How to evaluate counterfactuals in the quantum world

In the article I discuss possible amendments and corrections to Lewis’s semantics for counterfactuals that are necessary in order to account for the indeterministic and non-local character of the quantum world. I argue that Lewis’s criteria of similarity between possible worlds produce incorrect valuations for alternate-outcome counterfactuals in the EPR case. Later I discuss an alternative semantics which rejects the notion of miraculous events and relies entirely on the comparison of the agreement with respect to individual facts. However, a controversy exists whether to include future indeterministic events in the criteria of similarity. J. Bennett has suggested that an indeterministic event count toward similarity only if it is a result of the same causal chain as in the actual world. I claim that a much better agreement with the demands of the quantum-mechanical indeterminism can be achieved when we stipulate that possible worlds which differ only with respect to indeterministic facts that take place after the antecedent-event should always be treated as equally similar to the actual world. In the article I analyze and dismiss some common-sense counterexamples to this claim. Finally, I critically evaluate Bennett’s proposal regarding the truth-conditions for true-antecedent counterfactuals.     

A graded semantics for counterfactuals

This article presents an extension of Lewis’ analysis of counterfactuals to a graded framework. Unlike standard graded approaches, which use the probabilistic framework, we employ that of many-valued logics. Our principal goal is to provide an adequate analysis of the main background notion of Lewis’ approach—the one of the similarity of possible worlds. We discuss the requirements imposed on the analysis of counterfactuals by the imprecise character of similarity and concentrate in particular on robustness, i.e., the requirement that small changes in the similarity relation should not significantly change the truth value of the counterfactual in question. Our second motivation is related to the logical analysis of natural language: analyzing counterfactuals in the framework of many-valued logics allows us to extend the analysis to counterfactuals that include vague statements. Unlike previous proposals of this kind in the literature, our approach makes it possible to apply gradedness at various levels of the analysis and hence provide a more detailed account of the phenomenon of vagueness in the context of counterfactuals. Finally, our framework admits a novel way of avoiding the Limit Assumption, keeping the core of Lewis’ truth condition for counterfactuals unchanged.

     

A powers theory of modality: or,how I learned to stop worrying and reject possible worlds

Possible worlds, concrete or abstract as you like, are irrelevant to the truthmakers for modality—or so I shall argue in this paper. First, I present the neo-Humean picture of modality, and explain why those who accept it deny a common sense view of modality. Second, I present what I take to be the most pressing objection to the neo-Humean account, one that, I argue, applies equally well to any theory that grounds modality in possible worlds. Third, I present an alternative, properties-based theory of modality and explore several specific ways to flesh the general proposal out, including my favored version, the powers theory. And, fourth, I offer a powers semantics for counterfactuals that each version of the properties-based theory of modality can accept, mutatis mutandis. Together with a definition of possibility and necessity in terms of counterfactuals, the powers semantics of counterfactuals generates a semantics for modality that appeals to causal powers and not possible worlds.     

Counterfactuals and Spatiotemporal Events

One of the basic assumptions of David Lewis's formal semantics of counterfactuals is that the crucial relation of comparative similarity between possible worlds is a linear ordering.Yet there are arguments that when we take into account relativistic features of space-time, this relationshould be only a partial ordering. The first part of the paper deals with the question of how to formulate appropriatetruth conditions for counterfactuals under the supposition of a partial ordering of possible worlds. Such truthconditions will be put forward, and it will be argued that they are more general than those proposed in recentliterature, because they turn out to be applicable also when the so-called Limit Assumption is not met. The secondpart analyzes two relativistically invariant ways of interpreting spatiotemporal counterfactuals with antecedentsreferring to free-chance point events. After briefly examining key differences between these two approaches,the issue of their extension for a broader class of antecedents will be addressed. Following the approach of Finkelstein (1999), who has proposed a specifically designed similarity relation between possible worlds, servingas a generalization tool in the case of one of the above intuitions, the possibility of a similar extension forthe second interpretation will be considered. The main result of the paper is the theorem to the effect that thegeneralization of the second intuition is impossible to obtain. More specifically, the theorem proved in the paperstates that there is no similarity relation which together with the Lewis-style truth conditions for counterfactualswould imply the second of the above interpretations as a special case. Some consequences of thistheorem for the applicability of the Lewis logic of counterfactuals to quantum phenomena will be briefly mentionedat the end of the paper.     

Dispositional Essentialism and the Possibility of a Law-Abiding Miracle

An objection to dispositional essentialism has been that it yields an implausible semantics for counterfactuals if determinism happens to be true. I claim that an adequate remedy to this objection would be to endorse the metaphysical possibility of 'law-abiding miracles', i.e., uncaused spontaneous events. This proposal entails that determinism de jure is meta-physically impossible; I argue that this is not a serious disadvantage. I discuss consequences of the proposal for the logical form of laws, and consider a possible objection to the proposal on epistemic grounds.     

