Counterpossibles in Semantics and Metaphysics Timothy Williamson University of Oxford Abstract This paper defends from recent objections and misunderstandings the orthodox view that subjunctive conditionals with impossible antecedents are true. It ex- plains apparent counterexamples as cases where a normally reliable suppositional heuristic for assessing conditionals gives incorrect results, which some theorists take at face value. Keywords: conditionals, counterfactuals, counterpossibles, supposition, heuristics. If whales were fish, their behaviour would differ from what it actually is. If whales were fish, their behaviour would be just as it actually is. Those sound like genuine alternatives. Yet, since whales are by nature mammals, they pre- sumably could not have been fish; that would be contrary to their nature. Thus both conditionals are counterpossibles, counterfactual conditionals with impos- sible antecedents. Semantic orthodoxy makes all counterpossibles true. So the two conditionals were true, and not mutually exclusive after all. Is orthodoxy about counterpossibles correct? The problem is not just how best to tidy up an unimportant little corner of the logic and semantics of counterfactuals. It has significant theoretical and methodological ramifications in several directions. This paper defends orthodoxy against recent objections, and explains recalci- trantly unorthodox appearances by our pre-reflective reliance on a fallible heu- ristic in assessing conditionals. 1. What is at Stake A counterfactual is a conditional sentence like the so-called subjunctive ‘If this were so, that would be so’ and unlike the indicative ‘If this is so, that is so’. Typ- ically, we use counterfactuals to talk about what would have happened if some- thing had been different from how it actually was. Still, despite the etymology, a counterfactual may have a true antecedent; ‘If she were depressed, that would explain her silence’ does not imply that she is not depressed. But a counterpossi- ble is a counterfactual whose antecedent is impossible, so false. Argumenta 2, 2 (2017): 195-226 © 2017 University of Sassari ISSN 2465-2334 DOI 10.23811/42.arg2017.wil 196 Timothy Williamson What kind of impossibility is relevant? It is not epistemic. For consider this counterfactual: (1) If thinking had never occurred, science would have flourished. The antecedent of (1) is epistemically impossible, because it is incompatible with something we know: that we think. But that does not make the antecedent of (1) impossible in the relevant sense. Presumably, the universe could have been life- less and thoughtless forever. In that case, science would not have flourished. Defenders of orthodoxy should agree that (1) is false. The special theoretical problem in evaluating ‘If this had been so, that would have been so’ arises when this could not have been so. The relevant sort of possibility is objective rather than epistemic or subjective. Moreover, what matters is the most inclusive sort of objective possibility, which we may call metaphysical possibility. For the special theoretical problem in evaluating a counterfactual does not arise when, although the antecedent could not easily have been so, it could still have been so.1 Fortu- nately, the main issues about counterpossibles are not too sensitive to the precise shade of modality at issue. Some subjunctive conditionals arguably have an epistemic reading (Edging- ton 2008, Vetter 2016). This paper is mainly concerned with non-epistemic read- ings of subjunctive conditionals, though some of its conclusions may generalize to epistemic readings too. It is sometimes hard to tell whether one is reading a conditional epistemically or not; the reader may wish to bear that issue in mind. One view is that counterpossibles suffer some sort of presupposition failure (von Fintel 1998). If the presupposition were semantic, they might even lack a truth-value. However, that is too drastic. In explaining how one can tell that whales are not fish, someone might say ‘If whales were fish, they would behave quite differently’. That utterance is felicitous even if it is common ground in the conversation that whales could not really have been fish. Of course, once it is common ground that the antecedent of a counterfactual is impossible, the condi- tional will often lose any conversational relevance it might previously have had, and so be infelicitous to utter. Thus there may be a defeasible pragmatic presup- position that the antecedent is possible, but it would not enter the semantics. In discussing the semantics of the subjunctive conditional, I will treat it as a sentential operator, making a complex sentence α □→ β out of simpler sentences α and β. That may well be an over-simplification. On Kratzer’s influential ac- count (Kratzer 2012), ‘if’ restricts other sentential operators rather than being one in its own right. The arguments of this paper can be transposed to that more sophisticated setting. It is convenient, though not crucial, to put the problem of counterpossibles in terms of possible worlds. We read ‘possible’ as metaphysically possible, the rele- vant standard for present purposes. The orthodox evaluation of the counterfac- tual α □→ β depends on the truth-value of β at relevant possible worlds at which α is true. But what happens if α is true at no possible worlds? 