Many-Valued 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 FreeMany-Valued LogicFirst published Tue Apr 25, 2000; substantive revision Wed Nov 17, 2004Many-valued logics are non-classical logics. They are similar toclassical logic because they accept the principle oftruth-functionality, namely, that the truth of a compound sentence isdetermined by the truth values of its component sentences (and soremains unaffected when one of its component sentences is replaced byanother sentence with the same truth value). But they differ fromclassical logic by the fundamental fact that they do not restrict thenumber of truth values to only two: they allow for a larger setW of truth degrees.Just as the notion of ‘possible worlds’ in the semanticsof modal logic can be reinterpreted (e.g., as ‘moments oftime’ in the semantics of tense logic or as ‘states’in the semantics of dynamic logic), there does not exist a standardinterpretation of the truth degrees. How they are to be understooddepends on the actual field of application. It is general usage,however, to assume that there are two particular truth degrees, usuallydenoted by "0" and "1", respectively, which act like the traditionaltruth values "falsum" and "verum".The formalized languages for systems of many-valued logic(MVL) follow the two standard patterns for propositional and predicatelogic, respectively:there are propositional variables together with connectives and(possibly also) truth degree constants in the case of propositionallanguages,there are object variables together with predicate symbols,possibly also object constants and function symbols, as well asquantifiers, connectives, and (possibly also) truth degree constants inthe case of first-order languages.As usual in logic, these languages are the basis for semantically aswell as syntactically founded systems of logic.1. Semantics2. Proof Theory3. Systems of Many-Valued Logic4. Applications of Many-Valued Logic5. History of Many-Valued LogicBibliographyOther Internet ResourcesRelated Entries1. SemanticsThere are two kinds of semantics for systems of many-valuedlogic.1.1 Standard Logical Matrices1.2 Algebraic SemanticsWe discuss these in turn.1.1 Standard Logical MatricesThe most suitable way of defining a system S ofmany-valued logic is to fix the characteristic logical matrix for itslanguage, i.e. to fix:the set of truth degrees,the truth degree functions which interpret the propositionalconnectives,the meaning of the truth degree constants,the semantical interpretation of the quantifiers,and additionally,the designated truth degrees, which form a subset of theset of truth degrees and act as substitutes for the traditional truthvalue "verum".A well-formed formula A of a propositional language countsas valid under some valuation α (which maps the set ofpropositional variables into the set of truth degrees) iff it has adesignated truth degree under α. And A is logicallyvalid or a tautology iff it is valid under allvaluations.In the case of a first-order language, such a well-formed formulaA counts as valid under an interpretation α ofthe language iff it has a designated truth degree under thisinterpretation and all assignments of objects from the universe ofdiscourse of this interpretation to the object variables. Acounts as logically valid iff it is valid under allinterpretations.Like in classical logic, such an interpretation has to providea (non empty) universe of discourse,the meaning of the object constants of the language,the meaning of the predicate letters and the function symbols ofthe language.A model of some set Σ of well-formed formulas is avaluation α or an interpretation α such that all A∈ Σ are valid under α . That Σ entailsA means that each model of Σ is also a model ofA.1.2 Algebraic SemanticsThere is a second type of semantics for systems Sof many-valued logic which is based on a whole characteristic classK of (similar) algebraic structures. Each suchalgebraic structure has to provide all the data which have to beprovided by a characteristic logical matrix for the language ofS.The notion of validity of a formula A with respect to analgebraic structure from K is defined as if thisstructure would form a logical matrix. And logical validityhere means validity for all structures from the classK.The type of algebraic structures which may form such acharacteristic class K for some systemS of MVL is usually determined by the (syntactical orsemantical) Lindenbaum algebra of S, and often playsalso a crucial role within an algebraic completeness proof. Thealgebraic structures in K have a similar role forS as the Boolean algebras do for classical logic.For particular systems of MVL one has e.g. the followingcharacteristic classes of algebraic structures:for infinite valued Łukasiewicz logic the class ofMV-algebras,for infinite valued Gödel logic the class of allHeyting algebras which additionally satisfy prelinearity (x→ y) ∪ (y → x) = 1,for Hajek's