This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious program by Yli-Vakkuri and Hawthorne, who seek to establish that mathematics is committed to its own necessity. I demonstrate that their assumptions collapse the counterfactual conditional into the material conditional. This collapse entails the success of counterfactual strengthening, which is controversial within counterfactual logic, and which has counterexamples within pure and applied mathematics. I close (...) by discussing the dispensability of counterfactual conditionals within the language of mathematics. (shrink)

This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious program by Yli-Vakkuri and Hawthorne (2018), who seek to establish that mathematics is committed to its own necessity. I claim that their argument fails to establish this result for two reasons. First, their assumptions force our hand on a controversial debate within counterfactual logic. In particular, they license counterfactual strengthening— the inference from ‘If A (...) were true then C would be true’ to ‘If A and B were true then C would be true’—which many reject. Second, the system they develop is provably equivalent to appending Deduction Theorem to a T modal logic. It is unsurprising that the combination of Deduction Theorem with T results in necessitation; indeed, it is precisely for this reason that many logicians reject Deduction Theorem in modal contexts. If Deduction Theorem is unacceptable for modal logic, it cannot be assumed to derive the necessity of mathematics. (shrink)

It is widely held that counterfactuals, unlike attitude ascriptions, preserve the referential transparency of their constituents, i.e., that counterfactuals validate the substitution of identicals when their constituents do. The only putative counterexamples in the literature come from counterpossibles, i.e., counterfactuals with impossible antecedents. Advocates of counterpossibilism, i.e., the view that counterpossibles are not all vacuous, argue that counterpossibles can generate referential opacity. But in order to explain why most substitution inferences into counterfactuals seem valid, counterpossibilists also often maintain that counterfactuals (...) with possible antecedents are transparency‐preserving. I argue that if counterpossibles can generate opacity, then so can ordinary counterfactuals with possible antecedents. Utilizing an analogy between counterfactuals and attitude ascriptions, I provide a counterpossibilist‐friendly explanation for the apparent validity of substitution inferences into counterfactuals. I conclude by suggesting that the debate over counterpossibles is closely tied to questions concerning the extent to which counterfactuals are more like attitude ascriptions and epistemic operators than previously recognized. (shrink)

A counteridentical is a counterfactual with an identity statement in the antecedent. While counteridenticals generally seem non-trivial, most semantic theories for counterfactuals, when combined with the necessity of identity and distinctness, attribute vacuous truth conditions to such counterfactuals. In light of this, one could try to save the orthodox theories either by appealing to pragmatics or by denying that the antecedents of alleged counteridenticals really contain identity claims. Or one could reject the orthodox theory of counterfactuals in favor of a (...) hyperintensional semantics that accommodates non-trivial counterpossibles. In this paper, I argue that none of these approaches can account for all the peculiar features of counteridenticals. Instead, I propose a modified version of Lewis’s counterpart theory, which rejects the necessity of identity, and show that it can explain all the peculiar features of counteridenticals in a satisfactory way. I conclude by defending the plausibility of contingent identity from objections. (shrink)

Counterpossibles are counterfactuals with necessarily false antecedents. The problem of counterpossibles is easiest to state within the "nearest possible world" framework for counterfactuals: on this approach, a counterfactual is true (roughly) when the consequent is true in the "nearest" possible world where the antecedent is true. Since counterpossibles have necessarily false antecedents, there is no possible world where the antecedent is true. On the approach favored by Lewis, Stalnaker, Williamson, and others, counterpossibles are all trivially true. I introduce several arguments (...) against the trivial approach. First, it is counter-intuitive to think that all counterpossibles are true. Second, if all counterpossibles were true, then we could not make sense of their use in logical, philosophical, or mathematical arguments. Making sense of the role of sentences like these requires that they not have vacuous truth conditions. The account of counterpossibles I ultimately favor is an extension of the "nearest possible world" semantics discussed above. The Lewis/Stalnaker account is supplemented with the addition of impossible worlds, and the nearness metric is extended to range over these impossible worlds as well as possible worlds. Thus, a counterfactual is true when its consequent is true in the nearest world where the antecedent is true; if the counterfactual's antecedent is impossible, then the nearest world in question will be an impossible world. Once the framework of impossible worlds and similarity is in place, we can put it to use in the analysis of other philosophical phenomena. I examine one proposal that makes use of a theory of counterpossibles to develop an analysis of the notion of metaphysical dependence. (shrink)

This is the second part of a two-part series on the logic of hyperlogic, a formal system for regimenting metalogical claims in the object language (even within embedded environments). Part A provided a minimal logic for hyperlogic that is sound and complete over the class of all models. In this part, we extend these completeness results to stronger logics that are sound and complete over restricted classes of models. We also investigate the logic of hyperlogic when the language is enriched (...) with hyperintensional operators such as counterfactual conditionals and belief operators. (shrink)

A counterpossible is a counterfactual with an impossible antecedent. Counterpossibles present a puzzle for standard theories of counterfactuals, which predict that all counterpossibles are semantically vacuous. Moreover, counterpossibles play an important role in many debates within metaphysics and epistemology, including debates over grounding, causation, modality, mathematics, science, and even God. In this article, we will explore various positions on counterpossibles as well as their potential philosophical consequences.

Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional (...) obligation). Despite the close connections between conditional and modal logic, both the technical development and philosophical exploitation of the latter has outstripped that of the former, with the result that noticeable lacunae exist in the literature on conditional logic. My dissertation addresses a number of these underdeveloped frontiers, producing new technical insights and philosophical applications. -/- I contribute to the solution of a problem posed by Priest of finding sound and complete labeled tableaux for systems of conditional logic from Lewis' V-family. To develop these tableaux, I draw on previous work on labeled tableaux for modal and conditional logic; errors and shortcomings in recent work on this problem are identified and corrected. While modal logic has by now been thoroughly studied in non-classical contexts, e.g. intuitionistic and relevant logic, the literature on conditional logic is still overwhelmingly classical. Another contribution of my dissertation is a thorough analysis of intuitionistic conditional logic, in which I utilize both algebraic and worlds semantics, and investigate how several novel embedding results might shed light on the philosophical interpretation of both intuitionistic logic and conditional logic extensions thereof. -/- My dissertation examines deontic and connexive conditional logic as well as the underappreciated history of connexive notions in the analysis of conditional obligation. The possibility of interpreting deontic modal logics in such systems (via embedding results) serves as an important theoretical guide. A philosophically motivated proscription on impossible obligations is shown to correspond to, and justify, certain (weak) connexive theses. Finally, I contribute to the intensifying debate over counterpossibles, counterfactuals with impossible antecedents, and take—in contrast to Lewis and Williamson—a non-vacuous line. Thus, in my view, a counterpossible like "If there had been a counterexample to the law of the excluded middle, Brouwer would not have been vindicated" is false, not (vacuously) true, although it has an impossible antecedent. I exploit impossible (non-normal) worlds—originally developed to model non-normal modal logics—to provide non-vacuous semantics for counterpossibles. I buttress the case for non-vacuous semantics by making recourse to both novel technical results and theoretical considerations. (shrink)