On the infinite-valued Lukasiewicz logic that preserves degrees of truth
by Josep Maria Font, Àngel Gil, Antoni Torrens and Ventura Verdú
Archive for Mathematical Logic 45 (2006) 839--868
Abstract
Lukasiewicz's infinite-valued logic is commonly defined as the set of
formulas that take the value 1 under all evaluations in the
algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to
be semantically defined from Lukasiewicz's algebra by using a "truth-
preserving" scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there
exists no Gentzen calculus fully adequate for it. Another presentation of
the same deductive system can be obtained from a substructural Gentzen
calculus. In this paper we use the framework of abstract algebraic logic to
study a different deductive system which uses the aforementioned algebra
under a scheme of "preservation of degrees of truth". We characterize the
resulting deductive system in a natural way by using the lattice filters of
Wajsberg algebras, and also by using a structural Gentzen calculus, which
is shown to be fully adequate for it. This logic is an interesting example
for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a "graded" deduction theorem. Moreover, the Gentzen system is
algebraizable. The first deductive system mentioned turns out to be the
extension of the second by the rule of Modus Ponens.