Is "everything is true unless the opposite is proven" a fundamental math philosophy principle?

Question

I am not a mathematician, though I am aware that:

1. 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!
2. 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?

Answer 1

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.

Answer 2

First, truth and provability are two different properties. To check if something is true or not you need a model. For example, if you talk about addition - you need some structure on which addition is defined - like in natural numbers sentence $\forall x \forall y \exists z: x = y + z \vee y = x + z$ is true, while $\forall x: x + x = 0$ is false.

To check if something is provable, you need axioms and deduction rules. For example, if you have axioms $x$ and $x \to y$ and MP rule: $\{a, a \to b\}\vdash b$, then you can deduce $y$.

For "$\forall x\in X\colon P(x)$ is true if $X = \varnothing$" you can consider the following game: you you can choose $x \in X$ s.t. $P(x)$ is false then you win, otherwise I win; statement $\forall x\in X\colon P(x)$ is true if I win in this game assuming you play optimally. Of course if $X = \varnothing$ you can't win.

For $\bot \rightarrow \top$ is true, take for example statement "if number is greater than $4$, it's greater than $2$". We probably want to think this statement is true (for any number). Lets check cases:

1) if the number is $5$, then it's of form $\top \rightarrow \top$

2) if the number is $1$, then it's of form $\bot \rightarrow \bot$

3) if the number is $3$, then it's of form $\bot \rightarrow \top$