In our consideration of logic, we have talked about propositions and the ways in which they can be combined to form new propositions, but have given little or no attention to the way in which simple propositions are constructed. Discussion of words and phrases within propositions will, for the most part, be outside the scope of this text. However, to understand some very important types of mathematical propositions, it is essential to mention one such topic, namely,  the construction and meaning of propositions containing quantifiers. The following are examples of propositions which contain quantifiers:

some men are wealthy;

all men are prejudiced;

no man is patient.

The words some, all, and no are quantifiers. They tell us "how many" of a certain set of things is being considered. Many statements do not specifically contain quantifiers although quantification is implied. Consider the following mathematical propositions in the form of equations:

                                                                                          x²+4x=7,

x² - 4 = (x + 2) (x - 2).

The first is true in the sense that for at least one number x, x² + 4x = 7,  and the second is true in the more general sense that for every number x,  x² - 4 = (x + 2) (x - 2). The first equation could be proven false only by showing that no number x satisfies the equation, but the second could be shown to be false by exhibiting a single number x which fails to satisfy the equation. It is important, then, to distinguish carefully between these types of propositions.

We define the symbol ∀_xp to mean that for every x in a given set, the proposition p is true. ∀x is called the universal quantifier of the variable x and is usually read "for all x" or "for every x." We will define the symbol ∃_xp to mean that for one or more elements x of a certain set of elements, the proposition p is true. ∃x is called the existential quantifier of the variable x and is usually read "there exists an x such that p" or"for at least one x, p" or, less precisely, "for some x, p."

Since each quantifier refers to a particular set of permissible values for the variable x, this set must be mentioned. Frequently, this set is described in a sentence preceding the proposition that contains the quantifier, although the set may be included in the proposition itself. For instance,  the propositions above may be written in any of the following ways:

there exists a number x such that x² + 4x = 7;

for all numbers x, x² - 4 = (x + 2)(x - 2).

Alternatively, we may write :

∃_xp, where x belongs to the set of all numbers

and where p is the proposition x² + 4x = 7;

∀_xq, where x belongs to the set of all numbers

and where q is the proposition x² - 4 = (x + 2) (x - 2).

Still other forms are:

ifxisanumber, ∃x(x²+4x = 7);

if x is a number, ∀x[x² - 4 = (x + 2)(x - 2)].

It is important to know how the negations of propositions involving quantifiers are formed. A little reflection will justify the following formulas:

(∃_xp)' = ∀_xp'

(∀_xp)' = ∃_xp'.