Williamson on Counterpossibles

J Philos Logic DOI 10.1007/s10992-017-9446-x Williamson on Counterpossibles Francesco Berto1 · Rohan French2 · Graham Priest3 · David Ripley4 Received: 29 October 2016 / Accepted: 1 August 2017 © The Author(s) 2017. This article is an open access publication Abstract A counterpossible conditional is a counterfactual with an impossible antecedent. Common sense delivers the view that some such conditionals are true, and some are false. In recent publications, Timothy Williamson has defended the view that all are true. In this paper we defend the common sense view against Williamson’s objections. Keywords Impossible worlds · Counterpossible conditionals · Nonvacuism · Counterfactual modal epistemology  Francesco Berto Rohan French Graham Priest David Ripley 1 Institute for Logic, Language and Computation (ILLC), University of Amsterdam, Amsterdam, Netherlands 2 Department of Philosophy, School of Philosophical Historical and International Studies, Monash University, Melbourne, Australia 3 CUNY Graduate Center, New York, USA 4 Department of Philosophy, University of Connecticut, Storrs, CT, USA F. Berto et al. 1 Introduction A counterpossible conditional is a counterfactual conditional with an impossible antecedent. According to some theorists, who we will call vacuists, all counterpossibles are true. According to others, who we will call nonvacuists, some counterpossibles are true, and some are false.1 In recent work, Williamson [48, 50] has taken up the cause of vacuism. The purpose of this paper is to evaluate Williamson’s arguments. We will proceed as follows. In Section 2, we recall some motivations for both vacuism and nonvacuism, and sketch a sample nonvacuist semantics for counterfactuals using impossible worlds, to serve as a target for Williamson’s arguments. In Section 3, we present and rebut three arguments Williamson has given against nonvacuist semantics like the one we give. In Section 4, we present and rebut three attempts Williamson has made to undermine the intuitions that provide the most direct support for nonvacuism. In Section 5 we end by arguing that Williamson’s modal epistemology is not only compatible with nonvacuism, but actually leads in its direction. 2 Vacuism and Non-vacuism 2.1 The Consensus We begin by considering the orthodox treatment of counterfactuals, inherited from Kratzer [21], Lewis [24] and Stalnaker [42].2 To evaluate a counterfactual conditional like • If it hadn’t snowed last night, then John’s train wouldn’t have been late we consider the closest3 possible worlds in which it didn’t snow last night, and see whether those are worlds in which John’s train isn’t late. A counterfactual is true just in case all the closest A-worlds are B-worlds. Closeness is understood here as (largely contextually determined) similarity in the relevant respects, usually as minimal variation from the world of evaluation required to get the antecedent to come out true.4 The framework delivers the invalidity of certain (allegedly) intuitively invalid inferences involving counterfactuals, such as transitivity, contraposition, and antecedent strengthening. 1 This leaves out an option: that they are all false. We will not consider this possibility here, but see Kment [18] for discussion. 2 Precursors can be found in Sprigge [41] and Todd [44]. The debate between vacuists and nonvacuists has been with us from the beginnings of this orthodoxy; see for example Goddard and Routley [14, p. 454]’s nonvacuist criticism of Montague [29]’s vacuist treatment of conditionals. 3 This relies on what Lewis [24, pp. 19–20] calls the Limit Assumption: for no world do we have worlds that get closer and closer to it, endlessly. Nothing in the opposition between vacuism and nonvacuism hinges on this. 4 Which aspects of similarity need to count as relevant in order to deliver the intuitively good results is a complicated business: see e.g. chapters 12 and 13 of Bennett [4]. Williamson on Counterpossibles It also delivers vacuism. If A is impossible, there are no A-worlds. Thus, for any B, B is, vacuously, true at all the closest A-worlds; the counterpossible is true. 2.2 Why Nonvacuism? The issues we are to discuss arise when we consider conditionals like the following pair, essentially due to Nolan [30]: (1) If Hobbes had (secretly) squared the circle, all sick children in the mountains of South America at the time would have cared. (2) If Hobbes had (secretly) squared the circle, all sick children in the mountains of South America at the time would not have cared. Squaring the circle is impossible. The set of possible worlds in which Hobbes (secretly) squares the circle is empty. As a result, on the orthodox account, both (1) and (2) are true. This is a surprising result. It is intuitive, we take it, that (1) is false; it’s wrong to think that the children would have cared if Hobbes, per impossibile, had (secretly) squared the circle. Indeed, they wouldn’t even have known. But if (1) is false, then vacuism too is false. This has motivated the construction of nonvacuist semantic theories, which can deliver the intuitive verdict about (1).5 2.3 How Nonvacuism? One usual approach to nonvacuism (see for example [8, 10, 28, 30] amongst others) is to retain the contours of the orthodox account, while dropping the restriction to possible worlds. On such an approach, (1) can be false in the way any false counterfactual is: by having its consequent false at some of the closest worlds where its antecedent is true. Because it is impossible to square the circle, none of these worlds can be a possible world. So these approaches accept impossible worlds as well. In other respects, however, they match the orthodoxy.6 Here, we provide a simple nonvacuist semantic theory along these lines.7 We (the counterstart with a propositional language with connectives factual conditional), and modal operators  and ♦. Let  be the set of propositional parameters, and let  be the set of formulas. An interpretation is a tuple W, P , {RA : A ∈ }, ν, where: • W the set of worlds, 5 Some vacuists, e.g., [24, p. 25], have denied that there are any such intuitions of falsity. But all the arguments we consider here are compatible with the existence of these intuitions. (Of course, vacuists must hold that these intuitions are mistaken, but this is different from denying their existence.) 6 Counterfactual conditionals are hardly the only place where impossible worlds come in handy. Impossible worlds are also helpful in dealing with puzzles concerning content [14, 15] and intentionality [36, 39] generally. Just as with possible worlds, there are a range of views as to the nature and metaphysical status of impossible worlds. We do not enter into this debate here, but see [3, 6, 8, 15, 52]. 7 The following draws on Priest [35, Ch. 5]. F. Berto et al. • • • P ⊆ W is the set of possible worlds, so I = W \ P is the set of impossible worlds, for every formula, A ∈ , RA ⊆ W × W i (...truncated)