1,901 research outputs found
Propositional calculus for adjointness lattices
Recently, Morsi has developed a complete syntax for the class of all
adjointness algebras . There, is a partially ordered set with top element , is a
conjunction on for which is a left identity
element, and the two implication-like binary operations and on
are adjoints of .
In this paper, we extend that formal system to one for the class of
all 9-tuples , called \emph{%
adjointness lattices}; in each of which is a bounded lattice, and is an
adjointness algebra. We call it \emph{Propositional Calculus for Adjointness
Lattices}, abbreviated . Our axiom scheme for features four
inference rules and thirteen axioms. We deduce enough theorems and
inferences in to establish its completeness for ; by means of
a quotient-algebra structure (a Lindenbaum type of algebra). We study two
negation-like unary operations in an adjointness lattice, defined by means
of together with and . We end by developing complete syntax for
all adjointness lattices whose implications are -type implications
Fuzzy entropy from weak fuzzy subsethood measures
In this paper, we propose a new construction method for fuzzy and weak fuzzy subsethood measures based on the aggregation of implication operators. We study the desired properties of the implication operators in order to construct these measures. We also show the relationship between fuzzy entropy and weak fuzzy subsethood measures constructed by our method
A Deep Study of Fuzzy Implications
This thesis contributes a deep study on the extensions of the IMPLY operator in classical binary logic to fuzzy logic, which are called fuzzy implications. After the introduction in Chapter 1 and basic notations about the fuzzy logic operators In Chapter 2 we first characterize In Chapter 3 S- and R- implications and then extensively investigate under which conditions QL-implications satisfy the thirteen fuzzy implication axioms. In Chapter 4 we develop the complete interrelationships between the eight supplementary axioms FI6-FI13 for fuzzy implications satisfying the five basic axioms FI1-FI15. We prove all the dependencies between the eight fuzzy implication axioms, and provide for each independent case a counter-example. The counter-examples provided in this chapter can be used in the applications that need different fuzzy implications satisfying different fuzzy implication axioms. In Chapter 5 we study proper S-, R- and QL-implications for an iterative boolean-like scheme of reasoning from classical binary logic in the frame of fuzzy logic. Namely, repeating antecedents times, the reasoning result will remain the same. To determine the proper S-, R- and QL-implications we get a full solution of the functional equation , for all , . In Chapter 6 we study for the most important t-norms, t-conorms and S-implications their robustness against different perturbations in a fuzzy rule-based system. We define and compare for these fuzzy logical operators the robustness measures against bounded unknown and uniform distributed perturbations respectively. In Chapter 7 we use a fuzzy implication to define a fuzzy -adjunction in . And then we study the conditions under which a fuzzy dilation which is defined from a conjunction on the unit interval and a fuzzy erosion which is defined from a fuzzy implication to form a fuzzy -adjunction. These conditions are essential in order that the fuzzification of the morphological operations of dilation, erosion, opening and closing obey similar properties as their algebraic counterparts. We find out that the adjointness between the conjunction on the unit interval and the implication or the implication play important roles in such conditions
Fuzzy Sets and Formal Logics
The paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics. Our focus is on the historical side of these developments. © 2015 Elsevier B.V. All rights reserved.partial support by the Spanish projects EdeTRI (TIN2012-39348- C02-01) and 2014 SGR 118.Peer reviewe
The aggregation of propositional attitudes: towards a general theory
How can the propositional attitudes of several individuals be aggregated into overall collective propositional attitudes? Although there are large bodies of work on the aggregation of various special kinds of propositional attitudes, such as preferences, judgments, probabilities and utilities, the aggregation of propositional attitudes is seldom studied in full generality. In this paper, we seek to contribute to filling this gap in the literature. We sketch the ingredients of a general theory of propositional attitude aggregation and prove two new theorems. Our first theorem simultaneously characterizes some prominent aggregation rules in the cases of probability, judgment and preference aggregation, including linear opinion pooling and Arrovian dictatorships. Our second theorem abstracts even further from the specific kinds of attitudes in question and describes the properties of a large class of aggregation rules applicable to a variety of belief-like attitudes. Our approach integrates some previously disconnected areas of investigation.mathematical economics;
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