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Just as the notion of ‘possible worlds’ in the semantics of modal logic can be reinterpreted (e.g., as ‘moments of time’ in the semantics of tense logic or as ‘states’ in the semantics of dynamic logic), there does not exist a standard interpretation of the truth degrees. How they are to be understood depends on the actual field of application. It is general usage, however, to assume that there are two particular truth degrees, usually denoted by "0" and "1", respectively, which act like the traditional truth values "falsum" and "verum".

The formalized languages for systems of *many-valued logic*
(MVL) follow the two standard patterns for propositional and predicate
logic, respectively:

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

- Semantics
- Proof Theory
- Systems of Many-Valued Logic
- Applications of Many-Valued Logic
- History of Many-Valued Logic
- Bibliography
- Other Internet Resources
- Related Entries

- the set of truth degrees,
- the truth degree functions which interpret the propositional connectives,
- the meaning of the truth degree constants,
- the semantical interpretation of the quantifiers,

- the
*designated truth degrees*, which form a subset of the set of truth degrees and act as substitutes for the traditional truth value "verum".

In the case of a first-order language, such a well-formed formula
*A* counts as *valid* under an interpretation
of the language iff it has a designated truth
degree under this interpretation and all assignments of objects from
the universe of discourse of this interpretation to the object
variables. *A* counts as *logically valid* iff it is valid
under all interpretations.

Like in classical logic, such an interpretation has to provide

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

The notion of validity of a formula *A* with respect to an
algebraic structure from **K** is defined as if this structure
would form a logical matrix. And *logical validity* here means
validity for all structures from the class **K**.

The type of algebraic structures which may form such a
characteristic class **K** for some system **S** of MVL is
usually determined by the (syntactical or semantical) Lindenbaum
algebra of **S**, and often plays also a crucial role within an
algebraic completeness proof. The algebraic structures in **K**
have a similar role for **S** as the Boolean algebras do for
classical logic.

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

- for infinite valued
*Lukasiewicz logic*the class of MV-algebras, - for infinite valued
*Gödel logic*the class of all Heyting algebras which satisfy prelinearity (*x**y*) (*y**x*) = 1, - for Hajek's
*basic t-norm logic*the class of all divisible residuated lattices which satisfy prelinearity.

From a philosophical point of view, it would be preferable to have a semantic foundation for a system of MVL which uses a characteristic logical matrix. However, from a formal point of view, both approaches are equally important, and the algebraic semantics turns out to be the more general approach.

**Hilbert type calculi**. These calculi
are formed in the same way as the corresponding calculi for classical
logic: some set of *axioms* is used together with a set of
*inference rules*. The notion of derivation is the usual one.

**Gentzen type sequent calculi**. In
addition to the usual types of sequent calculi, researchers have also
recently started to discuss ‘hypersequent’ calculi for systems of MVL.
Hypersequents are finite sequences of ordinary sequents.

For finitely valued systems, particularly *m*-valued ones,
there are also sequent calculi which work with *generalized
sequents*. In the *m*-valued case, these are sequences of
length *m* of sets of formulas>.

**Tableau calculi**. The tree structure of
the tableaux remains the same in these calculi as in the tableau
calculi for classical logic. The labels of the nodes become more
general objects, namely, *signed formulas*. A signed formula is
a pair, consisting of a *sign* and a well-formed formula. A sign
is either a truth degree, or a set of truth degrees.

Tableau calculi with signed formulas are usually restricted to finite-valued systems of MVL, so that they can be dealt with in an effective way.

- Lukasiewicz logics
- Gödel logics
- t-Norm related systems
- 3-valued systems
- Dunn/Belnap's 4-valued system
- Product systems

of rationals within the real unit interval, or the whole unit intervalW_{m}= {k/m1 | 0km1}

as the truth degree set. The degree 1 is the only designated truth degree.W_{}= [0,1] = {xR | 0x1}

The main connectives of these systems are a strong and a weak conjunction, & and , respectively, given by the truth degree functions

a negation connective determined byu&v= max {0,u+v1},

uv= min {u,v},

and an implication connective with truth degree functionu= 1u,

uv= min {1, 1u+v}.

