Conditionals and Action Logics

The logic of conditionals (and, especially, of subjunctive or counterfactual conditionals) is a long-standing problem for the theory of common sense reasoning and philosophical logic. In this paper, I explore commonalities between the logical accounts of the conditional involving “closeness” relations over possible worlds and work in the logic of action and change that attempts to characterize the situation that results from the performance of an action. I will try to show that the latter approaches readily generalize to the case of conditionals (with suitably restricted antecedents), and that they fill a gap in the earlier, more abstract work on conditional logic. This is an abbreviated version for the CSR-07 workshop. Stalnaker’s and Lewis’ conditional logics Counterfactual conditionals are a prominent and pervasive part of common sense reasoning,1 but accounting for their meaning has been a long-standing problem in philosophy and logic. In the late 1960s (Stalnaker 1968; Stalnaker & Thomason 1970) and early 1970s (Lewis 1973), theories of conditionals were developed using the possible worlds approach to modal logic. These theories appealed to “closeness” relations over possible worlds. Stalnaker’s semantics in effect posited a well-ordering w over worlds with respect to an initial world w, with w as a least element. A conditional φ>ψ is true at w if and only if ψ is true at the closest world to w in which φ is true, if there is any such world; otherwise, φ > ψ is true at w. To put it another way, a Stalnaker frame includes a selection function s that for each world w and antecedent φ selects the world s(w, φ) that results from changingw so as to make φ true. The completeness theorem in (Stalnaker & Thomason 1970) shows how to recover appropriate well-orderings in canonical models of the axioms for conditional logic provided in that paper. Lewis’ semantics is more complicated; the chief difference between it and Stalnaker’s is that (φ>ψ) ∨ (φ>¬ψ) (Conditional Excluded Middle) is valid in Stalnaker’s logic, Copyright c © 2007, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. Thanks to the referees of this paper for helpful comments. (McCarthy & Costello 1999) makes this point well. but is invalid in Lewis’.2 Although there has been much discussion in the subsequent literature over the details of the semantics, and especially over the semantic differences between indicative and subjunctive conditionals, logics along these lines remain the standard approach to subjunctive conditionals. Many people (if not a majority) still defend Stalnaker’s original claim that there is one semantics for all conditionals, and that the differences between subjunctive and indicative conditionals are pragmatic. But even if these people are right, the differences between indicative (or epistemic) and subjunctive (or causal) conditionals are significant, and play an important role in reasoning. In this paper, I will work with Stalnaker’s semantics and will confine myself to subjunctive or causal conditionals.3 But a good deal of what I will say could be adapted to other logics of conditionals. An epistemological problem Work on causal conditionals prior to WW2 consisted, for the most part, of attempts to provide a philosophical analysis. In a landmark paper (Goodman 1955) dating to 1946, Nelson Goodman presented the problems confronting such an analysis so devastatingly that these attempts were largely abandoned. I don’t have space here to examine Goodman’s arguments. They deal mainly with a single example—‘If that match had been scratched it would have lighted’—and show that attempts to provide an analysis either bog down or prove to be circular. The problems that Goodman raises are, I believe, genuine, although his methodological assumptions are flawed in some fundamental ways, and he tends to exaggerate the difficulties. The logical solutions of the 1960s and 1970s end-run Goodman’s problem, rather than meeting it head-on. They provide abstract conditions on modal models for conditionals rather than analyses. Actually constructing the Stalnaker Therefore, Lewis uses a partial order over worlds, while Stalnaker uses a total order. The complexity of Lewis’ semantic conditions have to do with his treatment of the “limit assumption” and are not relevant to the present project. Although these conditionals are connected with subjunctive mood in some languages, the connection is tenuous in English. From here on, I will call them “causal conditionals.” selection function for a particular domain would raise problems like Goodman’s, but this construction is not needed to define a notion of logical consequence. You could say (and many of the people who adopted these theories did say) that they provide a solution to the logical problem of conditionals, but avoid the epistemological problem of conditionals. Despite the success of the logical theories, I believe that most of us involved in this episode realized that there was a gap. For instance, a number of papers were published criticizing the logical theories by pointing out that conditionals do not align with the closest world, if “closest” is measured by apparent overall similarity. (On this interpretation, worlds that could not be perceptually distinguished would be very close; worlds that are very different in many important ways would be far apart.) Since small actions can have sweeping effects, this crude notion of closeness will obviously not work for causal conditionals. The obvious answer to this objection is that the closeness that is appropriate for causal conditionals is not based on superficial similarity. But if you try to supplement this with a positive, specific, constructive account of the correct notion of “similarity,” you re-encounter Goodman’s problems. You can try (and people have tried) to produce such an account using philosophical methods, but I believe that this project exceeds the capacity of the methods that have been traditionally used in philosophy, and that still prevail in the field. The literature in AI and CSR AI has its own (much smaller) literature on conditionals; (Ginsberg 1986; Ortiz, Jr. 1999b; 1999a) are especially noteworthy. I do not have space here to discuss this work; it contains many good ideas, but I believe that it is flawed by a failure to carefully distinguish between causal and epistemological conditionals. And without making this distinction, it is hard to see the relevance of work on action and change to conditionals. As far as I know, the closest work in the AI literature to the approach I adopt here is (Pearl 2000). Pearl uses a very different, statistical framework, and the similarities lie mainly in the basic motivating ideas. Using an action logic to model simple