a temporal operator "either now or in the past or in the future.")4

3"Semantics for Tense and Time Reference in Natural Languages," unpublished manuscript.

4 The claims of this paragraph are subject to this important qualification. If, without ultimately invoking an ontology of times, we can make out a distinction between

A time such that John (now) believes that Fa at that time, is in the past and

John (now) believes that: in the past, Fa then '( will not be a logical constant. For, crudely speaking, John may not be presented with the time in question in such a way that the mode of presentation ensures that John recognizes that it is in the past. For precisely this reason, the complex quantifier '3t(t is before now & . t... )' is not a logical constant. There is a quite general issue of interest here: must expressions that interact with "believers" in a way that generates a certain kind of scope distinction necessarily



These remarks may encourage the view that any iterative expression given a homophonic axiom in the truth theory will be a logical constant. This is not so: extensional predicate modifiers provide a counterexample. Given a knowledge of which objects and sequences satisfy a (possibly complex) predicate 0 and a knowledge of the satisfaction clause for 'large,' one cannot know a priori which objects and sequences SATISFY5 rlarge q5: one needs also to know whether a certain object is large for a satisfier of q relative to a given sequence. Here knowledge, for every object x and sequence s, of whether x relative to s satisfies the embedded expression 0, in the presence of knowledge too of the satisfaction clause for 'large', is not sufficient for the required a priori inference. This example also serves incidentally to show that the enterprise of giving a criterion for logical-constant-hood is quite distinct from the project of characterizing a structurally valid inference, on which Gareth Evans has an interesting proposal.6 Indeed, if we accept Evans's suggestion, this should already have been obvious from the fact that, on the model-theoretic account of validity that is properly associated with the logical constants, the interpretation of the logical constants is held fixed through the models.

Not every expression handled by a truth theory is given a satisfaction condition; some, the terms, are assigned objects by an evaluation function, relative to a given sequence. (For brevity, I shall speak of a sequence assigning an object to an expression.) If we are not to exclude such expressions from the class of logical constants by decree, we must broaden the formulation of the criterion. One natural way to do so is as follows, where 'r/a1, .. ., an' specifies the syntactic category of an expression which when applied to n expressions of categories al, ..., an yields an expression of category r:

a is a logical constant iff a is noncomplex and, where the syntactic category of a is r/ai, ..., oa, for any expressions 13i, ..., 1% of

have a semantical treatment that invokes quantification over, or assignment of, objects of some kind?

6 I presuppose a truth theory for such modifiers along the lines of the following axioms and rule, which suffice for iteration of modifiers, but allow them to operate only upon atomic predicates in the innermost part of an abstract:

sats (s, rt1) -SATS (s*t, s, ) SATS(X, so rlarge 5-- large Xy(SATS(y, S, 40))X

and finitely many instances of SATS (x, S, r~xi (Fxi )) =- sats (s (i/x), rFxil)

where 'F' is replaced by an atomic predicate; plus the rule of inference: From Xv(A). and Vv(A = B), one can infer V(B) .

6 "Logical Form and Semantic Structure," in Evans and J. McDowell, eds., Truth and Meaning: Essays in Semantics (New York: Oxford, 1976).



categories 0r1, O.n, respectively, given knowledge of (a) which sequences satisfy those /3i which have satisfaction con-

ditions, and of (b) which object each sequence assigns to those f3i which are

input to the assignment function, and of (c) the satisfaction condition or assignment clause for expressions

of the form a(,yi, . . ., en) one can know a priori which sequences satisfy the expression a(f3,, .. ., #) of category T, or which object any given sequence assigns to a (fli, . . ., fln), in particular without knowing the properties and relations of the objects in the sequences.

"Knowing which object" is to be explained here in terms of the applicability of an attribution of belief that uses a proper name within the belief context. This can be made clear by an example. We will not count the functor "the father of t" as a logical constant merely because, if one knows that s assigns M\lozart to 3, knows that s assigns to any term rthe father of t1 the father of what it assigns to t, one can infer a priori that s assigns the father of Mozart to rthe father of p31. What is lacking is a proper name c of some language such that we can truly say in that language that the believer can infer that s assigns c to Fthe father of ,B1. (Similar remarks apply to the phrase 'knowing which sequences'.)

