Counterpossibles in Science: The Case of Relative Computability

I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as

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On counterpossibles

The traditional Lewis–Stalnaker semantics treats all counterfactuals with an impossible antecedent as trivially or vacuously true. Many have regarded this as a serious defect of the semantics. For intuitively, it seems, counterfactuals with impossible antecedents—counterpossibles—can be non-trivially true and non-trivially false. Whereas the counterpossible "If Hobbes had squared the circle, then the mathematical community at the time would have been surprised" seems true, "If Hobbes had squared the circle, then sick children in the mountains of Afghanistan at the time would have been thrilled" seems false. Many have proposed to extend the Lewis–Stalnaker semantics with impossible worlds to make room for a non-trivial or non-vacuous treatment of counterpossibles. Roughly, on the extended Lewis–Stalnaker semantics, we evaluate a counterfactual of the form "If A had been true, then C would have been true" by going to closest world—whether possible or impossible—in which A is true and check whether C is also true in that world. If the answer is "yes", the counterfactual is true; otherwise it is false. Since there are impossible worlds in which the mathematically impossible happens, there are impossible worlds in which Hobbes manages to square the circle. And intuitively, in the closest such impossible worlds, sick children in the mountains of Afghanistan are not thrilled—they remain sick and unmoved by the mathematical developments in Europe. If so, the counterpossible "If Hobbes had squared the circle, then sick children in the mountains of Afghanistan at the time would have been thrilled" comes out false, as desired. In this paper, I will critically investigate the extended Lewis–Stalnaker semantics for counterpossibles. I will argue that the standard version of the extended semantics, in which impossible worlds correspond to maximal, logically inconsistent entities, fails to give the correct semantic verdicts for many counterpossibles. In light of the negative arguments, I will then outline a new version of the extended Lewis–Stalnaker semantics that can avoid these problems.     

Remarks on counterpossibles

Since the publication of David Lewis’ Counterfactuals, the standard line on subjunctive conditionals with impossible antecedents (or counterpossibles) has been that they are vacuously true. That is, a conditional of the form ‘If p were the case, q would be the case’ is trivially true whenever the antecedent, p, is impossible. The primary justification is that Lewis’ semantics best approximates the English subjunctive conditional, and that a vacuous treatment of counterpossibles is a consequence of that very elegant theory. Another justification derives from the classical lore than if an impossibility were true, then anything goes. In this paper we defend non-vacuism, the view that counterpossibles are sometimes non-vacuously true and sometimes non-vacuously false. We do so while retaining a Lewisian semantics, which is to say, the approach we favor does not require us to abandon classical logic or a similarity semantics. It does however require us to countenance impossible worlds. An impossible worlds treatment of counterpossibles is suggested (but not defended) by Lewis (Counterfactuals. Blackwell, Oxford, 1973), and developed by Nolan (Notre Dame J Formal Logic 38:325–527, 1997), Kment (Mind 115:261–310, 2006a: Philos Perspect 20:237–302, 2006b), and Vander Laan (In: Jackson F, Priest G (eds) Lewisian themes. Oxford University Press, Oxford, 2004). We follow this tradition, and develop an account of comparative similarity for impossible worlds.     

Counterfactuals: Ambiguities,true premises,and knowledge

In this paper I explore the ambiguity that arises between two readings of the counterfactual construction, then–d and thel–p, analyzed in my bookA Theory of Counterfactuals. I then extend the analysis I offered there to counterfactuals with true antecedents, and offer a more precise formulation of the conception of temporal divergence points used in thel–p interpretation. Finally, I discuss some ramifications of these issues for counterfactual analyses of knowledge.     

The Analytic Truth and Falsity of Disjunctions

Disjunctive inferences are difficult. According to the theory of mental models, it is because of the alternative possibilities to which disjunctions refer. Three experiments corroborated further predictions of the mental model theory. Participants judged that disjunctions, such as Either this year is a leap year or it is a common year are true. Given a disjunction such as Either A or B, they tended to evaluate the four cases in its ‘partition’: A and B, A and not‐B, not‐A and B, not‐A and not‐B, as ‘possible’ or ‘impossible’ in ways that bore out the difference between inclusive disjunctions (‘or both’) and exclusive disjunctions (‘but not both’). Knowledge usually concerns what is true, and so when participants judge that a disjunction is false, or contingent, and evaluate the cases in its partition, they depend on inferences that yield predictable errors. They tended to judge that disjunctions, such as follows: Either the food is cold or else it is tepid, but not both, are true, though in fact they could be false. They tended to infer ‘mirror‐image’ evaluations that yield the same possibilities for false disjunctions as those for true disjunctions. The article considers the implications of these results for alternative theories based on classical logic or on the probability calculus.     

