45 research outputs found
Kripke completeness of strictly positive modal logics over meet-semilattices with operators
Our concern is the completeness problem for spi-logics, that is, sets of im- plications between strictly positive formulas built from propositional variables, conjunc- tion and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators provid- ing Birkhoff-style calculi, and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a complete- ness theory that aims to answer the question whether the two semantics define the same consequence relations for a given spi-logic
Modal meet-implication logic
We extend the meet-implication fragment of propositional intuitionistic logic
with a meet-preserving modality. We give semantics based on semilattices and a
duality result with a suitable notion of descriptive frame. As a consequence we
obtain completeness and identify a common (modal) fragment of a large class of
modal intuitionistic logics.
We recognise this logic as a dialgebraic logic, and as a consequence obtain
expressivity-somewhere-else. Within the dialgebraic framework, we then
investigate the extension of the meet-implication fragment of propositional
intuitionistic logic with a monotone modality and prove completeness and
expressivity-somewhere-else for it
Dualities in modal logic
Categorical dualities are an important tool in the study of (modal) logics. They offer conceptual understanding and enable the transfer of results between the different semantics of a logic. As such, they play a central role in the proofs of completeness theorems, Sahlqvist theorems and Goldblatt-Thomason theorems. A common way to obtain dualities is by extending existing ones. For example, Jonsson-Tarski duality is an extension of Stone duality. A convenient formalism to carry out such extensions is given by the dual categorical notions of algebras and coalgebras. Intuitively, these allow one to isolate the new part of a duality from the existing part. In this thesis we will derive both existing and new dualities via this route, and we show how to use the dualities to investigate logics. However, not all (modal logical) paradigms fit the (co)algebraic perspective. In particular, modal intuitionistic logics do not enjoy a coalgebraic treatment, and there is a general lack of duality results for them. To remedy this, we use a generalisation of both algebras and coalgebras called dialgebras. Guided by the research field of coalgebraic logic, we introduce the framework of dialgebraic logic. We show how a large class of modal intuitionistic logics can be modelled as dialgebraic logics and we prove dualities for them. We use the dialgebraic framework to prove general completeness, Hennessy-Milner, representation and Goldblatt-Thomason theorems, and instantiate this to a wide variety of modal intuitionistic logics. Additionally, we use the dialgebraic perspective to investigate modal extensions of the meet-implication fragment of intuitionistic logic. We instantiate general dialgebraic results, and describe how modal meet-implication logics relate to modal intuitionistic logics
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
Representations and Completions for Ordered Algebraic Structures
The primary concerns of this thesis are completions and representations for various classes of
poset expansion, and a recurring theme will be that of axiomatizability. By a representation we
mean something similar to the Stone representation whereby a Boolean algebra can be homomorphically
embedded into a field of sets. So, in general we are interested in order embedding
posets into fields of sets in such a way that existing meets and joins are interpreted naturally as
set theoretic intersections and unions respectively.
Our contributions in this area are an investigation into the ostensibly second order property
of whether a poset can be order embedded into a field of sets in such a way that arbitrary meets
and/or joins are interpreted as set theoretic intersections and/or unions respectively. Among
other things we show that unlike Boolean algebras, which have such a ‘complete’ representation
if and only if they are atomic, the classes of bounded, distributive lattices and posets with
complete representations have no first order axiomatizations (though they are pseudoelementary).
We also show that the class of posets with representations preserving arbitrary joins is
pseudoelementary but not elementary (a dual result also holds).
We discuss various completions relating to the canonical extension, whose classical construction
is related to the Stone representation. We claim some new results on the structure of
classes of poset meet-completions which preserve particular sets of meets, in particular that they
form a weakly upper semimodular lattice. We make explicit the construction of \Delta_{1}-completions
using a two stage process involving meet- and join-completions.
Linking our twin topics we discuss canonicity for the representation classes we deal with,
and by building representations using a meet-completion construction as a base we show that
the class of representable ordered domain algebras is finitely axiomatizable. Our method has
the advantage of representing finite algebras over finite bases