On evaluations of propositional formulas in countable structures

Abstract

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 F-valued evaluations of propositional formulas, where F is a C-valued L-structure. The prime examples of F are the field of rational numbers Q, its countable elementary extensions, its real and algebraic closures, the field of fractions Q(?), where ? is a positive infinitesimal and so on.