This paper is devoted to present Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method for multi-attribute group decision making under rough neutrosophic environment. The concept of rough neutrosophic set is a powerful mathematical tool to deal with uncertainty, indeterminacy and inconsistency. In this paper, a new approach for multi-attribute group decision making problems is proposed by extending the TOPSIS method under rough neutrosophic environment. Rough neutrosophic set is characterized by the upper and lower approximation operators and the (...) pair of neutrosophic sets that are characterized by truth-membership degree, indeterminacy membership degree, and falsity membership degree. In the decision situation, ratings of alternatives with respect to each attribute are characterized by rough neutrosophic sets that reflect the decision makers’ opinion. Rough neutrosophic weighted averaging operator has been used to aggregate the individual decision maker’s opinion into group opinion for rating the importance of attributes and alternatives. Finally, a numerical example has been provided to demonstrate the applicability and effectiveness of the proposed approach. (shrink)

Actualists of a certain stripe—dispositionalists—hold that metaphysical modality is grounded in the powers of actual things. Roughly: p is possible iff something has, or some things have, the power to bring it about that p. Extant critiques of dispositionalism focus on its material adequacy, and question whether there are enough powers to account for all the possibilities we intuitively want to countenance. For instance, it seems possible that none of the actual contingent particulars ever existed, but it is impossible to (...) explain this by appealing to the powers of some actual thing or things to bring it about. I argue instead that dispositionalism, in the simple form championed by its proponents, is formally inadequate. Dispositionalists interpret the modal operators as simple existential claims about powers, but if we interpret the operators that way, the resulting system of modal logic is too weak to capture metaphysical modality. I argue that we can modify the standard dispositionalist interpretations of the operators to secure formal adequacy, but at the cost of accepting that not all modality is grounded in powers. This, I shall suggest, is not a bad thing—the resulting theory still has powers at its core and has certain attractive features, in addition to formal adequacy, that the standard theory lacks. (shrink)

It is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with (...) the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$. This generalizes both Gelfand duality and Jónsson-Tarski duality. (shrink)

We present a class of normal modal calculi PFD, whose syntax is endowed with operators M r, one for each r [0,1] : if a is sentence, M r is to he read the probability that a is true is strictly greater than r and to he evaluated as true or false in every world of a F-restricted probabilistic kripkean model. Every such a model is a kripkean model, enriched by a family of regular probability evaluations with range in (...) a fixed finite subset F of [0,1] : there is one such a function for every world w, P F, and this allows to evaluate M ra as true in the world w iff p F r.For every fixed F as before, suitable axioms and rules are displayed, so that the resulting system P FD is complete and compact with respect to the class of all the F-restricted probabilistic kripkean models. (shrink)

Steve Awodey and Jesse Hughes. Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem.

A Hilbert algebra with supremum is a Hilbert algebra where the associated order is a join-semilattice. This class of algebras is a variety and was studied in Celani and Montangie . In this paper we shall introduce and study the variety of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras, which are Hilbert algebras with supremum endowed with a modal operator $${\Diamond}$$ ◊ . We give a topological representation for these algebras using the topological spectral-like representation for Hilbert algebras with supremum (...) given in Celani and Montangie . We will consider some particular varieties of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras. These varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic $${\mathbf{IntK}_{\Diamond}}$$ IntK ◊ . We also determine the congruences of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras in terms of certain closed subsets of the associated space, and in terms of a particular class of deductive systems. These results enable us to characterize the simple and subdirectly irreducible $${H_{\Diamond}^{\vee }}$$ H ◊ ∨ -algebras. (shrink)

Montague, Prior, von Wright and others drew attention to resemblances between modal operators and quantifiers. In this paper we show that classical quantifiers can, in fact, be regarded as S5-like operators in a purely propositional modal logic. This logic is axiomatized and some interesting fragments of it are investigated.

