I am not a mathematician, though I am aware that:
- Any forall-statement about empty set is (vacuously) true because $\neg{(\forall x \in \{\}: P)} \rightarrow \exists x \in \{\}: \neg P$, where $\exists x \in \{\} \equiv False$ by definition: empty set.. is empty!
- Implication has kind of useless "special case" – $False \rightarrow True$ – when precondition is false and yet the consequence holds. Technically, this particular situation has nothing to do with if-else because it is still unknown whether $True \rightarrow True$ will hold as well. Never the less, $False \rightarrow True \equiv True$.
It seems to me that math is driven by the following philosophical principle:
Everything is true unless the opposite is proven.
In the #1 it is necessary to find such $x \in \{\}$ that …, which is impossible. Being unable to prove "the opposite" implies undeniable truth. In the #2 it is necessary to show the case when precondition holds and consequence doesn't: unless it is shown, implication considered to be truthful.
Am I right?
Best Answer
I would rather state it as
Anything might be true until the opposite is proven.
If something is assumed to be true without a proof, someone else might say that the opposite is true, e.g. your claim is the opposite of his truth claim, so now you need to prove him wrong.