16 research outputs found
A tableau system for Quasi-hybrid logic
Hybrid logic is a valuable tool for specifying relational structures, at the same time that allows defining accessibility relations between states, it provides a way to nominate and make mention to what happens at each specific state. However, due to the many sources nowadays available, we may need to deal with contradictory information. This is the reason why we came with the idea of Quasi-hybrid logic, which is a paraconsistent version of hybrid logic capable of dealing with inconsistencies in the information, written as hybrid formulas.
In [5] we have already developed a semantics for this paraconsistent logic. In this paper we go a step forward, namely we study its proof-theoretical aspects. We present a complete tableau system for Quasi-hybrid logic, by combining both tableaux for Quasi-classical and Hybrid logics
Knowledge and ignorance in Belnap--Dunn logic
In this paper, we argue that the usual approach to modelling knowledge and
belief with the necessity modality does not produce intuitive outcomes
in the framework of the Belnap--Dunn logic (, alias
-- first-degree entailment). We then motivate and introduce a non\-standard
modality that formalises knowledge and belief in
and use to define and that
formalise the \emph{unknown truth} and ignorance as \emph{not knowing whether},
respectively. Moreover, we introduce another modality that stands
for \emph{factive ignorance} and show its connection with .
We equip these modalities with Kripke-frame-based semantics and construct a
sound and complete analytic cut system for and
-- the expansions of with
and . In addition, we show that as it is customarily defined
in cannot define any of the introduced modalities, nor,
conversely, neither nor can define . We also
demonstrate that and are not interdefinable and
establish the definability of several important classes of frames using
Zero-one laws with respect to models of provability logic and two Grzegorczyk logics
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems
Fuzzy bi-G\"{o}del modal logic and its paraconsistent relatives
We present the axiomatisation of the fuzzy bi-G\"{o}del modal logic
(formulated in the language containing and treating the
coimplication as a defined connective) and establish its PSpace-completeness.
We also consider its paraconsistent relatives defined on fuzzy frames with two
valuations and standing for the support of truth and falsity,
respectively, and equipped with \emph{two fuzzy relations} and used
to determine supports of truth and falsity of modal formulas. We establish
embeddings of these paraconsistent logics into the fuzzy bi-G\"{o}del modal
logic and use them to prove their PSpace-completeness and obtain the
characterisation of definable frames