III

We need to be careful about the knowledge permissible in the criterion: in particular, there are ways in which someone could know which sequences satisfy ,13, . . ., I3n, and the satisfaction clause for a, yet fail to be in a position to know which sequences satisfy a (31 ..., i3n). There are two main reasons for this. First, a being might know of every sequence whether or not it satisfied ,1, .. ., n without knowing that all the sequences he knows about are all the sequences there are; so that, for instance, he could not infer from the fact that all the sequences of which he knew failed to satisfy A that all of them failed to satisfy r3x2A 1. So let us suppose the being to know that all the sequences he knows of are all the sequences (of the relevant kind) that there are.

The second reason is that someone could have the knowledge we have so far attributed and yet not know whether the ith element of sequence s is identical with the ith element of s', and so not know whether s differed from s' in at most the jth place ("s Ha s"'), for any j. Without this knowledge, one cannot know which sequences satisfy r3xiAl for given i, even if one knows which sequences satisfy A. knows the satisfaction condition for the existential



quantifier, and knows that one knows of every sequence there is. It is not an end of the difficulties that arise here simply to stipulate that this gap too is closed. For since the move from

X knows of y that it's F and x knows of y that it's G to

X knows of y that it's F and G is invalid (since x may not realize that it is one and the same object that is F and is G) it is not valid to move from

x knows that s' ;i s and knows that s' satisfies A (where 's', s" are genuine variables) to

x knows that: s' i s and s' satisfies A Yet that is the knowledge that must be attainable if one is to be able to know which sequences satisfy r3xiA 1.

One way that appears adequate to meet these points is to require that, if the length of sequence s is n and that of s' is m and if the imagined being knows that s does (does not) satisfy A and that s' does (does not) satisfy A, then, for some language L that contains names for every element of s and for every element of s', where L contains no ambiguous names and coincides with English on predicates, variables, and connectives, it is true for some names cl, ..., Cn+m to say in L that the imagined person knows that (cl, ..., c) does (does not) satisfy A and that (Cn+11 * * ?i Cn+m) does (does not) satisfy A; where ci(1 ( i ( n) names in L the ith element of s('si') and Cn+j (1 < j ( m) names in L the jth element of s'('s'j') and where, finally, if si= s'i, then Ci = Cn+i. That is, if the ith elements of s and s' are identical, the person is in a position to infer that they are; he can also make the "conjunction" inference noted above. (Note that it does not follow from all this that, for any given sequence s, he knows whether si = sj, where i 5 j; this matters later.) In fact, even this condition must be enriched by the addition that the believer knows that he meets it. For nothing in the condition itself ensures that if si # s'i, then he knows that si # s'i. Without this, we still will not capture the quantifiers as logical constants: for suppose the believer

(i) knows that any sequence s' other than s" that differs from s in at most the fourth place satisfies A (where, say, A contains the second and fourth variables free); and

(ii) knows that s" does not satisfy A; and (iii) does not know whether S"2 = S2.

Then he will not be in a position to know whether or not s satisfies rFVx4A 1, unless we make this additional stipulation.



Instead of speaking of imaginary knowers, we could have written the criterion for logical-constanthood in terms of what could be deduced by a priori principles from premises specifying which sequences satisfied the expressions ,31, ..., n It is then even more evident that qualifications parallel to those just considered are still needed in such a formulation; for the quantificational constants, one needs in some form the premise that every relevant kind of sequence has been specified, and, for all sequences s and s', the premises must specify whether s and s' agree on their ith elements (for all i), either directly or by what can be deduced from the kind of specification given.

We cannot avoid the fact that, when the OL quantifiers range over an infinite domain, our imaginary knower is supposed to have knowledge of infinitely many sequences (albeit each of them finite in length). Even if we suppose the knower not to have relational beliefs of every sequence and instead specify his beliefs simply by description, he will be required to know infinitely many logically un- related truths; and anyway such descriptive knowledge will not help when the range of OL quantifiers is uncountable. Those who find it difficult to believe that any distinctions of importance to us between logical and nonlogical vocabulary could turn on the doubtfully intelligible conception of such a being are in any case likely to interpret OL quantification in an intuitionistic rather than a classical vein; those with such beliefs are referred to the adaptation of the criterion to the intuitionistic connectives which is suggested later. It is not at all clear that doubts on this score can be properly met by appealing to the formulation in terms of deducibility. For in that case the premises must either make use of a language with infinitely many primitive proper names, a language of which no finite being could ever have full mastery; or, if the sequences are specified by description, the whole set of initial premises embraces a body of truths that could not as a whole ever be known or even believed by a single finite being.

