Many-valued logic (also multi- or multiple-valued logic) refers to a
propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
in which there are more than two
truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some pro ...
s. Traditionally, in
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
's
logical calculus, there were only two possible values (i.e., "true" and "false") for any
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
. Classical
two-valued logic
In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is called ...
may be extended to ''n''-valued logic for ''n'' greater than 2. Those most popular in the literature are
three-valued (e.g.,
Łukasiewicz's and
Kleene's, which accept the values "true", "false", and "unknown"),
four-valued,
nine-valued, the
finite-valued (finitely-many valued) with more than three values, and the
infinite-valued (infinitely-many-valued), such as
fuzzy logic and
probability logic.
History
It is
wrong that the first known classical logician who did not fully accept the
law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
was
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
(who, ironically, is also generally considered to be the first classical logician and the "father of
wo-valuedlogic"). In fact, Aristotle did
not contest the universality of the law of excluded middle, but the universality of the bivalence principle: he admitted that this principle did not all apply to future events (''De Interpretatione'', ''ch. IX''), but he didn't create a system of multi-valued logic to explain this isolated remark. Until the coming of the 20th century, later logicians followed
Aristotelian logic, which includes or assumes the
law of the excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
.
The 20th century brought back the idea of multi-valued logic. The Polish logician and philosopher
Jan Łukasiewicz
Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic His work centred on philosophical logic, mathematical logic and history of logic. ...
began to create systems of many-valued logic in 1920, using a third value, "possible", to deal with Aristotle's
paradox of the sea battle. Meanwhile, the American mathematician,
Emil L. Post (1921), also introduced the formulation of additional truth degrees with ''n'' ≥ 2, where ''n'' are the truth values. Later, Jan Łukasiewicz and
Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
together formulated a logic on ''n'' truth values where ''n'' ≥ 2. In 1932,
Hans Reichenbach formulated a logic of many truth values where ''n''→∞.
Kurt Gödel in 1932 showed that
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
is not a
finitely-many valued logic, and defined a system of
Gödel logic In mathematical logic, a first-order Gödel logic is a member of a family of finite- or infinite-valued logics in which the sets of truth values ''V'' are closed subsets of the interval ,1containing both 0 and 1. Different such sets ''V'' in gene ...
s intermediate between
classical and intuitionistic logic; such logics are known as
intermediate logics In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate l ...
.
Examples
Kleene (strong) and Priest logic
Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
's "(strong) logic of indeterminacy" (sometimes
) and
Priest
A priest is a religious leader authorized to perform the sacred rituals of a religion, especially as a mediatory agent between humans and one or more deities. They also have the authority or power to administer religious rites; in partic ...
's "logic of paradox" add a third "undefined" or "indeterminate" truth value . The truth functions for
negation (¬),
conjunction
Conjunction may refer to:
* Conjunction (grammar), a part of speech
* Logical conjunction, a mathematical operator
** Conjunction introduction, a rule of inference of propositional logic
* Conjunction (astronomy), in which two astronomical bodies ...
(∧),
disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
(∨),
implication (), and
biconditional
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as t ...
() are given by:
The difference between the two logics lies in how
tautologies are defined. In only is a ''designated truth value'', while in both and are (a logical formula is considered a tautology if it evaluates to a designated truth value). In Kleene's logic can be interpreted as being "underdetermined", being neither true nor false, while in Priest's logic can be interpreted as being "overdetermined", being both true and false. does not have any tautologies, while has the same tautologies as classical two-valued logic.
Bochvar's internal three-valued logic
Another logic is Dmitry Bochvar's "internal" three-valued logic
, also called Kleene's weak three-valued logic. Except for negation and biconditional, its truth tables are all different from the above.
The intermediate truth value in Bochvar's "internal" logic can be described as "contagious" because it propagates in a formula regardless of the value of any other variable.
Belnap logic ()
Belnap's logic combines and . The overdetermined truth value is here denoted as ''B'' and the underdetermined truth value as ''N''.
Gödel logics ''Gk'' and ''G''∞
In 1932
Gödel defined a family
of many-valued logics, with finitely many truth values
, for example
has the truth values
and
has
. In a similar manner he defined a logic with infinitely many truth values,
, in which the truth values are all the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s in the interval