Studia Logica

, Volume 106, Issue 2, pp 345–370

# Interpolation in 16-Valued Trilattice Logics

• Reinhard Muskens
• Stefan Wintein
Open Access
Article

## Abstract

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.

## Keywords

Interpolation 16-Valued logic Trilattice $${SIXTEEN}_3$$ Multiple tree calculus

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