>Let's imagine I had shown that something was either one or two. You seem to be claiming I could not take positive proofs assuming each of them and turn it into a full proof.
When you break it down to the logical axioms, you can't use cases like that. Try it yourself. You get something like this:
Let A: (k > 0)
B: (k = 0)
k is a whole number, therefore A v B.
A -> X !=2
B -> X !=2
Okay, from here, you need to deduce X != 2. Traditionally we do this by applying proof by cases, but as I showed in my other comment, this requires the law of the excluded middle.
I don't see any other propositional logic axioms we can apply to achieve the same result.
When you break it down to the logical axioms, you can't use cases like that. Try it yourself. You get something like this:
Let A: (k > 0)
B: (k = 0)
k is a whole number, therefore A v B.
A -> X !=2
B -> X !=2
Okay, from here, you need to deduce X != 2. Traditionally we do this by applying proof by cases, but as I showed in my other comment, this requires the law of the excluded middle.
I don't see any other propositional logic axioms we can apply to achieve the same result.