basic t-norm logic the class of all divisibleresiduated lattices which satisfy prelinearity.From a philosophical point of view, it would be preferable to have asemantic foundation for a system of MVL which uses a characteristiclogical matrix. However, from a formal point of view, both approachesare equally important, and the algebraic semantics turns out to be themore general approach.2. Proof TheoryThe main types of logical calculi are all available for systems ofMVL:2.1 Hilbert type calculi2.2 Gentzen type sequent calculi2.3 Tableau calculiHowever, some of the above are available only for finitely valuedsystems.2.1 Hilbert type calculiThese calculi are formed in the same way as the corresponding calculifor classical logic: some set of axioms is used together witha set of inference rules. The notion of derivation is theusual one. 2.2 Gentzen type sequent calculiIn addition to the usual types of sequent calculi, researchers havealso recently started to discuss ‘hypersequent’ calculi forsystems of MVL. Hypersequents are finite sequences of ordinarysequents. For finitely valued systems, particularly m-valued ones,there are also sequent calculi which work with generalizedsequents. In the m-valued case, these are sequences oflength m of sets of formulas.2.3 Tableau calculiThe tree structure of the tableaux remains the same in these calculi asin the tableau calculi for classical logic. The labels of the nodesbecome more general objects, namely, signed formulas. A signedformula is a pair, consisting of a sign and a well-formedformula. A sign is either a truth degree, or a set of truth degrees. Tableau calculi with signed formulas are usually restricted tofinite-valued systems of MVL, so that they can be dealt with in aneffective way.3. Systems of Many-Valued LogicThe main systems of MVL often come as families which compriseuniformly defined finite-valued as well as infinite-valued systems.Here is a list:3.1 Łukasiewicz logics3.2 Gödel logics3.3 t-Norm based systems3.4 Three-valued systems3.5 Dunn/Belnap's 4-valued system3.6 Product systems3.1 Łukasiewicz logicsThe systems Lm and L∞ aredefined by the logical matrix which has either some finite setWm = {k/m−1 | 0 ≤ k ≤ m−1}of rationals within the real unit interval, or the whole unitintervalW∞ = [0,1] = {x ∈ R | 0 ≤ x ≤ 1}as the truth degree set. The degree 1 is the only designated truthdegree.The main connectives of these systems are a strong and a weakconjunction, & and ∧,respectively, given by the truth degree functionsu & v = max {0, u +v−1},u ∧ v = min{u, v},a negation connective ¬ determined by¬u = 1−u,and an implication connective → with truth degree functionu → v = min {1, 1−u +v}.Often, two disjunction connectives are also used. These are definedin terms of & and ∧,respectively, via the usual de Morgan laws using ¬. For thefirst-order Łukasiewicz systems one adds two quantifiers ∀,∃in such a way that the truth degree of∀xH(x) is the infimum of all therelevant truth degrees of H(x), and that the truthdegree of ∃xH(x) is the supremum ofall the relevant truth degrees of H(x).3.2 Gödel logicsThe systems Gm and G∞ aredefined by the logical matrix which has either some finite setWm = {k/m−1 | 0 ≤ k ≤ m−1}of rationals within the real unit interval, or the whole unitintervalW∞ = [0,1] = {x ∈ R | 0 ≤ x ≤ 1}as the truth degree set. The degree 1 is the only designated truthdegree.The main connectives of these systems are a conjunction ∧ and a disjunction ∨determined by the truth degree functionsu ∧ v = min{u, v},u ∨v = max{u, v},an implication connective → with truth degree functionu→vu≤v1u>vvand a negation connective ~ with truth degree function~uu=01u≠00For the first-order Gödel systems one adds two quantifiers∀, ∃in such a way that the truth degree of∀xH(x) is the infimum of all therelevant truth degrees of H(x), and that the truthdegree of ∃xH(x) is the supremum ofall the relevant truth degrees of H(x).3.3 t-Norm based systemsFor infinite valued systems with truth degree setW∞ = [0,1] = {x ∈ R | 0 ≤ x ≤ 1}the influence of fuzzy set theory quite recently initiated the studyof a whole class of such systems of MVL.These systems are basically determined by a (possiblynon-idempotent) strong conjunction connective &T whichhas as corresponding truth degree function a t-norm T, i.e. abinary operation T in the unit interval which is associative,commutative, non-decreasing, and has the degree 1 as a neutralelement:T(u,T(v,w)) =T(T(u,v),w),T(u,v) = T(v,u),u ≤ v ⇒ T(u,w) ≤T(v,w),T(u,1) = u.For all those t-norms which have the sup-preservationpropertyT(u, supi vi) =supi T(u,vi),there is a standard way to introduce a related implicationconnective →T with the truth degree functionu →T v = sup {z | T(u,z) ≤ v}.This implication connective is connected with the t-norm T by thecrucial adjointness conditionT(u,v) ≤ w ⇔ u ≤(v →T w),which determines →T uniquely for each T withsup-preservation property.The language is further enriched