In this paper we present a short but important summary of Pythagorean fuzzy sets. Especially we discuss several basic and important definitions of Pythagorean fuzzy sets, several operations on Pythagorean fuzzy sets, several algebraic laws of Pythagorean fuzzy sets. In the last we also present the concept of modal operators and normalization on Pythagorean fuzzy sets.

The logic S4LP combines the modal logic S4 with the justification logic LP, both axiomatically and semantically. We introduce a simple restriction on the behavior of constants in S4LP, having no effect on the LP sublogic. Under this restriction some powerful derived rules are established. Then these are used to show completeness relative to a semantics having what we call the local realizabilit...

: This paper provides a sound and complete axiomatisation for constant domain modal logics without Boolean negation. This is a simpler case of the difficult problem of providing a sound and complete axiomatisation for constant-domain quantified relevant logics, which can be seen as a kind of modal logic with a twoplace modal operator, the relevant conditional. The completeness proof is ...

within a language whose only modal operators are the box and the diamond; other modal idioms cannot be expressed within such a language at all. Nonetheless, quantified modal logic has enjoyed considerable success in uncovering and explaining ambiguities in modal sentences and fallacies in modal reasoning. A prime example of this success is the now standard analysis of the distinction between mo...

This note is a friendly amendment to Marti’s analysis of the failure of Føllesdal’s argument that modal distinctions collapse in Carnap’s logic S2. Føllesdal’s argument turns on the treatment of descriptions. Marti considers how modal descriptions, which Carnap banned, might be handled; she adopts an approach which blocks Føllesdal’s argument, but requires a separate treatment of non-modal desc...

This paper introduces a propositional modal model of Gödel-Löb Logic whose worlds are the models of Peano Arithemetic and whose box modality is equivalent to an operator satisfying the Hilbert Bernay’s conditions (e.g. provability.) The semantics of this model is extended to public announcement logic, and it is shown that announcing a formula is semantically equivalent to adding it as a new axi...

This paper argues for a modal view of probability. The syntax and semantics of one particularly strong probability logic are discussed and some examples of the use of the logic are provided. We show that it is both natural and useful to think of probability as a modal operator. Contrary to popular belief in AI, a probability ranging between 0 and 1 represents a continuum between impossibility a...

Kripke-style models with two accessibility relations, one intuitionistio and the other modal, are given for analogues of the modal system K based on Heyting's propositional logic. I t is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddi...

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula φ in the propositional modal language with A, there is a formula ψ not containing A such that φ and ψ are materially equivalent at the actual world in every Kripke model (respectively, nei...

In this paper, we study modal operator F α,β in intuitionistic fuzzy subgroup of a group and derive some results. 1. Introduction The idea of intuitionistic fuzzy sets (IFSs) was given by [1, 2] to generalize the notion of fuzzy sets (FSs). Intuitionistic fuzzy modal operator was defined by Atanassov in [3]. The modal operators have been known to be important tools for IFSs where the operators ...