Effectiveness in RPL, with Applications to Continuous Logic

TL;DR

This paper develops a computable model theory framework for rational Pavelka and continuous logic, enabling effective construction and analysis of models with uncountable truth values, extending classical computable model theory.

Contribution

It introduces effective versions of model theory theorems for continuous logic and reduces continuous logic to rational Pavelka logic, enabling effective model construction.

Findings

Provability degree of formulas is computable.

Effective Henkin construction for continuous logic.

Decidable models for certain separably categorical theories.

Abstract

In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of {\L}ukasiewicz logic) and continuous logic, and prove effective versions of some theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; show that provability degree of a formula w.r.t. a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory (such as that of a probability space or an $L^p$ Banach lattice) is decidable.