Rational Pavelka logic: the best among three worlds?

TL;DR

This survey compares three related fuzzy logics—{ extl}ukasiewicz, Rational Pavelka, and Graded Rational Pavelka—highlighting their differences, evolution, and the use of truth constants for reasoning with degrees of truth.

Contribution

It clarifies the relationships and historical development of these three formal systems within fuzzy logic, emphasizing their unique features and the role of truth constants.

Findings

RPL includes explicit rational truth constants as axioms.

GRPL employs graded formulas and proofs inspired by Pavelka's original ideas.

The systems naturally evolve from each other, with debates on their formal relationships.

Abstract

This comparative survey explores three formal approaches to reasoning with partly true statements and degrees of truth, within the family of {\L}ukasiewicz logic. These approaches are represented by infinite-valued {\L}ukasiewicz logic ({\L}), Rational Pavelka logic (RPL) and a logic with graded formulas that we refer to as Graded Rational Pavelka logic (GRPL). Truth constants for all rationals between $0$ and $1$ are used as a technical means to calibrate degrees of truth. {\L}ukasiewicz logic ostensibly features no truth constants except $0$ and $1$; Rational Pavelka logic includes constants in the basic language, with suitable axioms; Graded Rational Pavelka logic works with graded formulas and proofs, following the original intent of Pavelka, inspired by Goguen's work. Historically, Pavelka's papers precede the definition of GRPL, which in turn precedes RPL; retrieving these steps, we discuss how these formal systems naturally evolve from each other, and we also recall how this process has been a somewhat contentious issue in the realm of {\L}ukasiewicz logic. This work can also be read as a case study in logics, their fragments, and the relationship of the fragments to a logic.