IV Before we turn to some of the consequences of the criterion, we can acquire a firmer grasp of the way in which it is intended to operate by considering a certain kind of objection. Consider a two-place connective ... translated into English by the complex phrase

.. . &9 ) v - the earth moves It may be said that $ unwantedly fulfills our criterion for being a logical constant; for, regardless of whether the earth moves, if $ has



such satisfaction conditions, one can know a priori that a sequence will satisfy rA $ B1 if it satisfies A and satisfies B.

This objection rests on a misunderstanding of the criterion. It is intended that one be able to know whether s satisfies a (fl, . . ., AO) a priori given knowledge of the satisfaction condition for a and of which sequences satisfy f1, . . ./3kng regardless of the distribution of truth values over statements of the form 's satisfies pi'. $ fails this test because, if every sequence fails to satisfy A and also fails to satisfy B, one cannot know whether a given sequence satisfies FA $ B1 without knowing whether or not the earth moves-an a posteriori question.

The objector may naturally say at this point that this reply will be unavailable if he alters the translation of $ to

(... &_) v (7 + 5 = 12) Now we need to distinguish two cases. If $ is, relative to the given syntax and truth theory for its language, complex, then it fails the criterion anyway. (There is, no doubt, no objection to defining complex logical constants, but it is natural to require of them that they be built up using only logical vocabulary.) Indeed for this subcase we could have made this remark the first time round. On the other hand, suppose $ is, relative to the given syntax and truth theory, simple. But then there can be no possible warrant for not interpreting $ as conjunction tout court. Speakers of the language containing $ will have no dispositions beyond those appropriate to interpreting it as conjunction and it will be arbitrary to add any one a priori false disjunction to its translation rather than some other. It is hard to find an objection of this style that is not subject to a similar dilemma.

v

Some consequences of the criterion suggested are worth noting briefly. An obvious consequence is that expressions meeting it will be topic- neutral. For the criterion speaks of any expressions /1, .. ., A3n to which a is applied, regardless of what they speak of or are true of, and this in combination with the requirement of a priori knowledge ensures that there are no particular features, objects, properties, or relations on which the satisfaction (and so truth) of a (/3, .. ., 3n) depends, other than those already involved in the satisfaction conditions of /1 ,... ., In. This kind of topic neutrality is a necessary and sufficient condition for an expression to be a logical constant. This is not inconsistent with the definite earlier admission of 'At some time' (considered as an operator) as a logical constant; for the satisfaction relation for the expressions on which 'At some time'



operates requires temporal specification (as a feature), and so in this case temporal features are involved in a full specification of satisfaction for the As.

A second slightly less obvious consequence is that the notion of validity naturally generated by this account is the standard model- theoretic conception of validity, rather than any "substitutional" definition. For since in model theory one considers arbitrary domains and arbitrary assignments of sets of n-tuples of objects to n-place atomic predicates, but holds constant the interpretation of the logical constants, the model-theoretically valid sentences will be precisely those sentences which one can know a priori to be satisfied (absolutely) by all sequences (that is, true) regardless of which sequences satisfy the atomic nonlogical predicates, given knowledge of the satisfaction conditions of the logical constants (on this criterion).7 Hence there is no danger of counting as valid certain instances of invalid first-order quantificational schemata, even if the OL does not contain elementary number theory; nor does the possibility that the OL quantifiers in fact range over a finite domain produce the complications it entails for any substitutional conception.'

A third feature of the criterion is that it can explain the plausibility, where it is plausible, of this criterion9 (let us label it " (K)"): an expression of a language L is a logical constant just in case its meaning is wholly determined by the introduction and elimination

I Of course, when knowledge of the satisfaction and assignment conditions of other expressions is allowed too, then if one can know a priori that all bachelors are unmarried, not every expression one can know a priori to be satisfied by all sequences will be model-theoretically valid.

8 If there are just n objects in the range of the object-language quantifiers, then the familiar first-order sentences using identity that say there are at least m objects (m < n) cannot be turned false by substitution for their schematic letters, provided that identity is a logical constant; for they contain no schematic letters. Somewhat similar complications arise even if we do not count identity as a logical constant. If there are at most three objects in the range of the object-language quantifiers, then every instance of the schema

Vx3 yRxy D 3u3v3wVx (Rxu v Rxv V Rxw) is true.

On the philosophical motivation for model theory just given, it would be arbitrary to exclude interpretations with empty domains. If we want a justification for the standard practice of excluding them from consideration in the definition of validity, we must turn to some rationale such as that offered by W. V. Quine in "Meaning and Existential Inference" [From a Logical Point of View, 2nd ed., (New York: Harper & Row, 1961)]; this rationale is pragmatic in the non- derogatory sense that it does not follow from considerations about the nature of logical constants alone.

