A New Semantics for Positive Modal Logic

Abstract

The paper provides a new semantics for positive modal logic using Kripke frames having a quasi ordering $\leq$ on the set of possible worlds and an accessibility relation $R$ connected to the quasi ordering by the conditions (1) that the composition of $\leq$ with $R$ is included in the composition of $R$ with $\leq$ and (2) the analogous for the inverse of $\leq$ and $R$. This semantics has an advantage over the one used by Dunn in "Positive modal logic," Studia Logica (1995) and works fine for extensions of the minimal system of normal positive modal logic.