ON EVALUATIONS OF PROPOSITIONAL FORMULAS IN COUNTABLE STRUCTURES

Abstract

Let $L$ be a countable first-order language such that its set of constant symbols ${\rm Const}(L)$ 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 ${\rm Const}(L)$.
The purpose of such a formalism is to provide a general propositional framework for reasoning about $\mathbb F$-valued evaluations of propositional
formulas, where $\mathbb F$ is
a $C$-valued $L$-structures. 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 $\mathbb Q(\varepsilon)$, where $\varepsilon$ is a positive infinitesimal and so on.