with a negation connective, −T, determined by the truth degree function− T u = u →T0.This forces the language to have also a truth degree constant0 to denote the truth degree 0 because then − Tbecomes a definable connective.Usually one adds as two further connectives a (weak) conjunction ∧ and a disjunction ∨with truth degree functions.u ∧ v = min{u, v},u ∨ v = max{u, v}.For t-norms which are continuous functions (in the standard sense ofcontinuity for real functions of two variables) these additionalconnectives become even definable. Suitable definitions aremin {u,v} = T(u, (u→T v)) ,max {u,v} = min { ((u →Tv) →T v) , ((v→T u) →T u) }.Particular cases of such t-norm related systems are the infinitevalued Łukasiewicz and Gödel systems L∞,G∞, and also the product logic which has theusual arithmetic product as its basic t-norm.The class of all t-norms is very large, and up to now not reallywell understood. Even for those t-norms which have the sup-preservationproperty (and which are also called “left continuoust-norms”) the structural understanding is far from complete, butmuch better as for the general case: a discussion of the recent stateof the art is given by Jenei (2004). Sufficiently well understood isonly the further subclass of continuous t-norms: they are nicelycomposed out of isomorphic copies of the Łukasiewicz t-norm, theproduct t-norm, and the Gödel t-norm, i.e. the min-operation, asexplained e.g. in Gottwald (2001).Actually one is able to axiomatize t-norm based systems for someparticular classes of t-norms. As a fundamental result, Hájek(1998) has given an axiomatization of the logic which has, asconjectured by Hajek and proved in Cignoli/Esteva/Godo/Torrens (2000),as its algebraic semantics the class of all t-norm based structureswhose t-norm is a continuous function. Based upon this work, Esteva andGodo (2001) conjectured an axiomatization for the logic of all t-normswhich have the sup-preservation property, and Jenei/Montagna (2002)proved that this really is an adequate axiomatization.The axiomatization of further t-norm based systems, as well as thequestion for t-norm based quantifiers, are recent research problems.The main focus is given by the following two aspects which concernmodifications of the expressive power of these t-norm based systems:(i) strengthenings of this expressibility by forming systems withadditional negation operators or with multiple t-norm based conjunctionoperations; (ii) modifications of this expressibility e.g. by deletingthe truth degree constant 0 from the language, but adding animplication connective to the basic vocabulary, and (iii)generalizations which modify the basic t-norms into non-commutative“pseudo-t-norms” and thus lead to logics withnon-commutative conjunction connectives. A survey for thosedevelopments, restricted to the case of propositional systems, is givenby Gottwald/Hájek (200x).3.4 Three-valued systems3-valued systems seem to be particularly simple cases which offerintuitive interpretations of the truth degrees; these systems includeonly one additional degree besides the classical truth values.The mathematician and logician Kleene used a third truth degree for"undefined" in the context of partial recursive functions. Hisconnectives were the negation, the weak conjunction, and the weakdisjunction of the 3-valued Łukasiewicz system together with aconjunction ∧+ and animplication →+ determined by truth degree functionswith the following function tables:∧+0½100½0½½½½10½1→+0½10111½½½½10½1Here ½ is the third truth degree "undefined". In this Kleenesystem, the degree 1 is the only designated truth degree.Blau (1978) used a different system as an inherent logic of naturallanguage. In Blau's system, both degrees 1 and ½ are designated.Other interpretations of the third truth degree ½, for exampleas "senseless", "undetermined", or "paradoxical", motivated the studyof other 3-valued systems.3.5 Dunn/Belnap's 4-valued systemThis particularly interesting system of MVL was the result ofresearch on relevance logic, but it alsohas significance for computer science applications. Its truth degreeset may be taken asW* = {Ø, {⊥}, { }, {⊥, }},and the truth degrees interpreted as indicating (e.g. with respectto a database query for some particular state of affairs) that thereisno information concerning this state of affairs,information saying that the state of affairs fails,information saying that the state of affairs obtains,conflicting information saying that the state of affairs obtains aswell as fails.This set of truth degrees has two natural (lattice) orderings:a truth ordering which has {} on top of the incomparable degrees Ø , {⊥ , }, and has {⊥ } at the bottom;i.e.,  an information (or: knowledge) orderingwhich has {⊥ , } on top of theincomparable degrees {⊥ }, {}, andhas Ø at the bottom; i.e.,  Given the inf and the sup under the truth ordering, there are truthdegree functions for a conjunction and a disjunction