Abstract

This article provides an algebraic study of the propositional system$\mathtt {InqB}$of inquisitive logic. We also investigate the wider class of$\mathtt {DNA}$-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures,$\mathtt {DNA}$-varieties. We prove that the lattice of$\mathtt {DNA}$-logics is dually isomorphic to the lattice of$\mathtt {DNA}$-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$\mathtt {DNA}$-varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of$\mathtt {InqB}$is dually isomorphic to the ordinal$\omega +1$and give an axiomatisation of these logics via Jankov$\mathtt {DNA}$-formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1.