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Fuzzy Logic (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeFuzzy LogicFirst published Tue Sep 3, 2002; substantive revision Sun Jul 23, 2006The term "fuzzy logic" emerged in the development of the theory offuzzy sets by Lotfi Zadeh (1965). A fuzzy subset A of a (crisp) set X ischaracterized by assigning to each element x of Xthe degree of membership of x in A(e.g., X is a group of people, A the fuzzy set ofold people in X). Now if X is a set ofpropositions then its elements may be assigned their degree oftruth, which may be “absolutely true,” “absolutelyfalse” or some intermediate truth degree: a propositionmay be more true than another proposition. This is obvious in thecase of vague (imprecise) propositions like “this person isold” (beautiful, rich, etc.). In the analogy to variousdefinitions of operations on fuzzy sets (intersection, union,complement, …) one may ask how propositions can be combined byconnectives (conjunction, disjunction, negation, …) andif the truth degree of a composed proposition is determined by thetruth degrees of its components, i.e. if the connectives have theircorresponding truth functions (like truth tables of classicallogic). Saying “yes” (which is the mainstream of fuzzylogic) one accepts the truth-functional approach; this makes fuzzylogic to something distinctly different from probabilitytheory since the latter is not truth-functional (the probabilityof conjunction of two propositions is not determined by theprobabilities of those propositions). Two main directions in fuzzy logic have to be distinguished (cf. Zadeh 1994). Fuzzy logic in the broad sense(older, better known, heavily applied but not asking deep logicalquestions) serves mainly as apparatus for fuzzy control, analysis ofvagueness in natural language and several other applicationdomains. It is one of the techniques of soft-computing, i.e.computational methods tolerant to suboptimality and impreciseness(vagueness) and giving quick, simple and sufficiently goodsolutions. The monographs Novak 1989, Zimmermann 1991, Klir-Yuan 1996, Nguyen 1999 can serve as recommended sources of information. Fuzzy logic in the narrow sense is symbolic logic with acomparative notion of truth developed fully in the spirit ofclassical logic (syntax, semantics, axiomatization,truth-preserving deduction, completeness, etc.; both propositionaland predicate logic). It is a branch of many-valued logicbased on the paradigm of inference under vagueness. Thisfuzzy logic is a relatively young discipline, both serving as afoundation for the fuzzy logic in a broad sense and of independentlogical interest, since it turns out that strictly logicalinvestigation of this kind of logical calculi can go rather far. Abasic monograph is Hajek 1998,further recommended monographs are Turunen 1999, Novak et al. 2000;also recent monographs dealing withmany-valued logic (not specifically oriented to fuzziness), namely Gottwald 2001, Cignoli et al. 2000a;are highly relevant. The interested reader will find below some more information on fuzzyconnectives and a survey of a logical system called basic fuzzy(propositional and predicate) logic together with three strongersystems — Łukasiewicz, Gödel and product logic; ashort discussion on paradoxes and fuzzy logic; some comments on otherformal systems of fuzzy logic, complexity and, finally, a few remarkson fuzzy computing and bibliography.1. Fuzzy connectives2. Basic fuzzy propositional logic3. Basic fuzzy predicate logic4. Łukasiewicz, Gödel and product logic5. Fuzzy logic, paradoxes and probability6. Other systems of fuzzy logic7. On fuzzy computing8. Complexity9. GlossaryBibliographyOther Internet ResourcesRelated Entries1. Fuzzy connectives The standard set of truth degreesis the real interval [0,1] with its natural ordering ≤ (1 standing for absolute truth, 0 for absolute falsity); but one canwork with different domains, finite or infinite, linearly orpartially