H. P. Grice on P. F. Strawson on "if"

. The relations between ' if ' and * D ' have already, but only in part, been discussed. 1 The sign ' D 9 is called the Material Implication sign a name I shall consider later. Its meaning is given by the rule that any statement of the form A p D g ' is false in the case in which the first of its constituent statements is true and the second false, and is true in every other case considered in the system. That is to say, the falsity of the first constituent statement or the truth of the second are, equally, sufficient conditions of the truth of a statement of material implication ; the combination of truth in the first with falsity in the second is the single, necessary and sufficient, condition (1 Chapter 2, Section 7)of its falsity. The standard or primary -- the importance of this qualifying phrase can scarcely be overemphasized. There are uses of * if . . . then ... * which do not answer to the description given here,, or to any other descriptions given in this chapter -- use of an 4 If . . then . . .' sentence, on the other hand,, we saw to be in circumstances where, not knowing whether some statement which could be made by the use of a sentence corresponding in a certain way to the first clause of the hypothetical is true or not, or believing it to be false, we nevertheless consider that a step in reasoning from that statement to a statement related in a similar way to the second clause would be a sound or reasonable step ; the second statement also being one of whose truth we are in doubt, or which we believe to be false. Even in such circumstances as these we may sometimes hesitate to apply the word * true ' to hypothetical statements (i.e., statements which could be made by the use of * if ... then . .' in its standard significance), preferring to call them reasonable or well-founded ; but if we apply the word * true ' to them at all, it will be in such circumstances as these. Now one of the sufficient conditions of the truth of a statement of material implication may very well be fulfilled without the conditions for the truth (or reasonableness) of the corresponding hypothetical statement being fulfilled ; i.e., a statement of the form ** p 3q ' does not entail the corresponding statement of the form l if p9 then q \ But if we are prepared to accept the hypothetical statement, we must in consistency be prepared to deny the conjunction of the statement corresponding to the first clause of the sentence used to make the hypothetical statement with the negation of the statement corresponding to its second clause ; i.e., a statement of the form * if p then q ' does entail the corresponding statement of the form * p D q \ The force of the word * corresponding ' in the above paragraph needs elucidation. Consider the three following very ordinary specimens of hypothetical sentences : (1) If the Germans had invaded England in 1940, they would have won the war. (2) If Jones were in chaige, half the staff would have been dismissed. (3) If it rains, the match will be cancelled. The sentences which could be used to make statements corresponding in the required sense to the subordinate clauses can be ascertained by considering what it is that the speaker of each hypothetical sentence must (in general) be assumed either to be in doubt about or to believe to be not the case. Thus, for (1) to (8), the corresponding pairs of sentences are : (la) The Germans invaded England in 1940; they won the war. (2a) Jones is in charge ; half the staff has been dismissed- (Sa) It will rain ; the match will be cancelled. Sentences which could be used to make the statements of material implication corresponding to the hypothetical statements made by sentences (1) to (3) can now be framed from these pairs of sentences as follows : (Ml) The Germans invaded England in 1940 D they won the war. (M2) Jones is in charge D half the staff has been, dismissed. (M3) It will rain D the match will be cancelled. The very fact that these verbal modifications are necessary, in order to obtain from the clauses of the hypothetical sentence the clauses of the corresponding material implication sentence is itself a symptom of the radical difference between hypothetical statements and truth-functional statements. Some detailed differences are also evident from these examples. The falsity of a statement made by the use of 4 The Germans invaded England in 1940 ' or ' Jones is in charge ' is a sufficient condition of the truth of the corresponding statements made by the use of (Ml) and (M2) ; but not, of course, of the corresponding statements made by the use of (1) and (2). Otherwise, there would normally be no point in using sentences like (1) and (2) at all; for these sentences would normally l carry, in the tense or mood of the verb, an implication of the speaker's belief in the falsityof the statements corresponding to the clauses of the hypothetical. Its not raining is sufficient to verify a statement

(1 But not necessarily. One may use the pluperfect or the imperfect subjunctive when one is simply working out the consequences of an hypothesis which one may be prepared eventually to accept.)

