### 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.

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.

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