Elusive Counterfactuals

I offer a novel solution to the problem of counterfactual skepticism: the worry that all contingent counterfactuals without explicit probabilities in the consequent are false. I argue that a specific kind of contextualist semantics and pragmatics for would‐ and might‐counterfactuals can block both central routes to counterfactual skepticism. One, it can explain the clash between would‐ and might‐counterfactuals as in: (1) If you had dropped that vase, it would have broken. and (2) If you had dropped that vase, it might have safely quantum tunneled to China. Two, it can explain why counterfactuals like (1) can be true despite the fact that quantum tunneling worlds are among the most similar worlds. I further argue that this brand of contextualism accounts for the data better than other existing solutions to the problem.     

What are impossible worlds?

In this paper, I argue for a particular conception of impossible worlds. Possible worlds, as traditionally understood, can be used in the analysis of propositions, the content of belief, the truth of counterfactuals, and so on. Yet possible worlds are not capable of differentiating propositions that are necessarily equivalent, making sense of the beliefs of agents who are not ideally rational, or giving truth values to counterfactuals with necessarily false antecedents. The addition of impossible worlds addresses these issues. The kinds of impossible worlds capable of performing this task are not mysterious sui generis entities, but sets of structured propositions that are themselves constructed out of possible worlds and relations. I also respond to a worry that these impossible worlds are unable to represent claims about the shape of modal space itself.     

Anti-Normativism Evaluated

Abstract

I argue that recent attempts to show that meaning and content are not normative fail. The two most important arguments anti-normativists have presented are what I call the ‘argument from constitution’ and the ‘argument from guidance’. Both of these arguments suffer from the same basic problem: they overlook the possibility of focusing on assessability by norms, rather than compliance with norms or guidance by norms. Moreover, I argue that the anti-normativists arguments fail even if we ignore this basic problem. Thus, we have not been given good reasons to think that normativism is false.     

A Lewisian Trilemma

According to one reading of the thesis of Humean Supervenience, most famously defended by David Lewis, certain ‘fundamental’ (non‐modal) facts entail all there is but do not supervene on less fundamental facts. However, in this paper I prove that it follows from Lewis' possible world semantics for counterfactuals, in particular his Centring condition, that all non‐modal facts supervene on counterfactuals. Humeans could respond to this result by either giving up Centring or abandoning the idea that the most fundamental facts do not supervene on less fundamental facts. I argue that either response should in general be acceptable to Humeans: the first since there is nothing particularly Humean about Centring; the latter since Humeans should, independently of the result I present, be sceptical that the supervenience of one fact upon another by itself says anything about ‘fundamentality’. ¹      

Is Every Theory of Knowledge False?

Is knowledge consistent with literally any credence in the relevant proposition, including credence 0? Of course not. But is credence 0 the only credence in p that entails that you don't know that p? Knowledge entails belief (most epistemologists think), and it's impossible to believe that p while having credence 0 in p. Is it true that, for every value of ‘x,’ if it's impossible to know that p while having credence x in p, this is simply because it's impossible to believe that p while having credence x in p? If so, is it possible to believe that p while having (say) credence 0.4 in p? These questions aren't standard epistemological fare—at least in part because many epistemologists think their answers are obvious—but they have unanticipated consequences for epistemology. Let ‘improbabilism’ name the thesis that it's possible to know that p while having a credence in p below 0.5. Improbabilism will strike many epistemologists as absurd, but careful reflection on these questions reveals that, if improbabilism is false, then all of the most plausible theories of knowledge are also false. Or so I shall argue in this paper. Since improbabilism is widely rejected by epistemologists (at least implicitly), this paper reveals a tension between all of the most plausible theories of knowledge and a widespread assumption in epistemology.     

