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Following recent developments in the topic of generalized quantifiers, and also having in mind applications in the areas of security and artificial intelligence, a conservative enrichment of (two-sorted) first-order logic (FOL) with almost-everywhere quantification is proposed. The completeness of the axiomatization against the measure-heoretic semantics is carried out using a variant of the Lindenbaum–Henkin technique. The independence of the axioms is analysed, and the almost-everywhere quantifier is compared with related notions of generalized quantification. A suitable fragment of the logic is translated to FOL and validity is shown to be preserved.

Complete Axiomatization of Discrete-Measure Almost-Everywhere Quantification