I've previously written about presuppositionalist arguments regarding logic and knowledge. But another one of their arguments concerns the problem of induction. This is the logical problem posed by Hume of how to justify inferences from observed matters of fact to unobserved ones.
Briefly, the problem is this. Consider how you know or are justified in believing that if you (say) let go of a pencil in mid-air under normal circumstances, it will fall. Well, one thing you can point out is that in the past, unsupported pencils, as far as we know, have always fallen – and therefore, it is reasonable to suppose they will continue to do so. This is an inductive inference: from the fact that all observed A's have been B, one concludes that probably all A's – or at least that most A's – are B. However, the fact that pencils have always fallen is logically compatible with one not falling the next time the experiment is carried out, and even with none falling ever again. So how can one justify the inference?
In order to do so, one must maintain that nature is uniform in some way, so that future pencil drops will be like past ones. However, the problem is that the only evidence one has for such uniformity, it seems, is that that's what has been observed so far. In other words, the argument for the uniformity of nature itself appears to be based on an inductive inference. Induction, according to this argument, can therefore only be justified if one first assumes induction. And that, of course, is circular reasoning.
The presuppositionalists claim that atheists cannot justify their acceptance of induction – and that therefore they have no basis for believing in science, among other things. To believe in scientific reasoning on an atheist worldview, they say, one must engage in the above kind of circular reasoning. If on the other hand one starts out with the existence of a God who created a lawful world – a world that behaves in a uniform way – then one's belief in induction is justified. Therefore, only on the theistic worldview can science – as well as everyday inductive inferences – make sense.
This argument may not be quite as bad as the presuppositionalist one regarding deductive logic, but it is still a bad argument, as will now be demonstrated.
To begin with, merely claiming that the justifiability of induction presupposes some x is not so much to solve the problem as it is to ignore it. True, the presuppositionalist might argue that if induction is justified, then there must be a God. But that's not going to convince the induction skeptic. So even if the argument worked, it wouldn't be a solution to the problem.
But even worse, if the idea is that one has to accept whatever must be presupposed in order for inductive inference to be justified, then all we need to accept is the existence of a lawful world. The existence of a God who created such a world is superfluous. After all, there is no logical impossibility in the existence of a lawful yet uncreated universe. Thus, one might as well be a presuppositionalist, not with respect to God, but with respect to the uniformity of nature, and then proceed to argue in much the same way as above.
This is sufficient to show that the presuppositionalist argument is wrong. But so far nothing has been said as to why one is in fact justified in making inductive inferences. If a reasonable case can be made in favor of such inferences, it will show that not only is the presuppositionalist argument wrong, but in addition that the main premise it is based on is false.
I think a case logically justifying induction can be made. In fact, I think there are several ways to do this, but here I'll just present what is probably the simplest one.
Consider again the fact that every time a pencil (or any heavier-than-air object) has been let go in mid-air under normal circumstances it has fallen – that is, that all the observed instances of such an event have been like that. What is the best explanation of this fact? Certainly not that it is mere coincidence! It would be statistically impossible for that to be the case. It seems the best explanation is that there is some law of nature that makes such occurrences either naturally necessary or at the very least highly probable. But if that is the best explanation for what we've observed, then we are justified in believing that other instances will very probably be like the observed ones. For if there is an underlying law that explains why all observed pencils, etc., have behaved in this way, then we have good reasons for holding that other objects will do so as well.
Note that this argument does not depend on induction (in the narrow sense with which the problem of induction is concerned). Rather, it is an argument to the best explanation – an abductive argument. It is not that we have observed x many instances supporting uniformity in nature and inductively infer that there will probably be other such instances. It is that the best explanation of there having been that many instances displaying uniformity is that there is an underlying principle that causes such uniformity. Thus, there is a non-circular justification for accepting the existence of laws of nature. And it has nothing to do with God.
Now, presuppositionalists might object that the above argument does not guarantee the correctness of any inductive inference, for two reasons. First, the natural laws that explain the uniformity we've observed might only be probabilistic. They do not necessarily say that the next pencil I let go of will fall, but perhaps only that it is highly likely it will fall. Second, and worse, the existence of the laws themselves has not been proven with certainty, but again only with a high degree of probability. It is much more likely that the uniformity we've observed is a result of natural laws, but it is not strictly speaking impossible that it has all been mere coincidence.
All of that is true. However, that the uniformity we have observed has no explanation is so utterly unlikely that for all practical purposes it can be ruled out. So while it is the case that the argument, even if correct, does not strictly speaking guarantee the existence of laws, it makes the conclusion so probable that we can treat it as certain. And as to the laws themselves being probabilistic, observation shows that if they are (as the standard interpretation of quantum mechanics maintains), they result in events that at the level of everyday objects are so close to deterministic that once again it makes no practical difference. Thus, while it may be true (if the standard interpretation of QM is correct) that an unsupported medium-sized object near the surface of a planet might not fall towards it, the likelihood of such an event happening is so incredibly low that it almost certainly has never happened in the entire 14-billion-year history of our universe.
Now let's compare this with what presuppositionalists themselves can say based on their argument. They believe not only that there is a creator who has made a lawful universe, but that this creator is the God of the Bible – a work they interpret literally, since it is part of God's way of revealing things to us. But in that case, how confident can they be that an unsupported pencil will fall? It's clear that their confidence level should be far, far lower than that based on the above considerations. After all, their God is the author of the natural laws, and can therefore suspend them at any moment. Worse, according to the Bible, he has in fact done so on numerous occasions: among other things, he has turned water into wine, caused a man to walk on water, and stopped the earth's rotation without bringing about the worldwide catastrophe such an event should have caused – and all that in just the last few thousand years! It follows for this reason alone that the presuppositionalist should be considerably less confident than the atheist in making inductive inferences.
But in fact, the presuppositionalist's position is even weaker than that. For, as Michael Martin pointed out, if God can have a morally sufficient reason for allowing natural evils such as cancer and earthquakes, then he could very well have a morally sufficient reason for, say, making it the case that from this moment on pencils will no longer fall – or that they will behave in an entirely random manner. It follows that belief in the biblical God should actually create doubts regarding the justifiability of induction.
For Michael Martin's paper on this topic, see: "Does Induction Presume the Existence of the Christian God?"
Another online paper on this is Alex Malpass's "Induction, God and Begging the Question"