# Logical Quantification and Articles

• "Quantification" in logic indicates whether the set of objects possessing a particular property is a singleton, a proper subset, or the whole set.

• For example, the definition of continuity in calculus says

Given a function f, if for all ε > 0 there exists δ > 0 such that for all y if d(x,y) < δ, then d(f(x),f(y)) < ε, then we say that f is continuous at x.

• Similarly, the definition of inverse function says

If for all y, there is a unique x such that y = f(x), then the inverse f -1 of f exists, and f -1 (y) = x such that f(x) = y.

• The article "a" corresponds to the logical "there exists", and the article "the" corresponds to the logical "for all".

• Because articles in English are mandatory (if you omit them, you are speaking incorrectly), English speakers must be aware of quantification at all times. Native speakers do this automatically and effortlessly.

In this sense, some scholars like to say "English is more logical" because it forces you to use logical concepts to speak correctly.