Is there a deduction analog to the problem of induction?

Question

Aren't deductive and inductive reasoning equally unjustified? So, inductive reasoning is going from specifics to general, whilst deductive reasoning is going from general to specific. But in deductive reasoning, surely, forming 'general' opinions or laws is based on induction right? So why is there a problem with induction but deduction is provided this sense of superiority?

Answer 1

When it comes to justification there is indeed a symmetric problem of deduction. But forming general opinions or laws is not part of deduction, it is abductive (or in older terminology inductive), when it comes to science it is the "hypothetical" part of the hypothetico-deductive method, see Are “if smoke then fire” arguments deductive or inductive? for more on abduction in science.

The advantage of (formal) deduction is that at least we know that it is always truth preserving, even if we can not justify it. Induction, on the other hand, sometimes works and sometimes does not. So the problem of justifying it is much more substantive because the hope is that if we have an explicit justification at hand we will be able to better tell when is which. The problem of deduction is a particular case of the so-called Agrippa's trilemma/justificatory regress arguments, and is discussed by Dummett in Justification of Deduction:

"We should probably make use either of those very forms of inference which we were supposed to be justifying, or else of ones which we had already justified by reduction to our primitive rules. And, even if we did neither of these things, so that our proof was not strictly speaking circular, we should have used some principles of inference or other, and the question could then be raised what justified them: we should therefore either eventually be involved in circularity, or have embarked upon an infinite regress."

Dummett argues for a different kind of "justification", semantic justification, where interpretations (truth tables in simple cases) are used to show that deductive rules are indeed truth preserving. Of course, the same rules are implicitly used to reason about the truth tables, but this does not erase their explanatory (as opposed to justificatory) value, in place of a vicious circle we get a virtuous hermeneutic circle. An alternative, syntactic, approach to "justifying" deduction is Haack's Justification of Deduction. Here is Dummett on how deduction compares to induction on semantic approach:

"The situation is thus the reverse of what seems to be the case with induction. In the case of induction, we appear to have a quite unconvincing argument that there could not in principle be a justification, but we lack any candidate for a justification. In that of deduction, we have excellent candidates, in the soundness and completeness proofs, for arguments justifying particular logical systems; in the face of an apparently convincing argument that no such justification can exist.

The circularity that is alleged against any attempt to justify deduction, viz. to justify a whole system of deductive inference, is not of the usual kind. The validity of a particular form of inference is not a premiss for the semantic proof of its soundness; at worst, that form of inference is employed in the course of the proof. Now, clearly, a circularity of this form would be fatal if our task were to convince someone, who hesitates to accept inferences of this form, that it is in order to do so. But to conceive the problem of justification in this way is to misrepresent the position that we are in. Our problem is not to persuade anyone, not even ourselves, to employ deductive arguments: it is to find a satisfactory explanation of the role of such arguments in our use of language.

There is a second aspect to the problem of deduction that is specific to it, what Hintikka called the scandal of deduction (paraphrasing Hume's "scandal of induction"). The paradox, over which already Mill and Peirce puzzled, is that, on the one hand, deduction merely reveals what is already "contained" in the premises (as classically held by Locke and Kant) and hence produces no new information, but on the other hand, when mathematicians prove non-trivial theorems it seems like they do learn something new. The scandal of deduction is an active research topic in modern epistemic logic, see e.g. The Enduring Scandal of Deduction by Floridi and D'Agostino.

Answer 2

I typically present 'The Problem of Deduction' as a the following analogy to the better known 'Problem of Inductioon':

One of the workhorses of deduction surely is Modus Ponens ... but why do we trust Modus Ponens? Is there a proof for Modus Ponens?

Well, to demonstrate the validity of Modus Ponens, we typically do the following. We say that a truth-functional argument like Modus Ponens can be shown to be valid by putting it on a truth-table: if we find that there is no row where the premises of Modus Ponens are true and its conclusion is false, then Modus Ponens is valid. Well, when we put the Modus Ponens argument on a truth-table, we find that indeed there is no row where its premises are true and its conclusion is false. Hence, we conclude, Modus Ponens is valid ... Now, what deductive argument form did I just use to prove the validity of Modus Ponens?

Along the same lines, I think you may enjoy Lewis Caroll's story "What the tortoise said to Achilles"