Fuzzy Interval Inference Utilizing the Checklist Paradigm and BK-Relational Products

Chapter 20 - Automated Deduction for Many-Valued Logics

This study aimed to analyze the business impact of the catering sector using the fuzzy logic method. The research was conducted at a catering company in Istanbul, utilizing document review and participant observation methods to evaluate the business impact. The nominal prioritization method was used to identify critical business processes, and a model along with a mathematical formula was developed for calculating the business impact. The Fuzzy Logic Designer Toolbox in MATLAB was utilized for this calculation. The study identified eight critical business processes: (1) material supply, (2) material storage, (3) pre-preparation process, (4) cooking process, (5) portioning, (6) shipping, (7) hygiene and food safety, and (8) customer relationship management. The business impact was assessed using classical and fuzzy logic methods, and the results were compared. The fuzzy logic method provided a more flexible and comprehensive assessment, managing uncertainty and variability more effectively than classical logic. Overall, it proved to be more effective in optimizing business processes, offering a more dynamic and holistic approach to improving and prioritizing these processes.

Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving several reasoners, not just one as in the standard development.

We discuss a scheme for defining and reasoning about partial recursive functions within a classical two-valued logic in which all terms denote. We show how a total extension of the partial function introduced by a recursive declaration may be axiomatized within a classical logic, and illustrate by an example the kind of reasoning that our scheme supports. By presenting a naive set-theoretic semantics, we show that the system we propose is logically consistent. Our work is motivated largely by the pragmatic issues arising from mechanical theorem proving – we discuss some of the practical benefits and limitations of our scheme for mechanical verification of software and hardware systems.

Chapter 1 Knowledge Representation and Classical Logic

Quantum logic, usually interpreted as non-distributive but two-valued logic, is reinterpreted as many-valued logic. This allows to explain the logical background of the wave-particle duality exhibited in the double-slit experiment in a better way than it can be done with the use of classical two-valued logic.

By means of infinite product of uniformly distributed probability spaces of cardinal n, the concept of n-validity degrees and validity degree vectors of formulae in two-valued predicate logic are introduced. It is proved that the validity degree vectors of formulae can preserve the logical relation between formulae. Moreover, a consistency theorem is obtained which says that the n-validity degree τn(A) of the quantifierfree first-order formula A without any repeated predicate symbols or terms is independent of the natural number n, and is a constant equal to the validity degree τ (A0) of the corresponding proposition A0 in classical propositional logic.

Some find it plausible that a sufficiently long duration of torture is worse than any duration of mild headaches. Similarly, it has been claimed that a million humans living great lives is better than any number of worm-like creatures feeling a few seconds of pleasure each. Some have related bad things to good things along the same lines. For example, one may hold that a future in which a sufficient number of beings experience a lifetime of torture is bad, regardless of what else that future contains, while minor bad things, such as slight unpleasantness, can always be counterbalanced by enough good things. Among the most common objections to such ideas are sequence arguments. But sequence arguments are usually formulated in classical logic. One might therefore wonder if they work if we instead adopt many-valued logic. I show that, in a common many-valued logical framework, the answer depends on which versions of transitivity are used as premises. We get valid sequence arguments if we grant any of several strong forms of transitivity of ‘is at least as bad as’ and a notion of completeness. Other, weaker forms of transitivity lead to invalid sequence arguments. The plausibility of the premises is largely set aside here, but I tentatively note that almost all of the forms of transitivity that lead to valid sequence arguments seem intuitively problematic. Still, a few moderately strong forms of transitivity that might be acceptable lead to valid sequence arguments, although weaker statements of the initial value claims avoid these arguments at least to some extent.

Here we shall extract those minimal functional means of many-valued logic which are sufficient for the construction of paraconsistent logic. Thereby we shall define a rather wide class of many-valued logics which contain paraconsistent logic as a fragment. Thus one may speak of paraconsistent structure in terms of many-valued logic exactly as one may speak of, for example, modal structure in terms of many-valued logic. Also it will be shown that paraconsistent logic extracted from many-valued logic in its turn contains a fragment which is isomorphous to classical two-valued logic. As will be evidently found out the difference between constructed paraconsistent logic and the specified model for classical logic consists only in the properties of the operation of negation (and in the choice of the set of designated truth values for paraconsistent logic).

