Łukasiewicz–Tarski logic

In mathematics, Łukasiewicz logic (/luːkəˈʃɛvɪtʃ/; Polish pronunciation: [wukaˈɕɛvʲitʂ]) is a non-classical, many valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic; it was later generalized to n-valued (for all finite n) as well as infinitely-many-valued (ℵ₀-valued) variants, both propositional and first-order. The ℵ₀-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the Łukasiewicz-Tarski logic. It belongs to the classes of t-norm fuzzy logics and substructural logics.

This article presents the Łukasiewicz[-Tarski] logic in its full generality, i.e. as an infinite-valued logic. For an elementary introduction to the three-valued instantiation Ł₃, see three-valued logic.

The propositional connectives of Łukasiewicz logic are implication ${\displaystyle \rightarrow }$, negation ${\displaystyle \neg }$, equivalence ${\displaystyle \leftrightarrow }$, weak conjunction ${\displaystyle \wedge }$, strong conjunction ${\displaystyle \otimes }$, weak disjunction ${\displaystyle \vee }$, strong disjunction ${\displaystyle \oplus }$, and propositional constants ${\displaystyle {\overline {0}}}$ and ${\displaystyle {\overline {1}}}$. The presence of conjunction and disjunction is a common feature of substructural logics without the rule of contraction, to which Łukasiewicz logic belongs.

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