Often, two disjunction connectives are also used. These are defined
in terms of & and , respectively, via the usual de
Morgan laws using
. For the first-order Lukasiewicz systems one
adds two quantifiers
,
in such a way that the truth degree of
*xH*(*x*) is the *infimum* of
all the relevant truth degrees of *H*(*x*), and that the
truth degree of
*xH*(*x*) is the *supremum* of
all the relevant truth degrees of *H*(*x*).

of rationals within the real unit interval, or the whole unit intervalW_{m}= {k/m1 | 0km1}

Was the truth degree set. The degree 1 is the only designated truth degree._{}= [0,1] = {xR | 0x1}

The main connectives of these systems are a conjunction and a disjunction determined by the truth degree functions

an implication connective with truth degree functionuv= min {u,v},

uv= max {u,v},

and a negation connective with truth degree function

For 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 the
relevant truth degrees of *H*(*x*), and that the truth degree of
*xH*(*x*) is the *supremum* of
all the relevant truth degrees of *H*(*x*).

Wthe influence of fuzzy set theory quite recently initiated the study of a whole class of such systems of MVL._{}= [0,1] = {xR | 0x1}

These systems are basically determined by a (possibly
non-idempotent) strong conjunction connective &_{T} which
has as corresponding truth degree function a *t-norm* T, i.e. a
binary operation T in the unit interval which is associative,
commutative, non-decreasing, and has the degree 1 as a neutral
element:

- 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*.

T(there is a standard way to introduce a related implication connectiveu, sup_{i}v_{i}) = sup_{i}T(u,v_{i}),

This is connected with the t-norm T by the crucialu_{T}v= sup {z| T(u,z)v}.

T(which determinesu,v)wu(v_{T}w),

The language is further enriched with a negation connective,
_{T}, determined by the truth degree
function

This forces the language to have also a truth degree constant_{T}u=u_{T}0.

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

Particular cases of such t-norm related systems are the infinite valued Lukasiewicz and Gödel systems Luv= min {u,v},

uv= max {u,v}.

The class of all t-norms, even of those which have the sup-preservation property, is very large. Actually one is able to axiomatize t-norm based systems for some particular classes of t-norms. And Hajek (1998) has given an axiomatization of the logic which has as its algebraic semantics the class of all t-norm based structures whose t-norm is a continuous function.

The axiomatization of further t-norm based systems, as well as the question for t-norm based quantifiers, are recent research problems.

The mathematician and logician Kleene used a third truth degree
for "undefined" in the context of partial recursive functions. His
connectives were the negation, the weak conjunction, and the weak
disjunction of the 3-valued Lukasiewicz system together with a
conjunction
_{+} and an implication
_{+} determined by truth
degree functions with the following function tables:

_{+}0 ½ 1 0 0 ½ 0 ½ ½ ½ ½ 1 0 ½ 1

Here ½ is the third truth degree "undefined". In this Kleene system, the degree 1 is the only designated truth degree.

_{+}0 ½ 1 0 1 1 1 ½ ½ ½ ½ 1 0 ½ 1

Blau (1978) used a different system as an inherent logic of natural language. In Blau's system, both degrees 1 and ½ are designated. Other interpretations of the third truth degree ½, for example as "senseless", "undetermined", or "paradoxical", motivated the study of other 3-valued systems.

W* = {Ø, {}, {}, {, }},and the truth degrees interpreted as indicating (e.g. with respect to a database query for some particular state of affairs) that there is

- no 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 as well as fails.

- a
*truth ordering*which has {} on top of the incomparable degrees Ø , { , }, and has { } at the bottom; i.e.,

- an
*information*(or:*knowledge*)*ordering*which has { , } on top of the incomparable degrees { }, {}, and has Ø at the bottom; i.e.,

Actually, there is no standard candidate for a implication connective, and the choice of the designated truth degrees depends on the 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 truth degree functions over such *k*-tuples additionally
can 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-valued
*product system*.

