Many-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.


Through this value it is possible to define a negation \underset{\Pi}{\neg} and an additional conjunction \underset{\Pi}{\wedge} as follows: \begin{align} \underset{\Pi}{\neg} u &:= u \xrightarrow[\Pi]{}\overline{0} \\ u \underset{\Pi}{\wedge} v &:= u\odot (u \xrightarrow[\Pi]{} v) \end{align} === Post logics Pm === In 1921 Post defined a family of logics P_m with (as in L_v and G_k) the truth values 0, \tfrac 1 {m-1}, \tfrac 2 {m-1}, \ldots, \tfrac {m-2} {m-1}, 1.


In 1922 he developed a logic with infinitely many values L_\infty, in which the truth values spanned the real numbers in the interval [0,1].


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 == === Kleene (strong) and Priest logic === Kleene's "(strong) logic of indeterminacy" (sometimes K_3^S) and Priest's "logic of paradox" add a third "undefined" or "indeterminate" truth value .

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 G_k of many-valued logics, with finitely many truth values 0, \tfrac 1 {k-1}, \tfrac 2 {k-1}, \ldots, \tfrac {k-2} {k-1}, 1, for example G_3 has the truth values 0, \tfrac 1 2, 1 and G_4 has 0, \tfrac 1 3, \tfrac 2 3, 1.


Negation \underset{P}{\neg} and conjunction \underset{P}{\wedge} and disjunction \underset{P}{\vee} are defined as follows: \begin{align} \underset{P}{\neg} u &:= \begin{cases} 1, & \text{if }u=0\\ u-\frac {1}{m-1}, & \text{if }u\not= 0 \end{cases} \\ u \underset{P}{\wedge} v &:= \min\{u,v\} \\ u \underset{P}{\vee} v &:= \max\{u,v\} \end{align} === Rose logics === In 1951, Alan Rose defined another family of logics for systems whose truth-values form lattices.

123, December 1951, pp. 152–165; source). == Semantics == === Matrix semantics (logical matrices) === See Logical matrix ==Relation to classical logic== Logics are usually systems intended to codify rules for preserving some semantic property of propositions across transformations.

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