Normal forms for fuzzy relations and their contribution to universal approximation

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of "revenge Liar" arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's "useful four-valued logic", one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a "bi-and-a-half" lattice which determines seven-valued logics different from Priest's Logic of Paradox.

In the present chapter, we discuss the possibility of generalizing the very notion of a truth value by constructing truth values as complex units which may possess a ramified inner structure. We consider some approaches to truth values as structured entities and summarize this point in the notion of a generalized truth value conceived as a subset of some basic set of initial truth values of a lower degree. We are essentially led by the idea which is at the heart of Belnap and Dunn’s useful four-valued logic, where the set \({{\mathbf{2}}} = \{T,F\}\) of classical truth values is generalized to the set \({{\mathbf{4}}} = {{\fancyscript{P}}}({{\mathbf{2}}}) = \{\varnothing,\{T\},\{F\},\{T,F\}\}.\) We argue in favor of extending this process to the set \({{\mathbf{16}}} = {{\fancyscript{P}}}({{\mathbf{4}}}).\) It turns out that this generalization is well-motivated and leads to a notion of a truth value multilattice. In particular, we proceed from the bilattice \(FOUR_{2}\) with both an information and truth-and-falsity ordering to another algebraic structure, namely the trilattice \(SIXTEEN_{3}\) with an information ordering together with a truth ordering \(and\) a (distinct) falsity ordering. We also consider another exemplification of essentially the same structure based on the set of truth values one can find in various constructive logics.

Linguistic modelling based on semantic similarity relation among linguistic labels

We study the aggregation of individuals’ beliefs into their collective judgment on a certain proposition, called the conclusion , which is logically related to other preliminary propositions. While the literature has highlighted the crucial dependency of voting outcomes on the selection of an agenda —i.e., the set of propositions voted on—the structure of this dependency is not well understood. We introduce two novel premise-based rules, defined through the conjunctive normal form and disjunctive normal form of the conclusion. Our theorem shows that for any profile of voters’ beliefs, these rules respectively yield the highest and lowest voting outcomes, with respect to a given total order over the set of truth values, within the class of all premise-based rules. This result clarifies the extent to which a voting outcome can be manipulated through the selection of an agenda.

Fuzzy congruence relations

This paper introduces a fuzzy relational data base model (FRDB) that extends the conventional relational data base model with two key features: (1) the relations represent the set of data tuples to be the fuzzy relations; (2) selection conditions are associated with fuzzy set values to be able to query the fuzzy, imprecise information of objects in relations. An interpretation of the membership degree of tuples for fuzzy relations is proposed on the basis of the fuzzy set theory to develop the data model and data manipulating model of FRDB that consist of schemas, fuzzy relations and algebraic operations. Some properties of the fuzzy relational algebraic operations are also formulated and proven as extensions of the properties of relational algebraic operations in the conventional relational data base model.

Permutable fuzzy consequence and interior operators and their connection with fuzzy relations

Quantitative determination of locations vulnerable to ground subsidence at mining regions is necessary for effective prevention. In this paper, a method of constructing subsidence susceptibility maps based on fuzzy relations is proposed and tested at an abandoned underground coal mine in Korea. An advantage of fuzzy combination operators over other methods is that the operation is mathematically and logically easy to understand and its implementation to GIS software is simple and straightforward. A certainty factor analysis was used for estimating the relative weight of eight major factors influencing ground subsidence. The relative weight of each factor was then converted into a fuzzy membership value and integrated as a subsidence hazard index using fuzzy combination operators, which produced coal mine subsidence susceptibility maps. The susceptibility maps were compared with the reported ground subsidence areas, and the results showed high accuracy between our prediction and the actual subsidence. Based on the root mean square error and accuracy in terms of success rates, fuzzy γ-operator with a low γ value and fuzzy algebraic product operator, specifically, are useful for ground subsidence prediction. Comparing the results of a fuzzy γ-operator and a conventional logistic regression model, the performance of the fuzzy approach is comparative to that of a logistic regression model with improved computational. A field survey done in the area supported the method’s reliability. A combination of certainty factor analysis and fuzzy relations with a GIS is an effective method to determine locations vulnerable to coal mine subsidence.

