The more quantifiers a statement has the more complicated it is and the harder it is to understand. In this chapter we will look at how complicated statements can be made with quantifiers and give a method for seeing through this complexity.

We also see how to negate statements involving quantifiers. Fortunately, even for the most complicated of statements this is actually quite easy.

Complexity of quantifiers

The number of quantifiers in a mathematical statement gives a rough measure of the statement's complexity. Statements involving three or more quantifiers can be difficult to understand. This is the main reason why it is hard to understand the rigorous definitions of limit, convergence, continuity and differentiability in analysis as they have many quantifiers.

In fact, it is the alternation of the ∀ and ∃ that causes the complexity. For example, ∀x∀y∃zP(x, y, z) will, in general, be simpler than ∀x∃z∀yP(x, y, z). However, since we can replace ∀x∀y by ∀x, y, just counting the number of quantifiers gives a good measure of complexity.