Interpolation in 16-Valued Trilattice Logics
Reinhard Muskens, Stefan Wintein · Studia Logica · 2017
In a recent paper we have defined an analytic tableau calculus $${{\mathbf {\mathsf{{PL}}}}}_{\mathbf {16}}$$ for a functionally complete extension of Shramko and Wansing’s logic based on the trilattice $${SIXTEEN}_3$$ . This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic—such as the relations , , and that each correspond to a lattice order in $${SIXTEEN}_3$$ ; and , the intersection of and . It turns out that our method of characterising these semantic relations—as intersections of auxiliary relations that can be captured with the help of a single calculus—lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that , when restricted to $$\mathcal {L}_{tf}$$ , the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano.