arXiv subjects

Tomasz Witczak

Publications and source records attributed to Tomasz Witczak.

9 recordsLinked to original sources

arXiv2022

Some remarks on anti-topological spaces

This paper is devoted to a general presentation of anti-topological spaces. These structures have been initially proposed by Şahin, Kargın and M. Yücel in 2021. We analyse their basic definition, showing some of its subtleties and implications. The framework thus obtained is used to investigate anti-topological interpretation of some basic topological notions. For example, we discuss the idea of interior and closure and we show some results on door spaces. Moreover, we introduce two non-equivalent types of continuity. Finally, we investigate the idea of density and nowhere density. Finally, we give some preliminary suggestions concerning the modal logic of anti-topological spaces. It is noteworthy that the paper contains some additional remarks on infra-topological and weak spaces. They may be considered as a clarification or correction of some earlier results present in literature.

math.GN↗

arXiv2021

Negotiation sets: a general framework

It is well-known fact that there exists 1-1 correspondence between so-called double (or flou) sets and intuitionistic sets (also known as orthopairs). At first glance, these two concepts seem to be irreconcilable. However, one must remember that algebraic operations in these two classes are also defined differently. Hence, the expected compatibility is possible. Contrary to this approach, we combine standard definition of double set with operations which are typical for intuitionistic sets. We show certain advantages and limitations of this viewpoint. Moreover, we suggest an interpretation of our sets and operations in terms of logic, data clustering and multi-criteria decision making. As a result, we obtain a structure of discussion between several participants who propose their "necessary" and "allowable" requirements or propositions.

math.LO↗

arXiv2021

Infra-topologies revisited: logic and clarification of basic notions

In this paper we adhere to the definition of infra-topological space as it was introduced by Al-Odhari. Namely, we speak about families of subsets which contain empty set and the whole universe X, being at the same time closed under finite intersections (but not necessarily under arbitrary or even finite unions). This slight modification allows us to distinguish between new classes of subsets (infra-open, ps-infra-open and i-genuine). Analogous notions are discussed in the language of closures. The class of minimal infra-open sets is studied too, as well as the idea of generalized infra-spaces. Finally, we obtain characterization of infra-spaces in terms of modal logic, using some of the notions introduced above.

math.LO↗

arXiv2020

A note on the intuitionistic logic of false belief

In this paper we analyse logic of false belief in intuitionistic setting. This logic, studied in its classical version by Steinsvold, Fan, Gilbert and Venturi, describes the following situation: a formula F is not satisfied in a given world, but we still believe in it (or we think that it should be accepted). Another interpretations are also possible: e.g. that we do not accept F but it is imposed on us by a kind of council or advisory board. From the mathematical point of view, the idea is expressed by an adequate form of modal operator W which is interpreted in relational frames with neighborhoods. We discuss monotonicity of forcing, soundness, completeness and several other issues. We present also some simple systems in which confirmation of previously accepted formula is modelled.

math.LO↗

arXiv2020

Generalized topological semantics for weak modal logics

Generalized topological spaces are not necessarily closed under finite intersections. Moreover, the whole universe does not need to be open. We use modified version of this framework to establish certain models for non-normal modal logics. We consider at least two approaches to this topic, wherein one of them is based on the interplay of two operators, both reflecting the idea of necessity. The second one leads to the quite known logics MT4 and MNT4. We obtain some completeness results and we prove correspondence between different classes of frames. Also, three types of bisimulation are investigated.

math.LO↗

arXiv2019

Generalized topological spaces with associating function

Generalized topological spaces in the sense of Császár have two main features which distinguish them from typical topologies. First, these families of subsets are not closed under intersections. Second, we allow for the possibility that the whole space is not open. Hence, some points of the universe may be beyond any open set. In this paper we assume that such points are associated with certain open neighbourhoods by means of a special function F. We study various properties of the structures obtained in this way. We introduce the notions of F-interior and F-closure and we discuss issues of convergence and continuity in this new setting.

math.LO↗

arXiv2019

Multi-topological semantics for intuitionistic modal logic

We present three examples of \textit{multi-topological} semantics for intuitionistic modal logic with one modal operator $\Box$ (which behaves in some sense like necessity). We show that it is possible to treat neighborhood models, introduced earlier, as multi-topological. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. Also we propose transformation of multi-topological spaces into the neighborhood structures. Although our neighborhoods can be considered as bi-relational models, we believe that reasoning in terms of neighborhoods gives better topological intuitions.

math.LO↗

arXiv2018

Simple example of weak modal logic based on intuitionistic core

In this paper we present simple example of propositional logic which has one modal operator and is based on intuitionistic core. This system is very weak in modal sense - e.g. rules of regularity or monotonicity do not hold. It has complete semantics composed of possible worlds equipped with neighborhoods and pre-order relation. We discuss certain restrictions imposed on those structures. Also, we present characterization of axiom 4 known from logic S4.

math.LO↗

arXiv2018

Intuitionistic modal logic based on neighborhood semantics without superset axiom

In this paper we investigate certain systems of propositional intuitionistic modal logic defined semantically in terms of neighborhood structures. We discuss various restrictions imposed on those frames but our constant approach is to discard superset axiom. Such assumption allows us to think about specific modalities $Δ, \nabla$ and new functor $\rightsquigarrow$ depending on the notion of maximal neighborhood. We show how it is possible to treat our models as bi-relational ones. We prove soundness and completeness of proposed axiomatization by means of canonical model. Moreover, we show that finite model property holds. Then we describe properties of bounded morphism, behavioral equivalence, bisimulation and n-bisimulation. Finally, we discuss further researches (like some interesting classical cases and particular topological issues).

math.LO↗