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What is the precise difference between the phrases "when" and "just when"? Is this explanation correct?

  1. {A when B}. This means that if we have A, then must have B (i.e. A $\Rightarrow$ B); here, it is also possible to have B but not A (i.e. B $\nRightarrow$ A).
  2. {A just when B}. This means that if we have A we also have B (i.e. A $\Rightarrow$ B), and if we have B then we also have A (i.e. B $\Rightarrow$ A).
  3. Therefore:
    • $a \ge b$ when $a \ge b + 2,$ but not just when $a \ge b + 2.$
    • $a \ge b$ when $a \ge b - 1,$ but not just when $a \ge b - 1.$
ryang
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Sam OT
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2 Answers2

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Interpreting "A when B" as $A \Rightarrow B$ isn't really consistent with the english language. If I say "I get wet when it rains", that means (at least to me) that rain implies getting wet, not the other way around. I.e., whenever it rains I get wet, but I might get wet even if it doesn't rain (if I, say, jump into the pool). I'd thus read "A when B" as $B \Rightarrow A$.

For "A just when B", I agree - it means $A \Leftrightarrow B$. Just whenever A is the case, so is B, and conversely whenever B is the case, so is A.

fgp
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    I am more used to seeing "A just in case B" than "A just when B" but I do think they both mean "A if and only if B" (that is certainly what the first one means). – Carl Mummert Mar 13 '14 at 23:19
  • Oops, yeah, of course. I find it rather awkward to get my head around sometimes! But what about my example: $a \ge b$ when $a \ge b + 2$, but we can have $a = b$ which does not satisfy $a \ge b$. – Sam OT Mar 14 '14 at 08:36
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  1. In mathematics, although the assertions

    • $B$ if/when/whenever $A$
    • $A ⇒ B$
    • $A$ only if $B$

    are conventionally understood as synonymous, Bullet 3 fairly frequently triggers misunderstanding, confusion or even disagreement.

  2. In view of this, I'd refrain from using these similar-sounding phrasings in technical writing, treating them as ambiguous or poetic:

    • $A$ only when $B$

      (I'm inclined to interpret this like the above, as $A ⇒ B)$

    • $A$ just if/when $B$

      (I'm inclined to interpret this as $A ⇔ B,$ and would instead opt for $\text“A$ is equivalent to $B\text”$ or $\text“A$ is true if and only if $B$ is true”).

ryang
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