Conditionals and modularity in general logics

In this work in progress, we discuss independence and interpolation and related topics for classical, modal, and non-monotonic logics

Semantic interpolation

We treat interpolation for various logics

Interpolation and implicit definability in extensions of the provability logic

The provability logic GL was in the field of interest of A.V. Kuznetsov, who had also formulated its intuitionistic analog—the intuitionistic provability logic—and investigated these two logics and their extensions. In the present paper, different versions of interpolation and of the Beth property in normal extensions of the provability logic GL are considered. It is proved that in a large class of extensions of GL (including all finite slice logics over GL) almost all versions of interpolation and of the Beth property are equivalent. It follows that in finite slice logics over GL the three versions CIP, IPD and IPR of the interpolation property are equivalent. Also they are equivalent to the Beth properties B1, PB1 and PB2

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants

Modal logic of planar polygons

We study the modal logic of the closure algebra $P_2$, generated by the set of all polygons in the Euclidean plane $\mathbb{R}^2$. We show that this logic is finitely axiomatizable, is complete with respect to the class of frames we call "crown" frames, is not first order definable, does not have the Craig interpolation property, and its validity problem is PSPACE-complete

On PP-Interpolation in Local Theory Extensions and Applications to the Study of Interpolation in the Description Logics EL,EL+{\cal EL}, {\cal EL}^+

We study the problem of $P$-interpolation, where $P$ is a set of binary predicate symbols, for certain classes of local extensions of a base theory. For computing the $P$-interpolating terms, we use a hierarchic approach: This allows us to compute the interpolating terms using a method for computing interpolating terms in the base theory. We use these results for proving $\leq$-interpolation in classes of semilattices with monotone operators; we show, by giving a counterexample, that $\leq$-interpolation does not hold if by "shared" symbols we mean just the common symbols. We use these results for the study of $\sqsubseteq$-interpolation in the description logics ${\cal EL}$ and ${\cal EL}^+$.Comment: 33 page