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Timothy McGrew's solution to the problem of induction

Updated: May 24

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 (e²(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:

  1. 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].

  2. S is an n-fold sample of this population. [Image we toss a coin n times].

  3. S matches the population. [Assume the coin tosses we made are representative of a common coin toss with no outside factors influencing us].

  4. S has a proportion of 0.5 heads.

  5. The proportion of 0.5 heads lies in the interval [0.5-e, 0.5+e].

  6. x is a sample of the population. [I.e. an individual coin toss].

  7. 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 himself grants that we have experience of bread nourishing us and of the sun’s rising. If we may take our experience to be a sample, then it appears that we possess all the tools necessary to make a rational defence of everyday extrapolations against Humean scepticism." [13]


To be fair, he goes on to mention that philosophical battles are not so easily won and offers a few refutations to possible objections. However, I don't believe McGrew successfully answered Hume at all for the reason we indicated above: He does not escape the problem of brute facts.

Hume responds


Were Hume alive today, I believe that his response to McGrew would have been short and cogent.


"Your final paragraph, Dr McGrew, exposes the flaw in your reasoning. It is indeed the case that we experienced the nourishment of bread and the sun's rising day after day, but it remains to be shown why you would expect this to continue in the future. Brute facts don't form a population from which you can sample. A string of brute facts cannot produce a nicely contained population from which to extrapolate, to begin with. Chance must reign supreme.


One way out is to postulate causes that keep chance in check (so that we can say with confidence that bread will probably nourish us or that the sun will probably rise), but in doing so you'll simply beg the question. A universal principle of cause and effect that undergirds this sort of induction is the very thing you're supposed to prove. It seems you've tacitly presupposed it when offering your argument. You can't calculate probability against a backdrop of pure chance, Dr McGrew.


This much also becomes clear when I read the conclusion of your paper where you yourself admit that 'the universe can play some trick on us'. Your response to this admission is to quote Herodotus, the ancient Greek historian:


'There is nothing more profitable for a man than to take counsel with himself; for even if the event turns out contrary to one’s hope, still one’s decision was right, even though fortune has made it of no effect: whereas if a man acts contrary to good counsel, although by luck he gets what he had no right to expect, his decision was not any the less foolish'.


I see this as an admission of defeat. With this quote, you admit that you cannot solve my problem of induction. There is no, and can never be, necessary or logical connection between brute facts. It's all in the hands of fortune (or chance). There are no rational grounds to expect the future to resemble the past."


The author of nature


Now we've reached the climax of our journey. Can Hume not be refuted? How can the vast majority of philosophers believe Hume's problem to be insoluble? Our entire existence depends on induction. When we boil a kettle or enjoy a sunrise, it seems so trivial to us. Surely Hume missed something obvious.


As a Christian, I do believe that McGrew is a lot closer to the truth than Hume. When you boil a kettle and expect the water to become hot, and when you drive out earlier in one of the national parks to enjoy a sunset, I fully agree with the common-sense principle that we are justified in expecting the future to resemble the past with reasonable confidence. I use the law of large numbers in my everyday job as an actuarial analyst at the company I work for. I don't believe Hume was correct at all. Nor do I believe that Hume lived in a way that is consistent with the logical conclusions he reached.


However, Hume was more consistent than McGrew, and everyone that doesn't buy his arguments and yet proclaims to live by the principles of brute facts and chance. Given Hume's non-Christian starting point, he was correct.

So, what is the correct answer to Hume, then?


The matter may be put as follows: If an “Author of nature” is really presupposed it will and must in principle control the nature of reasoning that we employ. If we presuppose an “Author of nature,” then the facts are created by him. That means we cannot be empiricists, in the sense in which Hume takes empiricism. If an “Author of nature” is presupposed, all the facts of the “course and constitution of nature” are bound together by the mind of God. There are no brute facts. All facts are God-created and God-interpreted facts.


The same can be said for knowledge of the a priori sort. Our minds are also created by the "Author of nature". So we can never be a priorists of the Cartesian sense (that of Descartes), but this is an article for another time.


Future possibilities lie solely in the hands of God. If we start with brute facts, we've effectively negated the author of nature (begged the question) and his revelation from the start of our reasoning process. For the Christian, there is an entirely reasonable expectation that the constitution and course of nature will be the same in the future as it has been in the past because of the rationality of God that is back of it. There is no principle of chance that can subvert the will of God. The point is that only that will happen in the future which is in accord with the plan of God for history.

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