Łukasiewicz logic

In mathematics and philosophy, Łukasiewicz logic (/ˌluːkəˈʃɛvɪtʃ/; Polish: ) 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. In mathematics and philosophy, Łukasiewicz logic (/ˌluːkəˈʃɛvɪtʃ/; Polish: ) 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 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 areimplication → {displaystyle ightarrow } ,negation ¬ {displaystyle eg } ,equivalence ↔ {displaystyle leftrightarrow } ,weak conjunction ∧ {displaystyle wedge } ,strong conjunction ⊗ {displaystyle otimes } ,weak disjunction ∨ {displaystyle vee } ,strong disjunction ⊕ {displaystyle oplus } ,and propositional constants 0 ¯ {displaystyle {overline {0}}} and 1 ¯ {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.