It is a well-known fact that David Hume was the philosopher who presented the "problem of induction". The problem of induction refers to the impossibility of making inferences with regard to the future based on past experience in a world governed by Chance (the absence of cause).

In this article, I will summarise McGrew's solution, and critique it through the metaphorical pen of David Hume.

## The problem of brute facts (problem of induction introduced)

Van Til writes,

The whole point of Hume’s argument is that there is no rational presumption of any sort about future events happening in one way rather than in another. We may expect that they will, but if we do, we do so on non-rational grounds.

Our reasoning is based upon past experience. Past experience is nothing but an accumulation of brute facts which have been observed as happening in a certain order. Why should not the events of the future be entirely different in nature from the events of the past?

**Cornelius Van Til, ***Christian-Theistic Evidences***, 21.**

For Van Til, a brute fact is a fact with no pre-determined meaning. It is a fact divorced from a system. Brute facts are "discrete facts". So, if past experience is nothing but the accumulation of these brute facts, there are no rational grounds for believing the brute facts (which by their very nature exclude the idea of the system) will continue to act in the same way in the future.

## Timothy McGrew responds to Hume

Timothy McGrew is a professor of philosophy at * Western Michigan University* and the chair of the department of philosophy. He is considered a specialist in the philosophical applications of probability theory. McGrew is a Christian and has concerned himself with the rational defensibility of the Christian religion. We include McGrew in our discussion not because he managed to successfully refute Hume but to indicate that McGrew, although he believes to have answered Hume, still relies on brute facts to make his argument. His attempted answer will serve to further our purposes as we critique it.

Notable for our purposes then, is that McGrew wrote a paper in 2001 titled, * Direct Inference and the Problem of Induction* in which he sought to refute Hume's contention outlined above. If Hume's contention proves to be correct, then there would be no rational grounds for any belief let alone Christianity. So, if there is no answer forthcoming it would be concerning, to say the least, for the insurance industry (and the world in general). So, can McGrew save Butler's arguments (and insurers) by refuting Hume?

McGrew starts off his paper by noting that it would be difficult to overestimate the influence of Hume's problem of induction (making inferences with regard to the future based on past experience). He mentions that there exists the conviction in a considerable amount of modern philosophers that Hume's problem is insoluble. Despite the above, he aims to show that this pessimism is unfounded and to refute Humean scepticism on a theoretical, practical and modern level with regard to induction.

McGrew intended to accomplish this by using Bernoulli's law of large numbers but in reverse.

Basically, McGrew is seeking **the probability that the frequency with which a feature ***X*** occurs in a population** lies within a small interval, *e*, of the value *m/n, *that is* (m/n - e, m/n + e)*, given that an n-fold sample exhibits *X* with frequency *m/n* (where *m* is the number of members of the sample exhibiting *X*).

Or, more simply, you want to know what the split in proportion is between heads and tails on a coin toss. Imagine you sample 100 coin tosses (that is *n* = 100), and you observe that 50 of them are heads (that is *m* = 50). From this, we can determine that *m/n = *50/100 = 50%. Now, McGrew is asking, what is the probability the *actual* proportion (*p*) of heads lies in an interval *(m/n - e, m/n + e)*, where *e* is a sufficiently small number and* m/n = *50% (which is what we observed in our sample)?

According to McGrew, the main challenge is ensuring we have a sufficiently large sample size to accurately estimate the true proportion. Specifically, we want to make the inference that the actual proportion, *p*, lies within the range *m/n + e*, and *m/n - e*. This can be expressed with the following formula:

*| p - m/n | <= e*

The __central limit theorem__ supports this by stating that if the sample size is large enough, the sampling distribution of the sample mean will approximate a normal distribution. This allows us to use statistical methods to estimate the true population proportion with a given level of confidence.

The formula which McGrew uses to determine the proper sample size (*n*) is
*n >= 0.25 (e**²**(1-α))**ˉ** ¹ *(see Debora Mayo, *Error and the Growth of Knowledge*)

Being satisfied that his sample size is big enough to afford confidence that his sample is representative of the population, McGrew presents his argument:

For any property

*p*, at least*a%*of*n*-fold samples exhibit a proportion that matches the population. [This step merely means that given the above formula we glossed over, McGrew can be*a%*confident that his sample represents the population].*S*is an*n*-fold sample of this population. [Image we toss a coin n times].*S*matches the population. [Assume the coin tosses we made are representative of a common coin toss with no outside factors influencing us].*S*has a proportion of 0.5 heads.The proportion of 0.5 heads lies in the interval

*[0.5-e, 0.5+e].**x*is a sample of the population. [I.e. an individual coin toss].With probability

*[0.5-e, 0.5+e], x*is heads*.*

As McGrew indicated, he simply used the law of large numbers and a few direct inferences to make the above work. Hence, he believes to have solved the problem of induction. He indicates: "this solution to the problem of induction is of more than academic interest. *Prima facie*, is a cogent response to Hume’s challenge. **Hume**