In this way, the truth degrees of Dunn/Belnap's 4-valued system can be 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 SOA or not, and
- whether there is positive information about the falsity of this SOA or not.

In this case, the conjunction, disjunction, and negation of Dunn/Belnap's 4-valued system are componentwise definable by conjunction, disjunction, or negation, respectively, of classical logic, i.e. this 4-valued system is a product of two copies of classical two-valued logic.

- Applications to Linguistics
- Applications to Logic
- Applications to Philosophical Problems
- Applications to Hardware Design
- Applications to Artificial Intelligence
- Applications to Mathematics

**Applications to Linguistics**. A challenging
problem is the treatment of presuppositions in linguistics, i.e. of
assumptions that are only implicit in a given sentence. So, for
example, the sentence "The present king of Canada was born in Vienna"
has the *existential presupposition* that there is a present
king of Canada.

It is not a simple task to understand the propositional treatment of such sentences, e.g. to give criteria for forming their negation, or understanding the truth conditions of implications.

One type of solution for these problems refers to the use of many
truth degrees, e.g. to *product systems* with ordered pairs as
truth degrees: meaning that their components evaluate in parallel
whether the presupposition is met, and whether the sentence is true
or false. But 3-valued approaches have also been discussed.

**Applications to Logic**. A first type of
application of systems of MVL to logic itself is to use them to gain
a better understanding of other systems of logic. In this way the
Gödel systems arose out of an approach to test whether
intuitionistic logic may be understood as a finitely valued
logic. The introduction of systems of MVL by Lukasiewicz (1920) was
initially guided by the (finally unsuccessful) idea of understanding
the notion of possibility, i.e. modal logic, in a 3-valued way.

A second type of application to logic is the merging of different types of logical systems, e.g. the formulation of systems with graded modalities. Melvin Fitting (1991/92) considers systems that define such modalities by merging modal and many-valued logic, with intended applications to problems of Artificial Intelligence.

A third type of application to logic is the modeling of partial
predicates and truth value gaps. However, this is possible only in so
far as these truth value gaps behave "truth functionally", i.e. in so
far as the behavior of the truth value gaps in compound sentences can
be described by suitable truth functions. (This is not always the
case, e.g. it is not the case in formulations which use
*supervaluations*.)

** Applications to Philosophical Problems **.
How to understand the meaning of "truth" is an old philosophical
problem. A logical approach toward this problem consists in
enriching a formalized language *L* with a truth predicate
*T*, to be applied to sentences of *L* -- or, even better,
to be applied to sentences of the extension *L _{T}* of

Based upon this idea, a reasonable theory of such languages which
contain 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

Another approach toward such languages *L _{T}* which
contain their own truth predicate

A second application of MVL inside philosophy is to the old
paradoxes like the *Sorites* (heap) or the *falakros* (bald
man). (See the entry
Sorites paradox.) In the case of
the Sorites, the paradox is as follows:

(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 turn it into a heap. Hence (iii) a single grain of sand can never turn into a heap of sand, no matter how many grains of sand are added to it.Thus the true premise (i) gives a false conclusion (iii) via a sequence of inferences using (ii). A rather natural solution inside an extension of MVL with a graded notion of inference, often called

- (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.*

**Applications to Hardware Design**.
Classical propositional logic is used as a technical tool for the
analysis and synthesis of some types of electrical circuits built up
from "switches" with two stable states, i.e. voltage levels. A rather
straightforward generalization allows the use of an *m*-valued
logic to discuss circuits built from similar "switches" with *m*
stable states. This whole field of application of many-valued logic
is called many-valued (or even: fuzzy) switching. A good introduction
is Epstein (1993).

**Applications to Artificial Intelligence**. AI
is actually the most promising field of applications, which offers a
series of different areas in which systems of MVL have been used.

A first area of application concerns vague notions and commonsense reasoning, e.g. in expert systems. Both topics are modeled via fuzzy sets and fuzzy logic, and these refer to suitable systems of MVL. Also, in databases and in knowledge-based systems one likes to store vague information.