ordered. Truth functions of connectives have tobehave classically on the extremal values 0,1. It is broadly accepted that t-norms (triangular norms) arepossible truth functions of conjunction. (A binary operation* on the interval [0,1] is a t-norm if it is commutative,associative, non-decreasing and 1 is its unit element. Minimum(min(x,y) is the most popular t-norm. See theGlossary at the end.) Dually, t-conorms serve as truthfunctions of disjunction. See Klement et al. 2000for an extensive theory of t-norms. The truth functionof negation has to be non-increasing (and assign 0 to 1 andvice versa); the function 1 − x (Łukasiewicz negation) is thebest known candidate. Implication is sometimes disregarded but is of fundamentalimportance for fuzzy logic in the narrow sense. A straightforward butlogically less interesting possibility is to define implication fromconjunction and negation (or disjunction and negation) using thecorresponding tautology of classical logic; such implications arecalled S-implications. More useful and interesting areR-implications: an R-implication is defined as aresiduum of a t-norm; denoting the t-norm * and theresiduum ⇒ we have x ⇒ y =max{z| x*z ≤ y}. This iswell-defined only if the t-norm is left-continuous.2. Basic fuzzy propositional logic Basic fuzzy propositional logic is the logic of continuous t-norms (developed in Hajek 1998). Formulas are built from propositional variables using connectives& (conjunction), → (implication) and truth constant 0(denoting falsity). Negation ¬ φ is defined as φ → 0.Given a continuous t-norm * (and hence its residuum ⇒)each evaluation e of propositional variables by truth degreesfor [0,1] extends uniquely to the evaluatione*(φ) of each formula φ using * and ⇒as truth functions of & and →.A formula φ is a t-tautology or standardBL-tautology if e*(φ) = 1 for eachevaluation e and each continuous t-norm *. Thefollowing t-tautologies are taken as axioms of thelogic BL:(A1)(φ → ψ) → ((ψ → χ) → (φ → χ))(A2)(φ & ψ) → φ(A3)(φ & ψ) → (ψ & φ)(A4)(φ & (φ → ψ)) → (ψ & (ψ → φ))(A5a)(φ → (ψ → χ)) → ((φ & ψ) → χ)(A5b)((φ & ψ) → χ) → (φ → (ψ → χ))(A6)((φ → ψ) → χ) → (((ψ → φ) → χ) → χ)(A7) 0 → φ Modus ponens is the only deduction rule; this gives theusual notion of proof and provability of the logic BL.The standard completeness theorem (Cignoli et al. 2000b) says that a formula φ is a t-tautology iff it isprovable in BL.There is a more general semantics of BL, based on algebras calledBL-algebras (see Hajek 1998 for definition); eachBL-algebra can serve as the algebra of truth functions of BL. The general completeness theorem Hajek 1998 says that a formula φ is provable in BL iff it is a generalBL-tautology, i.e., a tautology for each (linearly ordered) BL-algebraL.3. Basic fuzzy predicate logic Basic fuzzy predicate logic has the same formulas as classicalpredicate logic (they are built from predicates of arbitrary arityusing object variables, connectives &, →, truth constant 0and quantifiers ∀, ∃. A standard interpretationis given by a non-empty domain M and for each n-arypredicate P by a n-ary fuzzy relation on M,i.e., a mapping assigning to each n-tuple of elements ofM a truth value from [0,1] — the degree in which then-tuple satisfies the atomic formulaP(x1,…,xn).Given a continuous t-norm, this defines uniquely (in Tarskistyle) the truth degree ||φ|| of each closed formula φ givenby the interpretation M and t-norm *. (Thedegree of an universally quantified formula ∀xφ isdefined as the infimum of truth degrees of instances of φ; similarly∃xφ and supremum. See the Glossary at the end of this entry.) This generalizes in an appropriate manner to a so called safeinterpretation over any linearly ordered BL-algebra anddefinition of the truth value ||φ||M,L given by theL-interpretation M. A formula is ageneral BL-tautology in the predicate logicBL∀ if its truth value is 1 in each safe interpretation. The following BL-tautologies are taken as axioms of BL∀: (a)axioms of the propositional