Symmetry had been well studied in classical logics and constraint programming since a decade. Early, Krishna-murthy showed that some tricky formulas admit short proofs when augmenting the propositional logic resolution proof system by the symmetry rule. However, in Artificial Intelligence, we usually manipulate incomplete information and need to include uncertainty to reason on knowledge with exceptions and non-monotonicity. Several non classic logics are introduced for that purpose, but as far as we know, symmetry for these frameworks had not been studied yet. Here, we are interested to extend the notion of symmetry to that non classical logics such as preferential logics, X-logics and default logics, then give a new symmetry inference rule for the X-logics and the default logics. Finally, we show how symmetry reasoning is profitable for these logics and how they handle some symmetries that do not exist in classical logics.

The present chapter charts the course of the acquisition of logical expressions in child language. The focus is on the meanings that child language learners initially assign to logical expressions, how children’s meanings compare with the meanings assigned by adults, and how the meanings of the logical expressions of natural language correspond to the meanings that are assigned to the corresponding vocabulary of classical logic. We review the findings of several cross-linguistic experimental studies investigating children’s interpretation of sentences that contain different combinations of logical expressions. In general, the findings of research indicate a strong overlap in the meanings that children assign to logical expressions and the meanings assigned to the corresponding expressions in classical logic.

It is well known that an important task of the artificial intelligence is to make a computer simulate a human being in dealing with certainty and uncertainty in information. Logic gives a technique for laying the foundations of this task. Information processing dealing with certain information is based on the classical logic. Nonclassical logic including many-valued logic and fuzzy logic takes the advantage of the classical logic to handle information with various facets of uncertainty, such as fuzziness and randomness. Therefore, nonclassical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information. For investigation of several properties, such logics have been represented as algebras, that is, sets with one, two, or more algebraic operations satisfying some conditions inspired by these logics. This inspiration is illustrated by the similarities between the names. We have BCK-algebras and a BCK positive logic, BCI-algebras and a BCI positive logic, and BL-algebras and a basic logic. In many cases, the connection between such algebras and their corresponding logics is much stronger. In this case, one can give a translation procedure which translates all well-formed formulas and all theorems of a given logic L into terms and theorems of the corresponding algebras. In some cases, one can give also an inverse translation. In this case, we say that the given logic and class of such algebras are isomorphic. Nevertheless, the study of algebras motivated by logics is interesting and very useful also in the case when these structures are not isomorphic. An important class of algebras inspired by logic is BL-algebras introduced by Hajek in order to provide an algebraic proof of the completeness theorem of Basic Logic. A classical example of a BL-algebra is the interval [0,1] endowed with the structure induced by a continuous t-norm. Other important examples of BL-algebras are MV-algebras introduced earlier as a model of some infinitely valued Łukasiewicz logic. All these algebras are also strongly connected with residuated lattices. Hence, the filter theory plays an important role in the study of these algebras. From logic point of view, various filters correspond to various sets of provable formulae in the respective logic. Below we present several papers on filters in various algebras. In one of these papers, it is proved that there is a one-to-one correspondence between the set of all vague filters and all vague congruence of a lattice implication algebra. In another, the relations among fuzzy t-filters on residuated lattices are described. Another paper is devoted to the filter topology on lattice implication algebras. In further articles is characterized the relationship between the main types of fuzzy filters in BE-algebras and EQ-algebras. The role of ideals and soft sets in BL-algebras is described in another two papers. In papers on pseudo-weak-R0 algebras, the most simplified axioms system is presented and it is proved that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL algebras. Pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Several presented papers are devoted to the possible applications of algebras inspired by logic and fuzzy sets. For example, a robust fuzzy logic controller is proposed for stabilization and disturbance rejection in nonlinear control systems of a particular type. The dynamic feedback controller is designed as a combination of a control law that compensates for nonlinear terms in a control system and a dynamic fuzzy logic controller that addresses unknown model uncertainties and an unmeasured disturbance. A mathematical derivation is used to prove that the controller is able to achieve asymptotic stability by processing state measurements. Properties of some hesitant triangular fuzzy aggregation operators based on Bonferroni means also are discussed. In one paper, the method of a construction of a ranking of the physical matches is proposed. This method is based on the fuzzy clustering analysis. Decisions are often made by many decision experts having preference of risks. In view of the grey situation group decision-making problems, a grey situation group decision-making method on the basis of prospect theory is shown. The method takes the positive and negative ideal situation distance as reference points, defines positive and negative prospect value function, and introduces decision experts' risk preference into grey situation decision-making to make the final decision more in line with decision experts' psychological behavior. One possibility of determination of the weight of each decision expert sets up comprehensive prospect value matrix for decision experts' evaluation, and finally determination of the optimal situation is presented. Jianming Zhan Bijan Davvaz Wieslaw A. Dudek Young Bae Jun Hee Sik Kim

