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I often see in papers something like:

1) This is in general not true

or

2) This is not true in general

Which I personally would consider to be written formally as something like

1) $\forall x: \neg p(x)$
2) $\exists x: \neg p(x)$

But I wonder whether this is generally what is meant and if the mathematical community is careful about how they use the word "general" or if it used in a more colloquial sense. Being somewhat of an outsider I find this hard to judge. Partly as it is often used as an aside and rarely a formalisation of the statement is present to check it against.

It's the sort of thing you just can't look up.

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  • $\begingroup$ Perhaps you're looking for \neg: $\neg$ $\endgroup$
    – JBL
    Commented Oct 14, 2010 at 17:58
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    $\begingroup$ You probably wanted to use \neg instead of \not. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Oct 14, 2010 at 17:58
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    $\begingroup$ Also, I think common usage is that "true in general" means "for all [whatever], [thing] is true" while "false in general" means "not (true in general)." $\endgroup$
    – JBL
    Commented Oct 14, 2010 at 18:01
  • $\begingroup$ I think I would parse both the 1st and 2nd phrases the same way, a la JBL's comment above mine. I would imagine the phrase preceding either of those to be "p is true given condition Y". In this case in particular, your interpretation (1) makes no sense, since $\exists Y: p(Y)$. $\endgroup$
    – Willie Wong
    Commented Oct 14, 2010 at 18:23
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    $\begingroup$ I would use "this is not true in general" to mean that it doesn't always hold. I would use "this is in general not true" to mean the same, but with the added suggestion that it tends not to be the case for your "average" non-degenerate situation. In terms of strict logical implication, I consider them to mean the same thing. $\endgroup$
    – Andrew D. King
    Commented Oct 14, 2010 at 18:25

2 Answers 2

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I don't know about what this means in general. I use it as a way of avoiding twisting my prose into horrendously convoluted statements whilst avoiding the possibility that some smart alec is going to pick up on a technicality.

More precisely, I use it when I wish to say something like "Not all snarks are boojums" but the sentence would work much better (either for grammatical reasons or to better convey the intended meaning) if I could just say, "snarks are not boojums". That's false as stated[1], so to avoid either saying anything actually incorrect or that someone's going to say, "But what about ...", I say "in general, snarks are not boojums".

What's important here is that I use it mostly in the prose section of a paper or seminar when I'm trying to focus the reader or listener's attention on the important facets of whatever it is that I'm explaining. So getting in to a long diversion of which snarkss are not boojumss (is it the lesser-spotted or the warbler variety?) would be counterproductive. Saying, "not all snarks are boojums" tends to draw ones attention to that class of snarks which are boojums. Saying "snarks are not boojums" is almost guaranteed to get some smart alec saying, "But what about greater-wrinkled snarks?" (especially in a lecture). So "in general, snarks are not boojums" has the triple benefit of 1) being true, 2) focussing the attention on the key point, and 3) not grammatically convoluted.

[1]: Banker and Carroll, Identifying subspecies of snark (1874)

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  • $\begingroup$ About half an hour after writing that, I found myself actually using "in general" in a paper to mean "I know it works for n = 1 but it doesn't work for any other n and that's the important bit.". (pause) Did I really write "snarkss" and "boojumss"? Gosh, maybe I am turning in to Gollum. $\endgroup$
    – Andrew Stacey
    Commented Oct 15, 2010 at 17:35
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In general the determinant of a sum is not the sum of the determinants. In that case I don't know that equality is interesting. In general we can not evaluate a path integral knowing only the endpoints (but conservative vector fields are useful). This arc length integral was fairly easy but in general numerical methods would be required. Those are some of the ways I might use it. So "usually" in some useful sense.

later perusing Math Reviews for the phrase "not true in general" reveals many cases of A implies B but the converse is not true in general Also A implies B however B is not true in general

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