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

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For every univariate formula χ we introduce a lattices of intermediate theories: the lattice of χ-logics. The key idea to define χlogics is to interpret atomic propositions as fixpoints of the formula χ 2 , which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of χ-logics and a special class of varieties of Heyting algebras. This approach allows us to build five distinct lattices-corresponding to the possible fixpoints of univariate formulas-among which the lattice of negative variants of intermediate logics. We describe these lattices in more detail.

For every univariate formula χ we introduce a lattices of intermediate theories: the lattice of χ-logics. The key idea to define χ-logics is to interpret atomic propositions as fixpoints of the formula χ 2 , which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of χ-logics and a special class of varieties of Heyting algebras. This approach allows us to build five distinct lattices-corresponding to the possible fixpoints of univariate formulas-among which the lattice of negative variants of intermediate logics. We describe these lattices in more detail.