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Many-valued logic

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Propositional calculus in which there are more than two truth values

Many-valued logic (also multi- or multiple-valued logic ) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., true and false) for any proposition. Classical two-valued logic 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, seven-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<br>[edit]

Aristotle, the "father of [two-valued] logic",[1] accepted the law of excluded middle but made an important distinction about the principle of bivalence. In (De Interpretatione, ch. IX),[2] he argued that statements about future events cannot always be definitively true or false. However, he didn't develop this insight into a systematic multi-valued logic — it remained a specific exception within his classical framework.<br>Logicians followed this Aristotelian logic tradition until the 20th century, consistently using the law of the excluded middle while acknowledging his concerns about future contingents. Systematic alternatives to classical logic only emerged in modern times.

The 20th century brought back the idea of multi-valued logic. The Polish logician and philosopher Jan Łukasiewicz 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 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 is not a finitely-many valued logic, and defined a system of Gödel logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics.

Examples<br>[edit]

Main articles: Three-valued logic, Four-valued logic, and Nine-valued logic

Kleene (strong) K3 and Priest logic P3<br>[edit]

Kleene's "(strong) logic of indeterminacy" K3 (sometimes

{\displaystyle K_{3}^{S}}

) and Priest's "logic of paradox" add a third undefined or indeterminate truth value I. The truth functions for negation (¬), conjunction (∧), disjunction (∨), implication (→K), and biconditional (↔K) are given by:[3]

→K

↔K

The difference between the two logics lies in how tautologies are defined. In K3 only T is a designated truth value, while in P3 both T and I are. (A logical formula is considered a tautology if it evaluates to a designated truth value). In Kleene's logic I can be interpreted as being underdetermined, being neither true nor false, while in Priest's logic I can be interpreted as being overdetermined, being both true and false. K3 does not have any tautologies, while P3 has the same tautologies as classical two-valued logic.[4]

Bochvar's internal three-valued logic<br>[edit]

Another logic is Dmitry Bochvar's internal three-valued logic

{\displaystyle B_{3}^{I}}

, also called Kleene's weak three-valued logic. Except for negation and biconditional, its truth tables are all different from the above.[5]

∧+

∨+

→+

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.[5]

Belnap logic (B4)<br>[edit]

Belnap's logic B4 combines K3 and P3. The overdetermined truth value is here denoted as B and the underdetermined truth value as N.

f¬

f∧

f∨

Gödel logics Gk and G∞<br>[edit]

In 1932 Gödel defined[6] a family

{\displaystyle G_{k}}

of many-valued logics, with finitely many truth values

{\displaystyle 0,{\tfrac {1}{k-1}},{\tfrac {2}{k-1}},\ldots ,{\tfrac {k-2}{k-1}},1}

, for example

{\displaystyle G_{3}}

has the truth values

{\displaystyle 0,{\tfrac {1}{2}},1}

and

{\displaystyle G_{4}}

has

{\displaystyle 0,{\tfrac {1}{3}},{\tfrac {2}{3}},1}

. In a similar manner he defined a logic with infinitely many truth values,

{\displaystyle G_{\infty }}

, in which the truth values are all the real numbers in the interval

{\displaystyle [0,1]}

. The designated truth value in these logics is 1.

The conjunction

{\displaystyle...