9 See, for instance, W. Kneale, "The Province of Logic," in H. D. Lewis, ed., Contemporary British Philosophy, Third Series (London: Allen & Unwin, 1956), pp. 254/5.


WHAT IS A LOGICAL CONSTANT? 23I

(or right and left introduction) rules for it in a sequent calculus for expressions of L. One aspect of (K) worth noting is that, if it is extensionally correct, one could hardly expect it to be fundamental. For a sequent calculus is intended to include only rules taking one from sequents with a certain semantic property (if all the antecedent formulas are true, so is at least one of the succedent formulas) to other sequents with that property. So, if what are intuitively central cases of logical constants satisfy (K), we should expect them to do so in virtue of some semantic property they possess; that is, in virtue of their meeting some requirement that speaks of their satisfaction conditions. That (K) is at least for the standard first- order constants implied by our criterion seems evident when we observe that introduction rules will be justified by sufficient conditions for the satisfaction of a (flu, . . ., An,) on the basis of the satisfaction or otherwise of /1, .. ., A3; that elimination rules will be justified by such necessary conditions; that if the rules fully capture these necessary and sufficient conditions, no aspect of the meaning of a will have been omitted; and that the a priori and universal ("for any /3, ..., Ant") elements of the criterion ensure what was surely intended to be part of (K), the a priori and universal character of the resulting introduction and elimination rules for anyone who knows the satisfaction condition for the constant in question. The qualification that "the rules fully capture those necessary and sufficient conditions" is needed. We shall later note some strong reasons for counting 'o' a logical constant. But the introduction and elimination rules for 'si' in a sequent calculus do not fully capture the necessary and sufficient conditions in question; for they are consistent with the supposition that rFLA 1, where A does not contain 'o', is true only if A is a logical truth.

I wish at this point to interpose a remark on the significance of cut-elimination theorems. In a paper entitled "What Is Logic?" in which he proposes a criterion similar to (K), Ian Hacking'0 writes that "Cut elimination is not something peripheral to logic. It is the guarantee that we have a system of logic at all. For deducibility is transitive" (9). It is clear, however, that the transitivity of deducibility in a given system could be ensured by explicit inclusion of a cut rule. The important question naturally arising is whether or not one can justify the stronger requirement that the cut rule actually be eliminable in the sense that no sequent ceases to be derivable when cut is omitted from the rules. I wish to argue at least that one property of a system that cut elimination implies, namely that every derivable sequent a has a proof in which occur only sub-

10 Mimeo of notes for lectures at the London School of Economics, 1971.



formulas of formulas in a (that is, has a proof with the subformula property), is indeed a requirement for a system to be logical. For suppose we have some sequent a all and only the logical constants of which occur in the set S. Then, if af is to be derivable, we shall want it to be derivable on the basis solely of rules relating to the constants in S and structural rules on the deducibility relation; we shall not want to make essential use of any rules on logical constants not in S. If o- is to be derivable, then for every assignment of truth values making all its antecedent formulas true, at least one succedent formula is made true. This will hold in virtue of the truth tables for the connectives in S. But the truth tables for any connectives are recoverable uniquely from proper introduction and elimination (or, better, antecedent and succedent introduction) rules for it, given the intended interpretation of the symbol linking antecedent with succedent.'1 (The same applies to the satisfaction conditions of the quantifiers.) So, if af is derivable, we should expect there to be some derivation of it using only rules for the constants in S plus structural rules. (Of course we should not in general expect the derivation to be effectively constructible from o- itself.) If there is no such derivation and if T- is plainly valid in the intended interpretation of the expressions in S, then we should conclude that either (i) the left and right introduction rules need strengthening, or (ii) the structural rules are incomplete, or (iii) at least one of the constants in S cannot have its whole sense determined by the left and right introduction rules in a sequent calculus. (This clause protects the claim from the charge that it attempts to settle serious technical questions by philosophical argument.)

