We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalization of Humberstone's inaccessibility logic. Viewing first-order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite variable fragments of first-order logic. We provide a simple condition stated only in terms of frames and bisimulations (...) which implies failure of interpolation. Its use is exemplified in a wide range of cases. (shrink)

Craig's interpolation theorem fails for the propositional logics E of entailment, R of relevant implication and T of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids.

An infinite sequence $\bgD=\ $ of possibly infinite sets of formulas in $n+1$ variables $\seq x0{n-1},y$ and a possibly infinite system of parameters $\vu$ is a \emph{parameterized graded deduction-detachment} \emph{system} for a deductive system $\bcS$ over a $\bcS$-theory $T$ if, for every $n $, where $\Fi_\bcS\sbA$ is the set of all $\bcS$-filters on $\sbA$.Theorem.Let $\bcS$ be a protoalgebraic deductive system over a countable language type. If $\bcS$ has a Leibniz-generating PGDD system over all Leibniz theories, then $\bcS$ has a fully (...) adequate Gentzen system.Theorem.Let $\bcS$ be a protoalgebraic deductive system. If $\bcS$ has a fully adequate Gentzen system, then $\bcS$ has a Leibniz-generating PGDD system over every Leibniz theory.Corollary.If $\bcS$ is a weakly algebraizable deductive system over a countable language type, then $\bcS$ has a fully adequate Gentzen system iff it has the multiterm deduction-detachment theorem.Corollary.If $\bcS$ is a finitely equivalential deductive system over a countable language type, then $\bcS$ has a fully adequate Gentzen system iff there is a finite Leibniz-generating \textup{GDD} system for $\bcS$ over all Leibniz $\bcS$-filters. (shrink)

Interpolation has become one of the standard properties that logicians investigate when designing a logic. In this paper, we provide strong evidence that the presence of interpolants is not only cogent for scientific reasoning but has also important practical implications in computer science. We illustrate that interpolation in general, and uniform splitting interpolants, in particular, play an important role in applications where formality and modularity are invoked. In recognition of the fact that common logical formalisms often lack uniform interpolants, we (...) advocate the need for developing general methods to engineer a specification logic so that some critical uniform interpolants become available. (shrink)