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If P, Then P! Or: a Crisis of Identity

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If P, Then P! Or: a Crisis of Identity If p, then p! Or: A crisis of Identity Matthew Mandelkern* Draft of July 22, 2019 All the dialecticians in common say that a conditional is sound when its finisher follows from its leader. But on the question of when it follows, and how, they disagree with one another. Sextus Empiricus: AM VIII.112 Abstract The Identity principle says that conditionals with the form pIf p, then pq are logical truths. Identity is overwhelmingly plausible, and has rarely been explicitly chal- lenged. But a wide range of conditionals nonetheless invalidate it. I explain the problem, and argue that the culprit is the principle known as Import-Export, which we must thus reject. I then explore how we can reject Import-Export in a way that still makes sense of the intuitions that support it, arguing that the differences between indicative and subjunctive conditionals play a key role in solving this puzzle. 1 Introduction Sextus Empiricus’s summation in the epigraph remains apt. Clearly a conditional in some sense says that the consequent follows from the antecedent; there remains a * All Souls College, Oxford, OX1 4AL, United Kingdom, [email protected]. Many thanks to audiences at Oxford, NYU, MCMP, and UCL, and to Andrew Bacon, Kyle Blumberg, Susanne Bobzien, David Boylan, Sam Carter, Ivano Ciardelli, Nilanjan Das, Kevin Dorst, Kit Fine, Branden Fitelson, Vera Flocke, Simon Goldstein, Jeremy Goodman, Ben Holguín, Justin Khoo, Cameron Domenico Kirk-Giannini, Angelika Kratzer, Vann McGee, Eliot Michaelson, Jonathan Phillips, Milo Phillips-Brown, Daniel Rothschild, Ian Rumfitt, Bernhard Salow, Philippe Schlenker, Ginger Schultheis, and Robert Stalnaker for very helpful feedback and discussion. Thanks especially to Cian Dorr for valuable discussion that helped shape key points in the second half of the paper. 1 great deal of controversy about what kind of following is involved.1 But plausibly, on any reasonable way of cashing out “following”, any sentence will follow from itself. And so, given any way of making precise this broad way of thinking about conditionals, sentences of the form pIf p, then pq should be logical truths. In the first part of this paper, I show that, despite the overwhelming plausibility of the principle that pIf p, then pq is always a logical truth—the Identity principle—a wide variety of theories of the conditional invalidate it. I then argue that the culprit behind this failure is the Import-Export principle, which says that pIf p, then if q, then rq and pIf p and q, then rq are always materially equivalent. I show that there is a deep and surprising tension between Import-Export, on the one hand, and, on the other, Identity (as well as its more general cousin Logical Implication, which says that pIf p, then qq is always true when p logically entails q). In light of the overwhelming plausibility of the latter principles, we should thus reject Import-Export. In the second part of the paper, I explore how to reject Import-Export while still accounting for the intuitive evidence that supports it. Surprisingly, intuitions concerning Import-Export seem to diverge for indicatives versus subjunctives: we find concrete counterexamples to Import-Export for subjunctives, but apparently not for indicatives. To account for this, I propose a local implementation of a widely accepted account of the difference between indicatives and subjunctives, on which indicatives, but not subjunctives, presuppose that the closest antecedent-world is epistemically possible. I show that on the resulting account, Import-Export is logically invalid for both indicatives and subjunctives—as desired—but it still holds for indicatives in a more limited sense: namely, in all the cases where an indicative’s epistemic presupposition is satisfied. I argue that this is the best way to account for intuitions in this area while still validating Logical Implication. 1 On the material view, the consequent must follow in the sense of being true at the actual world if the antecedent is (Grice 1989, Lewis 1976, Jackson 1987, Williamson 2019); on the strict view, in the sense of being true at all accessible antecedent worlds (Warmbrod 1981a,b, von Fintel 1999, 2001, Gillies 2007); on the variably strict view, in the sense of being true at all closest antecedent worlds (Stalnaker 1968, Stalnaker & Thomason 1970, Lewis 1973); on probabilistic views, in the sense of being suitably probable on the antecedent (Adams 1975, Edgington 1986). 