P. F. Strawson on H. P. Grice

It is perfectly possible to find interpretations for the A, E, I, and O forms for which all the laws of the traditional system hold good together. There are at least two distinct, though related, methods by which this can be done. One has only a limited and formalistic interest; the other illuminates some general features of our ordinary speech. But though they are very different in certain respects, the ways in which they operate to save the consistency of the system are closely related. I give the formalistic solution first, partly for the sake of completeness and partly for the light it casts on the second, or realistic, solution.

The first method consists simply in a farther elaboration of the kind of interpretation in class or quantificational terms which we have been considering. It is a kind of ad hoc patching up of the old system in order to represent it, in its entirety, as a fragment of the new. The method is to encounter every breakdown in a traditional law by amending the class or predicative interpretation suggested in such a way as to secure its validity. For example our second attempt at providing a translation in terms of positively and negatively existential formulae left us with three laws of the Square of Opposition invalid. So we begin with the ad hoc prevention of these breakdowns. Thus we want to make A and contradictories. Now, on Table 2, A is *~(*x)(fx m ~gx).(lx)(fxy and O is ' (3x)(fx . ~gx}\ The contradictory of an expression of the form * ~p . q 9 is the corresponding expression of the form ' p v ~q\ We accordingly decide to make O of this form by re-interpreting it as 4 (3x)(fx . ~gx) v ~(3x)(fx) '. A and O are now contradictories. Similarly, we re-interpret I as so that it is the contradictory of E, which is It is evident that we have, by this manoeuvre, saved the law that I and are subcontraries (i.e., that corresponding statements of these forms cannot both be false). This law broke down for the previous interpretations because corresponding statements of these forms could both be false, in the case where the corresponding statement of the form * ~(3x)(fx) ' was true. But on the new interpretation the truth of this statement is a sufficient condition of the truth of both I and statements ; since ' q Dp v q ' is analytic. Nor do we sacrifice any of the other laws of the Square of Opposition in saving these three. A and E have not been altered, so they remain contraries. The laws * A D I ' and 'EDO' remain valid. For the old form ofI entails the new form of I, and A entails the old form of I ; hence A entails the new form of I. Similarly, E entails the new 0.

Further amendments, however, are required. Although we have saved all the laws of the Square of Opposition, we have not altered E of Table 2 ; so its simple conversion remains invalid. Moreover, the amendments so far made render invalid the simple conversion of I. If we transpose the terms (i.e., the predicative variables) in the formula ' (3x)(fa .gx) v ~(~ix)(fx)* we obtain * (3aO(gff./aO v ~(3aOfeaO '* and these formulae are by no means equivalent. For 4 ~(3a?)(/k) , (3x)(gx) ' entails the first and is inconsistent with the second. The reason for the breakdown of the conversion of E was that 4 ~(3)(#&) ' [or ' p = '] was consistent with ~(3a)(/* #*) (3*)(/ar) [or P = O . a * 0] but not with its simple converse ~(3x)(gx .fa) . (3x)(gx) [or p* = O . (B * O]. The term-symmetry of E can obviously be restored, and the breakdown prevented, by adopting the interpretation ~(3)(/ **) (3*0(/aO . (3*)fea?) [or ap = . a 4= O . p 4= O] Similarly, the term-symmetry of I can be restored by re-interpreting it as (3x)(fx.gx) v ~(3aO(/fc) v ~0a0tor) which also maintains its status as the contradictory of E. Adopting these readings for E and I will obviously force us to make further alterations in the other forms in order to preserve their logical relations. Since, by the rule of obversion, ' xAy ' is equivalent to ' x&y' \ we can obtain the appropriate interpretation for A simply by negating the second term ( c g ' or 4 P ') throughout the latest form of E; which gives us or ap == . a 4= O . p 4= O] Finally, 0, as the contradictory of A, must be re-interpreted as PT.

