Modal Logics That Are Both Monotone and Antitone: Makinson’s Extension Results and Affinities between Logics

Abstract

A notable early result of David Makinson establishes that every monotone modal logic can be extended to $L_{\mathbf{I}}$, $L_{\mathbf{V}}$, or $L_{\mathbf{F}}$, and every antitone logic can be extended to $L_{\mathbf{N}}$, $L_{\mathbf{V}}$, or $L_{\mathbf{F}}$, where $L_{\mathbf{I}}$, $L_{\mathbf{N}}$, $L_{\mathbf{V}}$, and $L_{\mathbf{F}}$ are logics axiomatized, respectively, by the schemas $\square \mathit{\alpha }\leftrightarrow \mathit{\alpha }$, $\square \mathit{\alpha }\leftrightarrow \neg \mathit{\alpha }$, $\square \mathit{\alpha }\leftrightarrow \top$, and $\square \mathit{\alpha }\leftrightarrow \perp$. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: $L_{\mathbf{V}}$, $L_{\mathbf{F}}$, and the minimum amphitone logic $\mathsf{AM}$ axiomatized by the schema $\square \mathit{\alpha }\to \square \mathit{\beta }$. These logics, along with $L_{\mathbf{I}}$, $L_{\mathbf{N}}$, and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the nonmodal connectives. We close by discussing these divergences and conditions under which our results do carry over.