made by the use of (MS), but not a statement made by the use of (3). Its not raining Is also sufficient to verify a statement made by the use of (M4) 4 It will rain D the match will not be cancelled 5 . The formulae *j>Dg* and 4 j>D q' are consistent with one another, and the joint assertion of corresponding statements of these forms is equivalent to the assertion of the corresponding statement of the form * *-~p \ But * If it rains, the match will be cancelled ' is inconsistent with 4 If it rains, the match will not be cancelled ', and their joint assertion in the same context is self-contradictory. Suppose we call the statement corresponding to the first clause of a sentence used to make a hypothetical statement the antecedent of the hypothetical statement; and the statement corresponding to the second clause, its consequent. It is sometimes fancied that whereas the futility of identifying conditional statements with material implications is obvious in those cases where the implication of the falsity of the antecedent is normally carried by the mood or tense of the verb (e.g., (I) or (2)), there is something to be said for at least a partial identification in cases where no such implication is involved, i.e., where the possibility of the truth of both antecedent and consequent is left open (e.g., (3)). In cases of the first kind (* unfulfilled ' or * subjunctive ' conditionals) our attention is directed only to the last two lines of the truth-tables for * p D q ', where the antecedent has the truth-value, falsity ; and the suggestion that 4 ~p ' entails * if p, then q ' is felt to be obviously ,vrong. But in cases of the second kind we may inspect also the first two lines, for the possibility of the antecedent's being fulfilled is left open ; and the suggestion that ' p . q * entails * if p, then q * is not felt to be obviously wrong. This is an illusion, though engendered by a reality. The fulfilment of both antecedent and consequent of a hypothetical statement does not show that the man who made the hypothetical statement was right; for the consequent might be fulfilled as a result of factors unconnected with, or in spite of, rather than because of, the fulfilment of the antecedent. We should be prepared to say that the man who made the hypothetical statement was right only if we were also prepared to say that the fulfilment of the antecedent was, at least in part, the explanation of the fulfilment of the consequent. The reality behind the illusion is complex : 86 TRUTH-FUNCTIONS [en. 3 it is, partly, the fact that, in many cases, the fulfilment of both antecedent and consequent may provide confirmation for the view that the existence of states of affairs like those described by the antecedent is a good reason for expecting states of affairs like those described by the consequent ; and it is, partly, the fact that a man who says, for example, 4 If it rains, the match will be cancelled * makes a prediction (viz.. that the match will be cancelled) under a proviso (viz., that it rains), and that the cancellation of the match because of the rain therefore leads us to say, not only that the reasonableness of the prediction was confirmed, but also that the prediction itself was confirmed. Because a statement of the form * p D q ' does not entail the corresponding statement of the form ' if p9 then q ' (in its standard employment), we shall expect to find, and have found, a divergence between the rules for ' D ' and the rules for ' if J (in its standard employment). Because * if p, then q ' does entail * p D q *, we shall also expect to find some degree of parallelism between the rules; for whatever is entailed by * p "3 q ' will be entailed by * if p, then q ', though not everything which entails *pDg* will entail t if p, then q \ Indeed, we find further parallels than those which follow simply from the facts that 4 if p, then q ' entails ' p D q ' and that entailment is transitive. To laws (19)-(23) inclusive we find no parallels for 4 if*. But for (15) (pDj).JJD? (16) (P Dq).~qZ)~p (17) p'Dq s ~q1)~p (18) (?Dj).(?Dr)D(pDr) we find that, with certain reservations, 1 the following parallel laws hold good :

(1 The reservations are important. For example, it is often impossible to apply entailment-rule (iii) directly without obtaining incorrect or absurd results. Some modification of the structure of the clauses of the hypothetical is commonly necessary. But formal logic gives us no guide as to which modifications are required. If we apply rule (iii) to our specimen hypothetical sentences, without modifying at all the tenses or moods of the individual clauses, we obtain expressions which are scarcely English. If we preserve as nearly as possible the tense-mood structure, in the simplest way consistent with grammatical requirements, we obtain the sentences : If the Germans had not won the war, they would not have invaded England in 1940.) If half the staff had not been dismissed, Jones would not be in charge. If the match is not cancelled, it will not rain. But these sentences, so far from being logically equivalent to the originals, have in each case a quite different sense. It is possible, at least in some such cases, to frame sentences of more or less the appropriate pattern for which one can imagine a use and which do stand in the required logical relationship to the original sentences (e.g., * If it is not the case that half the staff has been dismissed, then Jones can't be in charge * ; or * If the Germans did not win the war, it's only because they did not invade England in 1940 * ; or even (should historical evidence become improbably scanty) * If the Germans did not win the war, it can't be true that they invaded England in 1940 *). These changes reflect differences in the circumstances in which one might use these, as opposed to the original, sentences. Thus the sentence beginning * If Jones were in charge . . . ' would normally (though not necessarily) be used by a man who antecedently knows that Jones is not in charge : the sentence beginning * If it's not the case that half the staff has been dismissed . . . ' by a man who is working towards the conclusion that Jones is not in charge. To say that the sentences are nevertheless logically equivalent is to point to the fact that the grounds for accepting either, would, in different circumstances, have been grounds for accepting the soundness of the move from * Jones is in charge ' to * Half the staff has been dismissed '.


