Łukasiewicz logic

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In mathematics, Łukasiewicz logic (/lkəˈʃɛvɪ/; 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]

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 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:

A \rightarrow (B \rightarrow A)
(A \rightarrow B) \rightarrow ((B \rightarrow C) \rightarrow (A \rightarrow C))
((A \rightarrow B) \rightarrow B) \rightarrow ((B \rightarrow A) \rightarrow A)
(\neg B \rightarrow \neg A) \rightarrow (A \rightarrow 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) \rightarrow (A \otimes (A \rightarrow B))
  • Double negation: \neg\neg A \rightarrow A.

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.

Real-valued semantics[edit]

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:

  • w(\theta \circ \phi)=F_\circ(w(\theta),w(\phi)) for a binary connective \circ,
  • w(\neg\theta)=F_\neg(w(\theta)),
  • w(\overline{0})=0 and w(\overline{1})=1,

and where the definitions of the operations hold as follows:

  • Implication: F_\rightarrow(x,y) = \min\{1, 1 - x + y \}
  • Equivalence: F_\leftrightarrow(x,y) = 1 - |x-y|
  • Negation: F_\neg(x) = 1-x
  • Weak Conjunction: F_\wedge(x,y) = \min\{x, y \}
  • Weak Disjunction: F_\vee(x,y) = \max\{x, y \}
  • Strong Conjunction: F_\otimes(x,y) = \max\{0, x + y -1 \}
  • Strong Disjunction: 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_\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].

Finite-valued and countable-valued semantics[edit]

Using exactly the same valuation formulas as for real-valued semantics Łukasiewicz (1922) also defined (up to isomorphism) semantics over

  • any finite set of cardinality n ≥ 2 by choosing the domain as { 0, 1/(n − 1), 2/(n − 1), ..., 1 }
  • any countable set by choosing the domain as { p/q | 0 ≤ pq where p is a non-negative integer and q is a positive integer }.

General algebraic semantics[edit]

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]

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).


  1. ^ 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 0-7204-2252-3
  2. ^ Hay, L.S., 1963, Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic 28:77–86.
  3. ^ a b Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
  4. ^ 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.