VI The connection of our criterion with (K) might prompt the suggestion that the criterion could be simplified by simply requiring that for a to be a logical constant it must be a truth function, in the

11 Consider, for instance, the following left- and right-introduction rules for' D ' (I follow Kleene's notation):

rot A *B, 0 r A DAB,0 Pr 0,A B,O 2 Or, 2, A D B m= 0,

The rules of inference are meant to preserve the property of a sequent of being such that if all its antecedent formulae are true, so is at least one succedent formula. From this and =D we conclude that, when either A is false or B is true, than A D B is true; for in the case in which all in r are true and all in 0 false, only then will the rule preserve the required property. From D== we can conclude that, if A is true and B false, A D B is false; for consider the case in which all in r and all in 2 are true, and all in 0 and all in Q are false. This has now settled the truth value of A D B for every line of its truth table.



sentential case, or an analogue of this when satisfaction is invoked. We could then drop the allegedly problematic concept of the a priori. The suggestion then is that, for the sentential connectives in a language L for whose truth conditions we need not invoke satisfaction, we could simply say:

There are no sentences 01, . . ., On, 71, ..., Yn of L such that, for all i (1 < i < n), 3i and yA have the same truth value and yet the truth value of a (fl, . . ., /Ln) differs from that of a(yi, . . ., ha).

The generalization of this to satisfaction is then straightforward: There are no (possibly open) sentences 31, .. ., fL2, 71, . . *, Yn of L such that, for all i (I < i < n) and all sequences s, s satisfies fd iff s satisfies yi and yet it is not the case that, for all sequences s, s satisfies a /1, * * * XAn) iff s satisfies a (71, * * *, en) - The difficulty with this proposal is that it does not as it is stated

imply that no empirical information about the world has to be used in determining which sequences satisfy a (I1, ..., I3n) on the basis of which satisfy the /i and the satisfaction condition for a. For the proposal is purely extensional, uses material conditionals, and speaks only of the actual truth values of sentences. It thus does not rule out "accidental" extensionality. Counterexamples for any interesting language with infinitely many sentences will of necessity be somewhat recherch6 and are hardly illuminating to construct; but they do exist. We do then need some extra element to do the work the a priori element performs.

It might, however, be asked why we should choose the a priori rather than necessity to perform this task. For we can, for instance, say, for a given sequence so, that w (sats (so, A) & Vs (sats (s, froA 1)

asats (s ,A)). D gsats (so , reA E (' sats(so, A) & Vs(sats(s, FrAm)

=- "" sats (s, A)) . D sats (so, ros A) and so on for the other sentential connectives at least. And if we accept the plausible doctrine that, if it is true a priori that p, then it is necessary that p, such a criterion cannot be narrower than the one originally suggested. There would be other alleged advantages too: since we can say that, for any terms ti, t2, where '*' is the assignment functor, E (s*tl = Hesperus & s*t2 = Phosphorus & Vs (sats (s, Ftl - t25)

S*tl = s*t2)) . D sats(s, rt1 = t21)) and we assert a similar formula with the consequent negated when the terms are assigned distinct objects relative to s, identity will be



decisively counted a logical constant. The difficulty with such a criterion lies rather in its width. If anything that is a donkey cannot but be a donkey and if anything that is not a donkey could not be a donkey, then, for any given sequence so of objects, we can say either that

o (soft = a & Vs (sats (s, Fdonkey (t1) 1) donkey (s*ts)). D sats (so, Fdonkey (t) 1))

or that

ci (so*t = a & Vs (sats (s, rdonkey (t,) 1) - donkey (s*t,)) . sats (so, rdonkey (t) 1))

for some proper name or variable a, according as it names or is assigned a donkey or not. But surely the predicate 'is a donkey' is not a logical constant. On the criterion that uses the concept of the a priori, it is not so counted, since one cannot ever know a priori of a given object whether or not it is a donkey. It would be circular to attempt to meet this objection by restricting the concept of necessity used to "logical" necessity. What to count as logical necessity should fall out as a consequence of a theory of the logical constants and logical truth, and not be a resource presupposed by the theory.

vii I now turn to consider four ways in which the suggested criterion pronounces on certain expressions whose inclusion with the logical constants has been considered problematic.

(a) Identity. If one accepts the Hilbert-Bernays definition of identity by exhaustion of lexicon which Quine advocates,'2 identity will not be a logical constant on the criterion simply because it will not be a primitive expression that is handled by some axiom of the truth theory for the language. However, instances of the laws of first-order quantification theory with identity will nevertheless count as logical truths on the concept associated with the criterion, since, when the identity sign is eliminated, model-theoretically valid sentences result.

