A Topos-Theoretic Approach to Counterfactual Logic

Topoi are known to be categories with extra properties that make them much alike the category of Sets. In a Topoi it is possible to define adequate notions of membership, elements and subobjects, power ”sets”, and finally, every Topoi has an internal logic able to justify any reasoning carried inside it. Most of the cases, this logic is not Classical (Boolean). The general logic for the Topoi is Intuitionistic Higher-order Logic. Topoi have their linguistic counter-part provided by Local Set Theories (LST). There is a deductive apparatus, in the style of Sequent Calculus, able to justify logical consequence inside any LST. Counterfactuals are subtle conditionals largely studied by the philosophical and logic community. Since Lewis, counterfactual have a uniform semantics provided by means of Neighborhood systems on top a possible world style semantics. In this article, taking into account the fundamental theorem on Topoi, we define, by means of the internal logic of Graphs, Lewis counterfactuals and show how to use the LST deductive apparatus to prove properties on counterfactual logic. This article can be also used as an initial step towards the definition of deductive systems for counterfactual logic, taken in an alternative way the ones already existent in the logic literature.