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. n] TRUTH-FUNCTIONAL CONSTANTS 87 (i) (if p, then q; and p)^q (ii) (if p, then qt and not-g) Dnot-j? (iii) (if p, then f) D (if not-0, then not-j?) (iv) (if p, then f ; and iff, then r)D(if j>, then r) (One must remember that calling the formulae (i)-(iv) is the same as saying that, e.g., in the case of (iii), c if p, then q ' entails 4 if not-g, then not-j> '.) And similarly we find that, for some steps which would be invalid for 4 if ', there are corresponding steps that would be invalid for * D '. For example : (p^q).q :. p are invalid inference-patterns, and so are if p, then q ; and q /. p if p, then ; and not-j? /. not-f . The formal analogy here may be described by saying that neither * p 13 q ' nor * if j?, then q * is a simply convertible formula. We have found many laws (e.g., (19)-(23)) which hold for * D ' and not for * if *. As an example of a law which holds for * if ', but not for c D *, we may give the analytic formula * ~[(if p, then q) * (if p, then not-g)] *. The corresponding formula

4 ~[(P 3 ?) * (j? 3 ~?}] * is not analytic, but (el (28)) is equivalent to the contingent formula 4 ~~p \ The rules to the effect that formulae such as (19)-{23) are analytic are sometimes referred to as & paradoxes of implication *. This is a misnomer. If 4 D * is taken as identical either with * entails * or, more widely, with * if * . . then . . .* in its standard use, the rules are not paradoxical, but simply incorrect. If * D * is given the meaning it has in the system of truthfunctions, the rules are not paradoxical, but simple and platitudinous consequences of the meaning given to the symbol. Throughout this section, I have spoken of a * primary or standard ? use of * if . . . then . . .*, or * if ? , of which the main characteristics were : that for each hypothetical statement made by this use of 4 if*, there could be made just one state* ment which would be the antecedent of the hypothetical and just one statement which would be its consequent; that the hypothetical statement is acceptable (true, reasonable) if the antecedent statement, if made or accepted, would, in the circumstances, be a good ground or reason for accepting the consequent statement; and that the making of the hypothetical statement carries the implication either of uncertainty about, or of disbelief in, the fulfilment of both antecedent and consequent. 1 Not all uses of * if ', however, exhibit all these characteristics. In particular, there is a use which has an equal claim to rank as standard and which is closely connected with the use described, but which does not exhibit the first characteristic and for which the description of the remainder must consequently be modified. I have in mind what are sometimes called 'variable' or 'general ' hypothetical : e.g., c lf ice is left in the sun, it melts ' ; * If the side of a triangle is produced, the exterior angle is equal to the sum of the two interior and opposite angles ' ; ' If a child is very strictly disciplined in the nursery, it will develop aggressive tendencies in adult life ' ,* and so on. To a statement made by the use of a sentence such as these there corresponds no single pair of statements which are, respectively, its antecedent and consequent. On the other 1 There is much more than this to be said about this way of using ' if * ; in particular, about the meaning of the question * whether the antecedent would be a good ground or reason for accepting the consequent ' and about the exact way in which this question is related to the question of whether the hypothetical is true {acceptable, reasonable) or not hand, for every such statement there is an Indefinite number of non-general hypothetical statements which might be called exemplifications, applications, of the variable hypothetical; e.g., a statement made by the use of the sentence 4 If this piece of ice is left in the sun, it will melt * To the subject of variable hypothetical 1 shall return later. 1 Two relatively uncommon uses of 4 if ' may be illustrated respectively by the sentences L If he felt embarrassed, he showed no signs of it * and c If he has passed his exam, Tm a Dutchman (I'll eat my hat, &c.) '. The sufficient and necessary condition of the truth of a statement made by the first is that the man referred to showed no sign of embarrassment. Consequently, such a statement cannot be treated either as a standard hypothetical or as a material implication. Examples of the second kind are sometimes erroneously treated as evidence that * if ' does, after all, behave somewhat as 4 D ' behaves. The evidence for this is, presumably, the facts (i) that there is no connexion between antecedent and consequent; (ii) that the consequent is obviously not (or not to be) fulfilled ; (iii) that the intention of the speaker is plainly to give emphatic expression to the conviction that the antecedent is not fulfilled either ; and (iv) the fact that '(p D q) . ~q ' entails 4 ~p '. But this is a strange piece of logic. For, on any possible interpretation, * if p then q ' has, in respect of (iv), the same logical powers as 4 p D q * ; and it is just these logical powers that we are jokingly (or fantastically) exploiting. It is the absence of connexion referred to in (i) that makes it a quirk, a verbal flourish, an odd use of * if. If hypothetical statements were material implications, the statements would be not a quirkish oddity, but a linguistic sobriety and a simple truth. Finally, we may note that 4 if ' can be employed not simply in making statements, but in, e.g., making provisional announcements of intention (e.g., * If it rains, I shall stay at home ') which, like unconditional announcements of intention, we do not call true or false but describe in some other way. If the man who utters the quoted sentence leaves home in spite of the rain, we do not say that what he said was false, though we might say that he lied (never really intended to stay in) ; or that he changed his mind. There are further uses of * if ' which I shall not discuss. 1 See Chapter 7, Part I.

The safest way to read the material implication sign is, perhaps, * not both . . . and not . . ,'. 10. The material equivalence sign * s ' has the meaning given by the following definition : *p ff'=D/'(pDff).(sOj)' and the phrase with which it is sometimes identified, viz., * if and only if ', has the meaning given by the following definition : 4 p if and only if q * = D/ * if p then g, and if q then p \ Consequently, the objections which hold against the identification of * p D q ' with * if p then q ' hold with double force against the identification of * p s q * with * p if and only if q '.