1 For more on the conception of metaphysical possibility as the most inclusive sort of objective possibility see Williamson 2016. One consequence discussed there is that, where ◊ expresses metaphysical possibility, the S4 principle ◊◊α É ◊α holds (contrary to Salmon 1989), for if ◊ expresses a sort of objective possibility, so does ◊◊. Corresponding- ly, □□α is no stronger than □α for this modality. Counterpossibles in Semantics and Metaphysics 197 The classic semantic accounts of counterfactuals, going back to Stalnaker (1968) and Lewis (1973), use the framework of compositional intensional se- mantics, and more specifically of possible worlds semantics (for simplicity in what follows, we hold fixed parameters such as the context and the assignment of values to variables, and leave them tacit). In this setting, we can explain why it is so natural to make all counterpossibles true. It is no optional peculiarity of Stalnaker’s semantics, or Lewis’s. We equate the intension |α| of a sentence α with the set of possible worlds at which α is true. By hypothesis, the intension of a counterfactual is a function of the intensions of its antecedent and consequent: (2) |α □→ β| = f(|α|, |β|) We can be more specific, for all that should matter about the consequent is at which possible worlds where the antecedent is true the consequent is also true, in other words, the intersection of the intension of the antecedent with the inten- sion of the consequent. That is just the restricting effect of the antecedent. On that assumption, (2) implies (3): (3) |α □→ β| = f(|α|, |α|∩|β|) The truth-value of the consequent at worlds where the antecedent is false should be irrelevant to the truth-value of the conditional, for it concerns only what would hold if its antecedent held. But if the antecedent is true at no possible worlds, then the consequent is true at no possible worlds at which the anteced- ent is true; (3) yields (4): (4) If |α| = {} then |α □→ β| = f({}, {}) Given (4), all counterpossibles have the same intension: they are indiscriminate. That does not yet decide between making them all true and making them all false. However, we want any counterfactual (counterpossible or not) whose consequent merely repeats its antecedent to be a trivial necessary truth, true throughout the set of all possible worlds W: (5) |α □→ α| = W Together, (4) (with β = α) and (5) require f({}, {}) to be W. Putting that back into (4), we get: (6) If |α| = {} then |α □→ β| = W In other words, all counterpossibles are necessary truths, and so truths. This is just the conclusion reached by Stalnaker, Lewis, and their successors in the mainstream of intensional semantics.2 Similar arguments can be made in the modal object-language, without ref- erence to worlds. For example, instead of (2) we can just require that counterfac- tuals with necessarily equivalent antecedents and necessarily equivalent conse- quents are themselves necessarily equivalent: (7) (□(α ≡ α*) ˄ □(β ≡ β*)) É □((α □→ β) ≡ (α* □→ β*)) 2 For Stalnaker (1968) there is a small wrinkle. For convenience, he postulates an impos- sible ‘absurd world’ λ where everything is true, and equates the truth-value of a counter- possible with the truth-value of its consequent at λ (the only world where its antecedent is true). However, the effect is the same, because the consequent is automatically true at λ. His semantics is still compositional with respect to intensions defined over possible worlds, because λ does not discriminate between sentences with the same intension. 198 Timothy Williamson A plausible auxiliary assumption to complete the argument is simply that con- junctions necessarily counterfactually imply their conjuncts (if this and that were so, this would be so): (8) □((α ˄ β) □→ α) Together, (7) and (8) imply (9), the analogue in the object-language of (6):3 (9) □¬α É □(α □→ β) A much simpler argument in the object-language for the truth of counterpossi- bles just uses a plausible and attractive assumption linking metaphysical modali- ty to counterfactuals. It is that strict implication materially implies counterfactu- al implication: (10) □(α É β) É (α □→ β) If α could not hold without β, α would not hold without β. Then, by elementary modal logic, an impossibility strictly implies anything: (11) □¬α É □(α É β) By transitivity, (10) and (11) entail (12): (12) □¬α É (α □→ β) If one assumes the necessitation of (10), one can also derive the necessitation of (12). Elsewhere, I have used (10) in deriving equivalents of metaphysical modali- ties in counterfactual terms, as part of an argument for understanding our cogni- tive capacities for handling metaphysical modalities as a by-product of our cog- nitive capacities for handling counterfactual conditionals (Williamson 2007: 156-77).