| > I think I might be confusing 'material implication' with 'logical implication' (also known as entailment, or logical consequence). 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. |