If we want to use proof by contradiction, shouldn't we first proof that a statement is either true or false?

Question

As an example how this could go wrong "This statement is false" We could proof this statement by contradiction as follows

Assume the statement is false $\implies$ The statement is false $\implies$ the statement is true. Which is a contradiction, hence the statement must be true.

Obviously, the statement being true leads to equally contradictory results, but in mathematics we I have never seen anyone continue to check also what happends if we assume the statement is true. And if we did, it would not be humanly possible to check all possible consequences of a statement being true.

Is there a mathematical axiom that claims that all statements are either true or false, and how can we be sure that this axiom itself does not lead to contradictions?

Answer 1

Is there a mathematical axiom that claims that all statements are either true or false?

Yes, it is called Law of excluded middle. This is always used for the vast majority of mathematics.

and how can we be sure that this axiom itself does not lead to contradictions?

We can't (provided we start with sufficiently strong axioms such as Peano arithmetic or ZFC). This is known as Goedel's second incompleteness theorem.

"This statement is false"

This is a self-referential statement. While this is possible in the english language, i am not aware of any formulation in classical mathematics of this statement. In my (and probably many other's) opinion this is therefore not a valid example to show that a mathematical statement can be neither true nor false.