Abstract

Structurally free logic LC was introduced in [4]. A natural extension of LC, in particular, in a sequent formulation, is by conjunction and disjunction that do not distribute over each other. We define a set theoretical semantics for these logics via constructing a representation of a lattice that we extend by intensional operations. Canonically, minimally overlapping filter-ideal pairs are used; this construction avoids the use of an equivalent of the axiom of choice and lends transparency to the structure. We also discuss adapting the present semantics for substructural logics, as well as, the question of cannnonicity of LC+