[From a short article written for the ILIAD Forum.]
The problem of induction is a notorious philosophical problem concerning inductive inferences; more specifically, whether that form of reasoning is generally reliable or rationally justified. An inductive inference aims to draw a general conclusion from a series of particular observations. For example, if I observe one thousand swans, and every one of those swans is white, I can infer inductively that probably all swans are white, and on that basis predict that any future swans I observe will (probably) be white. Unlike deductive inferences, in which the conclusion follows necessarily from the premises, inductive inferences cannot deliver absolute certainty—for example, the possibility of observing a non-white swan in the future cannot be decisively ruled out—but all else being equal, the greater the number of past observations confirming a general law or pattern, the stronger the inductive conclusion becomes.
Inductive inferences have been widely used in scientific research to discover laws of nature. To take one example, Newton’s universal law of gravitation was inferred inductively from empirical observations of the attractive forces between two masses. We haven’t observed the forces between every pair of masses in the universe at every point in time, of course, so we don’t have direct and infallible knowledge of a universal law. Nevertheless, we have made enough observations to be confident that they are instances of a universal law, and we can make reliable predictions about future events by positing that the universal law holds.
Hey, Dr Anderson.
I recognise, of course, that this is meant to be a concise article with undergraduate students as its main audience. But since the topic has been broached, I want to say that my worry is that the problem of induction is not quite as simple as presuppositional apologists routinely present it.
The assumption is that the problem of induction can be successfully resolved if the uniformity of nature is shown to be grounded in some further metaphysical fact (i.e., God’s providence). But providing a reasonable justification for the uniformity of nature doesn’t cut it. One of the most significant papers on the problem in recent literature was authored by J.D. Norton, titled: “A Demonstration of the Incompleteness of Calculi of Inductive Inference” (British Journal for the Philosophy of Science, 2018). Norton formally proves that no theory of probability will ever furnish a calculus that really justifies inductive inferences. Even if we freely grant the uniformity of nature, we will still crash into problems and paradoxes (eg., Nelson Goodman’s new riddle of induction, Carl Hempel’s raven paradox, or even Alan Chalmers’ arguments about unobservable entities like protons not being amenable to inductive reasoning).
Let’s focus on one. For those unaware, Marc Lange summarises Goodman’s paradox in this way:
“Hume’s argument, then, turns on the thought that every inductive argument is based on the same presupposition: that unobserved cases are similar to the cases that we have already observed. However, Nelson Goodman [1954] famously showed that such a ‘principle of the uniformity of nature’ is empty. No matter what the unobserved cases turn out to be like, there is a respect in which they are similar to the cases that we have already observed. Therefore, the ‘principle of the uniformity of nature’ (even if we are entitled to it) is not sufficient to justify making one prediction rather than another on the basis of our observations. Different possible futures would continue different past regularities, but any possible future would continue some past regularity. [Sober, 1988, pp. 63—69]
“For example, Goodman says, suppose we have examined many emeralds and found each of them at the time of examination to be green. Then each of them was also “grue” at that time, where: object x is grue at time t iff x is green at t where t is earlier than the year 3000 or x is blue at t where t is during or after the year 3000.
“Every emerald that we have found to be green at a certain moment we have also found to be grue at that moment. So if emeralds after 3000 are similar to examined emeralds in their grueness, then they will be blue, whereas if emeralds after 3000 are similar to examined emeralds in their greenness, then they will be green. Obviously, the point generalizes: no matter what the color(s) of emeralds after 3000, there will be a respect in which they are like the emeralds that we have already examined. The principle of the uniformity of nature is satisfied no matter how “disorderly” the world turns out to be, since there is inevitably some respect in which it is uniform. So the principle of the uniformity of nature is necessarily true; it is knowable a priori. The trouble is that it purchases its necessity by being empty.
“Thus, we can justify believing in the principle of the uniformity of nature. But this is not enough to justify induction. Indeed, by applying the “principle of the uniformity of nature” indiscriminately (both to the green hypothesis and to the grue hypothesis), we make inconsistent predictions regarding emeralds after 3000. So to justify induction, we must justify expecting certain sorts of past uniformities rather than others to continue.”
— Lange, “Hume and the Problem of Induction”, in Inductive Logic (2011): 58-59.
In other words, to resolve the problem of induction, the (otherwise vacuous) uniformity of nature must be supplied alongside much more specific assumptions regarding uniformity to provide a justification for our inductive inferences. In this case, we need to be able to distinguish lawlike from non-lawlike generalisations. An appeal to God’s providence, so far as I’m concerned, is not enough.
Hi Chris,
Thanks for the comments. I appreciate the feedback. Several things to say:
1. Yes, it was meant to be a concise article for an undergrad audience. So I have that excuse. :)
2. That said, it’s clear from the article that I’m focusing on the original, Humean problem of induction. Sure, there are other problems of induction (Goodman’s “new riddle,” etc.). Even if, for the sake of argument, Christian theism doesn’t furnish a ready answer to those further problems, that doesn’t really affect the point I’m making in the original article. Put it this way: even if the doctrine of divine providence isn’t sufficient to establish the rationality of inductive inferences, it is still (so I argue) necessary. And that’s enough for apologetical purposes.
3. To be clear, I didn’t appeal merely to divine providence, but to divine providence and a divine design plan for human cognitive faculties. The latter is just as important as the former for solving the problem.
4. Although I’ll have to keep this short, I think we can make some headway on Goodman’s paradox too. Richard Swinburne has argued that we can appeal to simplicity to justify the “green” inference over the “grue” inference; specifically, to the relative conceptual simplicity of the predicates involved. (See his discussion on pp. 88-89 of Epistemic Justification.) Now, I’m not an internalist like Swinburne, so I would want to couple this with a Plantingian proper-function account of rationality, according to which God has designed our cognitive faculties such that we favor the simpler (or perhaps we should say, more natural) predicates when it comes to inductive inferences. This isn’t an arbitrary preference, but rather a reliable, truth-directed preference that is underwritten by divine creation and providence, implemented through the cognitive design plan. (Plantinga talks about this in a few places including W&PF and WTCRL.)
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