If one does not accept the identification of indiscernibles-in-the- OL that the Hilbert-Bernays definition entails, so that where identity occurs it is primitive, identity is not a logical constant on the criterion as it now stands. Given knowledge that the ith element of sequence s is a, that the jth is b, and that any sequence satisfies rxi = xi1 iff its ith and jth elements are identical, one cannot always infer a priori whether or not s satisfies rxi = xj; for one needs also

12See for instance Set Theory and Its Logic, rev. ed. (Cambridge, Mass.: Harvard, 1969), pp. 13-15.



to know whether or not a = b. As we noted above, the extra stipulations on the believer's knowledge that we imposed earlier to accom- modate the quantifiers do not give him knowledge for a given sequence of whether its ith and jth elements are identical. Perhaps we could allow this extra knowledge and retain some important sense in which it is true that its admission does not involve a posteriori knowledge of properties and objects in the sequences. But even this addition would not make identity a logical constant if there were proper names in the language; for, once again, if one knows the satisfaction clause for identity, knows that s has Hesperus as its third element, and knows that s assigns Phosphorus to the name c, one is not thereby in a position to know whether or not s satisfies rx3 = c1.

Those who reject the Hilbert-Bernays definition could claim that this need for additional requirements on the imagined person's knowledge beyond those imposed to capture the quantifiers shows that there is some significant sense in which the standard connectives and quantifiers are logical constants and identity is not. But this fact, if it is a fact, should not make us reject the concept of truth in all models in which the identity sign is assigned the identity relation on the domain of the model ("normal" models in Mendelson's sense) as rather arbitrary; for, since the denotations of individual constants are allowed there to vary between models, rc = d- is not valid unless 'c' is 'd', and so all sentences valid on that notion can be known to be satisfied by all sequences a priori given knowledge of the satisfaction conditions of the components, without further requirements being placed on the imagined knower's knowledge.

Identity is indeed at one end of a spectrum of expressions, more and more of which will be included as logical constants the more we allow the imagined person to know about relations of identity and distinctness of objects in the sequences: consider 'a few', 'many', 'most'. Of course, if identity is counted a logical constant, so, derivatively, are the finite numerical quantifiers 'there are n objects (F's) x such that ... x... .'; but this does not yet settle the status of 'there are countably many objects (F's) x such that ... ... .' or that of 'there are uncountably many objects (F's) x such that ...x. . .'This is no defect in the criterion, provided that there is a corresponding spectrum in our intuitions. Insofar as Tarski placed emphasis on the word 'sharp' in the quotation with which we started, this might be taken as confirming his statement. But a distinction can be important and principled without being sharp. Moreover there remains a sharp distinction between those expres-



sions such that there is some amount of knowledge of identities that will make them logical constants, and those without this property. (b) Necessity. Homophonic truth theories for quantificational languages that contain the modal operator 'wo ' have been constructed using inter alia the axiom

a Vs n (satsL(s, foAli) -= satsL(s,A)) Is 'n' thus treated as a logical constant? One might be tempted to argue that it is, on the ground that although one cannot know a priori that Hesperus is Phosphorus, one can know independently of experience (by philosophical argument) that, if a sequence satisfies fX2 = Hesperus1, it satisfies rF (X2 = Hesperus) 1; and, similarly, that one can know a priori, given the satisfaction clause for 'o', that, if a sequence satisfies

rVx (water (x) D ]y (hydrogen (y) & x is partly composed of y))1 it satisfies the necessitation of that formula. These are errors. The criterion requires knowledge of which sequences satisfy a (81, . . ., An) on the basis only of knowledge of the satisfaction condition for a and of which sequences satisfy A3, ..., gin; knowledge was not given of the satisfaction conditions of A . . .,I . Without this, those reasons for counting 'w' a logical constant by the criterion lapse. Indeed, 'o' does not even fulfill the weaker "extensional" criterion purged of the a priori, considered a few pages earlier: there are many pairs of sentences A and B which are satisfied by exactly the same sequences and which are such that there are sequences satisfying FrEA1 which fail to satisfy rFub. Consider rFx = xl and Fx = x & pl for any true contingent sentence replacing 'p'; or

rFVx (water (x) D ]y (hydrogen (y) & x is partially composed of y)) I and the result of substituting 'the nth element to be isolated in a pure form in Europe' for 'hydrogen', for a substitution for 'n' making these coextensive predicates. Once again, though, it is perhaps worth noting that an expression some have wished to count as a logical constant is accommodated by only a slight change to the criterion: that of allowing knowledge of the satisfaction conditions of A , y . ns

