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In mathematics, Łukasiewicz logic (; Polish pronunciation: ) is a non-classical, many valued logic. It was originally defined in the early 20th-century by Jan Łukasiewicz as a three-valued logic;^{[1]} it was later generalized to n-valued (for all finite n) as well as infinitely-many-valued (ℵ_{0}-valued) variants, both propositional and first-order.^{[2]} The ℵ_{0}-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the Łukasiewicz-Tarski logic.^{[3]} It belongs to the classes of t-norm fuzzy logics^{[4]} and substructural logics.^{[5]}
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 Ł_{3}, see three-valued logic.
The propositional connectives of Łukasiewicz logic are implication \rightarrow, negation \neg, equivalence \leftrightarrow, weak conjunction \wedge, strong conjunction \otimes, weak disjunction \vee, strong disjunction \oplus, and propositional constants \overline{0} and \overline{1}. The presence of weak and strong conjunction and disjunction is a common feature of substructural logics without the rule of contraction, to which Łukasiewicz logic belongs.
The original system of axioms for propositional infinite-valued Łukasiewicz logic used implication and negation as the primitive connectives:
Propositional infinite-valued Łukasiewicz logic can also be axiomatized by adding the following axioms to the axiomatic system of monoidal t-norm logic:
That is, infinite-valued Łukasiewicz logic arises by adding the axiom of double negation to basic t-norm logic BL, or by adding the axiom of divisibility to the logic IMTL.
Finite-valued Łukasiewicz logics require additional axioms.
Infinite-valued Łukasiewicz logic is a real-valued logic in which sentences from sentential calculus may be assigned a truth value of not only zero or one but also any real number in between (e.g. 0.25). Valuations have a recursive definition where:
and where the definitions of the operations hold as follows:
The truth function F_\otimes of strong conjunction is the Łukasiewicz t-norm and the truth function F_\oplus of strong disjunction is its dual t-conorm. The truth function F_\rightarrow is the residuum of the Łukasiewicz t-norm. All truth functions of the basic connectives are continuous.
By definition, a formula is a tautology of infinite-valued Łukasiewicz logic if it evaluates to 1 under any valuation of propositional variables by real numbers in the interval [0, 1].
Using exactly the same valuation formulas as for real-valued semantics Łukasiewicz (1922) also defined (up to isomorphism) semantics over
The standard real-valued semantics determined by the Łukasiewicz t-norm is not the only possible semantics of Łukasiewicz logic. General algebraic semantics of propositional infinite-valued Łukasiewicz logic is formed by the class of all MV-algebras. The standard real-valued semantics is a special MV-algebra, called the standard MV-algebra.
Like other t-norm fuzzy logics, propositional infinite-valued Łukasiewicz logic enjoys completeness with respect to the class of all algebras for which the logic is sound (that is, MV-algebras) as well as with respect to only linear ones. This is expressed by the general, linear, and standard completeness theorems:^{[4]}
Font, Rodriguez and Torrens introduced in 1984 the Wajsberg algebra as an alternative model for the infinite-valued Łukasiewicz logic.^{[6]}
A 1940s attempt by Grigore Moisil to provide algebraic semantics for the n-valued Łukasiewicz logic by means of his Łukasiewicz–Moisil (LM) algebra (which Moisil called Łukasiewicz algebras) turned out to be an incorrect model for n ≥ 5. This issue was made public by Alan Rose in 1956. Chang's MV-algebra, which is a model for the ℵ_{0}-valued (infinitely-many-valued) Łukasiewicz-Tarski logic, was published in 1958. For the axiomatically more complicated (finite) n-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MV_{n}-algebras.^{[7]} MV_{n}-algebras are a subclass of LM_{n}-algebras, and the inclusion is strict for n ≥ 5.^{[8]} In 1982 Roberto Cignoli published some additional constraints that added to LM_{n}-algebras produce proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper Łukasiewicz algebras.^{[9]}
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