Quantifier Prefixes

In this chapter we restrict ourselves to quantifiers of type (1) and analyze quantifier prefixes. We already gave some basic properties for first order pre- fixes in subsection 2.3.1, but here we will study prefixes with Generalized Quantifiers, as well as other tuples of prefixes not present in FOL.

Since we establish in the previous chapter that all monadic quantifiers can be captured by cardinality properties, it would seem that quantifiers of type (1) can be trivially defined. Indeed, the definition 5.1.3 can be applied to these quantifiers and immediately shows that a quantifier QM(A) can be defined by two numbers, |M - A| and |A|. When the quantifier is domain independent, the first of these two numbers is irrelevant, and only |A| matters. Note, though, that not all type (1) quantifiers are domain independent: all(1) is not; quantifiers like $\textbf{n}\%\textbf{of}M(A) = \{A \subseteq M | |A| = \frac{|M| \times n}{100}\}$ is not domain independent either. It is arguable that in some of these cases, the relativization to a type (1, 1) quantifier, which yields a domain independent quantifier, is more meaningful. But in any case, quantifiers of type (1) can be easily captured by number properties, and hence the approach of the previous chapter applies. Why, then, study such quantifiers separately? The answer is that its type allows such quantifiers to be easily combined into quantifier prefixes, and that such prefixes exhibit interesting properties. Prefixes, not isolated quantifiers, are the real theme of this chapter.