Deduction theorem in congruential modal logics. (English) Zbl 1533.03014

Krawczyk, Krzysztof A.

Notre Dame J. Formal Logic 64, No. 2, 185-196 (2023).

Congruential modal logics are those satisfying the rule \(p \leftrightarrow q/\Box p \leftrightarrow\Box q\). A well-known interpretation for congruential modal logics is the so-called neighborhood semantics or Scott-Montague semantics.
A consequence relation \(\vdash\) is said to have a local deduction detachment theorem when for any \(\Sigma\cup\{\phi,\psi\}\) there is a finite set of formulas \(L(p,q)\) such that \( \Sigma\cup \{\phi\}\vdash\psi\) if and only if \( \Sigma\vdash L(\phi,\psi)\).
This work presents an algebraic proof of the theorem stating that there are continuum many axiomatic extensions of global consequence associated with the basic congruential modal system \(E\) that do not admit the local deduction detachment theorem. It also proves that all these logics lack the finite frame property and have exactly three proper axiomatic extensions, each of which admits the local deduction detachment theorem.

Reviewer: Ignacio Viglizzo (Bahía Blanca)

MSC:

03B45	Modal logic (including the logic of norms)
03G25	Other algebras related to logic
03G27	Abstract algebraic logic

Keywords:

modal logic; local deduction theorem; modal algebra; abstract algebra

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