So we have, as our final interpretation : Table 3 A ~(3*)(/ ~&) (3)(/*) - (3a?)(~#r) E ~(3*)(/* **) - (3)(/ff)'. (3*)te) I (3a?)(/aj .#*) v ~(3a>)(/0) v ~(3)fe) O (3o?)(/ . ~#r) v ~(3*)(/*0 v ~(3*)(~*}. For this interpretation, all the laws of the traditional logic hold good together ; and they hold good within the logic of classes or quantified formulae ; as a part of that logic. So the consistency of the system can be secured in this way. But the price paid for consistency will seem a high one, if we are at all anxious that the constants * all,' * some *, and * no ' of the system should faithfully reflect the typical logical behaviour of these words in ordinary speech. It is quite unplausible to suggest that if someone says ' Some students of English will get Firsts this year ', it is a sufficient condition of his having made a true statement, that no one at all should get a First. But this would be a consequence of accepting the above interpretation for I. Note that the dropping of the implication of plurality in * some ' makes only a minor contribution to the unplausibility of the translation. We should think the above suggestion no more convincing in the case of someone who said c At least one student of English will get a First this year '. The third table of translations, then, does, if anything, less than the other two to remove our sense of separation from the mother tongue. 7. So let us start again, taking the latter as our guide. And let us not be bound by the assumption from which all these difficulties have arisen; namely, that whatever interpretation we give to the four forms, it must be an interpretation in explicitly existential terms ; that all statements of the four forms must be positively or negatively existential, or both. Suppose someone says ' All John's children are asleep '. Obviously he will not normally, or properly, say this, unless he believes that John has children (who are asleep). But suppose he is mistaken. Suppose John has no children. Then is it true or false that all John's children are asleep ? Either answer would seem to be misleading. But we are not compelled to give either answer.

174 SUBJECTS, PREDICATES, EXISTENCE [CH. 6

We can, and normally should, say that, since John has no children,, the question does not arise.

But if the form of the statement were ~(*x)(fx.~gx) [Table 1]

the correct answer to the question, whether it is true, would be * Yes 5 ; for 4 ~ (3tf)(/#) * is a sufficient condition of the truth of * ~(3

And if the form of the statement were either ~(3XA ~&) *)(/) tTable *] or ~ (3x)(fx . ~gx) . (3x)(fx) . (3x)(~gx) [Table 3] the correct answer to the question would be that the statement was false; for * ~(3ff)(/#)' *s inconsistent with both these formulae. But one does -not happily give either answer simply on the ground that the subject-class is empty. One says rather that the question of the truth or falsity of the statement simply does not arise ; that one of the conditions for answering the question one way or the other is not fulfilled. The adoption of any of the explicitly existential analyses, whether it be a negatively existential one (Table 1) or a conjunction of negatively and positively existential components (Tables 2 and 3), forces us to conclude that the non-existence of any children of John's is sufficient to determine the truth or falsity of the general statement; makes it true for the first analysis, false for the other two. The more realistic view seems to be that the existence of children of John's is a necessary precondition not merely of the truth of what is said, but of its being either true or false. And this suggests the possibility of interpreting all the four Aristotelian forms on these lines : that is, as forms such that the question of whether statements exemplifying them are true or false is one that does not arise unless the subject-class has members. It is important to understand why people have hesitated to adopt such a view of at least some general statements. It is probably the operation of the trichotomy ' either true or false or meaningless ', as applied to statements, which is to blame. For this trichotomy contains a confusion : the confusion between sentence and statement (See Chapter 1, p. 4.).

Of course, the sentence * All John's children are asleep ' is not meaningless. It is perfectly significant. But it is senseless to ask, of the sentence, whether it is true or false. One must distinguish between what can be said about the sentence, and what can be said about the statements made, on different occasions, by the use of the sentence. It is about statements only that the question of truth or falsity can arise ; and about these it can sometimes fail to arise. But to say that the man who uses the sentence in our imagined case fails to say anything either true or false, is not to say that the sentence he pronounces is meaningless. Nor is It to deny that he makes a mistake. Of course, it is incorrect (or deceitful) for him to use this sentence unless (a) he thinks he is referring to some children whom he thinks to be asleep ; (b) he thinks that John has children ; (c) he thinks that the children he is referring to are John's. We might say that in using the sentence he commits himself to the existence of children of John's. It would prima facie be a kind of logical absurdity to say * All John's children are asleep ; but John has no children '. And we may be tempted to think of this kind of logical absurdity as a straightforward self-contradiction ; and hence be led once more towards an analysis like that of Table 2 ; and hence to the conclusion that the man who says * All John's children are asleep ', when John has no children, makes a false statement. But there is no need to be led, by noticing this kind of logical absurdity, towards this conclusion. For if a statement S presupposes a statement S' in the sense that the truth of S' is a precondition of the truth-orfalsity of S, then of course there will be a kind of logical absurdity in conjoining S with the denial of S'. This is precisely the relation, in our imagined case, between the statement that all John's children are asleep (S) and the statement that John has children, that there exist children of John's (S'). But we must distinguish this kind of logical absurdity from straightforward self-contradiction. It is self-contradictory to conjoin S with the denial of S' if S' is a necessary condition of the truth, simply, of S. It is a different kind of logical absurdity to conjoin S with the denial of S' if S' is a necessary condition of the truth orfalsity of S.

