| I think I might be confusing 'material implication' with 'logical implication' (also known as entailment, or logical consequence). Because from your last sentence, I thought, if A => B is True and A => not B is True, then A doesn't imply B, or B doesn't (necessarily) follow from A. I think this notion of "B follows from A" is something represented by entailment, not by material implication. (?) Would it be correct to say that material implication is just a formula (in which case it shouldn't even be called an implication or a conditional, but something like simply a 'material formula'), while entailment is the one that has a real world interpretation? (Also entailment cannot be encoded as a formula but has to be proven on a case-by-case basis?) edit: But this is again a problem. Because in order to prove entailment we'd invoke a logical proof, which would be a sequence (or a tree or a graph) of logical statements with the chain of reasoning connected by, surprise surprise, material implication, which we have already discarded as just a formula with no convincing logical interpretation! (hence our proof is not convincingly logical!) edit 2: And that is my main issue with how logicians try to justify material implication. On one hand, they try to convince you that MI is nothing but a formula. On the other hand, they use MI as a connecting glue in mathematical proofs which to me sounds like they're using it as 'entailment'. This feels like a double standard at best. |
Both are intended to work essentially identical, but when talking about logic, it might be helpful to make the distinction. Implication as a formula is usually written with →, semantic entailment between formulas with ⊨. That they are essentially identical is because, if A → B is entailed unconditionally ( ⊨ A → B ), then A entails B ( A ⊨ B ).
When you apply this distinction to
> if A => B is True and A => not B is True
you have ( ⊨ A → B ) and ( ⊨ A → ¬B ). That is not very different from ( A ⊨ B ) and ( A ⊨ ¬B ), or ( A ⊨ B ∧ ¬B ), where the consequence is contradictory. Unless logic is broken, you can only arrive at contradictions when you already started with them, so A itself must be contradictory. When you said that A is false, that's exactly what you required: A is contradictory.
Maybe the above doesn't help you much, but it should make clear that your problem isn't just with the difference between implication and entailment.
EDIT: I didn't see your edits. I think I already somewhat responded to edit 1 (implication does have an interpretation, as an encoding of entailment). Regarding edit 2: Implication is justified by mostly working like our intuition would expect. Maybe you should try coming up with alternatives and test how well they work.
MORE EDIT: I think what you're actually confused about is the difference between a formula being true only under certain circumstances vs. always. Implication being true when the premise is false is a bit like a broken clock being sometimes right: if you only focus on that one case, it's not what you expect, but when you take all cases into account, it's the only way it could be.