Łukasiewicz logic

In mathematics and philosophy, Łukasiewicz logic (ˌluːkəˈʃɛvɪtʃ , 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; 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. The א0-valued version was published in 1930 by Łukasiewicz and Alfred Tarski; consequently it is sometimes called the ŁukasiewiczTarski logic. It belongs to the classes of t-norm fuzzy logics and substructural logics. 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. The propositional connectives of Łukasiewicz 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 Łukasiewicz's original presentation: There are also defined modal operators, using the Tarskian Möglichkeit: The original system of axioms for propositional infinite-valued Łukasiewicz logic used implication and negation as the primitive connectives, along with modus ponens: Propositional infinite-valued Łukasiewicz logic can also be axiomatized by adding the following axioms to the axiomatic system of monoidal t-norm logic: Divisibility Double negation 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.