It is argued that two-valued logic and fuzzy logic both lack some representational and reasoning capability in modeling directly a social or natural world with a combination of positive, negative, and neutral relationships. This is due to the fact that the logic values in both lie in the interval (0, 1). An NPN (negative-positive-neutral) crisp logic and an NPN fuzzy logic are presented that assume, respectively, crisp or fuzzy values in the interval (-1, 1). The notions of NPN relations and NPN heuristic transitive closures are introduced. It is suggested that NPN logics can be embedded into NPN relations for cognitive process modeling. An application is given to illustrate the theory. >

Dung's argumentation framework is an abstract framework based on a set of arguments and a binary attack relation defined over the set. One instantiation, among many others, of Dung's framework consists in constructing the arguments from a set of propositional logic formulas. Thus an argument is seen as a reason for or against the truth of a particular statement. Despite its advantages, the argumentation approach for inconsistency handling also has important shortcomings. More precisely, in some applications what one is interested in are not so much only the conclusions supported by the arguments but also the precise explications of such conclusions. We show that argumentation framework applied to classical logic formulas is not suitable to deal with this problem. On the other hand, intuitionistic logic appears to be a natural alternative candidate logic (instead of classical logic) to instantiate Dung's framework. We develop constructive argumentation framework. We show that intuitionistic logic offers nice and desirable properties of the arguments. We also provide a characterization of the arguments in this setting in terms of minimal inconsistent subsets when intuitionistic logic is embedded in the modal logic S4.

Classical logic is not adequate to face the essential vagueness of human reasoning, which is approximate rather than precise in nature. The logical treatment of the concepts of vagueness and approximation is of increasing importance in artificial intelligence and related research. Consequently, many logicians have proposed different systems of many-valued logic as a formalization of approximate reasoning (see, for example, Goguen [G], Gerla and Tortora [GT], Novak [No], Pavelka [P], and Takeuti and Titani [TT]). As far as we know, all the proposals are obtained by extending the range of truth values of propositions. In these logical systems reasoning is still exact and to make a conclusion the antecedent clause of its rule must match its premise exactly. In addition. Wang [W] pointed out: “If we compare calculation with proving,... Procedures of calculation... can be made so by fairly well-developed methods of approximation; whereas... we do not have a clear conception of approximate methods in theorem proving.... The concept of approximate proofs, though undeniably of another kind than approximations in numerical calculations, is not incapable of more exact formulation in terms of, say, sketches of and gradual improvements toward a correct proof” (see pp, 224–225). As far as the author is aware, however, no attempts have been made to give a conception of approximate methods in theorem proving.The purpose of this paper is. unlike all the previous proposals, to develop a propositional calculus, a predicate calculus in which the truth values of propositions are still true or false exactly and in which the reasoning may be approximate and allow the antecedent clause of a rule to match its premise only approximately. In a forthcoming paper we shall establish set theory, based on the logic introduced here, in which there are ∣L∣ binary predicates ∈λ, λ ∈ L such that R(∈, ∈λ) = λ where ∈ stands for ∈1 and 1 is the greatest element in L, and x ∈λy is interpreted as that x belongs to y in the degree of λ, and relate it to intuitionistic fuzzy set theory of Takeuti and Titani [TT] and intuitionistic modal set theory of Lano [L]. In another forthcoming paper we shall introduce the resolution principle under approximate match and illustrate its applications in production systems of artificial intelligence.