logic BL, and(∀1)∀xφ(x) → φ(y)(∃1)φ(y) → ∃xφ(x)(∀2)∀x(χ→ψ) → (χ → ∀xψ)(∃2)∀x(φ → χ) → (∃xφ → χ)(∀3)∀x(φ ∨ χ) → (∀xφ ∨ χ)(where y is substitutable for x intoφ and x is not free in χ). Deduction rules are modus ponens and generalization as inclassical logic. The general completeness theorem says that a formula isprovable in the fuzzy predicate logic BL∀ iff it is a generalBL-tautology (of predicate logic). This generalizes in a natural wayto provability in a theory over BL∀ and truth in all models ofthe theory; see Hajek 1998 for details. But note that standard BL-tautologies,i.e. formulas true in all standard interpretations w.r.t. allcontinuous t-norms are not recursively axiomatizable (see Hajek 2001a, Montagna 2001for the final result). 4. Łukasiewicz, Gödel and product logicThe following table presents three most important continuoust-norms, their residua and the corresponding negation:  They define three corresponding notions of tautology (being true ineach evaluation w.r.t. the t-norm — standard Ł-tautologies,G-tautologies and Π-tautologies.) On the level of propositionallogic they are completely axiomatized as follows:Ł—BL plus the axiom ¬¬φ → φ of double negation,G—BL plus the axiom φ → (φ & φ) of idempotence of conjunction,Π—BL plus the axiom ¬¬φ → ((φ→ (φ & ψ)) → (ψ & ¬¬ψ)). This is standard completeness; we have also generalcompleteness with respect to BL-algebras satisfying thecorresponding additional conditions (making the additional axiomstrue): they are called MV-algebras (for Ł), G-algebras (for G)and product algebras (for Π) The corresponding predicate logicsŁ∀, G∀, Π∀ are extensions of the basicpredicate fuzzy logic BL∀ by the just formulated axiomscharacterizing Ł, G, Π. Analogously to BL∀ we have the general completenesstheorem for predicate logics: provability = general validity; forG∀ we have also standard completeness, but neitherstandard L∀-tautologies nor standard Π∀-tautologiesare recursively axiomatizable.5. Fuzzy logic, paradoxes and probability In classical logic, the liar paradox (sentence asserting its ownfalsity) relies on the fact that no formula can be equivalent to itsown negation. In Łukasiewicz logic this is not the case: ifφ has the value 0.5 then its negation ¬φ has the samevalue and is equivalent to φ. But one may ask if one can add to(classical) arithmetic a fuzzy truth predicate Tr satisfying,for formulas of this extended language, the disquotationschema φ ≡ Tr(φ), (where φ denotes the Gödel number of φ) The answer is “yes and no”: you get a theory which isconsistent but has no model expanding the standard naturalnumbers. This is discussed in Hajek et al. 2000; see also Grim et al. 1992. The Sorites paradox isrelated to notions like small, many etc.; considering them to be crisp(two-valued) leads to unnatural consequences. We shall sketch atreatment of the notion “small number” in fuzzylogic. (See Goguen 1968-69 for a “classic” analysis.) Without going into detail,imagine a theory inside fuzzy predicate calculus (BL∀ or other)containing crisp arithmetic of natural numbers (as above) and anadditional predicate Small with the axioms saying that 0 issmall (Small(0)), that Small respects ≤,i.e., ∀x,y (x≤y →(Small(y) → Small(x))), and that for all x, the implicationSmall(x)→Small(x+1) is almost true; finally that there is a nonsmall number, ∃x¬Small(x). The “induction” condition can be expressed in variousways, e.g., ∀x At(Small(x) → Small(x+1)) where At is an unary connective “almost true”.Its truth function has to satisfy some natural conditions, inparticular u→At(u). You can have At definable,introducing a new propositional constant r that should beinterpreted by a truth value near to 1 and defining Atφto be r→φ, thus the above formula becomes ∀x(r → (Small(x) → Small(x+1))), or equivalently ∀x((Small(x) & r) → Small(x+1)). You see that the theory admits many interpretations (and hence isconsistent). All interpretations satisfy in some sense the following:the truth degree of