(Joint work with Nick Bezhanishvili, Wesley Holliday, and Davide Quadrellaro)
In the first part of the talk, I will present a new algebraic semantics for inquisitive logic, which is based on special Heyting algebras—called inquisitive algebras—with propositional valuations ranging only over regular elements. In particular, inquisitive algebras arise from Boolean algebras via a free-algebra construction: this allows us to completely characterise the structure of these special algebras and, consequently, study the properties of inquisitive logic.
In the second part, I will sketch recent developments, generalising the algebraic semantics to the class of DNA-logics. In particular, the semantics allows us to define a duality between the lattice of DNA-logics and a sublattice of varieties of Heyting algebras. Combining this with the previous results, we will give a characterisation of the DNA-logics extending inquisitive logic.The first part of the talk is based on a joint work with N. Bezhanishvili and W. H. Holliday; the second part is based on the MsC thesis of D. E. Quadrellaro.