In proofs, are "for each" and "for any" synonyms?
In proofs, are "for each" and "for any" synonyms? Or some context is usually required to determine this?
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.]
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.
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 $X\subseteq S$"?