The "problem of induction"
Much is made by presuppositionalists of the "problem of induction". Materialists, it is said, cannot justify their use of induction. They cannot explain why we should expect that the Sun will "rise" tomorrow, for instance, even if it rose a million times before (I'm not going to keep putting quotation marks around "rose"- we all know it's a figure of speech). Only divine will, they say, can solve such problems.
Induction is nothing more than the act of deriving principles from specific instances. People in the past observed that the Sun always rose, and thus were justified in thinking it would keep doing so. Now that we understand how orbits work and the nature of the Sun, we have a much stronger basis for such inductive reasoning. Nevertheless, the basic principle is the same. The accumulation of scientific understanding does not change the fact of induction, but rather makes our inductive reasoning more accurate.
If we're going to examine this claim that we have a "problem of induction", we must first examine what is needed for induction to be valid, or any epistemic principle for that matter. Well, the first thing is that to discover principles or applications or instances of principles, those principles need to already exist. The name we give to these principles is "natural law". Natural law is the foundation of epistemology, science, and morality.
So what do we need to have natural law, and by extension, justify induction? We need:
1. A causal system. Things change in accordance with their nature.
2. A self-contained system. No transcendental agent can exist, and materialism must obtain.
These points may seem complicated, but they are in fact pretty simple : you need a universe where things work by cause and effect, and where there is nothing outside that universe that can change things without cause and effect. If you have these two things, you have natural law, because everything works by cause and effect and things that have a certain property will behave in a certain way.
The simplest example of a natural law is the law of gravity - anything that is material follows this law, because it is a causal fact about the basic property of "having mass". "Having mass" is expressed in having a gravitational field. Thus causality ensures that, for example, orbits obtain.
If we don't have a causal system, then there can be no principles. There would be no correlation between "having mass" and gravity, and therefore we would have no reason to expect orbits to obtain. In fact, we would have no reason for anything to obtain.
An even worse problem is that our sensory perception is in itself a causal chain. I can only "see" an object because photons are reflected off its surface and hit my photo-receptors. If causality does not obtain, then all of this falls apart, and sensory perception is no longer a valid cognitive tool. So eliminating point 1 makes all cognition an impossibility (which is itself a contradiction). Fortunately, we live in a causal universe!
What about point 2? If there exists a transcendent being - a being which can change material facts without using causality - then there is no more natural law. I'm not talking about a god here but any transcendent entity, say, an angel or a ghost.
Suppose an angel can change any material facts by magic. In this scenario, it could make it so that the Sun does not rise tomorrow. So now our inductive reasoning is confronted with an impenetrable barrier, and we would now live in a fantasy world. And in the fantasy world of the Bible, the Sun can stand still (Joshua 10:12-13)! A person, living in this fantasy world, who would make the inductive reasoning that the Sun would rise, would have been wrong in this instance. But what recourse would he have? No fact could possibly explain such an event. So all the consequences of the actions of this transcendent agent would throw up inductive reasoning, and we would have no reason to trust induction as a valid principle.
I can see an objection coming: what if a supernatural entity acts but is not transcendent? But this is plainly impossible. If this entity acts within cause and effect, then its effects are observable and measurable, making the entity material by definition.
Now, what ideologies deny 1 or 2? Certainly not ours. While atheism does not directly imply 1 or 2, they are perfectly compatible with a rational atheistic worldview. My naturalistic worldview demands that 1 and 2 both be true, for reasons I explained before.
If God is presupposed, however, 2 must be false! As I noted, a supernatural agent must necessarily be transcendent. Since God is a supernatural agent, it must be a transcendent agent. And if this is the case, then induction becomes invalid, and now we have a "problem of induction". But it belongs squarely to the Christian worldview!
But God is more than a transcendent agent- it is a sovereign agent. The existence of the universe and all material facts within it, is wholly subjective to God's will. Since God is Creator of everything, there can be no objective factors existing outside of God's will. And there is no doubt that in this scenario no knowledge is possible, including knowledge about God. This has the unfortunate (for Christians) consequence that the existence of God-believers is contradictory.
This can be easily seen in the fact that no believer, having accepted the existence of this transcendent agent, can reject the possibility that this agent is deluding him into believing what he believes. Once you open the door of subjectivity by allowing the existence of a transcendent Creator, everything goes, epistemically speaking.