connective. Anegation is, in a natural way, determined by a truth degree functionwhich exchanges the degrees {⊥ } and {}, and which leaves the degrees {⊥ , } and Ø fixed.Actually, there is no standard candidate for a implicationconnective, and the choice of the designated truth degrees depends onthe intended applications:for computer science applications it is natural to have {} as the only designated degree,for applications to relevance logic the choice of {}, {⊥ , }as designated degrees proved to be adequate.The choice of suitable entailment relations is still an openresearch topic.3.6 Product systemsThe general problem of finding an intuitive understanding of thetruth degrees occasionally has a nice solution: one can consider themas comprising different aspects of the evaluation of sentences. In sucha case of, say, k different aspects the truth degrees may bechosen as k-tuples of values which evaluate the singleaspects. (And these, e.g., may be standard truth values.)The truth degree functions over such k-tuples additionallycan be defined "componentwise" from truth degree (or: truth value)functions for the values of the single components. In this manner,k logical systems may be combined into one many-valuedproduct system.In this way, the truth degrees of Dunn/Belnap's 4-valued system canbe considered as evaluating two aspects of a state of affairs (SOA)related to a database:whether there is positive information about the truth of this SOAor not, andwhether there is positive information about the falsity of this SOAor not.Both aspects can use standard truth values for this evaluation.In this case, the conjunction, disjunction, and negation ofDunn/Belnap's 4-valued system are componentwise definable byconjunction, disjunction, or negation, respectively, of classicallogic, i.e. this 4-valued system is a product of two copies ofclassical two-valued logic.4. Applications of Many-Valued LogicMany-valued logic was motivated in part by philosophical goals whichwere never achieved, and in part by formal considerations concerningfunctional completeness. In the earlier years of development, thiscaused some doubts about the usefulness of MVL. In the meantime,however, interesting applications were found in diverse fields. Some ofthese shall now be mentioned.4.1 Applications to Linguistics4.2 Applications to Logic4.3 Applications to Philosophical Problems4.4 Applications to Hardware Design4.5 Applications to Artificial Intelligence4.6 Applications to Mathematics4.1 Applications to Linguistics. A challenging problem is the treatment of presuppositions inlinguistics, i.e. of assumptions that are only implicit in a givensentence. So, for example, the sentence "The present king of Canada wasborn in Vienna" has the existential presupposition that thereis a present king of Canada. It is not a simple task to understand the propositional treatment ofsuch sentences, e.g. to give criteria for forming their negation, orunderstanding the truth conditions of implications.One type of solution for these problems refers to the use of manytruth degrees, e.g. to product systems with ordered pairs astruth degrees: meaning that their components evaluate in parallelwhether the presupposition is met, and whether the sentence is true orfalse. But 3-valued approaches have also been discussed.4.2 Applications to LogicA first type of application of systems of MVL to logic itself is to usethem to gain a better understanding of other systems of logic. In thisway the Gödel systems arose out of an approach to test whetherintuitionistic logic may be understood as a finitely valued logic. Theintroduction of systems of MVL by Łukasiewicz (1920) was initiallyguided by the (finally unsuccessful) idea of understanding the notionof possibility, i.e. modal logic, in a 3-valued way. A second type of application to logic is the merging of differenttypes of logical systems, e.g. the formulation of systems with gradedmodalities. Melvin Fitting (1991/92) considers systems that define suchmodalities by merging modal and many-valued logic, with intendedapplications to problems of Artificial Intelligence.A third type of application to logic is the modeling of partialpredicates and truth value gaps. However, this is possible only in sofar as these truth value gaps behave "truth functionally", i.e. in sofar as the behavior of the truth value gaps in compound sentences canbe described by suitable truth functions. (This is not always the case,e.g. it is not the case in formulations which usesupervaluations.)4.3 Applications to Philosophical ProblemsHow to understand the meaning of "truth" is an old philosophicalproblem. A logical approach toward this problem consists in enriching aformalized language L with a truth predicate T, to beapplied to sentences of L — or, even better, to beapplied to sentences of the extension LT ofL with the predicate T. Based upon this idea, a reasonable theory of such languages whichcontain