A Logic for Vagueness

Logic and Logical Philosophy, 2017

In the common man reasoning the presence of vague predicates is pervasive and under the name "fuzzy logic in narrow sense" or "formal fuzzy logic" there are a series of attempts to formalize such a kind of phenomenon. This paper is devoted to discussing the limits of these attempts both from a technical point of view and with respect the original and principal task: to define a mathematical model of the vagueness. For example, one argues that, since vagueness is necessarily connected with the intuition of the continuum, we have to look at the order-based topology of the interval [0,1] and not at the discrete topology of the set {0, 1}. In accordance, in switching from classical logic to a logic for the vague predicates, we cannot avoid the use of the basic notions of real analysis as, for example, the ones of "approximation", "convergence", "continuity". In accordance, instead of defining the compactness of the logical consequence operator and of the deduction operator in terms of finiteness, we have to define it in terms of continuity. Also, the effectiveness of the deduction apparatus has to be defined by using the tools of constructive real analysis and not the one of recursive arithmetic. This means that decidability and semi-decidability have to be defined by involving effective limit processes and not by finite steps stopping processes.

International Journal of Computer Applications, 2013

Many-Valued logics have been developed to represent mathematical model of imprecision, vagueness, uncertainty and ambiguity in the information. In real world each and every species is vague, human knowledge and the natural languages have a bunch of vagueness or imprecise information. This paper attempts to present three main theories of many-valued logics to treat the vagueness: Fuzzy Logic, Vague Logic and Neutrosophic Logic. Author touches the various perspectives logical, algebraic operation, graphical representations and the practical usage. This paper addresses the modeling of vagueness. Author introduces the framework, Vague Inference System (VIS) for modeling the vagueness using vague logic.

2017

Vagueness is a phenomenon whose manifestation occurs most clearly in linguistic contexts. And some scholars believe that the underlying cause of vagueness is to be traced to features of language. Such scholars typically look to formal techniques that are themselves embedded within language, such as supervaluation theory and semantic features of contexts of evaluation. However, when a theorist thinks that the ultimate cause of the linguistic vagueness is due to something other than language-for instance, due to a lack of knowledge or due to the world's being itself vague-then the formal techniques can no longer be restricted to those that look only at within-language phenomena. If, for example a theorist wonders whether the world itself might be vague, it is most natural to think of employing many-valued logics as the appropriate formal representation theory. I investigate whether the ontological presuppositions of metaphysical vagueness can accurately be represented by (finitely) many-valued logics, reaching a mixed bag of results.

This paper shows that any propositional logic that extends a weak, non-distributive, non-associative, and non-commutative version of Full Lambek with a paraconsistent negation (FL) can be enriched with questions in the style of inquisitive semantics and logic. We introduce a relational semantic framework for substructural logics that enables us to define the notion of an inquisitive extension of λ, denoted as λ(?) , for any logic λ that is at least as strong as FL. A general theory of these " inquisitive extensions " is worked out. In particular, it is shown how to axiomatize λ(?) , given the axiomatization of λ. Furthermore, the general theory is applied to four prominent logical systems in the class: classical logic Cl, intuitionistic logic Int, relevant logic R, and Lukasiewicz fuzzy logic L. For the inquisitive extensions of these logics, axiomatization is provided, a suitable semantics found, and completeness proved.

Journal of Philosophical Logic

Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulation. Furthermore, the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation property holds for the relevant fragment of first order logic.

Communications in Computer and Information Science, 2012

In this paper we introduce a logic called FNG∼(Q) that combines the well-known Gödel logic with a strong negation, rational truthconstants and Possibilistic logic. In this way, we can formalize reasoning involving both vagueness and (possibilistic) uncertainty. We show that the defined logical system is useful to capture the kind of reasoning at work in the medical diagnosis system CADIAG-2, and we finish by pointing out some of its potential advantages to be developed in future work.

2001

The aim of the paper is to outline an idea of solving the problem of the vagueness of concepts. The starting point is a definition of the concept of vague knowledge. One of the primary goals is a formal justification of the classical viewpoint on the controversy about the truth and object reference of expressions including vague terms. It is proved that grasping the vagueness in the language aspect is possible through the extension of classical logic to the logic of sentences which may contain vague terms. The theoretical framework of the conception refers to the theory of Pawlak’s rough sets and is connected with Zadeh’s fuzzy set theory as well as bag (or multiset) theory. In the considerations formal logic means and the concept system of set theory have been used. The paper can be regarded as an outline of the logical theory of vague concepts.

Journal of Computer and System Sciences, 2010

In classical logics, the meaning of a formula is invariant with respect to the renaming of bound variables. This property, normally taken for granted, has been shown not to hold in the case of Independence Friendly (IF) logics. In this paper we argue that this is not an inherent characteristic of these logics but a defect in the way in which the compositional semantics given by Hodges for the regular fragment was generalized to arbitrary formulas. We fix this by proposing an alternative formalization, based on a variation of the classical notion of valuation. Basic metatheoretical results are proven. We present these results for Hodges' slash logic (from which these can be easily transferred to other IF-like logics) and we also consider the flattening operator, for which we give novel game-theoretical semantics.

Information Sciences, 2011

Fuzzy Description Logics are a formalism for the representation of structured knowledge affected by imprecision or vagueness. They have become popular as a language for fuzzy ontology representation. To date, most of the work in this direction has focused on the so-called Zadeh family of fuzzy operators (or fuzzy logic), which has several limitations. In this paper, we generalize existing proposals and show how to reason with a fuzzy extension of the logic SROIQ, the logic behind the language OWL 2, under finitely many-valued Łukasiewicz fuzzy logic. We show for the first time that it is decidable over a finite set of truth values by presenting a reasoning preserving procedure to obtain a non-fuzzy representation for the logic. This reduction makes it possible to reuse current representation languages as well as currently available reasoners for ontologies. be very useful as ontology languages . For instance, OWL Lite, OWL DL and OWL 2 are close equivalents to SHIF ðDÞ; SHOIN ðDÞ and SROIQðDÞ, respectively .