On the connection between the categorical and the modal logic approaches to Quantum Mechanics

This thesis aims at connecting the two research programs known as Categorical Quantum Mechanics and Dynamic Quantum Logic. This is achieved in three steps. First we define a procedure to extract a Modal Logic frame from a small category and a functor into the category of sets and relations. Second, we extend such methodology to locally small categories. Third, we apply it to the category of finite-dimensional Hilbert spaces to recover the semantics of Dynamic Quantum Logic. This process prompts new lines of research. At a general level, we study some logics arising from wide classes of small categories. In the case of Hilbert spaces, we investigate how to obtain richer semantics, containing probabilistic information. We design a logic for this semantics and prove that, via translation, it preserves the validities of Dynamic Quantum Logic.