Many-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 set\(W\) of truth degrees.

Just as the notion of ‘possible worlds’ in thesemantics of modal logic can be reinterpreted (e.g., as ‘momentsof time’ in the semantics of tense logic or as‘states’ in the semantics of dynamic logic), there doesnot exist a standard interpretation of the truth degrees. How they areto be understood depends on the actual field of application. It isgeneral usage, however, to assume that there are two particular truthdegrees, usually denoted by “0” and “1”. Theseparticular truth degrees act, respectively, like the traditional truthvalues “falsum” and “verum” – butsometimes also like “absolutely false” and“absolutely true”, particularly in cases in which thetraditional truth values of classical logic “split” into aseries of truth degrees.

Many-valued logics treat their truth degrees as technical tools, andintend to choose them suitably for particular applications. It is arather difficult philosophical problem to discuss the (possible,non-technical) nature of such “truth degrees” or“truth values”. The interested reader can consult themonograph Shramko/Wansing (2011) or the entry ontruth values.

The formalized languages for systems ofmany-valued logic(MVL) follow the two standard patterns for propositional and predicatelogic, respectively:

• there are propositional variablestogether with connectives and (possibly also) truth degree constants inthe case of propositional languages,
• there are object variables together with predicate symbols, possiblyalso object constants and function symbols, as well as quantifiers,connectives, and (possibly also) truth degree constants in the case offirst-order languages.

As usual in logic, these languages are the basis for semantically aswell as syntactically founded systems of logic.

• 1. Semantics
  • 1.1 Standard Logical Matrices
  • 1.2 Algebraic Semantics
  • 1.3 Game Semantics
• 2. Proof Theory
  • 2.1 Hilbert type calculi
  • 2.2 Gentzen type sequent calculi
  • 2.3 Tableau calculi
• 3. Systems of Many-Valued Logic
  • 3.1 Łukasiewicz logics
  • 3.2 Gödel logics
  • 3.3 t-Norm based systems
  • 3.4 Three-valued systems
  • 3.5 Dunn/Belnap’s 4-valued system
  • 3.6 Product systems
• 4. Applications of Many-Valued Logic
  • 4.1 Applications to Linguistics
  • 4.2 Applications to Logic
  • 4.3 Applications to Philosophical Problems
  • 4.4 Applications to Hardware Design
  • 4.5 Applications to Artificial Intelligence
  • 4.6 Applications to Mathematics
• 5. History of Many-Valued Logic
• Bibliography
  • Monographs and Survey Papers
  • Other Works Cited
• Academic Tools
• Other Internet Resources
• Related Entries

1. Semantics

There are three kinds of semantics for systems of many-valuedlogic.

• 1.1 Standard Logical Matrices
• 1.2 Algebraic Semantics
• 1.3 Game Semantics

We discuss these in turn.

1.1 Standard Logical Matrices

The most suitable way of defining a system \(\bS\) 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 whichinterpret the propositional connectives,
• the meaning of the truth degreeconstants,
• the semantical interpretation of the quantifiers,

and additionally,

• thedesignated truth degrees, which form a subset of theset of truth degrees and act as substitutes for the traditional truthvalue “verum”,
• and occasionally alsoanti-designated truth degrees, whichform a subset of the set of truth degrees and act as substitutes forthe traditional truth value “falsum”.

A well-formed formula \(A\) of a propositional language countsasvalid under some valuation \(\alpha\) (which maps the set ofpropositional variables into the set of truth degrees) iff it has adesignated truth degree under \(\alpha\). And \(A\) islogicallyvalid or atautology iff it is valid under allvaluations.

In the case of a first-order language, such a well-formed formula\(A\) counts asvalid under an interpretation \(\alpha\) 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. \(A\)counts aslogically valid iff it is valid under allinterpretations.

Like in classical logic, such an interpretation has to provide

• a (non empty) universe of discourse,
• the meaning of the object constants ofthe language,
• the meaning of the predicate letters and the function symbols of thelanguage.

Amodel of some set \(\Sigma\) of well-formed formulas is avaluation \(\alpha\) or an interpretation \(\alpha\) such that all \(A\)∈ \(\Sigma\) are valid under \(\alpha\) . That \(\Sigma\)entails\(A\) means that each model of \(\Sigma\) is also a model of\(A\).