Does Conceivability Entail Metaphysical Possibility?

In this paper, we argue that ‘Weak Modal Rationalism’, which is the view that ideal primary positive conceivability entails primary metaphysical possibility, is self‐defeating. To this end, we outline two reductio arguments against ‘Weak Modal Rationalism’. The first reductio shows that, from supposing that ‘Weak Modal Rationalism’ is true, it follows that conceivability both is and is not conclusive evidence for possibility. The second reductio shows that, from supposing that ‘Weak Modal Rationalism’ is true, it follows that it is possible that ‘Weak Modal Rationalism’ is necessarily false, and hence that ‘Weak Modal Rationalism’ is false. We then argue that adopting a weaker position according to which conceivability is merely prima facie evidence for possibility provides limited protection from our criticism of conceivability arguments. 1     

Beyond Existence and Non-Existence

Abstract

When Husserl speaks of the so-called ‘transcendental reduction’ or ‘phenomenological epochē’ many believe that he is eschewing the question of truth or existence. Two reasons are given for this: First, Husserl explicitly states that when we perform the reduction, we should no longer naively ‘accept [the world] as it presents itself to me as factually existing’ (Id I §30, p. 53) and should suspend our judgement with regard to ‘the positing of its actual being’ (Id I §88, p. 182). Second, Husserl seems to have no problem in referring to an ‘object’ of thought even when we refer to non-existent, hallucinatory or indeed impossible objects. This seems to suggest that he is not interested in the question whether or not there is a corresponding ‘ordinary’ object. The paper seeks to question this and will show that his inquiry never loses sight of the questions of truth and existence but rather brings them into the foreground.     

Buridan's Solution to the Liar Paradox

Jean Buridan has offered a solution to the Liar Paradox, i.e. to the problem of assigning a truth-value to the sentence ‘What I am saying is false’. It has been argued that either (1) this solution is ad hoc since it would only apply to self-referencing sentences [Read, S. 2002. ‘The Liar Paradox from John Buridan back to Thomas Bradwardine’, Vivarium, 40 (2), 189–218] or else (2) it weakens his theory of truth, making his ‘a logic without truth’ [Klima, G. 2008. ‘Logic without truth: Buridan on the Liar’, in S. Rahman, T. Tulenheimo and E. Genot, Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the Liar Paradox, Berlin: Springer, 87–112 (Chapter 5); Dutilh Novaes, C. 2011. ‘Lessons on truth from mediaeval solutions to the Liar Paradox’, The Philosophical Quarterly, 61 (242), 58–78]. Against (1), I will argue that Buridan's solution by means of truth by supposition does not involve new principles. Self-referential sentences force us to handle supposition more carefully, which does not warrant the accusation of adhoccery. I will also argue, against (2), that it is exaggerated to assert that this solution leads to a ‘weakened’ theory of truth, since it is consistent with other passages of the Sophismata, which only gives necessary conditions for the truth of affirmative propositions, but sufficient conditions for falsity.     

Book Reviews

Abstract

The aim of this paper is to identify what I take to be the main conceptual problem in Lewis’ semantics for counterfactuals when the Limit Assumption (LA) is not satisfied, what I call the Discrimination Problem (DP), and to present and discuss a modification of Lewis’ semantics that aims at solving DP. First, I outline Lewis’ semantics, highlighting the aspects that will be relevant for our discussion. Second, I present DP and discuss it with a heuristic example. Third, I present the new proposal and comment some formal consequences and the objections they give rise to. Fourth, I present an allegedly equivalent version of the proposal and discuss the difference between counterfactuals and strict conditionals. Finally, I defend that the two proposals are different and argue for the first one as the best way of capturing two intuitions that seem in conflict when the LA is not satisfied. I conclude with a dilemma: either the proposal here defended works or LA is inescapable. Thus, if the proposal doesn’t work, the paper must then be considered as an indirect defense of the Limit Assumption.     

