Outline of a Theory of Quantification

Abstract

Ramification¹ is fundamentally a theory of quantification. It says that no proposition can quantify over itself (or over propositions that can quantify over it, etc.).² Slightly more carefully, so as to not assume that propositions themselves contain quantifiers, it says that that there is an infinite hierarchy of orders of propositions, and that if a sentence (or, even more carefully, a formula P) denotes a proposition of order n, quantifiers in the sentence (P) can range over only orders m > n. I often speak loosely of propositions themselves quantifying with the understanding that such talk can be avoided if necessary. I also assume, contrary to Russell’s version of ramification but in line with Church’s (Church, 1976), that orders are cumulative, so that propositions of order n also appear in all orders m > n.