1.2 Algebraic Semantics

There is a second type of semantics for systems \(\bS\)of many-valued logic which is based on a whole characteristic class\(\bK\) 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 of\(\bS\).

The notion of validity of a formula \(A\) with respect to analgebraic structure from \(\bK\) is defined as if thisstructure would form a logical matrix. Andlogical validityhere means validity for all structures from the class\(\bK\).

The type of algebraic structures which form such a characteristicclass \(\bK\) for some system \(\bS\) of MVLusually may come from two different sources. A first source may bedetermined by extralogical considerations which distinguish some suchclass of algebraic structures. If a system \(\bS\) of MVLis, however, determined syntactically or by a single characteristicmatrix, such a class of algebraic structures often is determined by the(syntactical or semantical) Lindenbaum algebra of \(\bS\),and in such a case often plays also a crucial role within an algebraiccompleteness proof. The algebraic structures in \(\bK\) havea similar role for \(\bS\) as the Boolean algebras do forclassical logic.

For particular systems of MVL one has e.g. the followingcharacteristic classes of algebraic structures:

• for infinite valuedŁukasiewicz logic the class ofMV-algebras,
• for infinite valuedGödel logic the class of allHeyting algebras which additionally satisfy prelinearity \((x \rightarrow y) \cup (y \rightarrow x) = 1,\)
• for Hajek’sbasic t-norm logic BL the class of allt-algebras, i.e. all those algebraic structures which are formed by the real unit interval together with a left-continuous t-norm \(T\) and their residuation operation \(I_{T}\) defined as \(I_{T}(x,y) = \sup \{z\mid T(x,z) \le y\}.\)

For the first two of these examples one has, historically, the logicdetermined by a characteristic matrix, and the corresponding class ofalgebraic structures determined later on. For the third example thesituation is different: BL was designed to be the logic of allcontinuous t-norms, and from this extralogical approach the class ofall divisible residuated lattices which satisfy prelinearity wasfound.

From a philosophical point of view, however, it usually would bepreferable to have a semantic foundation for a system of MVL which usesa single characteristic logical matrix. But, from a formal point ofview, both approaches are equally important, and the algebraicsemantics turns out to be the more general approach.

1.3 Game Semantics

There are various ways in whichlogic and games can be related.Dialogical logic, e.g., offers a game-theoretic semantics for classical aswell as for intuitionistic logic: a formula counts as valid if aproponent who states this formula has a winning strategy over possibleattacks an opponent is allowed to realize.

In the context of the relationship between fuzzy sets and many-valuedlogic, an approach toward a game-oriented look at logical validity wasoffered by Robin Giles. Starting in 1975, he proposed in a series ofpapers Giles (1975,1976,1979), and again in Giles (1988), a generaltreatment of reasoning with vague predicates by means of a formalsystem based upon a convenient dialogue interpretation. He had alreadyused this dialogue interpretation in other papers, such as Giles 1974,which deals with subjective belief and the foundations of physics. Themain idea is to let “a sentence represent a belief byexpressing it tangibly in the form of a bet”. The bettingconcerns the actual outcomes of dispersive experiments with differentpossible results of known probability. In this setting then a“sentence \(\psi\) is considered to follow from sentences\(\phi_{1},\ldots,\phi_{n}\) just when he who acceptsthe bets \(\phi_{1},\ldots,\phi_{n}\) can at the sametime bet \(\psi\) without fear of loss”.

The (formal) language obtained in this way is closely related toŁukasiewicz’s infinite-valued logic \(\rL_{\infty}\): in factthe two systems coincide if one assigns to a sentence \(\phi\) the truthvalue \(1-\langle\phi\rangle\), with \(\langle\phi \rangle\) for the riskvalue of asserting \(\phi\). And he even adds the remark “that,with this dialogue interpretation, Łukasiewicz logic is exactlyappropriate for the formulation of the ‘fuzzy set theory’first described by L.A. Zadeh (1965); indeed, it is not too much toclaim that \(\rL_{\infty}\) is related to fuzzy set theory exactlyas classical logic is related to ordinary set theory”.

Different versions and generalizations of these dialogue games havebeen studied recently. Various aspects of these developments arediscuss, e.g., in Fermüller (2008) and Fermüller/Roschger(2014). Such approaches are not only able to provide game semanticsfor e.g. Gödel logics and product logic. There are also bridgeswhich connect such games with the design of sequent calculi formany-valued logics, cf. Fermüller/Metcalfe (2009).

