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Abstract: In this paper, we introduce a foundation for computable model theory ofrational Pavelka logic an extension of {\L}ukasiewicz logic and continuouslogic, and prove effective versions of some theorems in model theory. We showhow to reduce continuous logic to rational Pavelka logic. We also definenotions of computability and decidability of a model for logics withcomputable, but uncountable, set of truth values; show that provability degreeof a formula w.r.t. a linear theory is computable, and use this to carry out aneffective Henkin construction. Therefore, for any effectively given consistentlinear theory in continuous logic, we effectively produce its decidable model.This is the best possible, since we show that the computable model theory ofcontinuous logic is an extension of computable model theory of classical logic.We conclude with noting that the unique separable model of a separablycategorical and computably axiomatizable theory such as that of a probabilityspace or an $L^p$ Banach lattice is decidable.

Author: Farzad Didehvar, Kaveh Ghasemloo, Massoud Pourmahdian

Source: https://arxiv.org/

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