In Chapter 6, Making a statement, we called a sentence like ‘x is an odd number’ a conditional statement. Whether or not it was true depended on x. In this chapter we shall describe two fundamental ways of quantifying x so that we get statements rather than conditional statements.

We can exemplify them via the following. The sentence ‘x2 = 2’ is conditional on x. By combining such sentences with some description of the x we can form statements. For example, ‘For all x ∈ ℤ, x2 = 2’ or ‘There exists an x ∈ ℝ such that x2 = 2.’ Note that the latter is true and the former is false.

Since we are giving a description of how many of x we are talking about, i.e. assigning a quantity, we call these descriptions quantifiers. The fancy titles for the quantifiers we are interested in are universal quantifier and existential quantifier.

For all – the universal quantifier

Definition 10.1

The phrase ‘for all’ is the universal quantifier. It is denoted ∀. (This is an upside down A. Think of the A in ‘All’ to remember this.)