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In proofs, are "for each" and "for any" synonyms? Or some context is usually required to determine this?

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Compare

A1. If there’s a simple solution for each of the problems, the test is too easy.

A2. If there’s a simple solution for any of the problems, the test is too easy.

These are not equivalent. The first might be true without the second being true.

Compare also the plainly different

B1. There isn’t a simple solution for each of the problems.

B2. There isn’t a simple solution for any of the problems.

So it is important when formally regimenting English into the language of logic to note that, while many standalone or wide scope uses of ‘for any’ and ‘for each’ are equivalent, they do embed differently inside other logical operators.

[And before you rush to making synonymy claims at least for unembedded uses, it is worth remarking that there remain complicated differences between 'any' and 'each' even here. For example, 'any' can take plurals as well as singulars, so we can have both 'for any man' and 'for any men' (and these are different -- a table may be too heavy for any man to lift, but not too heavy for any men to lift); but we can't have 'for each men'. And 'any' can take mass nouns, while 'each' can't (so compare, ‘for any ice that doesn’t shift, try salt’ vs the ungrammatical ‘for each ice that doesn’t shift, try salt’). And so it goes. There's a good reason why we introduce formal quantifiers to avoid the vagaries of English usage.]

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  • $\begingroup$ +1 for emphasizing that, when used in the negative, these terms are not the same: your B1 is $\lnot \forall x. \ S(x)$, while your B2 is $\forall x. \ \lnot S(x)$, or equivalently $\lnot \exists x. \ S(x)$. Plainly different, indeed! $\endgroup$ – wchargin Mar 18 '18 at 21:15
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    $\begingroup$ Owing to its exceptional clarity, I believe this should be the accepted answer, and that the answer so marked missed the point almost entirely. $\endgroup$ – David A. Gray Mar 19 '18 at 0:37
  • $\begingroup$ @DavidA.Gray Changed. $\endgroup$ – Loli Mar 19 '18 at 5:04
  • $\begingroup$ Is this related to how 'or' in the English language is often meant exclusively ('xor' in maths) ? $\endgroup$ – BCLC Mar 19 '18 at 9:43
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    $\begingroup$ @BCLC I can't see that there is any connection between the complications with English quantifiers and e.g. the issue about whether English "or" has an exclusive sense (which is also complicated -- see my response to math.stackexchange.com/q/2653115) $\endgroup$ – Peter Smith Mar 19 '18 at 13:15
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They are synonymous, but may be used in different contexts.   Both declare that the predicate applies to every entity in the domain.   However, "for each" is more often used in an imperative sense: "for each entity make it so", where as "for any" is more often used in an assertive sense: "for any entity it will be so."   Yet that is more a suggestion than a hard rule; it would be acceptable to interchange with usage.

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    $\begingroup$ The first part of this answer is arguably basically right: but the second part seems very odd to me. Is that just a guess or is there linguistic evidence for that claim about imperative vs assertoric uses? $\endgroup$ – Peter Smith Mar 18 '18 at 21:25
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I would recommend avoiding "for any" because it can mean either "for some" or "for all", depending on context. That makes it easy to misunderstand, because you and your reader might disagree about which it does mean.

For example, compare:

  • Any person can downvote this answer;
  • If any person downvotes this answer, I will be sad.

The first of these is a universal statement: for all $x$, if $x$ is a person, $x$ can downvote. The second is existential: if there exists a person who downvotes, ... (If all people downvote then, well, I guess we get to discover if there's a negative rep cap, too!)

Those two examples are unambiguous but suppose somebody writes "A set $S$ is cromulent if, for any $X\subseteq S$, $f(X)=0$." Is that supposed to mean "for all $X\subseteq S$" or "for some $X\subseteq S$"?

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    $\begingroup$ You are basically right. But it is fun to note that you slide unannounced from the OP's "for any" to using plain "any" ("for any" can't replace "any" in your examples). Just more illustrations of the vagaries of English! $\endgroup$ – Peter Smith Mar 19 '18 at 16:06
  • $\begingroup$ @PeterSmith Good point. I've edited the mathematical example to use the "for any" phrasing and I'll have a think about the ordinary English ones. $\endgroup$ – David Richerby Mar 19 '18 at 16:15

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