caveman1917
Philosopher
- Joined
- Feb 26, 2015
- Messages
- 8,143
Whoever said it was the same? Not me.
That's what you did:
It is pretty easy to show that modus ponens depends on the axiom of non-contradiction.
Suppose you have
P1 p->q
P2 p
Conclusion q
But any conclusion C can only be a consequence of an argument A if there is no interpretation of A in which C is false. So we add the negation of C to A
P1 p->q
P2 p
P3 ~q
If q can be true and false at the same time then this is a valid interpretation of A in which the conclusion is false and therefore C cannot follow from A.
So modus ponens does depend upon the axiom of non-contradiction.
You haven't shown that modus ponens depends on non-contradiction, you've shown that it is inconsistent with the negation of non-contradiction. Big difference.
I said that a proof in those logics depended upon the axiom of non-contradiction and so any proof would have to assume this axiom and therefore any proof of the axiom would be circular.
Yes and this is false. Just because modus ponens is inconsistent with lack of non-contradiction doesn't mean it requires non-contradiction as an axiom. More generally, any set of axioms is inconsistent with the negation of one of its theorems, but that doesn't mean that said theorem needs to be assumed as an axiom in order to prove it. You actually got quite a bit of leeway to decide which things you will take as axioms and which you will take as theorems. You could take the excluded middle and De Morgan's laws as axiomatic and derive non-contradiction as a theorem, or you could take non-contradiction and De Morgan's laws as axiomatic and derive the excluded middle as a theorem, or...