truth predicates was developed in the mid-1930s by A. Tarski.One of the results was that such a language LT,which contains its own truth predicate T and has a certainrichness in expressive power, is necessarily inconsistent.Another approach toward such languages LT whichcontain their own truth predicate T was offered by S. Kripke(1975) and is essentially based upon the idea of considering Tas a partial predicate, i.e. as a predicate which has "truth valuegaps". In a case Kripke (1975) considers, these truth value gaps behave"truth functionally" and so can be treated like a third truth degree.Their propagation in compound sentences then becomes describable bysuitable truth degree functions of three-valued systems. In Kripke's(1975) approach this reference was to three-valued systems which S. C.Kleene (1938) had considered in the (mathematical) context of partialfunctions and predicates in recursion theory.A second application of MVL inside philosophy is to the oldparadoxes like the Sorites (heap) or the falakros(bald man). (See the entry Sorites paradox.) In the case of the Sorites, the paradox is asfollows:(i) One grain of sand is not a heap of sand. And (ii) adding onegrain of sand to something which is not a heap does not turn it into aheap. Hence (iii) a single grain of sand can never turn into a heap ofsand, no matter how many grains of sand are added to it.Thus the true premise (i) gives a false conclusion (iii) via asequence of inferences using (ii). A rather natural solution inside anextension of MVL with a graded notion of inference, often calledfuzzy logic, is to take the notion of heap as a vagueone, i.e. as a notion which may hold true of given objects only to some(truth) degree. Additionally it is suitable to consider premise (ii) asonly partially true, however to a degree which is quite near to themaximal degree 1. Then each single inference step is of the form:(a): k grains of sand do not make a heap.(ii): Adding one grain of sand to k grains does not make (k+1)grains into a heap.Hence (b): (k+1) grains of sand do not make a heap.However, this inference has to involve truth degrees for thepremises (a) and (ii), and has to provide a truth degree for theconclusion (b). The crucial idea for the modeling of this type ofreasoning inside MVL is to make sure that the truth degree for (b) issmaller than the truth degree for (a) in case the truth degree for (ii)is smaller than the maximal one. In effect, then, the sentence ngrains of sand do not make a heap tends toward being false for anincreasing number n of grains.4.4 Applications to Hardware DesignClassical propositional logic is used as a technical tool for theanalysis and synthesis of some types of electrical circuits built upfrom "switches" with two stable states, i.e. voltage levels. A ratherstraightforward generalization allows the use of an m-valuedlogic to discuss circuits built from similar "switches" with mstable states. This whole field of application of many-valued logic iscalled many-valued (or even: fuzzy) switching. A good introduction isEpstein (1993). 4.5 Applications to Artificial IntelligenceAI is actually the most promising field of applications, which offers aseries of different areas in which systems of MVL have been used. A first area of application concerns vague notions and commonsensereasoning, e.g. in expert systems. Both topics are modeled via fuzzysets and fuzzy logic, and these refer to suitable systems of MVL. Also,in databases and in knowledge-based systems one likes to store vagueinformation.A second area of application is strongly tied with this first one:the automatization of data and knowledge mining. Here clusteringmethods come into consideration; these refer via unsharp clusters tofuzzy sets and MVL. In this context one is also interested in automatedtheorem proving techniques for systems of MVL, as well as in methods oflogic programming for systems of MVL.4.6 Applications to MathematicsThere are three main topics inside mathematics which are related tomany-valued logic. The first one is the mathematical theory of fuzzysets, and the mathematical analysis of "fuzzy", or approximatereasoning. In both cases one refers to systems of MVL. The second topichas been approaches toward consistency proofs for set theory using asuitable system of MVL. And there is an — often only implicit --reference to the basic ideas of MVL in independence proofs (e.g. forsystems of axioms) which often refer to logical matrices with more thantwo truth degrees. However, here MVL is more a purely technical toolbecause in these independence proofs one is not interested in anintuitive understanding of the truth degrees at all. 5. History of Many-Valued LogicMany-valued logic as a separate subject was created by the Polishlogician and philosopher Łukasiewicz (1920), and developed firstin Poland. His first intention was to use a third, additional truthvalue for "possible", and to model in this way