I proved that x has an even number of factors of 2. Why can't I then split that into scenarios when that even number is 0, and cases where it isn't? I don't see any restrictions here.
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.
>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.
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.