Deduction theorem in congruential modal logics (Q6165006)

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 \textit{neighborhood semantics} or \textit{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.