Why does "if and only if" mean the exact same thing as "precisely when"?

Question

The proposition "A precisely when B" states that A has the same truth value as B. The proposition "A if and only if B" states that A is true if B is true and that A is true only if B is true.

Question: Can you intuitively explain why "if and only if" means the exact same thing as "precisely when"? (I've already checked the equivalence of "A precisely when B" and "A if and only if B" using a truth table, but unfortunatly this doesn't bring intuitive understanding of this issue.)

Answer 1

"$A$ is true if $B$ is true" means that when $B$ is true, then also $A$ is; but we do not know about the truth of $A$ when $B$ is false.

But then we add that "$A$ is true only if $B$ is true", that means that we cannot have $B$ false and $A$ true.

Conclusion: $A$ and $B$ must both be true "together" or false together.

Answer 2

Intuitively speaking, I think of both statements as having an implicit "over all possibilities" universal quantification. Let me explain:

• If I say "if $A$ then $B$", I mean that for all possibilities in which $A$ is true, $B$ is true. So let's say $A = p \land q$, $B = p \lor q$. Then "if $A$ then $B$" means for all possibilities of $p$ and $q$ that make $A$ true, they also make $B$ true.

• Similarly, therefore, "$A$ if and only if $B$" means that for all possibilities where $A$ is true, $B$ is true, and for all possibilities where $B$ is true, $A$ is true. (Both directions!)

• But that's just saying that $A$ and $B$ are true for exactly the same possibilities; on other words, we could say that $A$ holds precisely when $B$ holds, i.e. at precisely the same possibilities.

If you are curious, this can all be made formal. There is a notion of possible world, and if you like we can define a statement like "($p$ and $q$) if and only if ($q$ and $p$)" to be "true" just in case it is true in all possible worlds.

Answer 3

I'm taking this as a question about mathematical terminology.

Using "when" as a synonym for "if" is common mathematical English jargon that adds a cosy temporal flavour to what are really static assertions about mathematical objects. E.g., "$tu$ vanishes when $t = 0$" sounds more lively than "$tu = 0$ if $t = 0$" (I've thrown in the cosy term "$x$ vanishes" for $x = 0$ to add to the effect here.)

Then, because in the non-mathematical world we are used to talking about things happening at precise times, it feels right to say "precisely when" in place of "if and only if" or "iff": i.e., to write things like "if $u \neq 0$, then $tu$ vanishes precisely when $t = 0$" instead of "if $u \neq 0$, then $tu = 0$ iff $t = 0$". If you are reading mathematical English, you have to get to know these elegant variations. If you are writing mathematical English, I suggest you avoid them unless you are confident you will be understood.

Answer 4

"$A$ precisely when $B$" is defined as: when $B$ is true, then so is $A$, and when $B$ is not, neither is $A$.   So the truth state of $A$ and $B$ are equivalent; hence it is equivalent to: "$A$ if and only if $B$".

"The light is on precisely when the switch is on."

"The light is on if and only if the switch is on."