Universal Quantification - Basics

Basics

Suppose it is given that

2·0 = 0 + 0, and 2·1 = 1 + 1, and 2·2 = 2 + 2, etc.

This would seem to be a logical conjunction because of the repeated use of "and." However, the "etc." cannot be interpreted as a conjunction in formal logic. Instead, the statement must be rephrased:

For all natural numbers n, 2·n = n + n.

This is a single statement using universal quantification.

This statement can be said to be more precise than the original one. While the "etc." informally includes natural numbers, and nothing more, this was not rigorously given. In the universal quantification, on the other hand, the natural numbers are mentioned explicitly.

This particular example is true, because any natural number could be substituted for n and the statement "2·n = n + n" would be true. In contrast,

For all natural numbers n, 2·n > 2 + n

is false, because if n is substituted with, for instance, 1, the statement "2·1 > 2 + 1" is false. It is immaterial that "2·n > 2 + n" is true for most natural numbers n: even the existence of a single counterexample is enough to prove the universal quantification false.

On the other hand, for all composite numbers n, 2·n > 2 + n is true, because none of the counterexamples are composite numbers. This indicates the importance of the domain of discourse, which specifies which values n can take. In particular, note that if the domain of discourse is restricted to consist only of those objects that satisfy a certain predicate, then for universal quantification this requires a logical conditional. For example,

For all composite numbers n, 2·n > 2 + n

is logically equivalent to

For all natural numbers n, if n is composite, then 2·n > 2 + n.

Here the "if ... then" construction indicates the logical conditional.