Lattices of DNA-Logics and Algebraic Semantics of Inquisitive Logic
Davide Emilio Quadrellaro
Abstract:
This thesis studies algebraic semantics for the inquisitive logic InqB and for the related class of DNA-logics. DNA-logics were previously known in literature as negative variants of intermediate logics and have been studied only in syntactic terms. In this thesis, we show that there is a dual isomorphism between the lattice of DNA-logics and the lattice of suitable classes of Heyting algebras that we call DNA-varieties. We study several properties of DNA-logics and DNA-varieties and we prove a version of Tarski and Birkhoff Theorems for DNA-varieties. A special attention is then paid to introduce a notion of locally finiteness for this setting and to prove two key results concerning this property, i.e. that the DNA-variety of all Heyting algebras is not locally finite and that locally finite DNA-logics can be axiomatised by a version of Jankov formulas. Finally, we apply the general theory of DNA-logics to the case of inquisitive logic. We show that InqB is a DNA-logic and we use the method of Jankov formulas to prove that the sublattice Λ(InqB) of the extensions of InqB is dually isomorphic to ω + 1.