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prase comments on Algorithms as Case Studies in Rationality - Less Wrong
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Because in casual speech the question doesn't actually mean "does that imply that?", but rather "do we have a derivation of that from that, using our set of inference rules?" Not the same, but people seldom realise the distinction.
This is the "paradox of the material conditional", which is one of the primary motivations of relevance logic - to provide a sentential connective that corresponds to how we actually use "implies", as opposed to the material (truth-functional) implication.
http://plato.stanford.edu/entries/logic-relevance/
Good point! Perhaps you won't be surprised, though, if I say that my own preferred account of the conditional is the probabilistic conditional.
That is also a fair interpretation, especially for those students who just want to get the homework done with and don't really care about increasing their sureness in the theorem being re-proved.
If we additionally care about the argument and agree with all the inference rules, then I think there is a little more explaining to do.
Not only for the students, I think. Confusion between implication and inference was enough widespread to motivate Lewis Carroll to write an essay, and nothing much changed has since then. I didn't properly understand the distinction even after finishing university.