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In mathematical writings, one often encounters statements involving the phrase "in general" in the following sense:

After the number 2, the next few prime numbers (3,5,7) are each odd numbers, and in fact it is true in general that all prime numbers after 2 are odd. However, it is not generally true that all odd numbers are primes.

In other words, the phrase is being used to stress that there are no exceptions to the truth of the claim being proposed.

It just occurred to me how strange it is that most everybody else uses this phrase with precisely the opposite connotation, to indicate what is usually the case while acknowledging that it is may not always be the case. For example,

In general, Canadians tend to like ice hockey, though occasional exceptions do happen.

Is there any reason for how these apparently divergent meanings came to be in use?

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    You will find many, many cases where the language of mathematics differs from English (or any natural language). For me, it's reason enough that mathematics needs language that is precise and unambiguous. Natural language evolves organically and is used for a great variety of purposes, from basic communication to art, and serves a great variety of users, with varied backgrounds, skills, and sensibilities. – John Y Nov 11 '14 at 3:38
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    I have to disagree with the premise. Mathematicians do not in general use "in general" for things that are always true. One can find lots of counterexamples in math papers. Mathematicians do tend do use "true in general" for things that are always true. for example (The removal of one comparability does not in general result in a poset. Only a comparability which cannot be recovered by transitivity can be removed.) – Peter Shor Nov 11 '14 at 3:48
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    @PeterShor The premise is that Mathematicians use "in general" to mean "always". Your example, "The removal of one comparability does not in general result in a poset" translates as, "the removal of one comparability does not always result in a poset", which does not contradict the premise. – ChrisW Nov 11 '14 at 3:49
  • Here's another example: "Since the question of enumerating pattern avoiding linear extensions of posets in general is a very hard one, we focus instead on certain partially ordered sets called combs." A use of "in general" which means "usually, but not always". – Peter Shor Nov 11 '14 at 3:53
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    In general they do, but not in the general case. – Keith Nov 12 '14 at 4:20
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define:general gives,

affecting or concerning all or most people, places, or things; widespread

I suppose Mathematicians are interested in whether a rule is always true (for "all" things), whereas other people are satisfied to be interested in rules which are usually true (for "most" things).

Note that these are not (as your question alleges them to be) "precisely the opposite connotation".

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  • It's precisely the opposite in that it's "all" or "not-all," with exceptions or without. By either of the simplified definitions, the two definitions one (one for mathematicians (and logicians) and the other for the rest of us) comprise everything. – wys1wyg Mar 10 '15 at 23:22

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