Abstract
In this paper we study the logic IVC obtained by adding Lewis-style counterfactual conditionals to intuitionistic propositional logic. Building on recent work by Weiss [21], we first show how to introduce a Lewisian counterfactual operator into intuitionistic Kripke semantics. We then establish a complete axiomatization of the resulting logic.
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Notes
- 1.
This description of Lewis’s view presupposes the limit assumption, i.e., the assumption that for any w and any entertainable \(\varphi \) there be worlds where \(\varphi \) is true and which differ minimally from w in the relevant sense. In our study we will take this assumption for granted, for two reasons. First, this assumption allows for a nice characterization of the semantics in terms of selection functions, and does not affect the resulting propositional logic. Second, there are in fact good conceptual reasons to make the limit assumption: as [12] showed, this assumption is needed to guarantee that an entertainable antecedent has a consistent set of counterfactual consequences.
- 2.
- 3.
In the work of Lewis, the selection function takes formulas, rather than propositions, as its second argument. It would in principle be possible to do the same here. However, the presentation would become more complicated: some conditions in the definition of a model (in particular, the requirement that f should yield the same result when applied to intensionally equivalent formulas) appeal to the semantics of sentences, which in turn is defined with reference to the notion of a model. Letting selection functions take propositions allows us to avoid such seeming circularities.
- 4.
Our semantics departs from the one recently proposed by Weiss [21] in two ways: first, Weiss does not require f(w, X) to be upwards-closed; second, he requires f(w, X) to be defined for all subsets \(X\subseteq W\), not just for a designated set of such subsets. Both differences are important for our completeness result. At the same time, however, a Weiss model can be translated to one of our models, and vice versa, without affecting the satisfaction of formulas. A detailed comparison must be left for another occasion.
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Ciardelli, I., Liu, X. (2019). Minimal-Change Counterfactuals in Intuitionistic Logic. In: Blackburn, P., Lorini, E., Guo, M. (eds) Logic, Rationality, and Interaction. LORI 2019. Lecture Notes in Computer Science(), vol 11813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60292-8_4
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