A second area of application is strongly tied with this first one: the automatization of data and knowledge mining. Here clustering methods come into consideration; these refer via unsharp clusters to fuzzy sets and MVL. In this context one is also interested in automated theorem proving techniques for systems of MVL, as well as in methods of logic programming for systems of MVL.

**Applications to Mathematics**. There are
three main topics inside mathematics which are related to many-valued
logic. The first one is the mathematical theory of fuzzy sets, and
the mathematical analysis of "fuzzy", or approximate reasoning. In
both cases one refers to systems of MVL. The second topic has been
approaches toward consistency proofs for set theory using a suitable
system of MVL. And there is an -- often only implicit -- reference
to the basic ideas of MVL in independence proofs (e.g. for systems of
axioms) which often refer to logical matrices with more than two
truth degrees. However, here MVL is more a purely technical tool
because in these independence proofs one is not interested in an
intuitive understanding of the truth degrees at all.

Essentially parallel to the Lukasiewicz approach, the American mathematician Post (1921) introduced the basic idea of additional truth degrees, and applied it to problems of the representability of functions.

Later on, Gödel (1932) tried to understand intuitionistic logic in terms of many truth degrees. The outcome was the family of Gödel systems, and a result, namely, that intuitionistic logic does not have a characteristic logical matrix with only finitely many truth degrees. A few years later, Jaskowski (1936) constructed an infinite valued characteristic matrix for intuitionistic logic. It seems, however, that the truth degrees of this matrix do not have a nice and simple intuitive interpretation.

A philosophical application of 3-valued logic to the discussion of paradoxes was proposed by the Russian logician Bochvar (1938), and a mathematical one to partial function and relations by the American logician Kleene (1938). Much later Kleene's connectives also became philosophically interesting as a technical tool to determine fixed points in the revision theory of truth initiated by Kripke (1975).

The 1950s saw (i) an analytical characterization of the class of truth degree functions definable in the infinite valued propositional Lukasiewicz system by McNaughton (1951), (ii) a completeness proof for the same system by Chang (1958, 1959) introducing the notion of MV-algebra and a more traditional one by Rose/Rosser (1958), as well as (iii) a completeness proof for the infinite valued propositional Gödel system by Dummett (1959). The 1950s also saw an approach of Skolem (1957) toward proving the consistency of set theory in the realm of infinite valued logic.

In the 1960s, Scarpellini (1962) made clear that the first-order
infinite valued Lukasiewicz system is not (recursively)
axiomatizable. Hay (1963) as well as Belluce/Chang (1963) proved that
the addition of one infinitary inference rule leads to an
axiomatization of
L_{}. And Horn (1969) presented a
completeness 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 of vague notions by generalized set theoretic means,
which soon was related by Goguen (1968/69) to philosophical
applications, and which later on inspired also a lot of theoretical
considerations inside MVL.

The 1970s mark a period of restricted activity in pure many-valued logic. There was, however, a lot of work in the closely related area of (computer science) applications of vague notions formalized as fuzzy sets, initiated e.g. by Zadeh (1975, 1979). And there was an important extension of MVL by a graded notion of inference and entailment in Pavelka (1979).

In the 1980s, fuzzy sets and their applications remained a hot topic that called for theoretical foundations by methods of many-valued logic. In addition, there were the first complexity results e.g. concerning the set of logically valid formulas in first-order infinite valued Lukasiewicz logic, by Ragaz (1983). Mundici (1986) started a deeper study of MV-algebras.

These trends have continued since the 1980s. Research has included applications of MVL to fuzzy set theory and their applications, detailed investigations of algebraic structures related to systems of MVL, the study of graded notions of entailment, and investigations into complexity issues for different problems in systems of MVL. This research was complemented by interesting work on proof theory, on automated theorem proving, by different applications in artificial intelligence matters, and by a detailed study of infinite valued systems based on t-norms.

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*First published: April 25, 2000*

*Content last modified: April 25, 2000*