Similarity and cotenability

In this paper I present some difficulties for Lewis’s and similar theories of counterfactuals, and suggest that the problem lies in the notion of absolute similarity. In order to explain the problem, I discuss the relation between Lewis’s and Goodman’s theory, and show that the two theories are not related in the way Lewis thought they were.     

Counterfactual Discourse in Context

The classic Lewis‐Stalnaker semantics for counterfactuals captures that Sobel sequences are consistent sequences, for example:

• a. If Sophie had gone to the parade, she would have seen Pedro dance.
• b. But if Sophie had gone to the parade and been stuck behind someone tall, she would not have seen Pedro dance.

But reverse a sequence like this one and it no longer sounds so good, which is surprising on the classic semantics. This observation motivated Kai von Fintel (2001) and Thony Gillies (2007) to propose dynamic semantic accounts of counterfactual conditionals. Subsequently, Sarah Moss (2012) defended the classic semantics against the charge that it need be abandoned in the face of these order effects, arguing that the infelicity of the reverse sequences is pragmatic. I argue that both accounts are ultimately untenable, but each account has strengths. Seeing what works and what doesn't in each account points the way to the right positive view. With this in mind, I defend a contextualist account of counterfactuals that takes conversational relevance to play a central role.     

Counterfactuals and the fixity of the past

I argue that David Lewis’s attempt, in his ‘Counterfactual Dependence and Time’s Arrow’, to explain the fixity of the past in terms of counterfactual independence is unsuccessful. I point out that there is an ambiguity in the claim that the past is counterfactually independent of the present (or, more generally, that the earlier is counterfactually independent of the later), corresponding to two distinct theses about the relation between time and counterfactuals, both officially endorsed by Lewis. I argue that Lewis’s attempt is flawed for a variety of reasons, including the fact that his own theory about the evaluation of counterfactuals requires too many exceptions to the general rule that the past is counterfactually independent of the present. At the end of the paper, I consider a variant of Lewis’s strategy that attempts to explain the fixity of the past in terms of causal, rather than counterfactual, independence. I conclude that, although this variant avoids some of the objections that afflict Lewis’s account, it nevertheless seems to be incapable of giving a satisfactory explanation of the notion of the fixity of the past.     

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      

Estimating Conditional Chances and Evaluating Counterfactuals

The paper addresses a puzzle about the probabilistic evaluation of counterfactuals, raised by Ernest Adams as a problem for his own theory. I discuss Brian Skyrms’s response to the puzzle. I compare this puzzle with other puzzles about counterfactuals that have arisen more recently. And I attempt to solve the puzzle in a way that is consistent with Adams’s proposal about counterfactuals.     

Interventionist counterfactuals

A number of recent authors (Galles and Pearl, Found Sci 3 (1):151?C182, 1998; Hiddleston, No?s 39 (4):232?C257, 2005; Halpern, J Artif Intell Res 12:317?C337, 2000) advocate a causal modeling semantics for counterfactuals. But the precise logical significance of the causal modeling semantics remains murky. Particularly important, yet particularly under-explored, is its relationship to the similarity-based semantics for counterfactuals developed by Lewis (Counterfactuals. Harvard University Press, 1973b). The causal modeling semantics is both an account of the truth conditions of counterfactuals, and an account of which inferences involving counterfactuals are valid. As an account of truth conditions, it is incomplete. While Lewis??s similarity semantics lets us evaluate counterfactuals with arbitrarily complex antecedents and consequents, the causal modeling semantics makes it hard to ascertain the truth conditions of all but a highly restricted class of counterfactuals. I explain how to extend the causal modeling language to encompass a wider range of sentences, and provide a sound and complete axiomatization for the extended language. Extending the truth conditions for counterfactuals has serious consequences concerning valid inference. The extended language is unlike any logic of Lewis??s: modus ponens is invalid, and classical logical equivalents cannot be freely substituted in the antecedents of conditionals.     

Probabilities of counterfactuals and counterfactual probabilities

Probabilities figure centrally in much of the literature on the semantics of conditionals. I find this surprising: it accords a special status to conditionals that other parts of language apparently do not share. I critically discuss two notable ‘probabilities first’ accounts of counterfactuals, due to Edgington and Leitgeb. According to Edgington, counterfactuals lack truth values but have probabilities. I argue that this combination gives rise to a number of problems. According to Leitgeb, counterfactuals have truth conditions-roughly, a counterfactual is true when the corresponding conditional chance is sufficiently high. I argue that problems arise from the disparity between truth and high chance, between approximate truth and high chance, and from counterfactuals for which the corresponding conditional chances are undefined. However, Edgington, Leitgeb and I can unite in opposition to Stalnaker and Lewis-style ‘similarity’ accounts of counterfactuals.     