There is also a further type of dialogue games related to\(m\)-valued Łukasiewicz logic: a proponent is asking forinformation, and the answering opponent is allowed to lie up to\(m\) times. Such “Ulam games with lies” have beenintroduced by Mundici (1992).

2. Proof Theory

The main types of logical calculi are all available for systems ofMVL:

• 2.1 Hilbert type calculi
• 2.2 Gentzen type sequent calculi
• 2.3 Tableau calculi

However, some of the above are available only for finitely valuedsystems. The present state of the art for a wide class of infinitelyvalued logics is presented in Metcalfe/Olivetti/Gabbay (2009).

2.1 Hilbert type calculi

These calculi are formed in the same way as the correspondingcalculi for classical logic: some set ofaxioms is usedtogether with a set ofinference rules. The notion ofderivation is the usual one.

2.2 Gentzen type sequent calculi

In addition to the usual types of sequent calculi, researchers havealso recently started to discuss ‘hypersequent’ calculi forsystems of MVL. Hypersequents are finite multisets, i.e. finite unordered lists of ordinarysequents.

For finitely valued systems, particularly \(m\)-valued ones,there are also sequent calculi which work withgeneralizedsequents. In the \(m\)-valued case, these are sequences oflength \(m\) of sets of formulas.

2.3 Tableau calculi

The tree structure of the tableaux remains the same in these calculias in the tableau calculi for classical logic. The labels of the nodesbecome more general objects, namely,signed formulas. A signedformula is a pair, consisting of asign and a well-formedformula. A sign is either a truth degree, or a set of truthdegrees.

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 Logic

The 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 logics
• 3.2 Gödel logics
• 3.3 t-Norm based systems
• 3.4 Three-valued systems
• 3.5 Dunn/Belnap’s 4-valued system
• 3.6 Product systems

3.1 Łukasiewicz logics

The systems \(\rL_{m}\) and \(\rL_{\infty}\) aredefined by the logical matrix which has either some finite set

of rationals within the real unit interval, or the whole unitinterval

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, \(\amp\) and \(\wedge\), respectively, given by the truth degreefunctions

a negation connective \(\neg\) determined by

and an implication connective \(\rightarrow\) with truth degree function

Often, two disjunction connectives are also used. These are defined interms of \(\amp\) and \(\wedge\), respectively, via the usual deMorgan laws using \(\neg\). For the first-order Łukasiewicz systemsone adds two quantifiers \(\forall\), \(\exists\) in such a way that thetruth degree of \(\forall xH(x)\) is theinfimum of allthe relevant truth degrees of \(H(x)\), and that the truth degreeof \(\exists xH(x)\) is thesupremum of all the relevanttruth degrees of \(H(x)\).

3.2 Gödel logics

The systems \(\rG_{m}\) and \(\rG_{\infty}\) aredefined by the logical matrix which has either some finite set

of rationals within the real unit interval, or the whole unitinterval

as the truth degree set. The degree 1 is the only designated truthdegree.

The main connectives of these systems are a conjunction \(\wedge\) and adisjunction \(\vee\) determined by the truth degree functions

an implication connective \(\rightarrow\) with truth degree function

and a negation connective \(\sim\) with truth degree function

For the first-order Gödel systems one adds two quantifiers\(\forall\), \(\exists\) in such a way that the truth degree of\(\forall xH(x)\) is theinfimum of all therelevant truth degrees of \(H(x)\), and that the truthdegree of \(\exists xH(x)\) is thesupremum ofall the relevant truth degrees of \(H(x)\).

3.3 t-Norm based systems

For infinite valued systems with truth degree set

the influence of fuzzy set theory since the mid-1980s initiated the studyof a whole class of such systems of MVL.

These systems are basically determined by a (possiblynon-idempotent) strong conjunction connective \(\amp_{\rT}\) whichhas as corresponding truth degree function at-norm \(\rT\), i.e. abinary operation \(\rT\) in the unit interval which is associative,commutative, non-decreasing, and has the degree 1 as a neutralelement:

For all those t-norms which have thesup-preservationproperty

there is a standard way to introduce a related implicationconnective \(\rightarrow_{\rT}\) with the truth degree function

This implication connective is connected with the t-norm \(\rT\) by thecrucialadjointness condition

which determines \(\rightarrow_{\rT}\) uniquely for each \(\rT\) withsup-preservation property.

