But this is not true, because ¬(¬A ⟶ B) ⟶ A ∨ B. With what you’ve written you can get from the left side to the right side, but you can’t get from the right side to the left side.
What you need is: “Either Alice did it or Bob did it. If it wasn’t Alice, then it was Bob; and if it wasn’t Bob, then it was Alice.”
No, what he’s written is correct[0]. A ∨ B, ¬A ⟶ B, and ¬B ⟶ A are all equivalent. (Hence your last statement is also correct!) Note for instance that the last two are just contrapositives of one another and so equivalent.
[0]In classical logic, obviously, for the nitpickers; that’s all I’m going to consider here.
Note also you could easily see your initial comment to be wrong just by computing the truth tables! Equivalence in classical propositional logic is pretty easy—you don’t need to think about proofs, just write down the truth tables!
But this is not true, because ¬(¬A ⟶ B) ⟶ A ∨ B. With what you’ve written you can get from the left side to the right side, but you can’t get from the right side to the left side.
What you need is: “Either Alice did it or Bob did it. If it wasn’t Alice, then it was Bob; and if it wasn’t Bob, then it was Alice.”
Thus: A ∨ B ⟷ (¬A ⟶ B ∧ ¬B ⟶ A)
No, what he’s written is correct[0]. A ∨ B, ¬A ⟶ B, and ¬B ⟶ A are all equivalent. (Hence your last statement is also correct!) Note for instance that the last two are just contrapositives of one another and so equivalent.
[0]In classical logic, obviously, for the nitpickers; that’s all I’m going to consider here.
Ah, of course. Thanks.
Note also you could easily see your initial comment to be wrong just by computing the truth tables! Equivalence in classical propositional logic is pretty easy—you don’t need to think about proofs, just write down the truth tables!