The Connectives

7. “If”

Chapter 7 “If” The material conditional has an ingratiating perspicuity: we know just where we are with it. But where we are with it seems, on the whole, not where we ought to be. Nerlich [1971], p. 162.§7.1 CONDITIONALS 7.11 Issues and Distinctions This chapter addresses not only some of logical and semantic issues raised by ‘if then. . . ’, but also the behaviour of related connectives which have been proposed to formalize various special relations of implication and entailment. We associate the label conditionals with the former project – to be pursued in much of the present section – and tend to use implication for the latter. Such a division is not without problems, but will do well enough for imposing structure on the material in terms of motivating considerations. When these amount to an interest in how “if ” behaves in English, they issue in a logic of conditionals, while attempts at providing a connectival treatment of what in the first instance is most naturally thought of as a relation between propositions or sentences, we have an implicational logic: a logic for ‘that implies that. . . ’. As it happens, material implication as well as strict implication will, their names notwithstanding, receive attention in the present section (7.13, 7.15, respectively). Intuitionistic (or ‘positive’), relevant, and ‘contractionless’ forms of implication will occupy us in §7.2, the first also being touched on in 7.14 here. Many important topics under the general heading of implication will receive scant attention or no attention at all. These include connexive implication, mentioned only in passing in 7.19, and analytic implication (see Parry [1968], [1986], and Smiley [1962b]). (For comparative remarks on these – and the associated doctrines of ‘connexivism’connexive implication and ‘conceptivism’ – and relevant implication, see Routley, Plumwood, Meyer and Brady [1982].) Implicational statements and conditionals have converses; conjoining one with its converse gives an equivalence statement or biconditional, respectively, and these we shall discuss in §7.3. (“Statement”, here, means the linguistic expression of a proposition; as noted in 1.15, propositions themselves do not have converses.) 925 926 CHAPTER 7. “IF” The main stimulus for much of the formal work on conditionals has undoubtedly been over the claim that the inferential behaviour of at least indicative conditionals (to use a term whose sense will be explained presently) is adequately captured by the behaviour of → according to an →-classical consequence relation , to put the matter proof-theoretically. (A good deal of the opposition to a treatment of indicative conditions by means of → construed in terms of the broader class of →-intuitionistic consequence relations – as defined on p. 329 – would go through similarly; we will not take a special interest in IL as opposed to CL on this matter.) Speaking more semantically: that much if not all there is to be said about the interpretation of if then. . . is given by the familiar truth-table account (i.e., by the condition of →-booleanness on valuations ). These formulations are not quite equivalent but either would do as a way of making precise the contested claim – see the opening quotation above, from Nerlich, as well as that from Haiman below – that material implication gives the logic of the indicative conditional. The ‘paradoxes of material implication’ are much appealed to in opposition to such a claim, or indeed to a variant in which “→-classical” is replaced in the above formulation by “→-intuitionistic”. (These appear as (P+ ) and (P− ) in 7.13 below.) This replacement would lead us to a pattern of discussion of if along the lines of the discussion of and and or in the preceding chapters: looking at the natural deduction introduction and elimination rules for →. We do not pursue this strategy, however, because it turns out that many differences between distinct proposals hang not on the rules as such so much as on what the appropriate logical framework is taken to be, how careful one is with assumption-dependencies, and so on. (See 7.13 for more on this.) But let us recall that Modus Ponens has come under fire, as mentioned in the Digression on p. 528, as well as Conditional Proof in the relatively framework independent simple form that if one can derive B from A, one is entitled to assert “If A then B” outright (or as depending on such any additional assumptions used in that derivation). See 7.18.2, p. 1036, for one example of this. Many problematic inference patterns slightly less obvious than the paradoxes...