The meaning of quantifiers and the epsilon calculus : on the logical formalization of dependence relations

Abstract

This work argues that the expressive and inferential powers of quantified logic are not exhausted by the classical quantifiers ∀ and ∃. Indeed, based on both the Model and Proof theoretic semantics frameworks, the work will highlight two relations of dependence as arising within formulas and inferences which are not correctly represented by the standard interpretation of ∀ and ∃. Instead, I will claim that the quantifiers ‘for all’ and ‘there exists’ should be interpreted as choice functions − according to the ε-operator of the Epsilon Calculus. Finally, I will consider as a case study the Set theory formulated in the Epsilon Calculus so as to consider how the choice functions interpretation of quantifiers affects debates concerning impredicative definitions and the logical/combinatorial view of collections.