2024/03/20 by Holliday, Wesley H. · 1 citation
#03B45 #03G10 #06B15 #F.4.1 #FOS: Computer and information sciences #FOS: Mathematics #I.2.4 #Logic (math.LO) #Logic in Computer Science (cs.LO)
paper · doi:10.48550/arxiv.2403.14043
Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices L equipped with an antitone operation ¬ sending 1 to 0, a completely multiplicative operation \Box, and a completely additive operation \Diamond. Such lattice expansions can be represented by means of a set X together with binary relations \vartriangleleft, R, and Q, satisfying some first-order conditions, used to represent (L,¬), \Box, and \Diamond, respectively. Indeed, any lattice L equipped with such a ¬, a multiplicative \Box, and an additive \Diamond embeds into the lattice of propositions of a frame (X,\vartriangleleft,R,Q). Building on our recent study of "fundamental logic", we focus on the case where ¬ is dually self-adjoint (a≤ ¬ b implies b≤¬ a) and \Diamond ¬ a≤¬\Box a. In this case, the representations can be constrained so that R=Q, i.e., we need only add a single relation to (X,\vartriangleleft) to represent both \Box and \Diamond. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures (X,\vartriangleleft, R).