Some Instances of Graded Consequence in the Context of Interval-Valued Semantics (original) (raw)
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On Cautiousness and Expressiveness in Interval-Valued Logic
Lecture Notes in Computer Science, 2019
In this paper, we study how cautious conclusions should be taken when considering interval-valued propositional logic, that is logic where to each formula is associated a real-valued interval providing imprecise information about the penalty incurred for falsifying this formula. We work under the general assumption that the weights of falsified formulas are aggregated through a non-decreasing commutative function, and that an interpretation is all the more plausible as it is less penalized. We then formulate some dominance notions, as well as properties that such notions should follow if we want to draw conclusions that are at the same time informative and cautious. We then discuss the dominance notions in light of such properties.
Fuzzy Sets and Systems, 2010
In this paper some basic aspects of the theory of graded consequence are reconsidered, based on many results that appeared since the inception of graded consequences in 1986. Relationships of the notion of graded consequence with the notions of consequence in many-valued and fuzzy logics are discussed. Following this, a detailed study is carried out on the necessary conditions that follow from axioms generalizing the rules "ex falso quodlibet" and "reasoning by cases" added to the basic three axioms of graded consequence of reflexivity, monotonicity and cut inspired from Tarski and Gentzen. Towards the converse direction, in the process of finding non-trivial examples satisfying the five graded consequence axioms some sufficient conditions on the algebraic structures and the set of valuations are obtained. The significance of these conditions holding simultaneously is investigated.
Some technical features of the graded consequence
Some technical features of the graded consequence, 2011
This paper is devoted to examine some mathematical features of Chakraborty's theory of graded consequence (see [3], [4], [5], [6]). Namely we emphasize the suitability of analyzing the connections of such a fundamental approach to fuzzy logic with the notions of canonical extension of a deduction apparatus, closure operator, compactness, recursive enumerability (see [1], [2], [8], [9], [10]). 2 Preliminaries on fuzzy logic We denote by U the interval [0, 1] and we look this interval as a complete lattice in which λ ∧ µ = inf{λ, µ} and λ ∨ µ = sup{λ, µ}. Given a nonempty set S we call fuzzy subset of S any map s : S → U. The class U S of all fuzzy subsets of S defines a complete lattice whose join and meet operations we call union and intersection, respectively. We define the complement −s of s by setting −s(x) = 1 − s(x) for every x ∈ S. Let's call continuous chain an order-reversing family (S λ) λ∈U of subsets of S such that S µ = ∩ λ<µ S λ. Then we can identify the fuzzy subsets of S with the continuous chains of subsets of S. Indeed, every fuzzy subset s is associated with the continuous chain C(s, λ)) λ∈U of its cuts, where C(s, λ) = {x ∈ S : s(x) ≥ λ}. Since for every x ∈ S s(x) = sup{λ ∈ U : x ∈ C(s, λ)}, such a correspondence is injective. Conversely, given any continuous chain (S λ) λ∈U of subsets of S, define s by setting s(x) = sup{λ ∈ U : x ∈ S λ }. Then s is a fuzzy subset whose family of cuts coincides with (S λ) λ∈U. This proves that the correspondence is one-to-one. Let F be a set whose elements we call formulas, then an Hilbert deduction system, in brief an H-system, is a pair Σ = (LA, IR) such that LA is a subset of F , the set of logical axioms, and IR a set of inference rules. In turn, an inference rule is a partially defined n-ary map r : F n → F. We denote by Dom(r) the domain of r. Given X ⊆ F , a proof π of a formula α under the hypotheses X is any sequence α 1 , ..., α m of formulas such that α m = α and, for any i = 1, .
