Characterization of Lattice Properties Within Modal Extensions
Alfredo R. Freire, Manuel A. Martins
TL;DR
This work investigates modal extensions of lattice-based logics, focusing on interpreting the necessity operator $\Box$ as a meet across accessible worlds and exploring regular NAW semantics where $\Box \varphi$ holds exactly when $\varphi$ is true in all accessible worlds. It develops a formal framework of $A$-valuations and normal modal valuations, and analyzes how lattice properties (completeness, linearity outside a designated set $F$, down-distribution, and implicativity) determine the validity of modal principles such as axiom $K$, including disjunction distribution $(\Box \varphi \lor \Box \psi) \rightarrow \Box(\varphi \lor \psi)$. The paper proves that, for complete lattices with an implicative $F$, every frame satisfies $K$, and it discusses how non-implicative or non-linear structures can fail $K$; it also presents a twist-algebra example to model paraconsistent modal semantics and demonstrates how the choice of $F$ governs regular vs non-regular behavior. Through these results, the authors illuminate how algebraic lattice properties govern modal reasoning across worlds with potentially different logical semantics, with implications for cross-world communication and potential finite-model considerations for applications.
Abstract
This paper investigates the extension of lattice-based logics into modal languages. We observe that such extensions admit multiple approaches, as the interpretation of the necessity operator is not uniquely determined by the underlying lattice structure. The most natural interpretation defines necessity as the meet of the truth values of a formula across all accessible worlds -- an approach we refer to as the \textitnormal interpretation. We examine the logical properties that emerge under this and other interpretations, including the conditions under which the resulting modal logic satisfies the axiom K and other common modal validities. Furthermore, we consider cases in which necessity is attributed exclusively to formulas that hold in all accessible worlds.