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When using quantifiers it is probably important to pick up certain habits that Veterans agree upon as early as possible.

Since it was pointed out to me by a highly esteemed member that it's sometimes better style to avoid quantifiers I was wondering what the convention is with respect to when to use them and when to avoid them.

Since they are logically equivalent to the words spelled out in plain english but do it in less space I was under the impression it would never really hurt to use them, but that is probably a naive view to take, so I'm looking for some advice there.

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I think quantifiers are used more when working things out manually as a shortcut, but in literature in statements of theorems and in proofs the full phrases tend to get used. – Paul Slevin Feb 8 '12 at 1:01
I added the soft-question tag because this does not admit a definitive answer, though I think it's a good question. I believe it might depend on the area in math. Generally, though, it's acceptable and maybe even a good idea to forego the symbols $\forall$ and $\exists$ altogether (unless you're in formal logic or something related). – anon Feb 8 '12 at 1:03
Use whichever is easiest to understand. – Chris Taylor Feb 8 '12 at 1:06
up vote 7 down vote accepted

I think one cannot do better than to quote Halmos on How to write Mathematics (page 142):

The symbolism of formal logic is indispensable in the discussion of the logic of mathematics, but used as a means of transmitting ideas from one mortal to another it becomes a cumbersome code. The author had to code his thoughts in it (I deny that anybody thinks in terms of $\exists$, $\forall$, $\wedge$, and the like), and the reader has to decode what the author wrote; both steps are a waste of time and an obstruction to understanding. Symbolic presentation, in the sense of either the modern logician or the classical epsilontist, is something that machines can write and few but machines can read.

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Logical equivalence isn't the issue. The issue is that symbols are hard to read quickly, and using them when they're not necessary slows down the reader. You can find similar advice (avoiding $\forall, \exists$, and so forth) in Knuth, Larrabee, and Roberts' Mathematical Writing on the very first page.

It seems to me that many students think writing using formal symbols makes what they write more rigorous. This is generally not true.

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Nice article! I'll definitely read it. – Pedro Tamaroff Feb 8 '12 at 2:53

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