Logics for at most countable first-order structures

Abstract

In this paper we present two extensions of ω-logic with infinitary inference rules, denoted Arch-ω-logic and non-Arch-ω-logic. We provide the corresponding Hilbert-style axiomatizations and prove their strong completeness with respect to countable Archimedean and non-Archimedean fields, respectively. Through several examples we illustrate a natural representation of various weight functions within the proposed framework and applications to non-monotonic reasoning and neuro-symbolic computing.