Fuzzy relations and fuzzy relational databases

A bipolar fuzzy set theory is presented for cognitive modeling and multiagent decision analysis. Firstly, notions of bipolar fuzziness are introduced. Secondly an interval-based bipolar fuzzy logic is defined which generalizes a real-valued bipolar fuzzy logic by allowing interval-based linguistic variables x and y to be substituted into (S, =, V, /spl otimes/) or (S, =, /spl cup/, /spl odot/) where S={/spl forall/(x,y)|(x,y)/spl isin/([-1,0]/spl times/[0,+1])}. Thirdly, a fuzzy number-based bipolar logic is presented which. Further generalizes the interval-based model by allowing /spl alpha/-level fuzzy number-based linguistic variables x and y to be substituted into (S, =, V, /spl otimes/) or (S, =, /spl cup/, /spl odot/), S={/spl forall/(x,y)|(x,y) maps ([-1,0]/spl times/[0,+1]) to [0,1]}. Bipolar fuzzy set operations of disjunction composition (V-/spl otimes/), union-composition (/spl cup/, /spl odot/), are proved commutative and associative; V respect to /spl otimes/ and /spl cup/ respect to /spl odot/ are proved distributive. It is shown that a interval-based bipolar variable is a nesting of a real-valued bipolar variable; a trapezoidal-fuzzy number-based bipolar variable is an 2-level nesting of an interval-based bipolar variable; and an /spl alpha/-level (/spl alpha/ is an integer) fuzzy number-based bipolar variable is an /spl alpha/+1 level nesting of an interval-based bipolar variable. Based on the nesting features, it is proved that /spl alpha/-level fuzzy number-based bipolar operations can be converted to interval-based and then real-valued bipolar operations. The conversions lead to significant computational simplification on bipolar fuzzy relations. Major advantages of the bipolar fuzzy set theory include: (1) it formalizes a unified approach to polarity and fuzziness; (2) it captures die bipolar or double-sided (negative and positive, or effect and side effect) nature of human perception and cognition; and (3) it provides a basis for bipolar cognitive modeling and multiagent decision analysis. >

Construction is a highly dynamic environment with numerous interacting factors that affect construction processes and decisions. Uncertainty is inherent in most aspects of construction engineering and management, and traditionally, it has been treated as a random phenomenon. However, there are many types of uncertainty that are not naturally modelled by probability theory, such as subjectivity, ambiguity and vagueness. Fuzzy logic provides an approach for handling such uncertainties. However, fuzzy logic alone has some limitations, including its inability to learn from data and its extensive reliance on expert knowledge. To address these limitations, fuzzy logic has been combined with other techniques to create fuzzy hybrid techniques, which have helped solve complex problems in construction. In this chapter, a background on fuzzy logic in the context of construction engineering and management applications is presented. The chapter provides an introduction to uncertainty in construction and illustrates how fuzzy logic can improve construction modelling and decision-making. The role of fuzzy logic in representing uncertainty is contrasted with that of probability theory. Introductory material is presented on key definitions, properties and methods of fuzzy logic, including the definition and representation of fuzzy sets and membership functions, basic operations on fuzzy sets, fuzzy relations and compositions, defuzzification methods, entropy for fuzzy sets, fuzzy numbers, methods for the specification of membership functions and fuzzy rule-based systems. Finally, a discussion on the need for fuzzy hybrid modelling in construction applications is presented, and future research directions are proposed.

Hedge algebras: An algebraic approach to structure of sets of linguistic truth values

In the article [4], a new constructive set theoretic principle called Refinement was introduced and analysed. While it seemed to be significantly weaker than its alternative, the more established axiom of Fullness (a constructive version of the Powerset axiom from classical set theory), it was shown to suffice to imply many of the mathematically important consequences. In this article, we will define for each set A a set of truth values which measures the complexity of the equality relation on A. Using these sets we will show that Refinement is actually equivalent to Fullness on the basis of the other axioms of constructive Zermelo‐Fraenkel set theory (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

It is shown that the infinite-valued first-order Godel logic G° based on the set of truth values {1/k: k e w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kroger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal logic of linear discrete time with gaps follows.

We explore different ways to generalize Hintikka’s classic game theoretic semantics to a many-valued setting, where the unit interval is taken as the set of truth values. In this manner a plethora of characterizations of Łukasiewicz logic arise. Among the described semantic games is Giles’s dialogue and betting game, presented in a manner that makes the relation to Hintikka’s game more transparent. Moreover, we explain a so-called explicit evaluation game and a ‘bargaining game’ variant of it. We also describe a recently introduced backtracking game as well as a game with random choices for Łukasiewicz logic.