2 2 A crisis of Identity Identity, again, says that sentences with the form pIf p, then pq are logical truths. Identity is a corollary of a more general principle, Logical Implication, which says that, when p logically entails q, then pIf p, then qq is a logical truth (‘logical entailment’ is, here and throughout, understood in its classical sense: preservation of truth in all intended models). In this section I will focus on Identity; later in the paper I will return to the more general Logical Implication principle. Identity is one of the most natural, and least controversial, principles in the logic of the conditional. Arló-Costa & Egré(2016) call it ‘constitutive of the very notion of conditional’. This seems correct: it seems beyond doubt that, if p holds, then p holds. I will begin by showing that, despite Identity’s plausibility—and the lack of explicit challenges to it in the literature—Identity is invalidated by a wide range of current theories of the conditional, in particular those (apart from the material conditional) which validate the Import-Export principle. I will henceforth work with a simple propositional language with atoms A;B;C :::, Boolean connectives ‘^’, ‘:’, ‘_’; the material conditional ‘⊃’ (so p ⊃ q abbreviates :p_q), and the material biconditional ‘≡’ (so p ≡ q abbreviates (p ⊃ q)^(q ⊃ p)). Finally, we have a conditional connective ‘>’; later in the paper I will distinguish the indicative conditional connective ‘>i’ from the subjunctive one ‘>s’, but for now I use just one connective ‘>’ which ranges over both indicatives and sub- junctives. Lower-case italics range over sentences; for readability I use sentences autonymously. Where G is a set of sentences of our language, ‘G j= p’ means that G logically entails p, in the standard sense that p is true in every intended model where all the elements of G are true. We write ‘p j= q’ for ‘fpg j= q’; and ‘j= p’ for ‘? j= p’, i.e. to say that p is a logical truth. I emphasize that the use of a formal language is just to facilitate readability, so p > q is just an abbreviation of pIf p, then 2 qq. With this notation in hand, I restate the three key principles which will play a central role in what follows, using ‘!’ as a meta-language material conditional: 2 Kratzer(1986) famously argues that it is a mistake to treat ‘if’ as a two-place connective, as I will here. But Khoo(2013) convincingly showed that this question about the syntax of conditionals does not bear on results of the kind I will be discussing here; a parallel argument to Khoo’s shows that my points go through regardless of the syntax of conditionals. 3 • Logical Implication (LI): p j= q ! j=p > q • Identity: j= p > p • Import-Export (IE): j= (p > (q > r)) ≡ ((p ^ q) > r) LI and Identity are self-explanatory. IE is a bit more complicated: it says that what we do with two successive conditional antecedents is the same as what we do with the corresponding conjunctive antecedent. So, for instance, IE says that pairs like the following are generally equivalent: (1) a. If the coin is flipped, then if it lands heads, we’ll win. b. If the coin is flipped and it lands heads, we’ll win. And likewise for the subjunctive version: (2) a. If the coin had been flipped, then if it had landed heads, we would have won. b. If the coin had been flipped and it had landed heads, we would have won. All three of these principles are prima facie plausible. The plausibility of LI and Identity is, I take it, manifest; the plausibility of IE comes from the felt equivalence of the pairs in (1) and (2). Later on we will explore in more detail the case for and against each of these principles. With this background on the table, let me turn to the central claim of this section: that all existing theories of the conditional, apart from the material conditional, which validate IE also invalidate Identity (and thus LI). To explain this point, start by thinking about what it takes to validate IE. IE says, in essence, that information in subsequent antecedents is agglomerated: a conditional with two antecedents is evaluated in the same way as a conditional with one corresponding conjunctive antecedent. That means that, to validate IE, we need some way of “remembering” successive conditional antecedents. To see why, suppose instead we adopt a classic variably strict view like that of Stalnaker 1968, which does not have a mechanism to do this. On Stalnaker’s view, p > q is true just in case q is true at the closest p-world (see §6 for more exposition). So p > (q > r) says that r is true at the closest q-world 4 to the closest p-world. By contrast, (p ^ q) > r says that r is true at the closest p ^ q-world. A little reflection shows that these truth-conditions are orthogonal, and so IE is invalid. What we need to validate IE, instead, is some way of keeping track of successive conditional antecedents, and then using these together to evaluate the most deeply-embedded consequent. Different IE-validating theories of the conditional have different mechanisms for doing this.
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