UC Irvine

Two-dimensional semantics is one of the main theories of meaning in contemporary analytic philosophy. Much of what makes it interesting is the unified manner it has of providing a semantic analysis of statements involving the metaphysical notions of necessity, possibility, and actuality, and the epistemic notions of a priori and a posteriori knowledge or knowability. This unified account allows for the development of modal logics endowed with semantic structures which are inspired by two-dimensional theories of meaning, in which the formal languages contain operators corresponding to the aforementioned metaphysical and epistemic notions.

In this dissertation, I develop first-order two-dimensional modal logics for different modal languages and present sound and complete semantic tableaux for all of them. Additionally, I argue that the first-order extensions call for new actuality operators so that certain statements in natural language can be formalized. I argue that this brings new issues for two-dimensional modal logics, such as the problem of omniscience with respect to actual truths. Finally, from the perspective of mathematical logic, I show that extensions of certain propositional two-dimensional modal logics to the n-dimensional case are finitely axiomatizable. Decidability of n-dimensional modal logics is proved by systematic constructions of semantic tableaux.