One of the standard axioms for Boolean contact algebras says that if a region __x__ is in contact with the join of __y__ and __z__, then __x__ is in contact with at least one of the two regions. Our intention is to examine a stronger version of this axiom according to which if __x__ is in contact with the supremum of some family __S__ of regions, then there is a __y__ in __S__ that is in contact with __x__. We study (...) a modal possibility operator which is definable in complete algebras in the presence of the aforementioned axiom, and we prove that the class of complete algebras satisfying the axiom is closely related to the class of modal KTB-algebras. We also demonstrate that in the class of complete extensional contact algebras the axiom is equivalent to the statement: _every region is isolated_. Finally, we present an interpretation of the modal operator in the class of the so-called _resolution contact algebras_. (shrink)

Given a 1-ary sentence operator , we describe L - another 1-ary operator - as as a left inverse of in a given logic if in that logic every formula is provably equivalent to L. Similarly R is a right inverse of if is always provably equivalent to R. We investigate the behaviour of left and right inverses for taken as the operator of various normal modal logics, paying particular attention to the conditions under which these (...) logics are conservatively extended by the addition of such inverses, as well as to the question of when, in such extensions, the inverses behave as normal modal operators in their own right. (shrink)

In the paper [8], the first author developped a topos- theoretic approach to reference and modality. (See also [5]). This approach leads naturally to modal operators on locales (or spaces without points). The aim of this paper is to develop the theory of such modal operators in the context of the theory of locales, to axiomatize the propositional modal logics arising in this context and to study completeness and decidability of the resulting systems.

In the framework of the topos approach to quantum mechanics we give a representation of physical properties in terms of modal operators on Heyting algebras. It allows us to introduce a classical type study of the mentioned properties.

A philosophical discussion of the intuitive meaning of the formalism of modal propositional logics.

We investigate the join semilattice of modal operators on a Boolean algebra B. Furthermore, we consider pairs ⟨f,g⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle f,g \rangle $$\end{document} of modal operators whose supremum is the unary discriminator on B, and study the associated bi-modal algebras.

Chang algebras as algebraic models for Chang's modal logics [1] are defined. The main result of the paper is a representation theorem for these algebras.

The minimal weakening \ of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of \ with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of \. These logics have the finite model property and they are decidable.

Łukasiewicz presented two different analyses of modal notions by means of many-valued logics: the linearly ordered systems Ł3,..., Open image in new window,..., \; the 4-valued logic Ł he defined in the last years of his career. Unfortunately, all these systems contain “Łukasiewicz type paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following Łukasiewicz’s strategy for (...) defining truth-functional modal logics. The systems we define lack “Łukasiewicz type paradoxes”. Following Brady, we endow them with Belnap–Dunn type bivalent semantics. (shrink)

The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation (...) for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it. (shrink)

Departing from basic concepts in abstract logics, this paper introduces two concepts: conjunctive and disjunctive limits. These notions are used to formalize levels of modal operators.

In this paper, we show that all semisimple varieties of bounded weak-commutative residuated lattices with an S4-like modal operator are discriminator varieties. We also give a characterization of discriminator and EDPC varieties of bounded weak-commutative residuated lattices with an S4-like modal operator follows.

Consider two standard quantified modal languages A and P whose vocabularies comprise the identity predicate and the existence predicate, each endowed with a standard S5 Kripke semantics where the models have a distinguished actual world, which differ only in that the quantifiers of A are actualist while those of P are possibilist. Is it possible to enrich these languages in the same manner, in a non-trivial way, so that the two resulting languages are equally expressive-i.e., so that for each (...) sentence of one language there is a sentence of the other language such that given any model, the former sentence is true at the actual world of the model iff the latter is? Forbes (1989) shows that this can be done by adding to both languages a pair of sentential operators called Vlach-operators, and imposing a syntactic restriction on their occurrences in formulas. As Forbes himself recognizes, this restriction is somewhat artificial. The first result I establish in this paper is that one gets sameness of expressivity by introducing infinitely many distinct pairs of indexed Vlach-operators. I then study the effect of adding to our enriched modal languages a rigid actuality operator. Finally, I discuss another means of enriching both languages which makes them expressively equivalent, one that exploits devices introduced in Peacocke (1978). Forbes himself mentions that option but does not prove that the resulting languages are equally expressive. I do, and I also compare the Peacockian and the Vlachian methods. In due course, I introduce an alternative notion of expressivity and I compare the Peacockian and the Vlachian languages in terms of that other notion. (shrink)

