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# Łukasiewicz logic

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 Title: Łukasiewicz logic Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Łukasiewicz logic

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; 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[-Tarski] 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.

## Language

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.

## Axioms

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

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

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

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).

Font, Rodriguez and Torrens introduced in 1984 the Wajsberg algebra as an alternative model for the infinite-valued Łukasiewicz logic.

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 ℵ0-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 MVn-algebras. MVn-algebras are a subclass of LMn-algebras, and the inclusion is strict for n ≥ 5. In 1982 Roberto Cignoli published some additional constraints that added to LMn-algebras produce proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper Łukasiewicz algebras.