Note on extending congruential modal logics (Q5963201)

If \(A\) iff \(B\) implies \(\square A\) iff \(\square B\), then this modal logic is called congruential. The current paper studies some model theoretical properties of congruential modal logics. In particular, it studies under which conditions which formulae produce sublogics of identity logic, which is a system containing \(\square A\) iff \(A\) for any \(A\). This research agenda was initiated by \textit{D. Makinson} [Notre Dame J. Formal Logic 12, 252--254 (1971; Zbl 0193.29301); J. Philos. Log. 2, 193--196 (1973; Zbl 0266.02016)], and the current paper improves upon it.NEWLINENEWLINEThis digestible paper identifies various other formulas extending congruential logics which are sublogics of identity logic. The author achieves this by considering quite understudied models of modal logic, that is neighborhood models.