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I saw in one place the phrase "A which means B" being used as "A is equivalent to B", and in another instance the same phrase being used to mean "A if only B".

Which is the correct usage? My feeling is that it means "A is equivalent to B".

ryang
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Hilbert
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    That sounds informal to me. I agree it is ambiguous as stated. Even as a phrase in English it is ambiguous. "That person is a lawyer, which means they went to law school." or "that person has heterochromia, which means that their eyes aren't the same color." – lulu Jun 04 '23 at 00:09
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    It's like the difference between a mathematician's "or" and a layperson's "or". – Shaun Jun 04 '23 at 00:10
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    $a$ is contained in $b$. Am I talking about membership of inclusion? Natural language is natural, it has inherent ambiguity. Words are overloaded and used for multiple meanings. That's just how language works. The way to make sense of it is context. Keep reading, figure out which meaning makes sense. Eventually you develop a feel for it, and sometimes you're still wrong (sometimes through no fault of your own). – Asaf Karagila Jun 04 '23 at 07:12

2 Answers2

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The word 'means' technically has an 'iff' (⇔) sense like how definitions assign meaning; as such, which means is synonymous with which is equivalent to.

However, even in mathematical writing, which means and which implies are frequently used interchangeably; that is, 'means' is informally frequently treated as having an 'only if' (⇒) sense.

Reproducing lulu's example:

  • it is ambiguous: "that person is a lawyer, which means they went to law school" versus "that person has heterochromia, which means that their eyes aren't the same color."

For what it's worth: I frequently catch myself writing "which means" when I merely mean '⇒'; when I do, I prefer to just change the phrasing.

Does "which means" mean if or iff?

"A which means B" being used to mean "A if only B".

Correction: "A, which means B" informally sometimes means "A, which is only if B" (i.e., "A is true, which implies that B is true"), never "A, which is if B" or "A, which is if only B".

ryang
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  • "The word 'means' technically has an 'iff' (⇔) sense" Only when the subject is the term itself, not a statement. If you say "The word 'lawyer' means 'person who went to law school'", then you are using it as 'iff'. If you say "They are a lawyer, which means they went to law school", then it's just 'if'. – Acccumulation Jun 04 '23 at 05:31
  • @Acccumulation 1. The final phrase in your comment really ought to be 'which is only if' (as pointed out in my answer). $\quad$ 2. I've never heard of a formal rule in the English language that forbids reading <statement A>, which means <statement B> in an 'iff' sense when not referencing terms; in any case, in practice, this is indeed typically read in an 'only if' sense. – ryang Jun 04 '23 at 05:53
  • That's just what "means" means. – Acccumulation Jun 04 '23 at 18:57
  • @Acccumulation Not sure why you are merely repeating your point; I have been referring to the phrase ‘which means’. $\quad$ To be clear: A is true, which means that B is true conveys precisely A is true, and A is true means that B is true; even here, the English language surely has no rule forbidding using/reading 'means' in an 'iff' sense; the context can help disambiguate. – ryang Jun 05 '23 at 05:36
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"A, which means B", means "A only if B" (not "A if only B"). Without the comma, it is incorrect grammar. The only way it means "A is equivalent to B" is if A is mentioned rather than used. For instance, "This is a 'triangle', which means 'three-sided polygon'".

Acccumulation
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    Sorry, but I disagree. Whether it's an "only if" vs. an "iff" depends entirely on the context, and that's a matter better suited for the English language SEs rather than MSE. Consider for example "a number is divisible by $a$ and $b$, which means it's divisible by $\operatorname{lcm}(a,b)$", which both means and is intended to convey an "iff". Or "a triangle with all angles equal is twice isosceles, which means equilateral". – dxiv Jun 04 '23 at 05:57
  • @dxiv You cite cases where iff holds, but that doesn't mean that "means" means iff. – Acccumulation Jun 04 '23 at 18:58
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    An "iff" always implies "only if", so there is no case where "iff" can be the only correct interpretation. This is a matter of second-guessing what the author meant in the context, and my reading of those examples is that the author would have intended them to be read as an "iff". All that said, this is about linguistics more than math. – dxiv Jun 04 '23 at 20:06