16 research outputs found
Sufficient conditions for local tabularity of a polymodal logic
On relational structures and on polymodal logics, we describe operations
which preserve local tabularity. This provides new sufficient semantic and
axiomatic conditions for local tabularity of a modal logic. The main results
are the following.
We show that local tabularity does not depend on reflexivity. Namely, given a
class of frames, consider the class of
frames, where the reflexive closure operation was applied to each relation in
every frame in . We show that if the logic of
is locally tabular, then the logic of is
locally tabular as well.
Then we consider the operation of sum on Kripke frames, where a family of
frames-summands is indexed by elements of another frame. We show that if both
the logic of indices and the logic of summands are locally tabular, then the
logic of corresponding sums is also locally tabular.
Finally, using the previous theorem, we describe an operation on logics that
preserves local tabularity: we provide a set of formulas such that the
extension of the fusion of two canonical locally tabular logics with these
formulas is locally tabular
Admissibility of Ξ <sub>2</sub>-inference rules: Interpolation, model completion, and contact algebras
We devise three strategies for recognizing admissibility of non-standard inference rules via interpolation, uniform interpolation, and model completions. We apply our machinery to the case of symmetric implication calculus S2IC, where we also supply a finite axiomatization of the model completion of its algebraic counterpart, via the equivalent theory of contact algebras. Using this result we obtain a finite basis for admissible Ξ 2-rules
Polyatomic Logics and Generalised Blok-Esakia Theory
This paper presents a novel concept of a Polyatomic Logic and initiates its
systematic study. This approach, inspired by Inquisitive semantics, is obtained
by taking a variant of a given logic, obtained by looking at the fragment
covered by a selector term. We introduce an algebraic semantics for these
logics and prove algebraic completeness. These logics are then related to
translations, through the introduction of a number of classes of translations
involving selector terms, which are noted to be ubiquitous in algebraic logic.
In this setting, we also introduce a generalised Blok-Esakia theory which can
be developed for special classes of translations. We conclude by showing some
systematic connections between the theory of Polyatomic Logics and the general
Blok-Esakia theory for a wide class of interesting translations.Comment: 48 pages, 2 figure
An Algebraic Approach to Inquisitive and DNA-Logics
This article provides an algebraic study of the propositional system InqB of inquisitive logic. We also investigate the wider class of DNA-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, DNA -varieties. We prove that the lattice of DNA-logics is dually isomorphic to the lattice of DNA -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff's classic variety theorems. We also introduce locally finite DNA -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of InqB is dually isomorphic to the ordinal omega + 1 and give an axiomatisation of these logics via Jankov DNA -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].(1)Peer reviewe
On decidable extensions of Propositional Dynamic Logic with Converse
We describe a family of decidable propositional dynamic logics, where atomic
modalities satisfy some extra conditions (for example, given by axioms of the
logics K5, S5, or K45 for different atomic modalities). It follows from recent
results (Kikot, Shapirovsky, Zolin, 2014; 2020) that if a modal logic
admits a special type of filtration (so-called definable filtration), then its
enrichments with modalities for the transitive closure and converse relations
also admit definable filtration. We use these results to show that if logics
admit definable filtration, then the propositional dynamic
logic with converse extended by the fusion has the finite
model property
Aczel-Mendler Bisimulations in a Regular Category
Aczel-Mendler bisimulations are a coalgebraic extension of a variety of
computational relations between systems. It is usual to assume that the
underlying category satisfies some form of axiom of choice, so that the theory
enjoys desirable properties, such as closure under composition. In this paper,
we accommodate the definition in general regular categories and toposes. We
show that this general definition 1) is closed under composition without using
the axiom of choice, 2) coincide with other types of coalgebraic formulations
under milder conditions, 3) coincide with the usual definition when the
category has the regular axiom of choice. In particular, the case of toposes
heavily relies on power-objects for which we recover some nice properties on
the way. Finally, we describe several examples in Stone spaces, toposes for
name-passing, and modules over a ring.Comment: Submission to the CALCO 2023 special LMCS issu