I recently listened to a discussion during which an apologist advanced a particular argument about the problem of induction. It was being used as part of a dialectic in which an apologist was pinning a sceptic on the topic of induction. The claim being advanced was that inductive inferences are instances of the informal fallacy ‘begging the question’, and thus irrational. This was being said in an attempt to get the sceptic to back down from the claim that induction was justified.
However, the apologist’s claim was a mistake; it was a mistake to call inductive inferences instances of begging the question. Unwrapping the error is instructive in seeing how the argument ends up when repaired. I argue that the apologetic technique used here is unsuccessful, when taken to its logical conclusion.
Broadly speaking, the problem of induction is how to provide a general justification for inferences of the type:
All observed a’s are F.
Therefore, all a’s are F.
This sort of inference is not deductively valid; there are cases where the conclusion is false even though the premises are true. So, why do we think these are good arguments to use if they are deductively invalid? How do we justify using inductive inferences?
Usually, when we justify a claim, we either present some kind of deductive argument, or we provide some kind of evidential material. These are each provided because they raise the probability of the claim being true. So if I say that lead pipes are dangerous, I could either provide an argument (along the lines of ‘Ingesting lead is dangerous, lead pipes cause people to ingest lead, therefore lead pipes are dangerous’), or I could appeal to some evidence (such as the number of people who die of lead poisoning in houses with lead pipes), etc.
Given this framework, when we are attempting to justify the general use of inductive inferences, we can either provide a deductive justification (i.e. an argument) or an inductive justification (i.e. some evidence).
A deductive justification would be an argument which showed that inductive inference was in some sense reliable. But with any given inductive inference, the premises are always logically compatible with the negation of their conclusion. With any given inference, there is no a priori deductive argument which could ever show that the inference leads from true premises to true conclusion. You cannot tell just by thinking about it a priori that bread will nourish you or that water will drown you, etc. No inductive inference can be known a priori to be truth preserving. Thus, there can be no hope of a deductive justification for induction.
Let’s abandon trying to find a deductive justification. All that is left is an inductive justification. Any inductive inferences in support of inductive inference in general is bound to end up begging the question. Let’s go through the steps.
Imagine you are asked why it is that you think it is that inductive inferences are often rational things to make. You might want to reply that they are justified because they have worked in the past; after all, you might say, inductive inferences got human kind to the moon and back. The idea is that induction’s success is some evidential support for induction.
However, this is not so, and we should not be impressed by induction’s track record. In fact, it is a red herring, for suppose (even though it is an overly generous simplification) that every past instance of any inductive inference made by anyone ever went from true premises to a true conclusion, i.e. that induction had a perfectly truth-preserving track record. Even if the track record of induction was perfect like this, we would still not be able to appeal to this as a justification for my next inductive inference without begging the question. If we did, then we would be making an inductive inference from the set of all past inductions (which we suppose for the sake of argument to be perfectly truth-preserving) to the next future induction (and the claim that it is also truth-preserving). However, moving from the set of past inductive inferences to the next one is just the sort of thing we are trying to justify in the first place, i.e. an inductive inference. It is just a generalisation from a set of observed cases to unobserved cases. To assume that we can make this move is to assume that induction is justified already.
So if someone offers the (even perfect) past success of induction as justification for inductive inferences in general, then this person is assuming that it is justified to use induction when they make their argument. Yet, the justification of this sort of move is what the argument is supposed to be establishing. Thus, the person arguing in this way is assuming the truth of their conclusion in their argument, and this is to beg the question.
Thus, even in the most generous circumstances imaginable, where induction has a perfect track record, there can be no non-question begging inductive justification for future inductive inferences.
2. Does induction beg the question?
We have seen above that when trying to provide a justification for induction, there can be no deductive justification, and no non-question begging inductive justification. Does this mean that inductive inferences themselves beg the question? The answer to that question is quite clearly: no.
Inductive inferences are an instance of an informal fallacy, and that fallacy is called (not surprisingly): the fallacy of induction. The fallacy is in treating inductive arguments like deductive arguments. The irrationality that is being criticised by the fallacy of induction is the irrationality of supposing that because ‘All observed a‘s are F’ is true, this means that ‘All a‘s are F’ is true. Making that move is a fallacy.