We should note, however, that there may be an unqualified sense in which 'o' is a logical constant. It was argued earlier that temporal operators are logical constants if the quantifiers are. Modal operators are clearly very similar to temporal operators; and if, intuitively speaking, we allowed the imagined believer knowledge, for every possible state of affairs, of whether a given sequence s satisfied A there, combined with knowledge of the satisfaction clause for 'w',



he could infer a priori whether or not s satisfied rFA m. So 'w' would be a logical constant. Whether modal and temporal operators are given relevantly parallel semantical treatments must depend on the resolution of issues over the possibility of such expressions operating on complete sentences that can stand alone to make complete assertions and of issues over the possibility of treating 'Iz metalinguistically. (c) Set Membership. If all the apparently set-theoretical discourse of a given language can be construed virtually, in the sense of Quine, then 'E' will not be a primitive expression treated by some axiom of the truth theory for the language, and so will not strictly be a logical constant by the criterion. But the laws governing such a use of 'E', when definitionally expanded in context, will be valid theorems of quantification theory. Discourse using apparently set-theoretical vocabulary may not be purely virtual and still not require the use of a genuine set-membership predicate in the metalanguage of the truth theory for the language: such would be the case if substitutional quantification were permitted into the positions occupied by set- abstract expressions. Even when the OL requires a genuine ontology of sets and 'E' is treated like any other atomic predicate, it will not count as a logical constant under the criterion: if one knows that the ith element of sequence s is object x and that the jth is set y, one will not know whether s satisfies rFx E x9n unless one knows whether x E y. For familiar reasons, the set y and object x may not be presented to one in such a way that one can tell a priori whether or not x e y (though if it is, it necessarily is). The concept of relational belief of a set is riddled with difficulties, and indeed inherits all the philosophical problems of set-theoretical discourse that is neither virtually nor substitutionally construable.

Those who have wished to include 'E' among the logical constants have probably been influenced by the feeling that it satisfies some intuitive requirement of topic neutrality. When the discourse involving 'E' has not required an ontology of sets for its interpretation, we can make good sense of this feeling by noting that, provided they talk of objects, there are no restrictions on the kinds of predicates, functors, and names that occur in a sentence, even a true sentence, after 'E' has been eliminated in context. (It is worth noting that a parallel claim holds for identity under the Hilbert-Bernays definition.) When, however, a genuine ontology of sets is required for interpretation (if in fact there are any intelligible such cases) the topic-neutrality claim for 'CE seems indefensible: though it is indeed the case that any object whatsoever may be a member of a set, talk involving 'e' has a genuine subject matter-the sets.



(d) Intuitionistic Constants. The very concept of "knowing which" is inappropriate to assessing whether the intuitionistic constants are logical constants. For (on the way I have been using the concept) if one knows which F's are G's and knows of all the F's there are, then one knows of every F either that it is G or that it is not G; and in the case in which 'G' here is replaced by 'satisfies A o', where Ao is an undecidable sentence of an intuitionistic language, we shall simply beg the question against intuitionism if we make use of the concept of knowing which sequences satisfy a formula. But the original criterion that applied to the logical constants can be re- garded as an instance of a schema that is applicable in a way that does not prejudice the case for or against any particular theory of meaning. For, whatever the correct form of a theory of meaning for a particular language, it is not conceivable that it fails to specify recursively the application conditions of some (possibly relational) predicate 4 to the sentences of the language. The criterion for logical- constanthood used earlier operated with the classical satisfaction relation, but there will be analogous criteria for other conceptions of a theory of meaning, using whatever predicate 4 they favor. Intuitionists, or more expecially those who wish to extend their account of meaning in mathematics to all areas, specify the meaning of A by stating the conditions under which we have a proof of A, or, more generally, under which A is assertible. Is it enough, then, to say the following for the case of sentential connectives?

a is a logical constant iff, given that for any sequence s that assertibly satisfies fhi, one knows that it does, and given that for any sequence s for which the supposition that it satisfies pli can be reduced to decidable absurdity, one knows that it can, and given knowledge of the (intuitionistic) satisfaction clause for a, then, for any s that assertibly satisfies a0jti, ..., fn), one can infer a priori that s assertibly satisfies a(QliB ... in)*

We will certainly need to make some provision to exclude "dis- junctive" a priori features from the satisfaction condition for a, but the condition just given will indeed capture the intuitionist's conjunction, alternation, and negation; his conditional, however, is not definable in terms of any of these,13 and in fact it will not be counted as a logical constant unless we enrich the believer's antecedent knowledge. This is simply because the intuitionist is prepared to assert rA -* B1 in certain cases even though neither B is assertible nor A reducible to decidable absurdity; he will still assert rA -+ B1

13 J. C. C. McKinsey, "Proof of the Independence of the Primitive Symbols of Heyting's Calculus of Propositions," Journal of Symbolic Logic, iv, 4 (December 1939): 155-158.



provided there assertibly is a way of transforming any given proof of A into a proof of B. So we must attribute to the believer some grasp of the totality of possible (forms of) proof of A and of B; here there is of course an important issue, that of the necessity of specifying this totality without impredicativity, an issue it would take us too far afield to pursue at this point.

