Ten questions and one problem on fuzzy logic

Recently, I had a very interesting friendly e-mail discussion with Professor Parikh on vagueness and fuzzy logic. Parikh published several papers concerning the notion of vagueness. They contain critical remarks on fuzzy logic and its ability to formalize reasoning under vagueness [10,11]. On the other hand, for some years I have tried to advocate fuzzy logic (in the narrow sense, as Zadeh says, i.e. as formal logical systems formalizing reasoning under vagueness) and in particular, to show that such systems (of many-valued logic of a certain kind) offer a fully fledged and extremely interesting logic [4, 5]. But this leaves open the question of intuitive adequacy of many-valued logic as a logic of vagueness. Below I shall try to isolate eight questions Parikh asks, add two more and to comment on all of them. Finally, I formulate a problem on truth (in)definability in Łukasiewicz logic which shows, in my opinion, that fuzzy logic is not just “applied logic” but rather belongs to systems commonly called “philosophical logic” like modal logics, etc.     

Interval-valued level cut sets of fuzzy set

The connections between Zadeh fuzzy set and three-valued fuzzy set are established in this paper. The concepts of interval-valued level cut sets on Zadeh fuzzy set are presented and new decomposition theorems and representation theorems of Zadeh fuzzy set are established based on new cut sets. Firstly, four interval-valued level cut sets on Zadeh fuzzy set are defined as three-valued fuzzy sets and it is shown that the interval-valued level cut sets of Zadeh fuzzy set are generalizations of normal cut sets on Zadeh fuzzy set, and have the same properties as those of normal cut sets of Zadeh fuzzy set. Secondly, the new decomposition theorems are established based on these new cut sets. It is pointed out that each kind of interval-valued level cut sets corresponds to two decomposition theorems. Thus eight decomposition theorems are obtained. Finally, the definitions of three-valued inverse order nested sets and three-valued order nested sets are presented with eight representation theorems based on new nested sets.     

Logics for belief functions on MV-algebras

In this paper we present a generalization of belief functions over fuzzy events. In particular we focus on belief functions defined in the algebraic framework of finite MV-algebras of fuzzy sets. We introduce a fuzzy modal logic to formalize reasoning with belief functions on many-valued events. We prove, among other results, that several different notions of belief functions can be characterized in a quite uniform way, just by slightly modifying the complete axiomatization of one of the modal logics involved in the definition of our formalism.     

On the algebraic structure of linear, relevance, and fuzzy logics

 Substructural logics are obtained from the sequent calculi for classical or intuitionistic logic by suitably restricting or deleting some or all of the structural rules (Restall, 2000; Ono, 1998). Recently, this field of research has come to encompass a number of logics - e.g. many fuzzy or paraconsistent logics - which had been originally introduced out of different, possibly semantical, motivations. A finer proof-theoretical analysis of such logics, in fact, revealed that it was possible to subsume them under the previous definition (see e.g. Aguzzoli and Ciabattoni, 2000). Although proof systems for substructural logics are currently being investigated with remarkable success, their algebraic models do not seem equally satisfactory. In fact: (i) such structures are often very weak, i.e. they do not possess many interesting algebraic properties; (ii) as a consequence, their theories of ideals, congruences, and representation are as a rule scarcely developed, or even lacking. In this paper, we address these difficulties. Received: 18 February 2000 / Published online: 12 December 2001     

Residuated fuzzy logics with an involutive negation

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant , namely is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that is an involutive negation. However, for t-norms without non-trivial zero divisors, is G?del negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation. Received: 14 April 1998     

Arithmetical complexity of fuzzy predicate logics — A survey II

Results on arithmetical complexity of important sets of formulas of several fuzzy predicate logics (tautologies, satisfiable formulas, …) are surveyed and some new results are proven.     

Expressivity in chain-based modal logics

We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and (appropriately defined) modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including ?ukasiewicz, Gödel, and product modal logics.     

Normal forms for fuzzy logics: a proof-theoretic approach

A method is described for obtaining conjunctive normal forms for logics using Gentzen-style rules possessing a special kind of strong invertibility. This method is then applied to a number of prominent fuzzy logics using hypersequent rules adapted from calculi defined in the literature. In particular, a normal form with simple McNaughton functions as literals is generated for ?ukasiewicz logic, and normal forms with simple implicational formulas as literals are obtained for Gödel logic, Product logic, and Cancellative hoop logic.     

