An algebraic approach to inquisitive and DNA-logics

This article provides an algebraic study of the propositional system InqB of inquisitive logic. We also investigate the wider class of DNA-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, DNA-varieties. We prove that the lattice of DNA-logics is dually isomorphic to the lattice of DNA-varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety theorems. We also introduce locally finite DNA-varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of InqB is dually isomorphic to the ordinal ω + 1 and give an axiomatisation of these logics via Jankov DNA-formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].