Small(x+1) is only slightly lessthan (or equal to) the truth degree ofSmall(x). Thus the paradox can be handled in theframe of fuzzy logic in an axiomatic way, not enforcing any uniquesemantics. The semantics need not be numerical and the truth valuesneed not be linearly ordered (there are BL algebras whose order is notlinear). Several other notions can be handled similarly; for example the fuzzynotion probably can be axiomatized as a fuzzymodality. Having a probability on Boolean formulas, define foreach such formula φ a new formula Pφ, read “probablyφ”, and define the truth value of Pφ to be theprobability of φ. One gets a reasonably elegant bridge betweenfuzziness and probability, with a simple axiom system overŁukasiewicz logic. See Hajek 1998;for an axiomatization of “very true” see Hajek 2001b.6. Other systems of fuzzy logic We mention a few: Pavelka's logic. (Łukasiewicz with rational truthconstants; see Pavelka 1979, Hajek 1998, Novak et al. 2000; V. Novak systematically develops this logic as a logic with evaluatedsyntax (working with pairs (formula, truth value)), fuzzytheories (sets of evaluated formulas) and fuzzy modus ponens[from (φ,u), (φ→ψ,v)derive (ψ,u*v) where * is Łukasiewiczt-norm]. This has excellent properties thanks to the factthat Łukasiewicz t-norm is the only continuous t-norm whoseresiduum is continuous. Expansions of other logics with truthconstants were studied in Esteva et al. 2000, and recently in Esteva et al. 2006 and Savicky et al. 2006.Expansions of basic logic BL by aditionalconnectives. These include logics with an additional involutivenegation (Esteva et al. 2000), and logics putting Łukasiewicz and product logic together (Esteva & Godo 1999, Cintula 2001, Cintula 2003, Horcik & Citula 2004).The monoidal t-norm based logic MTL. Introduced in Esteva & Godo 2001 as well as its predicate variant MTL∀. This is ageneralization of the logic BL — a logic of left continuoust-norms. It has stronger variants IMTL and ΠMTL generalizing theŁukasiewicz and product logic. These logics are (strongly)complete with respect to corresponding algebras. For results onstandard completeness of these logics, see Jenei & Montagna 2002 and (for ΠMTL) Horcik 2005.Fuzzy logics with non-commutative conjunction.(φ&ψ not necessarily equivalent to ψ&φ). Fordetails see di Nola et al. 2002, Hajek 2003, and for standard completeness, Jenei & Montagna 2003.Logics with an additional involutive negation. See Esteva et al. 2000. For a logic putting Łukasiewicz product and Gödellogic together (mentioned above), see Esteva et al. 1999 and Cintula 2001. To close this section let us mention a very general treatment offuzzy logics in the frame of the so-called weakly implicative logicspresented in Cintula 2006 and two recent survey papers: on t-norm based propositional logics Gottwald & Hajek 2005 and on t-norm based predicate logics Cintula and Hajek 2006. 7. On fuzzy computing We briefly comment on so-called fuzzy IF-THEN rules as an example offuzzy logic in a broad sense. They may be understood as partialimprecise knowledge on some crisp function and have (in the simplestcase) the form IF x is Ai THENy is Bi. They shouldnot be immediately understood as implications; thinkof a table relating values of a (dependent) variabley to values of an (independent variable) x:xA1…AnyB1…Bn Ai, Bi maybe crisp (concrete numbers) or fuzzy (small, medium, …) It may beunderstood in two, in general non-equivalent ways: (1) as a listing of n possibilities, called Mamdani's formula:MAMD(x,y)≡n i=1 (Ai(x) & Bi(y)). (where x is A1 and y isB1 or x isA2 and y is B2 or…). (2) as a conjunction of implications:RULES(x,y)≡n i=1 (Ai(x) → Bi(y)).