2,253 research outputs found
Intuitionistic Non-Normal Modal Logics: A general framework
We define a family of intuitionistic non-normal modal logics; they can bee
seen as intuitionistic counterparts of classical ones. We first consider
monomodal logics, which contain only one between Necessity and Possibility. We
then consider the more important case of bimodal logics, which contain both
modal operators. In this case we define several interactions between Necessity
and Possibility of increasing strength, although weaker than duality. For all
logics we provide both a Hilbert axiomatisation and a cut-free sequent
calculus, on its basis we also prove their decidability. We then give a
semantic characterisation of our logics in terms of neighbourhood models. Our
semantic framework captures modularly not only our systems but also already
known intuitionistic non-normal modal logics such as Constructive K (CK) and
the propositional fragment of Wijesekera's Constructive Concurrent Dynamic
Logic.Comment: Preprin
Intuitionistic non-normal modal logics: A general framework
International audienceWe define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic
A fully labelled proof system for intuitionistic modal logics
Labelled proof theory has been famously successful for modal logics by mimicking their relational seman-tics within deductive systems. Simpson in particular designed a framework to study a variety of intuitionisticmodal logics integrating a binary relation symbol in the syntax. In this paper, we present a labelled sequentsystem for intuitionistic modal logics such that there is not only one, but two relation symbols appearingin sequents: one for the accessibility relation associated with the Kripke semantics for normal modal logicsand one for the preorder relation associated with the Kripke semantics for intuitionistic logic. This putsour system in close correspondence with the standard birelational Kripke semantics for intuitionistic modallogics. As a consequence it can be extended with arbitrary intuitionistic Scott-Lemmon axioms. We showsoundness and completeness, together with an internal cut elimination proof, encompassing a wider array ofintuitionistic modal logics than any existing labelled system
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
Ecumenical modal logic
The discussion about how to put together Gentzen's systems for classical and
intuitionistic logic in a single unified system is back in fashion. Indeed,
recently Prawitz and others have been discussing the so called Ecumenical
Systems, where connectives from these logics can co-exist in peace. In Prawitz'
system, the classical logician and the intuitionistic logician would share the
universal quantifier, conjunction, negation, and the constant for the absurd,
but they would each have their own existential quantifier, disjunction, and
implication, with different meanings. Prawitz' main idea is that these
different meanings are given by a semantical framework that can be accepted by
both parties. In a recent work, Ecumenical sequent calculi and a nested system
were presented, and some very interesting proof theoretical properties of the
systems were established. In this work we extend Prawitz' Ecumenical idea to
alethic K-modalities
Goldblatt-Thomason Theorems for Modal Intuitionistic Logics
We prove Goldblatt-Thomason theorems for frames and models of a wide variety
of modal intuitionistic logics, including ones studied by Wolter and
Zakharyaschev, Goldblatt, Fischer Servi, and Plotkin and Sterling. We use the
framework of dialgebraic logic to describe most of these logics and derive
results in a uniform way
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
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