Suppose we're looking at two objects:
(i) the conditional statement (A --> B)
(ii) the inference A |- B
In propositional logic, we have that (i) is true if and only if (ii) is valid. My question is if there exist any systems of logic where this does NOT hold? If so, is anyone familiar with arguments as to why these two things could ever disentangle? It seems intuitively true that conditional statements (at least indicative conditional statements) are just arguments in disguise. In other words, there doesn't seem to be any difference whatsoever in someone uttering "If A, then B" and "An argument with premise A entails B". Or is there?