The relation between S and S' in the first case is that S entails S'. We need a different name for the relation between S and S' in the second case ; let us say, as above, that S presupposes S'.

Underlying the failure to distinguish sentence and statement, and the bogus trichotomy * true, false, or meaningless ', we may detect a further logical prejudice which helps to blind us to the facts of language. We may describe this as the belief or, perhaps better, as the wish, that if the uttering of a sentence by one person, at one time, at one place, results in a true statement, then the uttering of that sentence by any other person, at any other time, at any other place, results in a true statement. It is, of course, incredible that any formal logician should soberly believe this. It is, however, very natural that they should wish it were so; and hence talk as if it were so. And to those tempted to talk as if it were so, the distinction I have insisted upon between sentence and statement will not occur or will seem unimportant. Why this wish-belief should be natural to formal logicians, and what further effects it has, I shall discuss later. What I am proposing, then, is this. There are many ordinary sentences beginning with such phrases as ' All . . .', fc All the . . .', 4 No . . .', 4 None of the ...',' Some ...',' Some of the ...',' At least one ..,',' At least one of the . . .' which exhibit, in their standard employment, parallel characteristics to those I have just described in the case of a representative ' All . . .' sentence. That is to say, the existence of members of the subject-class is to be regarded as presupposed (ih the special sense described) by statements made by the use of these sentences; to be regarded as a necessary condition, not of the truth simply, but of the truth or falsity, of such statements. I am proposing that the four Aristotelian forms should be interpreted as forms of statement of this kind. Will the adoption of this proposal protect the system from the charge of being inconsistent when interpreted? Obviously it will. For every case of invalidity, of breakdown in the laws, arose from the non-existence of members of some subject-class being compatible with the truth, or with the falsity, of some statement of one of the four forms. So our proposal, which makes the non-existence of members of the subject-class incompatible with either the truth or the falsity of any statement of these forms, will cure all these troubles at one stroke. We are to imagine that every logical rule of the system, when expressed in terms of truth and falsity, is preceded by the phrase c Assuming that the statements concerned are either true or false, then . . .' Thus the rule that A is the contradictory of O states that, if corresponding statements of the A and forms both have truth-values, then they must have opposite truth-values; the rule that A entails I states that, if corresponding statements of these forms have truth-values, then if the statement of the A form is true, the statement of the I form must be true ; and so on. The suggestion that entailment-rules should be understood in this way is not peculiar to the present case (Compare the discussion of the truth-functional system, Chapter 3, pp. 68-69.) What is peculiar to the present case is the requirement that, in order for any statement of one of the four forms to have a truth-value, to be true or false, it is necessary that the subject-class should have members. That the adoption of this suggestion will save the rules of the traditional system from breakdown is obvious enough for all the rules except, perhaps, those permitting, or involving the validity of, the simple conversion of E and of I. That the subjectclass referred to in a statement of either of these forms must be non-empty in order for the statement to be true or false does not guarantee, in the case of the truth of an E statement or the falsity of an I statement, the non-emptiness of the predicateclass. This was the reason why the final interpretations of Table 3 required three components for each form instead of two. But, whilst this is true, it does not constitute an objection, nor lead to the breakdown of the rules as we are now to understand them. Thus perhaps a statement of the form * xEy ' might be true while the corresponding statement of the form * 7/Eo? ' was neither true nor false. But all that we require is that so long as corresponding statements of the forms ' xl&y ' and fc yE*x r are both either true or false, they must either be both true or both false. This is secured to us by interpreting * x&y ' as the form of hosts of ordinary statements, beginning with * No . . .' or ' None of the . . .', of the kind described in this section. Similar considerations hold for I; though mention of I reminds us of one not unimportant reservation we must make, before simply concluding that the constants ' all *, ' some ', * no ' of the traditional system can be understood, without danger to any of the rules, as having just the sense which these words have in the hosts of ordinary statements of the kind we are discussing. And this is a point already made : viz., that ' some ', in its most common employment as a separate word, carries an implication of plurality which is inconsistent with the requirement that should be the strict contradictory of A, and I of E. So * some ', occurring as a constant of the system, is to be interpreted as ' At least one . . .' or ' At least one of the . . .', while * all ' and * no ', so occurring, can be read as themselves.

