When I was scholar demonstrations were done with equivalences, but reasonings seem to say that using implications would be enough?

Question

When I was scholar, quite every demonstration I saw was made using equivalence $\iff$ signs.

But I came on a course teaching the reasoning available in mathematics, well named: "the toolbox for demonstrations". And among them, were:

• demonstration by a counter example (to prove that $\forall x, P(x)$ is true, demonstrate that $\exists x, \lnot P(x)$ is false)
• demonstration by contraposition (instead of demonstrating: $P(x) \implies Q(x)$, demonstrate $\lnot Q(x) \implies \lnot P(x)$)
• demonstration by absurdity (use the fact that the only relation that is false, in $P \implies Q$ is when $P$ is true while $Q$ is false, to demonstrate that a case where $P(x) \implies Q(x)$ having $P$ true, $Q$ false would lead to a nonsense)

...
and others reasonings.

Things are going as if, to demonstrate something, ensuring that whole equivalence wasn't mandatory.
As if, testing the reciprocity of a relation wouldn't be truly needed to complete a demonstration.
Equivalences ($\iff$) would be no more needed, and implications ($\implies$) would be enough.

I feel a bit strange in front of that. Can you explain me why I am seeing this?

Answer 1

It would be nice to see those proofs that you learned earlier. But, I am guessing that the proofs you are referring to were boolean algebra proofs (i.e. proofs where you took a statement and transformed it through principles like DeMorgan or Double Negation into some other statement). And yes, every boolean algebra rule is an equivalence.

However, now you are doing inference. And yes, the rules here no longer reflect equivalences. For example, one of the rules in a system of inference is that the statement $P \land Q$ implies statement $P$. But note: $P \land Q$ and $P$ are of course not equivalent.

Rules of inference can also be applied to multiple statements. For example, $P \to Q$ and $Q \to R$ together imply $P \to R$. But in this case, it technically doesn't even make sense to say that these are equivalent, because we are no longer talking about two statements, but three. And it is also not true that $P \to R$ implies either $P \to Q$ or $Q \to R$, let alone both. So there is again really no equivalence here. But I think you will agree that as an inference, it makes total sense and is totally useful for doing proofs.

Also: when doing inferences, you can still make good use of equivalences, so it is not true that we are throwing out all those equivalence rules.