The algebra of fuzzy logic

In the following, human thinking based on premises with no complete truth value is reviewed for controlling the algebra of fuzzy sets operations. Assuming a system may be developed in this sphere, it should be considered as the algebra of fuzzy sets, as the same algebra is satisfied by classical logic and sets. As will be proved, this algebra is not a lattice and consequently the Zadeh definitions do not constitute an adequate representation. The binary operations of my algebra are “interactive” types. An axiom system is given that, in my opinion, is the foundation of the conception, adequately and without redundancy. The agreement of the theorems deduced from the axiom system with the intuitive expectations is shown. A special arithmetical structure satisfying this algebra is given, and the relation between this structure and the theory of probability is analyzed.Adapting a process of classical logics, fuzzy quantifiers are defined on the basis of the operations of propositional algebra. A “qualifier” is also defined. The qualifier is functional; applying it to Ax we get the statement “usually Ax” s a middle cource between the statements “at least once Ax” and “always Ax”. The concept of entailment of fuzzy logics is introduced. This concept is an innovative generalization of the classical deduction theory, opposite to the concept of entailment of classical multi-valued logics. An important error of the abbreviated system of notation of the fuzzy theory [e.g. m(x, AvB)] appears: the functional type operations (e.g. quantifiers) cannot be interpreted in propositional calculus. Therefore a new system of symbols is proposed in this paper.     

Fuzzy events and fuzzy logics in classical information systems

Classical information systems are introduced in the framework of measure and integration theory. The measurable characteristic functions are identified with the exact events while the fuzzy events are the real measurable functions whose range is contained in the unit interval. Two orthogonality relations are introduced on fuzzy events, the first linked to the fuzzy logic and the second to the fuzzy structure of partial a Baer¹-ring. The fuzzy logic is then compared with the “empirical” fuzzy logic induced by the classical information system. In this context, quantum logics could be considered as those empirical fuzzy logics in which it is not possible to have preparation procedures which provide physical systems whose “microstate” is always exactly defined.     

Spatial reasoning under imprecision using fuzzy set theory,formal logics and mathematical morphology

In spatial reasoning, in particular for applications in image understanding, structure recognition and computer vision, a lot of attention has to be paid to spatial relationships and to the imprecision attached to information and knowledge to be handled. Two main components are knowledge representation and reasoning. We show in this paper that the fuzzy set framework associated to the formalism provided by mathematical morphology and formal logics allows us to derive appropriate representations and reasoning tools.     

A normal form which preserves tautologies and contradictions in a class of fuzzy logics

Most of the normal forms for fuzzy logics are versions of conjunctive and disjunctive classical normal forms. Unfortunately, they do not always preserve tautologies and contradictions which is important, for example, for automated theorem provers based on refutation methods.De Morgan implicative systems are triples like the De Morgan systems, which consider fuzzy implications instead of t-conorms. These systems can be used to evaluate the formulas of a propositional language based on the logical connectives of negation, conjunction and implication. Therefore, they determine different fuzzy logics, called implicative De Morgan fuzzy logics.In this paper, we will introduce a normal form for implicative De Morgan systems and we will show that for implicative De Morgan fuzzy logics whose t-norms are strict, this normal form preserves contradictions as well as tautologies.     

Fuzzy propositional logics

Fuzzy logic $\text{L}$_∞9 considered in connection with fuzzy sets theory, is a special theory, is a special many valued logic with truth-value sets [0, 1], which has been studied already by Lukasiewicz. We consider also his versions $\text{L}$_m for m ? 2 with finite truth-value sets. In all cases we add two further propositional connectives, one conjunction and one disjunction. For these logics we give a list of tautologies, consider relations between their sets of tautologies, prove their compactness, and mention some further results.     

Strict core fuzzy logics and quasi-witnessed models

In this paper we prove strong completeness of axiomatic extensions of first-order strict core fuzzy logics with the so-called quasi-witnessed axioms with respect to quasi-witnessed models. As a consequence we obtain strong completeness of Product Predicate Logic with respect to quasi-witnessed models, already proven by M.C. Laskowski and S. Malekpour in [19]. Finally we study similar problems for expansions with ??, define ??-quasi-witnessed axioms and prove that any axiomatic extension of a first-order strict core fuzzy logic, expanded with ??, and ??-quasi-witnessed axioms are complete with respect to ??-quasi-witnessed models.     

Very true on CBA fuzzy logic

CBA logic was introduced as a non-associative generalization of the Łukasiewicz many-valued propositional logic. Its algebraic semantic is just the variety of commutative basic algebras. Petr Hájek introduced vt-operators as models for the “very true” connective on fuzzy logics. The aim of the paper is to show possibilities of using vt-operators on commutative basic algebras, especially we show that CBA logic endowed with very true connective is still fuzzy.     

Fuzzy measure of fuzzy events defined by fuzzy integrals

We define the concept of fuzzy measure of a fuzzy event by using a general form of fuzzy integral proposed by Murofushi, called fuzzy t-conorm integral, encompassing previous definitions. Zadeh defined the probability measure of a fuzzy event, and later the possibility measure of fuzzy event. Using a duality property of fuzzy t-conorm integral, we propose a general definition of fuzzy measure of fuzzy events, which is compatible with previous definitions of Zadeh, and possesses all properties of a fuzzy measure, in particular the duality property. Using our definition, we examine the case of decomposable measures and belief functions. A comparison with previous works is provided.