(if x is A1 then y isB1 and …). Both MAMD and RULES define a binary fuzzy relation(given the interpretation of Ai's,Bi's and truth functions ofconnectives). Now given a fuzzy inputA*(x) one can consider the imageB* of A*(x) underthis relation, i.e., B*(y) ≡∃x(A(x) &R(x,y)), where R(x,y) isMAMD(x,y) (most frequent case) orRULES(x,y). Thus one gets an operatorassigning to each fuzzy input set A* acorresponding fuzzy output B*. Usually this iscombined with some fuzzifications converting a crisp inputx0 to some fuzzyA*(x) (saying something as "x issimilar to x0") and a defuzzificationconverting the fuzzy image B* to a crisp outputy0. Thus one gets a crisp function; its relationto the set of rules may be analyzed. For detailed information on fuzzycontrol see Driankov et al. 1993. (But be sure not to call minimum "Mamdani implication"— minimum is not an implication at all! For logical analysis,see e.g., Hajek 2000.)8. Complexity For propositional logics it is always a natural question whether alogic is decidable, i.e., whether its set of tautologies is recursive,and if it is, whether it is in co-NP (its complement beingnon-neterministically computable in polynomial time). Similarly forthe set of satisfiable formulas and NP. (Also sets of positivetautologies, i.e. formulas having a positive value in each evaluationand positively satisfiable formulas are discussed.) It has been shownthat for our logics tautologies are co-NP-complete (of maximalcomplexity in co-NP) and satisfiable formulas are NP-complete. See Baaz et al. 2002 and Hanikova 2002 for final results. The corresponding predicate logics are undecidable (as is theclassical predicate logic) but of various degree of undecidability inthe sense of so-called arithmetical hierarchy ofΣn-sets and Πn-sets. For the readerknowing this hierarchy we mention that for example the set of standardpredicate tautologies of Gödel logic is Σ1-complete,for Łukasiewicz it is Π2-complete and for productlogic it is non-arithmetical (outside the arithmetical hierarchy).Not surprisingly, the set of general predicate tautologies of each ofthese logics is Σ1-complete (due to completenesstheorem). Much more is known; see Hajek 2005 for a survey of known results. Most difficult results onnon-arithmeticity were obtained by Montagna 2001 and Montagna 2005.9. GlossaryTo help the reader not familiar with the basic notions of highermathematics I comment here on two notions used:Continuous t-norm. A t-norm is a particular operationx*y with arguments and values in the real unit interval[0,1]. Such an operation is continuous, intuitivelyspeaking, if small changes of the arguments lead only to smallchanges of the result of the operation. Precisely, for each ε > 0 there is a δ > 0such that wherever |x1 − x2| < δ and |y1 − y2| < δ then|(x1*y1) −(x2*y2)| < ε. Infimum and supremum of a subset of the real unit interval[0,1]. Let A be a set of truth values, hence a subset of[0,1]. A truth value x is a lower bound ofA if x ≤ y for each element yof A; it is the infimum of A if it is thelargest lower bound (notation: x = inf(A)).Clearly, if A has a least element then this element is itsinfimum; but if A has no least element then its infimum isnot its element. For example if A is the set of all positivetruth values (x > 0) then inf(A)=0. Dually,x is an upper bound of A if x ≥y for all y in A; the supremum ofA is its least upper bound.BibliographyBaaz, M., Hajek, P., Montagna, F., and Veith,H. (2002), "Complexity of t-tautologies", Annals of Pure andApplied Logic 113: 3-11. Cignoli, R., D'Ottaviano, I., and Mundici,D. (2000a), Algebraic foundations of many-valuedreasoning, Dordrecht: Kluwer. 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Other Internet ResourcesFuzzy Sets and Systems, maintained by Robert Fullér (Operations Research, Eötvös LorándUniversity)The Fuzzy Logic ArchivesFuzzy Logic Frequently Asked Questions (Carnegie Mellon/AI)Related Entries logic: classical | logic: many-valued | Sorites paradoxAcknowledgments Support of the grant No. A1030004/00 of the Grant Agency of theAcademy of Sciences of the Czech Republic is acknowledged. Thanks aredue to David Coufal for his help in converting this entry into html. Copyright © 2006 byPetr Hajek<hajek@cs.cas.cz> |
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