Lattice logic as a fragment of (2-sorted) residuated modal logic

摘要

Correspondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure F = (W,R) is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components, W = X|Y, and R ⊆ X × Y, then frames C = (X,R, Y) are at the same time the basis for models of non-distributive lattice logic and of twosorted, residuated modal logic. This suggests that a reduction of the first to the latter may be possible, encoding Positive Lattice Logic (PLL) as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Goedel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions.