# Łukasiewicz logic

In mathematics and philosophy, Łukasiewicz logic (/ˌlkəˈʃɛvɪ/ LOO-kə-SHEV-itch, Polish: [wukaˈɕɛvitʂ]) 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 (0-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 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:

{\begin{aligned}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).\end{aligned}} 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 $\circ ,$ • $w(\neg \theta )=F_{\neg }(w(\theta )),$ • $w\left({\overline {0}}\right)=0$ and $w\left({\overline {1}}\right)=1,$ 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. Obviously, $F_{\otimes }(.5,.5)=0$ and $F_{\oplus }(.5,.5)=1$ , so if $T(p)=.5$ , then $T(p\wedge p)=T(\neg p\wedge \neg p)=0$ while the respective logically-equivalent propositions have $T(p\vee p)=T(\neg p\vee \neg p)=1$ .

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 each 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. C. C. 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.