129 research outputs found
Slanted canonicity of analytic inductive inequalities
We prove an algebraic canonicity theorem for normal LE-logics of arbitrary
signature, in a generalized setting in which the non-lattice connectives are
interpreted as operations mapping tuples of elements of the given lattice to
closed or open elements of its canonical extension. Interestingly, the
syntactic shape of LE-inequalities which guarantees their canonicity in this
generalized setting turns out to coincide with the syntactic shape of analytic
inductive inequalities, which guarantees LE-inequalities to be equivalently
captured by analytic structural rules of a proper display calculus. We show
that this canonicity result connects and strengthens a number of recent
canonicity results in two different areas: subordination algebras, and transfer
results via G\"odel-McKinsey-Tarski translations.Comment: arXiv admin note: text overlap with arXiv:1603.08515,
arXiv:1603.0834
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
Non normal logics: semantic analysis and proof theory
We introduce proper display calculi for basic monotonic modal logic,the
conditional logic CK and a number of their axiomatic extensions. These calculi
are sound, complete, conservative and enjoy cut elimination and subformula
property. Our proposal applies the multi-type methodology in the design of
display calculi, starting from a semantic analysis based on the translation
from monotonic modal logic to normal bi-modal logic
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