In mathematics, Łukasiewicz logic (English pronunciation: /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;[1] it was later generalized to n-valued (for all finite n) as well as infinitely-many-valued variants, both propositional and first-order.[2] It belongs to the classes of t-norm fuzzy logics[3] and substructural logics.[4]
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The propositional connectives of Łukasiewicz logic are implication , negation , equivalence , weak conjunction , strong conjunction , weak disjunction , strong disjunction , and propositional constants and . 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.
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 of strong conjunction is the Łukasiewicz t-norm and the truth function of strong disjunction is its dual t-conorm. The truth function 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].
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:[3]