Łukasiewicz logic

Łukasiewicz logic

In mathematics, Łukasiewicz logic is a non-classical, many valued logic. It was originally defined by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in Polish). Ruch filozoficzny 5:170–171. English translation: On three-valued logic, in L. Borkowski (ed.), "Selected works by Jan Łukasiewicz", North–Holland, Amsterdam, 1970, pp. 87–88. ISBN 0720422523] it was later generalized to "n"-valued (for all finite "n") as well as infinitely-many-valued variants, both propositional and first-order.Hay, L.S., 1963, Axiomatization of the infinite-valued predicate calculus. "Journal of Symbolic Logic" 28:77–86.] It belongs to the classes of t-norm fuzzy logicsHájek P., 1998, "Metamathematics of Fuzzy Logic". Dordrecht: Kluwer.] and substructural logics.Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, "Trends in Logic" 20: 177–212.]

Language

The propositional connectives of Łukasiewicz logic are"implication" ightarrow,"negation" eg,"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, among which Łukasiewicz logic belongs.

Axioms

The original system of axioms for propositional infinite-valued Łukasiewicz logic used implication and negation as the primitive connectives:: A ightarrow (B ightarrow A): (A ightarrow B) ightarrow ((B ightarrow C) ightarrow (A ightarrow C)): ((A ightarrow B) ightarrow B) ightarrow ((B ightarrow A) ightarrow A): ( eg B ightarrow eg A) ightarrow (A ightarrow B)

Propositional infinite-valued Łukasiewicz logic can also be axiomatized by adding the following axioms to the axiomatic system of monoidal t-norm logic:
* "Divisibility:" (A wedge B) ightarrow (A otimes (A ightarrow B))
* "Double negation:" eg eg A ightarrow AThat 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.

Real-valued semantics

Infinite-valued Łukasiewicz logic is a real-valued logic in which sentences from sentential calculus are assigned a truth value of arbitrary precision between 0 and 1. Valuations have a recursive definition, with
* w( heta circ phi)=F_circ(w( heta),w(phi)) for a binary connective circ,
* w( eg heta)=F_ eg(w( heta)),
* w(overline{0})=0 and w(overline{1})=1, where
*F_ ightarrow(x,y) = min{1, 1 - x + y }
*F_leftrightarrow(x,y) = 1 - |x-y|
*F_ eg(x) = 1-x
*F_wedge(x,y) = min{x, y }
*F_vee(x,y) = max{x, y }
*F_otimes(x,y) = max{0, x + y -1 }
*F_oplus(x,y) = min{1, x + y }

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_ ightarrow 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] .

General algebraic semantics

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::The following conditions are equivalent::* A is provable in propositional infinite-valued Łukasiewicz logic:* A is valid in all MV-algebras ("general completeness"):* A is valid in all linearly ordered MV-algebras ("linear completeness"):* A is valid in the standard MV-algebra ("standard completeness")

References


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