On the Treatment of Incomparability in Ordering Semantics and Premise Semantics

In his original semantics for counterfactuals, David Lewis presupposed that the ordering of worlds relevant to the evaluation of a counterfactual admitted no incomparability between worlds. He later came to abandon this assumption. But the approach to incomparability he endorsed makes counterintuitive predictions about a class of examples circumscribed in this paper. The same underlying problem is present in the theories of modals and conditionals developed by Bas van Fraassen, Frank Veltman, and Angelika Kratzer. I show how to reformulate all these theories in terms of lower bounds on partial preorders, conceived of as maximal antichains, and I show that treating lower bounds as cutsets does strictly better at capturing our intuitions about the semantics of modals, counterfactuals, and deontic conditionals.     

Antecedent-Relative Comparative World Similarity

In “Backward Causation and the Stalnaker–Lewis Approach to Counterfactuals,” Analysis 62:191–7, (2002), Michael Tooley argues that if a certain kind of backward causation is possible, then a Stalnaker–Lewis comparative world similarity account of the truth conditions of counterfactuals cannot be sound. In “Tooley on Backward Causation,” Analysis 63:157–62, (2003), Paul Noordhof argues that Tooley’s example can be reconciled with a Stalnaker–Lewis account of counterfactuals if the comparative world similarity relation on which the Stalnaker–Lewis account relies is allowed to be antecedent-relative. In this paper I show that taking comparative world similarity to be antecedent-relative results in a formal semantics which is a comparative world similarity semantics in name only.     

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.     

A Peculiarity in Pearl’s Logic of Interventionist Counterfactuals

We examine a formal semantics for counterfactual conditionals due to Judea Pearl, which formalizes the interventionist interpretation of counterfactuals central to the interventionist accounts of causation and explanation. We show that a characteristic principle validated by Pearl’s semantics, known as the principle of reversibility, states a kind of irreversibility: counterfactual dependence (in David Lewis’s sense) between two distinct events is irreversible. Moreover, we show that Pearl’s semantics rules out only mutual counterfactual dependence, not cyclic dependence in general. This, we argue, suggests that Pearl’s logic is either too weak or too strong.     

Quasi-miracles,typicality, and counterfactuals

If one flips an unbiased coin a million times, there are 2^1,000,000 series of possible heads/tails sequences, any one of which might be the sequence that obtains, and each of which is equally likely to obtain. So it seems (1) ‘If I had tossed a fair coin one million times, it might have landed heads every time’ is true. But as several authors have pointed out, (2) ‘If I had tossed a fair coin a million times, it wouldn’t have come up heads every time’ will be counted as true in everyday contexts. And according to David Lewis’ influential semantics for counterfactuals, (1) and (2) are contradictories. We have a puzzle. We must either (A) deny that (2) is true, (B) deny that (1) is true, or (C) deny that (1) and (2) are contradictories, thus rejecting Lewis’ semantics. In this paper I discuss and criticize the proposals of David Lewis and more recently J. Robert G. Williams which solve the puzzle by taking option (B). I argue that we should opt for either (A) or (C).     

In Defense of a Probability Based Semantics for Counterfactuals

In a recent paper Lee Walters criticizes a number of philosophers – including Gundersen – for committing a ‘failure in the argumentative strategy’ when they attempt to amend the standard Lewis semantics for counterfactuals in order to avoid the so‐called principle of Conjunction Conditionalization. In this article we defend a Gundersen‐style probability‐based semantics against Walter's major misgivings: that it is not logically conservative, that it is committed to the Connection Hypothesis, and that it cannot deal satisfactory with irrelevant semi‐factuals.     

An Extended Lewis/Stalnaker Semantics and the New Problem of Counterpossibles

Abstract

Closest-possible-world analyses of counterfactuals suffer from what has been called the ‘problem of counterpossibles’: some counterfactuals with metaphysically impossible antecedents seem plainly false, but the proposed analyses imply that they are all (vacuously) true. One alleged solution to this problem is the addition of impossible worlds. In this paper, I argue that the closest possible or impossible world analyses that have recently been suggested suffer from the ‘new problem of counterpossibles’: the proposed analyses imply that some plainly true counterpossibles (viz., ‘counterlogicals’) are false. After motivating and presenting the ‘new problem’, I give reasons to think that the most plausible objection to my argument is not compelling.     

Semantic Externalism and Presentism

Abstract

In this paper I discuss an unconventional form of presentism which, I claim, captures better than all other versions of the doctrine the fundamental notion underpinning it, namely, the notion that ‘only what is present is real’. My proposal is to take this maxim as stating, not the rather uncontroversial view that past things are not real now, but the more radical idea that they never were. This rendition of presentism is, I argue, the only one that is neither trivial nor absurd. I examine this proposal by considering it against a sceptical hypothesis that bears similarities to it, viz., the hypothesis that the world was created five minutes ago. On this hypothesis, the past, all but five minutes of it, is unreal, in precisely the sense in which the presentism I discuss claims it is. I show that, assuming semantic externalism, this sceptical hypothesis cannot be sustained, but that a somewhat weaker hypothesis, the Creationist hypothesis that the world is 5,768 years old, cannot be refuted. Together, these conclusions enable a demarcation of those presentist intuitions that language and thought tolerate and those they do not.