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I've been reading about the problem of induction and I have trouble understanding the argument for nature's uniformity being impossible to prove deductively.

Stanford Encyclopedia of Philosophy on the principle of nature's uniformity:

The principle can't be proved deductively, for whatever can be proved deductively is a necessary truth, and the principle is not necessary; its antecedent is consistent with the denial of its consequent.

Although I intuitively agree with the conclusion that this principle cannot be proved deductively, the explanation seems kind of lacking. Why can't it be a necessary a truth ?

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    They are using a more stringent criterion for necessity than is the common use. For the standard logical notion of necessity, something is necessary if it is true in every possible world. It is clearly possible to contrive a world in which one part of the physical world obeys different rules than another, because in modes other than the natural, e.g. in morality, we see this happen all the time. – user9166 Oct 20 '17 at 22:41
  • A thought. The conformity of Nature to laws in the local world we know may not extend to all of existence but this in itself would not mean it is not uniform. It may be uniform in a deeper way, So I'd say the Nature is bound to be uniform in that it must obey the laws of Nature at all times, and if the local laws vary this would not mean that Nature is not uniform. Does this thinking stand up? – PeterJ Nov 23 '17 at 12:43
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The quoted passage is part of an exposition of Hume's original argument. One of the previous paragraphs explains what "deductively" meant to Hume:

"The deductive system that Hume had at hand was just the weak and complex theory of ideas in force at the time, augmented by syllogistic logic. His ‘demonstrations’ rather than structured deductions are often founded on the principle that conceivable connections are possible, inconceivable connections impossible, and necessary connections are those the denials of which are impossible or inconceivable. That said, and though we should today allow contingent connections that are neither probabilistic nor causal, there are few points at which the distinction is not clear".

Hume certainly did not have in mind something like analytic necessity of Carnap (although it might be the closest modern analog) or possible worlds necessity of Kripke et al. Non-uniform nature is certainly "conceivable", so by Hume's lights the uniformity (and hence induction) can not be necessary.

Of course, syllogistic is a pretty weak system of logic, by which not much can be deduced even in elementary mathematics, and conceivability and uniformity are pretty vague notions. So is the notion of possibility, one can certainly come up with collections of possible worlds that contain non-"uniform" worlds, as well as those that do not contain them because, arguably, uniformity is required for intelligent life to evolve (anthropic reasoning), say. But there is a bigger problem with deducing "uniformity" under any notion of deduction or necessity. Namely, the "uniformity" itself can not be too uniform because induction does not apply universally, and it is unclear which aspects are supposed to be "uniform", and which are not. The entire concept dissolves when one attempts to state it more precisely, and it is unclear what it even means for it to be "necessary", see What are the critiques of the “we might as well assume it” solution to the problem of induction? And of course something so vague can hardly be deducible on any view of deduction.

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