In mathematics and philosophy, Łukasiewicz logic (/ˌwʊkəˈʃɛvɪtʃ/ WUUK-ə-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 modal logic;^[1] 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.^[2] The ℵ₀-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the Łukasiewicz–Tarski logic.^[3] It belongs to the classes of t-norm fuzzy logics^[4] and substructural logics.^[5]

Łukasiewicz logic was motivated by Aristotle's suggestion that bivalent logic was not applicable to future contingents, e.g. the statement "There will be a sea battle tomorrow". In other words, statements about the future were neither true nor false, but an intermediate value could be assigned to them, to represent their possibility of becoming true in the future.

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 ${\displaystyle \rightarrow }$ ("implication"), and the constant ${\displaystyle \bot }$ ("false"). Additional connectives can be defined in terms of these:

${\displaystyle {\begin{aligned}\neg A&=_{def}A\rightarrow \bot \\A\vee B&=_{def}(A\rightarrow B)\rightarrow B\\A\wedge B&=_{def}\neg (\neg A\vee \neg B)\\A\leftrightarrow B&=_{def}(A\rightarrow B)\wedge (B\rightarrow A)\\\top &=_{def}\bot \rightarrow \bot \end{aligned}}}$

The ${\displaystyle \vee }$ and ${\displaystyle \wedge }$ connectives are called weak disjunction and conjunction, because they are non-classical, as the law of excluded middle does not hold for them. In the context of substructural logics, they are called additive connectives. They also correspond to lattice min/max connectives.

In terms of substructural logics, there are also strong or multiplicative disjunction and conjunction connectives, although these are not part of Łukasiewicz's original presentation:

${\displaystyle {\begin{aligned}A\oplus B&=_{def}\neg A\rightarrow B\\A\otimes B&=_{def}\neg (A\rightarrow \neg B)\end{aligned}}}$

There are also defined modal operators, using the Tarskian Möglichkeit:

${\displaystyle {\begin{aligned}\Diamond A&=_{def}\neg A\rightarrow A\\\Box A&=_{def}\neg \Diamond \neg A\end{aligned}}}$

Axioms

This section needs expansion with: additional axioms for finite-valued logics. You can help by adding to it. (August 2014)

The original system of axioms for propositional infinite-valued Łukasiewicz logic used implication and negation as the primitive connectives, along with modus ponens:

${\displaystyle {\begin{aligned}A&\rightarrow (B\rightarrow A)\\(A\rightarrow B)&\rightarrow ((B\rightarrow C)\rightarrow (A\rightarrow C))\\((A\rightarrow B)\rightarrow B)&\rightarrow ((B\rightarrow A)\rightarrow A)\\(\neg B\rightarrow \neg A)&\rightarrow (A\rightarrow B).\end{aligned}}}$

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

Divisibility

${\displaystyle (A\wedge B)\rightarrow (A\otimes (A\rightarrow B))}$

Double negation

${\displaystyle \neg \neg A\rightarrow A.}$

That is, infinite-valued Łukasiewicz logic arises by adding the axiom of double negation to basic fuzzy logic (BL), or by adding the axiom of divisibility to the logic IMTL.

Finite-valued Łukasiewicz logics require additional axioms.

Proof Theory

This section needs expansion with: discussion of sequent calculi and natural deduction systems needed. You can help by adding to it. (June 2022)

A hypersequent calculus for three-valued Łukasiewicz logic was introduced by Arnon Avron in 1991.^[6]

Sequent calculi for finite and infinite-valued Łukasiewicz logics as an extension of linear logic were introduced by A. Prijatelj in 1994.^[7] However, these are not cut-free systems.

Hypersequent calculi for Łukasiewicz logics were introduced by A. Ciabattoni et al in 1999.^[8] However, these are not cut-free for ${\displaystyle n>3}$ finite-valued logics.

A labelled tableaux system was introduced by Nicola Olivetti in 2003.^[9]

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 0 or 1 but also any real number in between (e.g. 0.25). Valuations have a recursive definition where:

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

and where the definitions of the operations hold as follows:

• Implication: ${\displaystyle F_{\rightarrow }(x,y)=\min\{1,1-x+y\}}$
• Equivalence: ${\displaystyle F_{\leftrightarrow }(x,y)=1-|x-y|}$
• Negation: ${\displaystyle F_{\neg }(x)=1-x}$
• Weak conjunction: ${\displaystyle F_{\wedge }(x,y)=\min\{x,y\}}$
• Weak disjunction: ${\displaystyle F_{\vee }(x,y)=\max\{x,y\}}$
• Strong conjunction: ${\displaystyle F_{\otimes }(x,y)=\max\{0,x+y-1\}}$
• Strong disjunction: $$Zdroj:https://en.wikipedia.org?pojem=Łukasiewicz_logic
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