A related problem with the traditional picture is its treatment of deductive principles like (D) as rules of inference. In fact they are rules of about what follows from what. Recall what (D) says:
(D) From premises of the form “all F are G” and “a is F” the corresponding conclusion of the form “a is G” follows.
(D) says that a certain conclusion follows from certain premises. It is not a rule of inference. It does not say, for example, that if you believe “All F are G” and also believe “a is F” you may or must infer “a is G.” That putative rule of inference is not generally correct, whereas the rule about what follows from what holds necessarily and universally. The alleged rule of inference is not generally correct because, for example, you might already believe “a is not G” or have good reason to believe it. In that case, it is not generally true that you may or must also infer and come to believe “a is G.“
Perhaps you should instead stop believing “All F are G” or “a is F.” Perhaps you should put all your energy into trying to figure out the best response to this problem, which may involve getting more data. Or perhaps you should go have lunch and work out how to resolve this problem later!
From inconsistent beliefs, everything follows. But it is not the case that from inconsistent beliefs you can infer everything.
Deductive logic is a theory of what follows from what, not a theory of reasoning. It is a theory of deductive consequence. Deductive rules like (D) are absolutely universal rules, not default rules, they apply to any subject matter at all, and are not specifically principles about a certain process. Principles of reasoning are specifically principles about a particular process, namely reasoning. If there is a principle of reasoning that corresponds to (D), it holds only as a default principle, other things being equal.
Another thing to be careful about in logic class. It took a while to train myself to avoid pronouncing “p → q” as “p implies q,” which is a very common thing to do in math/CS logic circles. I’m now very careful to distinguish the conditional from implication, and try to get my students to do the same.
* Gillian’s post happened a while ago, I know. I’m slow. Do go and read it and the comments.