Implicit definability of truth constants in {\L}ukasiewicz logic

TL;DR

This paper investigates how truth constants can be implicitly defined within { extL}ukasiewicz logic, exploring rational and irrational elements, and connecting definability to computational complexity and fuzzy logic semantics.

Contribution

It introduces a notion of implicit definability tailored to { extL}ukasiewicz logic, explores rational and irrational element definability, and links these to complexity and fuzzy logic interpretations.

Findings

Rational elements are explicitly definable in standard semantics.

Irrational elements are definable in infinitary { extL}ukasiewicz logic.

Definability results inform computational complexity of Rational Pavelka logic.

Abstract

In the framework of propositional {\L}ukasiewicz logic, a suitable notion of implicit definability, tailored to the intended real-valued semantics and referring to the elements of its domain, is introduced. Several variants of implicitly defining each of the rational elements in the standard semantics are explored, and based on that, a faithful interpretation of theories in Rational Pavelka logic in theories in {\L}ukasiewicz logic is obtained. Some of these results were already presented in H\'ajek's "Metamathematics of Fuzzy Logic" as technical statements. A connection to the lack of (deductive) Beth property in {\L}ukasiewicz logic is drawn. Moreover, while irrational elements of the standard semantics are not implicitly definable by finitary means, a parallel development is possible for them in the infinitary {\L}ukasiewicz logic. As an application of definability of the rationals, it is shown how computational complexity results for Rational Pavelka logic can be obtained from analogous results for {\L}ukasiewicz logic. The complexity of the definability notion itself is studied as well. Finally, we review the import of these results for the precision/vagueness discussion for fuzzy logic, and for the general standing of truth constants in {\L}ukasiewicz logic.