Interpolation in 16-Valued Trilattice Logics

In a recent paper we have defined an analytic tableau calculus (Formula presented.) for a functionally complete extension of Shramko and Wansing’s logic based on the trilattice (Formula presented.). This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic—such as the relations [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], and [InlineEquation not available: see fulltext.] that each correspond to a lattice order in (Formula presented.); and [InlineEquation not available: see fulltext.], the intersection of [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. 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 [InlineEquation not available: see fulltext.], when restricted to (Formula presented.), the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano.

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