Logic in Asia: Studia Logica Library, 2019
Logic in Asia: Studia Logica Library This book series promotes the advance of scientific research within the field of logic in Asian countries. It strengthens the collaboration between researchers based in Asia with researchers across the international scientific community and offers a platform for presenting the results of their collaborations. One of the most prominent features of contemporary logic is its interdisciplinary character, combining mathematics, philosophy, modern computer science, and even the cognitive and social sciences. The aim of this book series is to provide a forum for current logic research, reflecting this trend in the field's development. The series accepts books on any topic concerning logic in the broadest sense, i.e., books on contemporary formal logic, its applications and its relations to other disciplines. It accepts monographs and thematically coherent volumes addressing important developments in logic and presenting significant contributions to logical research. In addition, research works on the history of logical ideas, especially on the traditions in China and India, are welcome contributions. The scope of the book series includes but is not limited to the following: • Monographs written by researchers in Asian countries. • Proceedings of conferences held in Asia, or edited by Asian researchers. • Anthologies edited by researchers in Asia. • Research works by scholars from other regions of the world, which fit the goal of "Logic in Asia".
Interval-valued preference structures
European Journal of Operational Research, 1998
Different languages that are offered to model vague preferences are reviewed and an interval-valued language is proposed to resolve a particular difficulty encountered with other languages. It is shown that interval-valued languages are well defined for De Morgan triples constructed by continuous triangular norms, conorms and a strong negation function. A new transitivity condition for vague preferences is suggested and its relationships to known transitivity conditions are established. A complete characterization of intervalvalued preference structures is also provided.
Theory of graded consequence and fuzzy logics
In one of the papers by Pelta et.al in Mathware and soft computing, the authors made a remark that Until now the construction of superflcial many- valued logics, that is, logics with an arbitrary number (bigger than two ) of truth values but always incorporating a binary consequence relation, has prevailed in investigations of logical many-valuedness. This remark seems to be a genuine observation although calling the entire tradition of many valued logics superflcial is unwarranted. However, in 1986 one of the present authors made an attempt to address this issue and subsequently some other researchers continued work on this idea. Prior to this, fuzzy logic was already magniflcently developed by Pavelka and others. Apparently it may seem that fuzzy logic deals with many-valuedness in the notion of consequence. In fact, there are evidences of such claims in the existing literature. The main objective of this paper is to show that this is not the case , that is fuzzy logics have never...
TEMA - Tendências em Matemática Aplicada e Computacional, 2010
The aim of this work is to analyze the relationship between interval QLimplications and their contrapositions named interval D-implications.In order to achieve this aim, the commutative classes relating to these concepts are studied.We also analyze under which conditions the main properties corresponding to punctual D-implications and QL-implications are still valid when an interval-based fuzzy approach,on the best interval representation, is considered.
Algebras of Intervals and a Logic of Conditional Assertions
Journal of Philosophical Logic, 2004
Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, Łukasiewicz's threevalued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates.
A BILATTICE-BASED FRAMEWORK FOR HANDLING GRADED TRUTH AND IMPRECISION
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2007
We present a family of algebraic structures, called rectangular bilattices, which serve as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and interval-valued fuzzy sets (IVFSs). These structures are useful on one hand to clarify the exact nature of the relationship between the above two common extensions of fuzzy sets, and on the other hand provide an intuitively attractive framework for the representation of uncertain and potentially conflicting information. We also provide these structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships. Application potential and intuitive appeal of the proposed framework are illustrated in the context of preference modeling.
2012
In the traditional fuzzy logic, we use numbers from the interval [0, 1] to describe possible expert's degrees of belief in different statements. Comparing the resulting numbers is straightforward: if our degree of belief in a statement A is larger than our degree of belief in a statement B, this means that we have more confidence in the statement A than in the statement B. It is known that to get a more adequate description of the expert's degree of belief, it is better to use not only numbers a from the interval [0, 1], but also subintervals [a, a] ⊆ [0, 1] of this interval. There are several different ways to compare intervals. For example, we can say that [a, a] ≤ [b, b] if every number from the interval [a, a] is smaller than or equal to every number from the interval [b, b]. However, in interval-valued fuzzy logic, a more frequently used ordering relation between interval truth values is the relation [a, a] ≤ [b, b] ⇔ a ≤ b & a ≤ b. This relation makes mathematical sense-it make the set of all such interval truth values a lattice-but, in contrast to the above relation, it does not have a clear logical interpretation. Since our objective is to describe logic, it is desirable to have a reasonable logical interpretation of this lattice relation. In this paper, we use the notion of modal intervals to provide such a logical interpretation.