On Evaluations of Propositional Formulas in Countable Structures

Aleksandar Perovicć, Dragan Doder, Zoran Ognjanović, Miodrag Rašković

Let $L$ be a countable first-order language such that its set of constant symbols Const($L$) is countable. We provide a complete infinitary propositional logic (formulas remain finite sequences of symbols, but we use inference rules with countably many premises) for description of $C$-valued $L$-structures, where $C$ is an infinite subset of Const($L$). The purpose of such a formalism is to provide a general propositional framework for reasoning about $\mathbb F$-valued evaluations of propositional formulas, where $F$ is a $C$-valued $L$-structure. The prime examples of $\mathbb F$ are the field of rational numbers $\mathbb Q$, its countable elementary extensions, its real and algebraic closures, the field of fractions $\mathbb Q(\epsilon)$, where " is a positive infinitesimal and so on.