Łukasiewicz logic

In mathematics, Łukasiewicz logic is a non-classical, many valued logic. It was originally defined 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. ISBN 0720422523] it was later generalized to "n"-valued (for all finite "n") as well as infinitely-many-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.] It belongs to the classes of t-norm fuzzy logicsHájek 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.]

Language

The propositional connectives of Łukasiewicz logic are"implication" $ightarrow$,"negation" $eg$,"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 weak and strong conjunction and disjunction is a common feature of substructural logics without the rule of contraction, among which Łukasiewicz logic belongs.

Axioms

The original system of axioms for propositional infinite-valued Łukasiewicz logic used implication and negation as the primitive connectives:: $A\; ightarrow\; (B\; ightarrow\; A)$: $(A\; ightarrow\; B)\; ightarrow\; ((B\; ightarrow\; C)\; ightarrow\; (A\; ightarrow\; C))$: $((A\; ightarrow\; B)\; ightarrow\; B)\; ightarrow\; ((B\; ightarrow\; A)\; ightarrow\; A)$: $(\; eg\; B\; ightarrow\; eg\; A)\; ightarrow\; (A\; ightarrow\; B)$

Propositional infinite-valued Łukasiewicz logic can also be axiomatized by adding the following axioms to the axiomatic system of monoidal t-norm logic:
* "Divisibility:" $(A\; wedge\; B)\; ightarrow\; (A\; otimes\; (A\; ightarrow\; B))$
* "Double negation:" $eg\; eg\; A\; ightarrow\; A$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.

Real-valued semantics

Infinite-valued Łukasiewicz logic is a real-valued logic in which sentences from sentential calculus are assigned a truth value of arbitrary precision between 0 and 1. Valuations have a recursive definition, with
* $w(\; heta\; circ\; phi)=F\_circ(w(\; heta),w(phi))$ for a binary connective $circ,$
* $w(\; eg\; heta)=F\_\; eg(w(\; heta)),$
* $w(overline\{0\})=0$ and $w(overline\{1\})=1,$ where
*$F\_\; ightarrow(x,y)\; =\; min\{1,\; 1\; -\; x\; +\; y\; \}$
*$F\_leftrightarrow(x,y)\; =\; 1\; -\; |x-y|$
*$F\_\; eg(x)\; =\; 1-x$
*$F\_wedge(x,y)\; =\; min\{x,\; y\; \}$
*$F\_vee(x,y)\; =\; max\{x,\; y\; \}$
*$F\_otimes(x,y)\; =\; max\{0,\; x\; +\; y\; -1\; \}$
*$F\_oplus(x,y)\; =\; min\{1,\; x\; +\; y\; \}$

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 $F\_\; ightarrow$ 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] .

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")

References