An Algebraic Approach to Inquisitive and -Logics

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

Algebraizable Weak Logics

“…(1) both facts were proved by Ciardelli in [8]. (2) follows from (1) together with Proposition 4.17 from [2]. (3) From Theorem 3.32 in [2].…”

“…The next theorem collects together some previous result by Ciardelli in [8] on the schematic variant of InqB and the characterization of regularly generated varieties from [2].…”

“…Clearly A ¬ is polynomially definable. Let ML be the variety of all Medvedev algebras, then ML is generated by its subclass of core-generated Heyting algebras A with core A ¬ , see [1,2] and Theorem 39 later.…”

“…Given a weak logic, a natural question to ask is whether we can associate it to some standard logical system. The study of negative variants of intermediate logics has lead to the notion of schematic fragment, which was defined in [8,26] and further investigated in [2,29]. Here we generalize it to arbitrary weak logics.…”

“…The behaviour of these logical systems prevented so far a uniform and general study of these logics, and did not allow for immediately applying the facts and results from abstract algebraic logic, which hold only for logical systems in Tarski's strong sense. For example, algebraic semantics for several versions of inquisitive and dependence logics were introduced in [1,2,27,30], but it remained an open question whether these semantics are in any sense unique. In fact, it is a seminal result from abstract algebraic logic that the equivalent 1 algebraic semantics of a logic is unique, e.g.…”