difference between "let" and "for all"

Question

What is the difference between "let" and "for all"?

Consider the following example

1. For all natural numbers n, if n is even, then n squared is even.

2. Let n be a natural number. If n is even, then n squared is even.

Answer 1

"For all" means just what it says. It means that the rest of the statement is true when $n$ is replaced by any natural number.

"Let" is used for introducing a "supplemental hypothesis". That is, a statement that we are assuming is true for the rest of the discussion. That statement can introduce a new constant value, such as it does here for $n$. Or it can used to add additional conditions on existing values (though usually we say "suppose" in that case). This is a common technique in proofs. You introduce the supplemental hypothesis $A$, then prove some other statement $B$ making use of it. The theorem is then "$A \implies B$". So in case (2). you are really stating

"If $n$ is a natural number, then (if $n$ is even, then $n^2$ is even)."

Which is an awkward way of stating what you mean. It also has an implicit quantization. I.e., it really means:

"For all $n$, if $n$ is a natural number, then (if $n$ is even, then $n^2$ is even)."

Fortunately, logic and grammar allow us to reduce this to the form of (1). The important thing to note, though, is that (1) is the proper statement of the "theorem", while (2) is appropriate only in limited discussions, where the supplemental hypothesis is only assumed to hold within that discussion.