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# Lukasiewicz fuzzy logic

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### Lukasiewicz fuzzy logic

In mathematics, Łukasiewicz logic (; 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.

## 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 $\overline\left\{0\right\}$ and $\overline\left\{1\right\}$. 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:

$A \rightarrow \left(B \rightarrow A\right)$
$\left(A \rightarrow B\right) \rightarrow \left(\left(B \rightarrow C\right) \rightarrow \left(A \rightarrow C\right)\right)$
$\left(\left(A \rightarrow B\right) \rightarrow B\right) \rightarrow \left(\left(B \rightarrow A\right) \rightarrow A\right)$
$\left(\neg B \rightarrow \neg A\right) \rightarrow \left(A \rightarrow B\right).$

Propositional infinite-valued Łukasiewicz logic can also be axiomatized by adding the following axioms to the axiomatic system of monoidal t-norm logic:

• Divisibility: $\left(A \wedge B\right) \rightarrow \left(A \otimes \left(A \rightarrow B\right)\right)$
• 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:

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

and where the definitions of the operations hold as follows:

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

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:[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).

## References

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