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Minds and Machines

Erschienen in:

01.03.2013

A Lewisian Logic of Causal Counterfactuals

verfasst von: Jiji Zhang

Erschienen in: Minds and Machines | Ausgabe 1/2013

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Abstract

In the artificial intelligence literature a promising approach to counterfactual reasoning is to interpret counterfactual conditionals based on causal models. Different logics of such causal counterfactuals have been developed with respect to different classes of causal models. In this paper I characterize the class of causal models that are Lewisian in the sense that they validate the principles in Lewis’s well-known logic of counterfactuals. I then develop a system sound and complete with respect to this class. The resulting logic is the weakest logic of causal counterfactuals that respects Lewis’s principles, sits in between the logic developed by Galles and Pearl and the logic developed by Halpern, and stands to Galles and Pearl’s logic in the same fashion as Lewis’s stands to Stalnaker’s.

Vorheriger Artikel Preferential Semantics for Plausible Subsumption in Possibility Theory

Nächster Artikel Variable-Centered Consistency in Model RB

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The general approach has also been followed by prominent philosophers to illuminate the epistemology of causation (Spirtes et al. 2000) and the nature of causal explanation (Woodward 2003; Woodward and Hitchcock 2003).

As Lewis (1973, p. 3) noted, this slightly ungrammatical reading of φ \(\square\hskip-.35pc\rightarrow\) ψ (instead of if it had been the case that φ, it would have been the case that ψ) was deliberate, in order not to interfere with tense. Moreover, the antecedent is not required to be actually false, so some conditionals under the radar are, properly speaking, not contrary-to-fact conditionals. Like Lewis, I will tolerate this terminological inaptness.

This formulation assumes that there are closest φ-worlds, which Lewis labels the Limit Assumption (1973, pp. 19–20). Lewis’s semantics can be formulated without this assumption, but the present, slightly less general formulation fits our purposes well.

In Lewis’s language, there is no restriction on the form of antecedent, so Ant(L) is just the set of all sentences. But as we will see, the form of antecedent is restricted in the language of causal counterfactuals. In that language, Ant(L) is a proper subset of the set of all sentences.

Stalnaker’s (1968) own formulation is in terms of world-selection function instead of set-selection function, where an absurd world is included to play the role of the empty set.

This approach to modeling interventions originated in econometrics (Strotz and Wold 1960; Fisher 1970), and was masterfully articulated and developed by Pearl (2009).

The notation of [] (and 〈〉 to be introduced later) is obviously borrowed from dynamic logic (e.g., Harel 1979).

It might be tempting to simply dismiss the problem, on the ground that counterfactuals with disjunctive antecedents concern consequences of interventions that are not well defined, and so should not bear truth values under the causal semantics. But some counterfactuals of this sort seem to have as determinate a truth value as any counterfactual can. Consider the firing squad example. Suppose, as it happened, the court decided not to order execution. As a result, neither rifleman shot, and the prisoner survived. The following counterfactual seems clearly true: if rifleman 1 had shot or rifleman 2 had shot, the prisoner would have died.

GP (as well as Halpern) also considered the class of recursive models, a subclass of M _un(S). That class, though important for other purposes, does not need a special attention for the purpose of this paper.

Halpern (2000, p. 326) used a slightly different but equivalent expression in the first conjunct of this schema (his D9).

Halpern (2000, p. 326) seemed to remark in passing that the [] version of composition is also valid in his logic, which, if I understood his remark correctly, was a mistake.

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Titel: A Lewisian Logic of Causal Counterfactuals
verfasst von: Jiji Zhang
Publikationsdatum: 01.03.2013
Verlag: Springer Netherlands
Erschienen in: Minds and Machines / Ausgabe 1/2013
Print ISSN: 0924-6495
Elektronische ISSN: 1572-8641
DOI: https://doi.org/10.1007/s11023-011-9261-z

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