In mathematics and philosophy, à Âukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan à Âukasiewicz as a three-valued modal logic; it was later generalized to n-valued (for all finite n) as well as infinitely-many-valued (âµ<sub>0</sub>-valued) variants, both propositional and first order. The âµ<sub>0</sub>-valued version was published in 1930 by à Âukasiewicz and Alfred Tarski; consequently it is sometimes called the à ÂukasiewiczTarski logic. It belongs to the classes of t-norm fuzzy logics and substructural logics.
à Âukasiewicz 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 à Âukasiewicz(âÂÂTarski) logic in its full generality, i.e. as an infinite-valued logic. For an elementary introduction to the three-valued instantiation à Â<sub>3</sub>, see three-valued logic.
The propositional connectives of à Âukasiewicz logic are ("implication"), and the constant ("false"). Additional connectives can be defined in terms of these:
The and 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 à Âukasiewicz's original presentation:
There are also defined modal operators, using the Tarskian Möglichkeit:
The original system of axioms for propositional infinite-valued à Âukasiewicz logic used implication and negation as the primitive connectives, along with modus ponens:
Propositional infinite-valued à Âukasiewicz logic can also be axiomatized by adding the following axioms to the axiomatic system of monoidal t-norm logic:
That is, infinite-valued à Âukasiewicz 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 à Âukasiewicz logics require additional axioms.
A hypersequent calculus for three-valued à Âukasiewicz logic was introduced by Arnon Avron in 1991.
Sequent calculi for finite and infinite-valued à Âukasiewicz logics as an extension of linear logic were introduced by A. Prijatelj in 1994. However, these are not cut-free systems.
Hypersequent calculi for à Âukasiewicz logics were introduced by A. Ciabattoni et al in 1999. However, these are not cut-free for finite-valued logics.
A labelled tableaux system was introduced by Nicola Olivetti in 2003.
A hypersequent calculus for infinite-valued à Âukasiewicz logic was introduced by George Metcalfe in 2004.
Infinite-valued à Âukasiewicz 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:
and where the definitions of the operations hold as follows:
The truth function of strong conjunction is the à Âukasiewicz t-norm and the truth function of strong disjunction is its dual t-conorm. Obviously,
and , so if , then
while the respective logically-equivalent propositions have .
The truth function is the residuum of the à Âukasiewicz t-norm. All truth functions of the basic connectives are continuous.
By definition, a formula is a tautology of infinite-valued à Âukasiewicz logic if it evaluates to 1 under each valuation of propositional variables by real numbers in the interval [0, 1].
Using exactly the same valuation formulas as for real-valued semantics à Âukasiewicz (1922) also defined (up to isomorphism) semantics over
The standard real-valued semantics determined by the à Âukasiewicz t-norm is not the only possible semantics of à Âukasiewicz logic. General algebraic semantics of propositional infinite-valued à Âukasiewicz 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 à Âukasiewicz 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:
Here valid means necessarily evaluates to 1.
Font, Rodriguez and Torrens introduced in 1984 the Wajsberg algebra as an alternative model for the infinite-valued à Âukasiewicz logic.
A 1940s attempt by Grigore Moisil to provide algebraic semantics for the n-valued à Âukasiewicz logic by means of his à ÂukasiewiczâÂÂMoisil (LM) algebra (which Moisil called à Âukasiewicz algebras) turned out to be an incorrect model for n âÂÂ¥ 5. This issue was made public by Alan Rose in 1956. C. C. Chang's MV-algebra, which is a model for the âµ<sub>0</sub>-valued (infinitely-many-valued) à ÂukasiewiczâÂÂTarski logic, was published in 1958. For the axiomatically more complicated (finite) n-valued à Âukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MV<sub>n</sub>-algebras. MV<sub>n</sub>-algebras are a subclass of LM<sub>n</sub>-algebras, and the inclusion is strict for n âÂÂ¥ 5. In 1982 Roberto Cignoli published some additional constraints that added to LM<sub>n</sub>-algebras produce proper models for n-valued à Âukasiewicz logic; Cignoli called his discovery proper à Âukasiewicz algebras.
à Âukasiewicz logics are co-NP complete.
à Âukasiewicz logics can be seen as modal logics, a type of logic that addresses possibility, using the defined operators,
A third doubtful operator has been proposed, .
From these we can prove the following theorems, which are common axioms in many modal logics:
We can also prove distribution theorems on the strong connectives:
However, the following distribution theorems also hold:
In other words, if , then , which is counter-intuitive. However, these controversial theorems have been defended as a modal logic about future contingents by A. N. Prior. Notably, .