Abstract not found

Logical definability is investigated for certain classes of coalgebras related to state-transition systems, hidden algebras and Kripke models. The filter enlargement of a coalgebra A is introduced as a new coalgebra A + whose states are special "observationally rich" filters on the state set of A. The ultrafilter enlargement is the subcoalgebra A of A + whose states are ultrafilters. Boolean combinations of equations between terms of observable (or output) type are identified as a natural class of formulas for specifying properties of coalgebras. These observable formulas are permitted to have a single state variable, and form a language in which modalities describing the effects of state transitions are implicitly present. A and A + validate the same observable formulas. It is shown that a class of coalgebras is de nable by observable formulas iff the class is closed under disjoint unions, images of bisimulations, and (ultra) lter enlargements. (Closure under images of bisimulations is equivalent to closure under images and domains of coalgebraic morphisms.) Moreover, every set of observable formulas has the same models as some set of conditional equations.

There exist modal logics that are validated by their canonical frames but are not sound and complete for any elementary class of frames. Continuum many such bimodal logics are exhibited, including one of each degree of unsolvability, and all with the finite model property. Monomodal examples are also constructed that extend K4 and are related to the proof of non-canonicity of the McKinsey axiom. We dedicate this paper to Max Cresswell, a pioneer in the study of canonicity, on the occasion of his 65th birthday. 1

A rst-order sentence is quasi-modal if its class of models is closed under the modal validity preserving constructions of disjoint unions, inner substructures and bounded epimorphic images.

This is an exposition and analysis of van Benthem's original proof, hitherto unpublished, that if the class of structures (frames) validating a modal formula is closed under elementary equivalence, then it is the class of all models of a single first-order sentence.

A contravariant duality is constructed between the category of coalgebras of a given signature, and a category of Boolean algebras with operators, including modal operators corresponding to state transitions in coalgebras, and distinguished elements abstracting the sets of states defined by observable equations. This duality is used to give a new proof that a class of coalgebras is definable by Boolean combinations of observable equations if it is closed under disjoint unions, domains and images of coalgebraic morphisms, and ultrafilter enlargements. The proof reduces the problem to a direct application of Birkhoff's variety theorem characterising equational classes of algebras.

This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems.

abstract. A structural semantics is developed for a first-order logic, with infinite disjunctions and conjunctions, that is characterised algebraically by quantales. The model structures involved combine the “covering systems” approach of Kripke-Joyal intuitionistic semantics from topos theory with the ordered groupoid structures used to model various connectives in substructural logics. The latter are used to interpret the noncommutative quantal conjunction & (“and then”) and its residual implication connectives. The completeness proof uses the MacNeille completion and the theory of quantic nuclei to first embed a residuated semigroup into a quantale, and then represent the quantale as an algebra of subsets of a model structure. The final part of the paper makes some observations about quantal modal logic, giving in particular a structural modelling of the logic of closure operators on quantales.

Algebraic Logic. In preparation. Manuscript.