Proper Use of Quantifier _ \forall($\forall$) \for each \for every

Question

In our natural language \forall\forevery\foreach are pretty much well discernable for me. However, in mathematical description I always confusing whether we actually needs all these 3 of them.

Question is,

$1$. Do mathematician need above three all?

$1-1$. If so, what kind of difference do they hold for which functional use?

Answer 1

The short answer: the phrases "for each", "for every" and "for all" all mean exactly the same thing.

Answer 2

These statements are equivalent, and any confusion is arising as more of an English issue, then a logic issue. $\forall$ is the universal quantifier. It is stated, in English, generally as "for all," but this is just an English phrase for the universal quantifier. The phrases "for each" and "for every" are less common, but still valid ways of prononcing $\forall$ that sometimes are motivated by context.

Consider: $\forall_i \in \mathbb{R}$ in English, it would by odd to pronounce the universal quantifier as "for each" or "for every." "For each /every Real number" implies a sort of discrete aspect that is not a part of the Real numbers. Saying "for all", captures the continuous nature of the set in English more readily.

Consider: $\forall_x \in A = \text{{x|x $\in \mathbb{N} \land x \lt 10$}}$ in this case all of the English pronunciations of the universal quantifier make sense, this is a finite discrete set, so "each element," "every element," and "all elements" convey the correct idea in English.

Again though, this is interjecting confusion from the world of English that ideally does not exist because of the clarity of symbols like $\forall$. How you pronounce this symbol is ultimately meaningless, the symbol notes that some propositional function is satisfied by every member of a domain of discourse.