24 research outputs found
Paraconsistent Modal Logics
AbstractWe introduce a modal expansion of paraconsistent Nelson logic that is also as a generalization of the Belnapian modal logic recently introduced by Odintsov and Wansing. We prove algebraic completeness theorems for both logics, defining and axiomatizing the corresponding algebraic semantics. We provide a representation for these algebras in terms of twist-structures, generalizing a known result on the representation of the algebraic counterpart of paraconsistent Nelson logic
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
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
On Definability of Connectives and Modal Logics over FDE
The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics
The lattice of Belnapian modal logics: Special extensions and counterparts
Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on:• introducing the classes (or rather sublattices) of so-called explosive, complete and classical Belnapian modal logics;• assigning to every normal modal logic three special conservative extensions in these classes;• associating with every Belnapian modal logic its explosive, complete and classical counterparts.We investigate the relationships between special extensions and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK
Quantitative Hennessy-Milner Theorems via Notions of Density
The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes;
it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can
be distinguished by a modal formula. Numerous variants of this theorem have since been established
for a wide range of logics and system types, including quantitative versions where lower bounds on
behavioural distance (e.g. in weighted, metric, or probabilistic transition systems) are witnessed
by quantitative modal formulas. Both the qualitative and the quantitative versions have been
accommodated within the framework of coalgebraic logic, with distances taking values in quantales,
subject to certain restrictions, such as being so-called value quantales. While previous quantitative
coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo)metric spaces,
in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more
widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known
Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given
by Borel measures on metric spaces, as an instance of such a general result. In the process, we also
relax the restrictions imposed on the quantale, and additionally parametrize the technical account
over notions of closure and, hence, density, providing associated variants of the Stone-Weierstraß
theorem; this allows us to cover, for instance, behavioural ultrametrics.publishe