Against Fallibilism

In this paper I argue for a doctrine I call ‘infallibilism’, which I stipulate to mean that If S knows that p, then the epistemic probability of p for S is 1. Some fallibilists will claim that this doctrine should be rejected because it leads to scepticism. Though it's not obvious that infallibilism does lead to scepticism, I argue that we should be willing to accept it even if it does. Infallibilism should be preferred because it has greater explanatory power than fallibilism. In particular, I argue that an infallibilist can easily explain why assertions of ‘p, but possibly not-p’ (where the ‘possibly’ is read as referring to epistemic possibility) is infelicitous in terms of the knowledge rule of assertion. But a fallibilist cannot. Furthermore, an infallibilist can explain the infelicity of utterances of ‘p, but I don't know that p’ and ‘p might be true, but I'm not willing to say that for all I know, p is true’, and why when a speaker thinks p is epistemically possible for her, she will agree (if asked) that for all she knows, p is true. The simplest explanation of these facts entails infallibilism. Fallibilists have tried and failed to explain the infelicity of ‘p, but I don't know that p’, but have not even attempted to explain the last two facts. I close by considering two facts that seem to pose a problem for infallibilism, and argue that they don't.     

Intrinsic Value and the Last Last Man

Even if you were the last person on Earth, you should not cut down all the trees—or so goes the Last Man thought experiment, which has been taken to show that nature has intrinsic value. But ‘Last Man’ is caught on a dilemma. If Last Man is too far inside the anthropocentric circle, so to speak, his actions cannot be indicative of intrinsic value (a Local problem). If Last Man is cast too far outside the anthropocentric circle, though, then value terms lose their cogency (a Global problem). The experiment must satisfy conditions in a seemingly impossible ‘goldilocks’ zone. To this end I propose a new version, the Ultramodal Last Man, which appeals to Routley's work in metaphysics and non‐classical logic. With this ‘Last Last Man’, I argue that the Local /Global dilemma is resolved: impossible equations balance in ultramodal space. For defenders and critics alike, this helps to clarify the demands of intrinsic value, and renews a role for non‐standard logics in value theory.     

How to do philosophy of religion: towards a possible speaking about the impossible

Abstract

It is postulated from different philosophical traditions, and explicitly in recent literature, that there is no further need for doing philosophy of religion – it has become an impossible task. I argue, however, that there remains a philosophical space for this practice and that this space determines greatly how philosophy of religion can be done. The starting point of my argument is the current discussion in the SAJP between De Wet and Giddy and the significance of my article is that it puts this debate within the broader international philosophical context by engaging the work of Trakakis and Desmond to resolve some of the apparently intractable issues raised. Trakakis discusses the divide between the analytic and continental philosophical traditions in which De Wet and Giddy’s work is further contextualized and clarified. Desmond’s work is seminal in its search for a metaxology wherein he advocates a new ‘in between’ position for doing philosophy of religion. I take this view of Desmond further by applying it to the current debate in South Africa and also using it to indicate some possibilities of speaking about the impossible.     

Can Counterfactuals Really Be about Possible Worlds?

The standard view about counterfactuals is that a counterfactual (A > C) is true if and only if the A‐worlds most similar to the actual world @ are C‐worlds. I argue that the worlds conception of counterfactuals is wrong. I assume that counterfactuals have non‐trivial truth‐values under physical determinism. I show that the possible‐worlds approach cannot explain many embeddings of the form (P > (Q > R)), which intuitively are perfectly assertable, and which must be true if the contingent falsity of (Q > R) is to be explained. If (P > (Q > R)) has a backtracking reading then the contingent facts that (Q > R) needs to be true in the closest P‐worlds are absent. If (P > (Q > R)) has a forwardtracking reading, then the laws required by (Q > R) to be true in the closest P‐worlds will be absent, because they are violated in those worlds. Solutions like lossy laws or denial of embedding won't work. The only approach to counterfactuals that explains the embedding is a pragmatic metalinguistic approach in which the whole idea that counterfactuals are about a modal reality, be it abstract or concrete, is given up.     

Counterfactuals as Short Stories

We present an analysis of counterfactuals in terms of stories and combine it with an account similar to Walton’s account of truth in fiction to yield truth conditions for counterfactuals. We discuss unusual features of this account, and compare it to other main approaches. In particular, we argue that our analysis succeeds in accounting for counterpossibles and counterfactuals with true antecedents while the other two main approaches fail, and we give reasons for thinking that it is important to have an adequate account of these two areas.