I'm not really sure if this is the right part of SE for this question or if it should be in the Meta part - if it should be, then let me know and I'll change it!
Anyway, here's my question:
What is the precise difference between the statements "When" and "Just When".
I think that I have the answer, but would value confirmation.
Statement 1: {We have A when B}. So we can have A but not B (ie, B $\nRightarrow$ A), but if we have A, then must have B (ie A $\Rightarrow$ B - this would be consistent with the usual contrapositive)?
Eg, let $a,b \in \Bbb N$: $a \ge b$ when $a \ge b + 2$, but not just when $a \ge b + 2$; but we can have $a = b$ which does not satisfy $a \ge b$.
Statement 2: {We have A just when B}. So now whenever we have A, we also have B (ie, A $\Rightarrow$ B), but if we have B, then we also have A (ie B $\Rightarrow$ A).
However, consider an adaptation of the the example above: $a \ge b$ just when $a \ge b - 1$; however, we do not necessarily have $a \ge b$ when $a \ge b - 1$. Is this (adaptation) correct?
Thanks! :)
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.