Bi-approximation semantics for substructural logic at work

In this paper, we introduce bi-approximation semantics, a two-sorted relational semantics, via the canonical extension of lattice expansions. To characterise Ghilardi and Meloni's parallel computation, we introduce doppelgänger valuations which allow us to evaluate sequents and not only formulae. Moreover, by introducing the bi-directional approximation and bases, we track down a connection to Kripke-type semantics for distributive substructural logics through a relationship between basis and the existential quantifier. Based on the framework, we give a possible interpretation of the two sorts, and prove soundness via bi-approximation and completeness via an algebraic representation theorem plus invariance of validity along a back-and-force correspondences.