The language is further enriched with a negation connective,\(-_{\rT}\), determined by the truth degree function

This forces the language to have also a truth degree constant\(\uO\) to denote the truth degree 0 because then \(-_{\rT}\)becomes a definable connective.

Usually one adds as two further connectives a (weak) conjunction\(\wedge\) and a disjunction \(\vee\) with truth degree functions.

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 are

Particular cases of such t-norm related systems are the infinitevalued Łukasiewicz and Gödel systems \(\rL_{\infty}\),\(\rG_{\infty}\), and also theproduct logic which has theusual arithmetic product as its basic t-norm.

From an analytical point of view, for a t-norm \(\rT\) theirsup-preservation property is the left-continuity of this binaryfunction \(\rT\), i.e. the property that each one of the unaryfunctions \(\rT_{a}(x)=\rT(a,x)\) is left-continuous. And thecontinuity of such a t-norm T can be characterized through thealgebraicdivisibility condition

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 the structural understanding is far from complete, but muchbetter as for the general case: a discussion of the recent state of theart is given by Jenei (2004). Sufficiently well understood is only thefurther subclass of continuous t-norms: they are nicely composed out ofisomorphic copies of the Łukasiewicz t-norm, the product t-norm,and the Gödel t-norm, i.e. the min-operation, as explained e.g. inGottwald (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 BL of all continuoust-norms. Besides the previously mentioned algebraic semantics thislogic has, as conjectured by Hajek and proved inCignoli/Esteva/Godo/Torrens (2000), as another algebraic semantics theclass of all t-norm based structures whose t-norm is a continuousfunction. Based upon this work, Esteva and Godo (2001) conjectured anaxiomatization for the logic MTL of all t-norms which have thesup-preservation property, and Jenei/Montagna (2002) proved that thisreally is an adequate axiomatization. And Esteva/Godo/Montagna (2004)offer a method to axiomatize the logic of each single continuous t-norm: they provide an algorithm which gives for each particular continuous t-norm\(\rT\) a finite list of axiom schemata which, if added to the logic BLof all continuous t-norms, yield an adequate axiomatization of the particular t-norm based logic for \(\rT\).

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 \(\uO\) 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. Surveys for those developmentshad been given by Gottwald/Hájek (2005), Gottwald (2008), andCintula/Hájek (2010).

A nearly complete presentation of the state of the art in 2011 isthe monograph Cintula/Hájek/Noguera (2011). And the particularcontributions of P. Hájek to these developments are honored inthe book Montagna (2015).

3.4 Three-valued systems

3-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 recursivefunctions. His connectives were the negation, the weak conjunction,and the weak disjunction of the 3-valued Łukasiewicz systemtogether with a definable conjunction \(\wedge_{+}\) and a definableimplication \(\rightarrow_{+}\) determined by truth degree functionswith the following function tables (these latter have truth degree½ iff one of their constituents has truth degree ½):

Here ½ is the third truth degree “undefined”. Inthis Kleene system, the degree 1 is the only designated truthdegree.

Blau (1978) used a different system as an inherent logic ofnatural language. In Blau’s system, both degrees 1 and ½ aredesignated. Other interpretations of the third truth degree ½,for example as “senseless”, “undetermined”, or“paradoxical”, motivated the study of other 3-valuedsystems.

3.5 Dunn/Belnap’s 4-valued system

This particularly interesting system of MVL was the result ofresearch onrelevance logic, but italso has significance for computer science applications. Its truthdegree set may be taken as

and the truth degrees interpreted as indicating (e.g. with respectto a database query for some particular state of affairs) that thereis

• no information concerning this state ofaffairs,
• information saying that the state ofaffairs fails,
• information saying that the state ofaffairs obtains,
• conflicting information saying that the state of affairs obtains aswell as fails.

This set of truth degrees has two natural (lattice) orderings:

• atruth ordering which has \(\{\top\}\) on top of theincomparable degrees \(\varnothing\), \(\{\bot , \top\}\), and has\(\{\bot\}\) at the bottom; i.e.,
  
• aninformation (or:knowledge)orderingwhich has \(\{\bot , \top\}\) on top of the incomparable degrees \(\{\bot\}, \{\top\}\), and has \(\varnothing\) 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 \(\{\bot\}\) and \(\{\top\}\), and which leavesthe degrees \(\{\bot, \top\}\) and \(\varnothing\) 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 isnatural to have \(\{\top\}\) as the only designated degree,
• for applications to relevance logic the choice of \(\{\top\}\), \(\{\bot, \top\}\) as designated degrees proved to be adequate.

