It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued modal 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.
[5] 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.
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.
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:
Finite-valued Łukasiewicz logics require additional axioms.
A hypersequent calculus for three-valued Łukasiewicz logic was introduced by Arnon Avron in 1991.
Hypersequent calculi for Łukasiewicz logics were introduced by A. Ciabattoni et al in 1999.
A labelled tableaux system was introduced by Nicola Olivetti in 2003.
[9] A hypersequent calculus for infinite-valued Łukasiewicz logic was introduced by George Metcalfe in 2004.
[10] Infinite-valued Łukasiewicz 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).
of strong conjunction is the Łukasiewicz t-norm and the truth function
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 [0, 1].
General algebraic semantics of propositional infinite-valued Łukasiewicz logic is formed by the class of all MV-algebras.
This is expressed by the general, linear, and standard completeness theorems:[4] 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 Łukasiewicz logic.
[11] 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 MVn-algebras.
[12] MVn-algebras are a subclass of LMn-algebras, and the inclusion is strict for n ≥ 5.
[13] In 1982 Roberto Cignoli published some additional constraints that added to LMn-algebras produce proper models for n-valued Łukasiewicz logic; Cignoli called his discovery proper Łukasiewicz algebras.
[17] 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:
[18][19] However, these controversial theorems have been defended as a modal logic about future contingents by A. N.