Łukasiewicz Logic

In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;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. 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.Hay, L.S., 1963
Axiomatization of the infinite-valued predicate calculus
''Journal of Symbolic Logic'' 28:77–86. The ℵ₀-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the ŁukasiewiczTarski logic. citing Łukasiewicz, J., Tarski, A.
Untersuchungen über den Aussagenkalkül
Comp. Rend. Soc.
Sci. et Lettres Varsovie Cl. III 23, 30–50 (1930). It belongs to the classes of t-norm fuzzy logics Hájek P., 1998, ''Metamathematics of Fuzzy Logic''. Dordrecht: Kluwer. and substructural logics.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.

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 $\overline$ and $\overline$.
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:
: $\begin A &\rightarrow (B \rightarrow A) \\ (A \rightarrow B) &\rightarrow ((B \rightarrow C) \rightarrow (A \rightarrow C)) \\ ((A \rightarrow B) \rightarrow B) &\rightarrow ((B \rightarrow A) \rightarrow A) \\ (\neg B \rightarrow \neg A) &\rightarrow (A \rightarrow B). \end$

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 $\circ,$
* $w(\neg\theta) = F_\neg(w(\theta)),$
* $w\left(\overline\right) = 0$ and $w\left(\overline\right) = 1,$

and where the definitions of the operations hold as follows:
* Implication: $F_\rightarrow(x,y) = \min\$
* Equivalence: $F_\leftrightarrow(x, y) = 1 - , x - y,$
* Negation: $F_\neg(x) = 1 - x$
* Weak conjunction: $F_\wedge(x, y) = \min\$
* Weak disjunction: $F_\vee(x, y) = \max\$
* Strong conjunction: $F_\otimes(x, y) = \max\$
* Strong disjunction: $F_\oplus(x, y) = \min\.$

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 , 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
* any countable set by choosing the domain as .

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 MV_''n''-algebras. citing Grigolia, R.S.: "Algebraic analysis of Lukasiewicz-Tarski’s n-valued logical systems". In: Wójcicki, R., Malinkowski, G. (eds.) Selected Papers on Lukasiewicz Sentential Calculi, pp. 81–92. Polish Academy of Sciences, Wroclav (1977) MV_''n''-algebras are a subclass of LM_''n''-algebras, and the inclusion is strict for ''n'' ≥ 5. In 1982 Roberto Cignoli published some additional constraints that added to LM_''n''-algebras produce proper models for ''n''-valued Łukasiewicz logic; Cignoli called his discovery ''proper Łukasiewicz algebras''.R. Cignoli, Proper n-Valued Łukasiewicz Algebras as S-Algebras of Łukasiewicz n-Valued Propositional Calculi, Studia Logica, 41, 1982, 3-16,

References

Further reading

* Rose, A.: 1956, Formalisation du Calcul Propositionnel Implicatif ℵ₀ Valeurs de Łukasiewicz, C. R. Acad. Sci. Paris 243, 1183–1185.
* Rose, A.: 1978, Formalisations of Further ℵ₀-Valued Łukasiewicz Propositional Calculi, Journal of Symbolic Logic 43(2), 207–210.
* Cignoli, R., “The algebras of Lukasiewicz many-valued logic - A historical overview,” in S. Aguzzoli et al.(Eds.), Algebraic and Proof-theoretic Aspects of Non-classical Logics, LNAI 4460, Springer, 2007, 69-83.

{{DEFAULTSORT:Lukasiewicz logic
Many-valued logic
Fuzzy logic