The interpretation which I propose for the traditional forms has, then, the following merits : (a) it enables the whole body of the laws of the system to be accepted without inconsistency ; (b) with the reservation noted above, it gives the constants of the system just the sense which they have in a vast group of statements of ordinary speech ; (c) it emphasizes an important general feature of statements of that group, viz., that while the existence of members of their subject-classes is not a part of what is asserted in such statement, it is, in the sense we have examined, presupposed by them. It is this last feature which makes it unplausible to regard assertions of existence as either the whole, or conjunctive or disjunctive parts, of the sense of such ordinary statements as * All the men at work on the scaffolding have gone home ' or * Some of the men are still at work '. This was the reason why we were unhappy about regarding such expressions as ' (x)(fxDgx) ' as giving the form of these sentences ; and why our uneasiness was not to be removed by the simple addition of positively or negatively existential formulae. Even the resemblance between * There is not a single book in his room which is not by an English author J and the negatively existential form ' ~ (3x)(fx . ~ gx) ' was deceptive. The former, as normally used, carries the presupposition * booksin-his-room ' and is far from being entailed by 4 not-a-book-inhis-room ' ; whereas the latter is entailed by l ~(3x)(fx) \ So it is that if someone, with a solemn face, says 4 There is not a single foreign book in his room ' and then later reveals that there are no books in the room at all, we have the sense, not of having been lied to, but of having been made the victim of a sort of linguistic outrage. Of course he did not say there were any books in the room, so he has not said anything false. Yet what he said gave us the right to assume that there were, so he has misled us. For what he said to be true (or false) it is necessary (though not sufficient) that there should be books in the room. Of this subtle sort is the relation between ' There is not a book in his room which is not by an English author s and * There are books in his room.’  (Some will say these points are irrelevant to logic (are ' merely pragmatic *). If to call them * irrelevant to logic * is to say that they are not considered in formal systems, then this is a point I should wish not to dispute, but to emphasize. But to logic as concerned with the relations between general classes of statements occurring in ordinary use, with the general conditions under which such statements are correctly called * true * or ' false % these points are not irrelevant. Certainly a * pragmatic ' consideration, a general rule of linguistic conduct, may perhaps be seen to underlie these points : the rule, namely, that one does not make the (logically) lesser, when one could truthfully (and with equal or greater linguistic economy) make the greater, claim. Assume for a moment that the form * There is not a single . . . which is not . . . f were introduced into ordinary speech with the same sense as * ~(3x)(fx . ~gx) *. Then the operation of this general rule would inhibit the use of this form where one could truly say simply * There is not a single . . . ' (or * ~(3a?)(/a?) '). And the operation of this inhibition would tend to confer on the introduced form just those logical presuppositions which I have described ; the form would tend, if it did not remain otiose, to develop just those differences I have emphasized from the logic of the symbolic form it was introduced to represent. The operation of this * pragmatic rule ' wjw first pointed out to me, in a different connexion, by Mr. H. P. Grice.). What weakens our resistance to the negatively existential analysis in this case more than in the case of the corresponding * All '-sentence is the powerful attraction of the negative opening phrase * There is not . . .'. To avoid misunderstanding I must add one point about this proposed interpretation of the forms of the traditional system. I do not claim that it faithfully represents the intentions of its principal exponents. They were, perhaps, more interested in formulating rules governing the logical relations of more imposing general statements than the everyday ones I have mostly considered ; were interested, for example, in the logical powers of scientific generalizations, or of other sentences which approximate more closely to the desired conditions that if their utterance by anyone, at any time, at any place, results in a true statement, then so does their utterance by anyone else, at any other time, at any other place. We have yet to. consider how far the account here given of certain general sentences of common speech is adequate for all generalizations.