We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of (...) completeness for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system. (shrink)

We generalize the modal interpretation of quantum mechanics so that it may be applied to composite systems represented by arbitrary density operators. We discuss the interpretation these density operators receive and relate this to the discussion about the interpretation of proper and improper mixtures in the standard interpretation.

The modal interpretation of quantum mechanics allows one to keep the standard classical definition of realism intact. That is, variables have a definite status for all time and a measurement only tells us which value it had. However, at present modal dynamics are only applicable to situations that are described in the orthodox theory by projective measures. In this paper we extend modal dynamics to include positive operator measures. That is, for example, rather than using a (...) complete set of orthogonal projectors, we can use an overcomplete set of nonorthogonal projectors. We derive the conditions under which Bell's stochastic modal dynamics for projective measures reduce to deterministic dynamics, showing that Brown and Hiley's generalization of Bohmian mechanics [quant-ph/0005026, ] cannot be thus derived. We then show how deterministic dynamics for positive operators can also be derived. As a simple case, we consider a Harmonic oscillator, and the overcomplete set of coherent state projectors. We show that the modal dynamics for this POM in the classical limit correspond to the classical dynamics, even for the nonclassical number state |n>. This is in contrast to the Bohmian dynamics, which for energy eigenstates, the dynamics are always non-classical. (shrink)

In many applications we have a set of objects together with their properties. Since the available information is usually incomplete and/or imprecise, the true knowledge about subsets of objects can be determined approximately only. In this paper, we discuss a fuzzy generalisation of two pairs of relation-based operators suitable for fuzzy set approximations, which have been recently investigated by Düntsch and Gediga. Double residuated lattices, introduced by Orlowska and Radzikowska, are taken as basic algebraic structures. Main properties of these operators (...) are presented. (shrink)

In this paper we develop the notion of formal precover in a topos by defining a relation between elements and sets in a local set theory. We show that such relations are equivalent to modalities and to universal closure operators. Finally we prove that these relations are well characterized by a convenient restriction to a particular set.

In this paper the language of first-order modal logic is enriched with an operator @ ('actually') such that, in any model, the evaluation of a formula @A at a possible world depends on the evaluation of A at the actual world. The models have world-variable domains. All the logics that are discussed extend the classical predicate calculus, with or without identity, and conform to the philosophical principle known as serious actualism. The basic logic relies on the system K, (...) whereas others correspond to various properties that the actual world may have. All the logics are axiomatized. (shrink)

In this paper, I claim that two ways of defining validity for modal languages (“real-world” and “general” validity), corresponding to distinction between a correct and an incorrect way of defining modal valid- ity, correspond instead to two substantive ways of conceiving modal truth. At the same time, I claim that the major logical manifestation of the real- world/general validity distinction in modal propositional languages with the actuality operator should not be taken seriously, but simply as (...) a by-product of the way in which the semantics of such an operator is usually given. (shrink)

Operators can be compared in at least two respects: expressive strength and deductive strength. Inspired by Hintikka’s treatment of question embedding verbs, the variations of noncontingency operator, and also the various combinations of modal operators and Boolean connectives, we propose a logic with strong noncontingency operator as the only primitive modality. The novel operator is deductively but not expressively stronger than both noncontingency operator and essence operator, and expressively but not deductively weaker than the (...) necessity operator. The frame-definability power of this new logic is in between standard modal logic and noncontingency logic. A notion of bisimulation is proposed to characterize this logic within standard modal logic and first-order logic. Axiomatizations over various frame classes are presented, among which the minimal logic is related to the treatment of an alternative semantics of the agreement operator proposed by Lloyd Humberstone. (shrink)