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AN ALGEBRAIC APPROACH TO INQUISITIVE AND $\mathtt {DNA}$-LOGICS

Part of: Boolean algebras (Boolean rings) Varieties General logic Distributive lattices

Published online by Cambridge University Press:  02 December 2021

NICK BEZHANISHVILI ,
GIANLUCA GRILLETTI Open the ORCID record for GIANLUCA GRILLETTI [Opens in a new window]  and
DAVIDE EMILIO QUADRELLARO
NICK BEZHANISHVILI
Affiliation:
INSTITUTE FOR LOGIC LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM
GIANLUCA GRILLETTI
Affiliation:
MUNICH CENTRE FOR MATHEMATICAL PHILOSOPHY LUDWIG MAXIMILLIAN UNIVERSITY
DAVIDE EMILIO QUADRELLARO
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF HELSINKI
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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

Keywords

inquisitive logicintermediate logicsnegative variantsHeyting algebrasalgebraic semantics

MSC classification

Primary: 03B55: Intermediate logics
Secondary: 03B60: Other nonclassical logic 06D20: Heyting algebras 08B15: Lattices of varieties
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 4 , December 2022 , pp. 950 - 990
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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Footnotes

1

This article is based on [41].

References

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