Universal Quantifiers

Abstract

This paper refutes the traditional approach that translates all, every, and each uniformly into a universal quantifier and claims that every and each correspond to ∀∃ in logic whereas all simply amounts to ∀. This contrast directly accounts for the grammatical number that the quantifiers are associated with. Furthermore, to address various different properties observed between each and every, I claim that each is necessarily translated into ∀∃ but every is optionally so. In arguing so, I further show that, in the case of each, what has been described as a collective reading has the ∃∀∃-structure while the collective interpretation of every can be expressed as ∃∀.