The choice of suitable entailment relations is still an openresearch topic.

This 4-valued system has an interesting interpretation in thecontext of information bases stored in a computer which was explained byBelnap (1977). A more recent generalization by Shramko/Wansing (2005) toknowledge bases in computer networks leads to 16-valued systems, whichare e.g. also studied by Odintsov (2009).

These 16-valued systems are also of interest from a philosophicalpoint of view and extensively presented in the monographShramko/Wansing (2011).

3.6 Product systems

The 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:

1. whether there is positive informationabout the truth of this SOA or not, and
2. whether 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 Logic

Many-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 Linguistics
• 4.2 Applications to Logic
• 4.3 Applications to Philosophical Problems
• 4.4 Applications to Hardware Design
• 4.5 Applications to Artificial Intelligence
• 4.6 Applications to Mathematics

4.1 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 theexistential 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. toproduct 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.

Another type of ideas to use MVL tools in linguistics consists inapproaches toward the modeling of natural language phenomena. Basicideas and some applications are offered e.g. inNovák/Perfilieva/ Močkoř (1999) and Novák(2008).

4.2 Applications to Logic

A first type of application of systems of MVL to logic itself is touse them to gain a better understanding of other systems of logic. Inthis way the Gödel systems arose out of an approach to testwhether intuitionistic logic may be understood as a finitely valuedlogic. The introduction of systems of MVL by Łukasiewicz (1920)was initially guided by the (finally unsuccessful) idea ofunderstanding the notion of possibility, i.e. modal logic, in a3-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 Problems

How 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 \(L_{T}\)of \(L\) 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\(L_{T}\), which contains its own truth predicate\(T\) and has a certain richness in expressive power, isnecessarily inconsistent.

Another approach toward such languages \(L_{T}\) which contain theirown truth predicate \(T\) was offered by S. Kripke (1975) and isessentially based upon the idea of considering \(T\) as a partialpredicate, i.e. as a predicate which has “truth valuegaps”. In a case Kripke (1975) considers, these truth value gapsbehave “truth functionally” and so can be treated like athird truth degree. Their propagation in compound sentences thenbecomes describable by suitable truth degree functions of three-valuedsystems. In Kripke’s (1975) approach this reference was tothree-valued systems which S. C. Kleene (1938) had considered in the(mathematical) context of partial functions and predicates inrecursion theory.

A second application of MVL inside philosophy is to the old paradoxeslike theSorites (heap) or thefalakros (baldman). (See the entrySorites 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 one grain of sand to something which is not a heap does not turnit into a heap. Hence (iii) a single grain of sand can never turn intoa heap of sand, no matter how many grains of sand are added toit.

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 avagueone, 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:

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\(n\)grains of sand do not make a heap tends toward being false for anincreasing number \(n\) of grains.

4.4 Applications to Hardware Design

Classical 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. voltagelevels. A rather straightforward generalization allows the use ofan \(m\)-valued logic to discuss circuits built from similar“switches” with \(m\) stable states. This whole fieldof application of many-valued logic is called many-valued (or even:fuzzy) switching. A good introduction is Epstein (1993).

4.5 Applications to Artificial Intelligence

AI is actually the most promising field of applications, whichoffers a series of different areas in which systems of MVL have beenused.

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: theautomatization of data and knowledge mining. Here clustering methodscome into consideration; these refer via unsharp clusters to fuzzysets and MVL. In this context one is also interested in automatedtheorem proving techniques for systems of MVL, as well as in methodsof logic programming for systems of MVL. Part of this trend is therecent development of generalized description logics, so-called fuzzydescription logics, which allow the inclusion of technical tools(truth degrees, connectives, graded predicates) from the field of MVLinto – from the point of view of full first-order logics:rudimentary – systems of logic, so-called description logics, seeStraccia (2001), Hájek (2005), Stoilos et al. (2008).

4.6 Applications to Mathematics

There 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 Logic

Many-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 themodalities “it is necessary that” and “it ispossible that”. This intended application to modal logic did notmaterialize. The outcome of these investigations are, however, theŁukasiewicz systems, and a series of theoretical resultsconcerning 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 \(\rL_{\infty}\). 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.

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