the modalities "it isnecessary that" and "it is possible that". This intended application tomodal logic did not materialize. The outcome of these investigationsare, however, the Łukasiewicz systems, and a series of theoreticalresults concerning these systems.Essentially parallel to the Łukasiewicz approach, the Americanmathematician Post (1921) introduced the basic idea of additional truthdegrees, and applied it to problems of the representability offunctions.Later on, Gödel (1932) tried to understand intuitionistic logicin terms of many truth degrees. The outcome was the family ofGödel systems, and a result, namely, that intuitionistic logicdoes not have a characteristic logical matrix with only finitely manytruth degrees. A few years later, Jaskowski (1936) constructed aninfinite valued characteristic matrix for intuitionistic logic. Itseems, however, that the truth degrees of this matrix do not have anice and simple intuitive interpretation.A philosophical application of 3-valued logic to the discussion ofparadoxes was proposed by the Russian logician Bochvar (1938), and amathematical one to partial function and relations by the Americanlogician Kleene (1938). Much later Kleene's connectives also becamephilosophically interesting as a technical tool to determine fixedpoints in the revision theory of truth initiated by Kripke (1975).The 1950s saw (i) an analytical characterization of the class oftruth degree functions definable in the infinite valued propositionalŁukasiewicz system by McNaughton (1951), (ii) a completeness prooffor the same system by Chang (1958, 1959) introducing the notion ofMV-algebra and a more traditional one by Rose/Rosser (1958), as well as(iii) a completeness proof for the infinite valued propositionalGödel system by Dummett (1959). The 1950s also saw an approach ofSkolem (1957) toward proving the consistency of set theory in the realmof infinite valued logic.In the 1960s, Scarpellini (1962) made clear that the first-orderinfinite valued Łukasiewicz system is not (recursively)axiomatizable. Hay (1963) as well as Belluce/Chang (1963) proved thatthe addition of one infinitary inference rule leads to anaxiomatization of L∞. And Horn (1969) presented acompleteness proof for first-order infinite valued Gödel logic.Besides these developments inside pure many-valued logic, Zadeh (1965)started an (application oriented) approach toward the formalization ofvague notions by generalized set theoretic means, which soon wasrelated by Goguen (1968/69) to philosophical applications, and whichlater on inspired also a lot of theoretical considerations insideMVL.The 1970s mark a period of restricted activity in pure many-valuedlogic. There was, however, a lot of work in the closely related area of(computer science) applications of vague notions formalized as fuzzysets, initiated e.g. by Zadeh (1975, 1979). And there was an importantextension of MVL by a graded notion of inference and entailment inPavelka (1979).In the 1980s, fuzzy sets and their applications remained a hot topicthat called for theoretical foundations by methods of many-valuedlogic. In addition, there were the first complexity results e.g.concerning the set of logically valid formulas in first-order infinitevalued Łukasiewicz logic, by Ragaz (1983). Mundici (1986) starteda deeper study of MV-algebras.These trends have continued since the 1980s. Research has includedapplications of MVL to fuzzy set theory and their applications,detailed investigations of algebraic structures related to systems ofMVL, the study of graded notions of entailment, and investigations intocomplexity issues for different problems in systems of MVL. Thisresearch was complemented by interesting work on proof theory, onautomated theorem proving, by different applications in artificialintelligence matters, and by a detailed study of infinite valuedsystems based on t-norms – which now often are called(mathematical) fuzzy logics.BibliographyMonographs and Survey PapersAckermann, R. (1967): An Introduction to Many-ValuedLogics. Routledge and Kegan Paul, London.Bolc, L. and Borowik, P. (1992): Many-Valued Logics, 1.Theoretical Foundations. Springer, Berlin.Bolc, L. and Borowik, P. (2003): Many-Valued Logics, 2.Automated Reasoning and Practical Applications. Springer, Berlin.Cignoli, R., d'Ottaviano, I. and Mundici, D. (2000): AlgebraicFoundations of Many-Valued Reasoning. Kluwer Acad. Publ.,Dordrecht.Epstein G. (1993): Multiple-Valued Logic Design. Instituteof Physics Publishing, Bristol.Fitting, M. and Orlowska, E. (eds.) (2003): Beyond Two.Physica Verlag, Heidelberg.Gottwald, S. 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Simovici and Ivan Stojmenovic, managing editors.Related Entries logic: classical | logic: fuzzy | logic: paraconsistent | logic: relevance | Prior, Arthur | Sorites paradox Copyright © 2004 bySiegfried Gottwald<gottwald@rz.uni-leipzig.de> |
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