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.
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:
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.
Using exactly the same valuation formulas as for real-valued semantics Łukasiewicz (1922) also defined (up to isomorphism) semantics over
Łukasiewicz logics can be seen as modal logics using the defined operators,
From these we can prove the following theorems, which are common axioms in many modal logics: