Łukasiewicz logic

In mathematics, Łukasiewicz logic (/luːkəˈʃɛvɪtʃ/; 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 (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]

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:

$A \rightarrow (B \rightarrow A)$

$(A \rightarrow B) \rightarrow ((B \rightarrow C) \rightarrow (A \rightarrow C))$

$((A \rightarrow B) \rightarrow B) \rightarrow ((B \rightarrow A) \rightarrow A)$

$(\neg B \rightarrow \neg A) \rightarrow (A \rightarrow B).$

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 ≤ p ≤ q 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:^[4]

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.^[6]

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.^[7] MV_n-algebras are a subclass of LM_n-algebras, and the inclusion is strict for n ≥ 5.^[8] 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.^[9]

References

1. 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. ISBN 0-7204-2252-3
2. Hay, L.S., 1963, Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic 28:77–86.
3. Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. p. vii. ISBN 978-3-319-01589-7. citing Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. Comp. Rend. Soc. Sci. et Lettres Varsovie Cl. III 23, 30–50 (1930).
4. Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
5. 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.
6. http://journal.univagora.ro/download/pdf/28.pdf citing J. M. Font, A. J. Rodriguez, A. Torrens, Wajsberg Algebras, Stochastica, VIII, 1, 5-31, 1984
7. Lavinia Corina Ciungu (2013). Non-commutative Multiple-Valued Logic Algebras. Springer. pp. vii–viii. ISBN 978-3-319-01589-7. 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)
8. Iorgulescu, A.: Connections between MV_n-algebras and n-valued Łukasiewicz–Moisil algebras—I. Discrete Math. 181, 155–177 (1998) doi:10.1016/S0012-365X(97)00052-6
9. R. Cignoli, Proper n-Valued Łukasiewicz Algebras as S-Algebras of Łukasiewicz n-Valued Propositional Calculi, Studia Logica, 41, 1982, 3-16, doi:10.1007/BF00373490

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. doi:10.2307/2272818
• 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. doi:10.1007/978-3-540-75939-3_5