76 research outputs found
Coalgebraic completeness-via-canonicity for distributive substructural logics
We prove strong completeness of a range of substructural logics with respect
to a natural poset-based relational semantics using a coalgebraic version of
completeness-via-canonicity. By formalizing the problem in the language of
coalgebraic logics, we develop a modular theory which covers a wide variety of
different logics under a single framework, and lends itself to further
extensions. Moreover, we believe that the coalgebraic framework provides a
systematic and principled way to study the relationship between resource models
on the semantics side, and substructural logics on the syntactic side.Comment: 36 page
Algebraic proof theory for LE-logics
In this paper we extend the research programme in algebraic proof theory from
axiomatic extensions of the full Lambek calculus to logics algebraically
captured by certain varieties of normal lattice expansions (normal LE-logics).
Specifically, we generalise the residuated frames in [16] to arbitrary
signatures of normal lattice expansions (LE). Such a generalization provides a
valuable tool for proving important properties of LE-logics in full uniformity.
We prove semantic cut elimination for the display calculi D.LE associated with
the basic normal LE-logics and their axiomatic extensions with analytic
inductive axioms. We also prove the finite model property (FMP) for each such
calculus D.LE, as well as for its extensions with analytic structural rules
satisfying certain additional properties
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
Canonical extension and canonicity via DCPO presentations
The canonical extension of a lattice is in an essential way a two-sided
completion. Domain theory, on the contrary, is primarily concerned with
one-sided completeness. In this paper, we show two things. Firstly, that the
canonical extension of a lattice can be given an asymmetric description in two
stages: a free co-directed meet completion, followed by a completion by
\emph{selected} directed joins. Secondly, we show that the general techniques
for dcpo presentations of dcpo algebras used in the second stage of the
construction immediately give us the well-known canonicity result for bounded
lattices with operators.Comment: 17 pages. Definition 5 was revised slightly, without changing any of
the result
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