Even by allowing grasp of the totality of possible proofs of A and of B we have not settled the question of whether -+ is a logical constant when the language deals with an empirical subject matter. -* will be a logical constant only if no a posteriori truths enter the methods of transformation of proofs of A that one admitted. If rVx (- Ex v Gx)l were assertible but a posteriori (whatever that combination may mean for a universal sentence) and if this were taken as sufficient for asserting that r3xFx -* 3xGx1 even when neither r3xGx1 is assertible nor rFxFxl reducible to decidable falsity, then -* would not be a logical constant (without a much more radical extension of the antecedent knowledge allowed to the believer). It is noteworthy then that, on the interpretation that does make -+ a logical constant, there will be no sentences of the form rA -* B1 dealing with purely a posteriori subject matter that both are assertible and are not logical consequences of decidable sentences that are assertible.

The discussion of the previous two paragraphs can be precisely paralleled for the intuitionist's universal quantifier. But it is again worth noting how sharp the dilemma is over universal quantification when the account is applied to empirical subject matter. In Heyting arithmetic, the only reason that universally quantified sentences that are not logical consequences of decidable sentences are provable is that we already have in the axioms some such sentences; and it is plausible that satisfaction of these axioms is involved in what it is to be, or to function as, a natural number. But with respect to con- tingently existing objects knowledge of which is a posteriori, there seem to be simply no universally quantified truths of corresponding status. How, then, without allowing a posteriori methods within the range of methods spoken of in the intuitionist's assertibility condition for 'V', are we to be in a position to assert any universally quantified sentences that are not logical consequences of decidable sentences? Yet, if such methods are allowed in, it is hard to see how, without making the assertibility condition for FVxiAl involve other universally quantified conditions that we cannot effectively recognize as obtaining when they do so, the assertibility condition for universally quantified sentences could involve anything other than what would be called the "evidence" (from a classical point of view)



for a general truth. But then, of course, unrestricted universal elimination (Quine's "universal instantiation") is not a rule that is strictly faithful to the sense of that quantifier. Perhaps when the intuitionist's conception is extended beyond mathematics, he must be prepared for the possibility that no quantifier of his can stand to the classical realist's universal quantifier as the intuitionistic universal quantifier in arithmetic stands to the classical one there.

VIII My final observation on the general criterion for logical-constanthood is to observe how it can help to explain the plausibility of the operation of the Principle of Charity in a way that (at least) guarantees that there will be no assent to the negations of certain uncontentious logical truths. For the logical constants, the Principle of Charity involves identifying where we can as negation, conjunction, . . ., devices of another language on the basis solely of the conditions under which wholes containing these devices are assented to or dissented from by speakers of the other language on the basis of assent or dissent to the expressions on which they operate. Now if, given knowledge of the satisfaction conditions of a, one cannot tell a priori whether or not a (1, ..., 3in) is satisfied by a sequence on the basis of which sequences satisfy 31, ..., /3n, then assent or dissent to a(#,, ..., i3n) will not depend solely on assent and dissent patterns to /1, ..., An: it will depend also on the way the world is (or is believed to be). We can expect our predictions about assent and dissent to be wrong in empirically possible cases if we apply Charity in the form both Quine and Davidson envisage for the logical constants, to an expression that is not on our criterion a logical constant. I conclude that, in this particular case, as perhaps else- where, Charity and Constancy are inseparable.

CHRISTOPHER PEACOCKE All Souls College, Oxford University of California, Berkeley

BOOK REVIEWS

Res Cogitans: An Essay in Rational Psychology. ZENO VENDLER. Ithaca, N.Y., and London: Cornell University Press, 1972. ix, 225 p. $9.75. Vendler's principal goal in this book is to develop and defend a systematic and predominantly rationalist account of speech and thought, and of the relationship between them. Central to rational-


Article Contentsp. 221p. 222p. 223p. 224p. 225p. 226p. 227p. 228p. 229p. 230p. 231p. 232p. 233p. 234p. 235p. 236p. 237p. 238p. 239p. 240

Issue Table of ContentsThe Journal of Philosophy, Vol. 73, No. 9 (May 6, 1976), pp. 221-252Front MatterWhat is a Logical Constant? [pp. 221 - 240]Book Reviewsuntitled [pp. 240 - 252]

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