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AN ALGEBRAIC APPROACH TO INQUISITIVE AND
$\mathtt {DNA}$-LOGICS
- Part of
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- Journal:
- The Review of Symbolic Logic / Volume 15 / Issue 4 / December 2022
- Published online by Cambridge University Press:
- 02 December 2021, pp. 950-990
- Print publication:
- December 2022
-
- Article
- Export citation
-
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