Begging the question is when an argument is such that the truth of the conclusion is assumed in the premises. Inductive inferences do not assume the truth of the conclusion in the premises. For example, when you decide to get into a commercial plane and fly off on holiday somewhere, you are making an inductive inference. This is the inference from all the safe flights that have happened in the past, to the fact that this flight will be safe. The premise is that most flights in the past have been safe. Because (as an inductive argument) the premise is logically compatible with the falsity of its conclusion, the premise clearly does not assume that the next flight will be safe, and so the argument does not beg the question.
In fact, this actually shows that no argument can be both a) an inductive argument and b) guilty of the fallacy of begging the question. So technically, the claim apologists that inductive inferences beg the question is provably false.
Of course, if we tried to justify induction in general by pointing to the past success of induction, that would be begging the question. But to justify the claim that the next flight will be safe by pointing out the previous record of safe flights is not begging the question, it is just an inductive inference.
So the apologist who made the claim that induction begs the question is just wrong about that. He was getting confused by the fact that justifying induction inductively is begging the question. But when we keep the two things clear, it is obvious that inductive inferences themselves do not, and indeed cannot, beg the question.
3. But what if it did?
Induction does not beg the question. That much is pretty clear. But what would be the case if induction was guilty of some other fallacy? Well, if each inductive inference itself was an instance of, say, a fallacy like circular reasoning (like begging the question) then it would mean that people act irrationally when they make inductions, like deciding it is safe to fly on a plane. Yet, it seems like people are not irrational when they make decisions like this. Sure, there are irrational inductive inferences, like that from the fact that the last randomly selected card was red that the next card will be red. But not all inductive inferences are like this, such as the plane example. So the person who wants to claim that inductive inferences are circular has to say something which explains this distinction between the paradigmatic rational inference (like flying) and less rational (or irrational) inductive inferences. Saying that they are all circular would leave no room to distinguish between the good and bad inductive inferences.
So the apologist owes us something about how it is that we can make apparently irrational inductive inferences which seem otherwise perfectly rational. In response to this, they could make the radical move and reject inductive inferences altogether. This would mean that they have doubled down on the claim that induction is circular; ‘Yes, it is circular’, they will say, ‘throw the whole lot out!’.
Yet they are unlikely to make this move. Each day, everyone makes inductive inferences all the time. Every time you take a breath of air, or a drink of water, you are inductively inferring about what will result from the previous experiences you had about those activities. You are inductively inferring that water will quench your thirst because it has done so in the past. So if the apologist wants to reject induction altogether then he must not also rely on it like this, or else be hypocritical.
More likely than outright rejection, they will try to maintain that although induction is irrational in some sense, it can still be done rationally nonetheless. After all, there is a big difference between inferring that the next plane will land safely, or that the next glass of water will nourish, than that the next card will be red. The former are well supported by the evidence, whereas the latter is not. This is what allows us to distinguish between rational and irrational inductive inferences. Not all inductive inferences are on par; some have lots of good evidence backing them up, and some have none.
So, if the apologist wants to maintain that all inductive inferences are guilty of begging the question, then (assuming they don’t deny the rationality of all induction) they would still owe us an account of what makes the difference between a rational inductive inference and an irrational inductive inference. And the account would have to be something along the evidential lines I have just sketched above. How else does one figure out what inductive inferences are rational and which are not, if not by appeal to the evidence? If some new fruit were discovered, you would not want to be the first person to try it for fear of it being poisonous. But if you see 100 people eat of the fruit without dying, you would begin to feel confident that it wasn’t poisonous. This is perfectly rational. Thus, even if the apologist’s claim were correct, if they do not want to reject induction altogether, they end up in the same situation as the atheists, having to distinguish between good and bad inductive inferences based on the available evidence in support of them.
Even if the charge of irrationality stood (which it does not), it would have to be relegated to the status of not actually playing any role in distinguishing good inductive inferences from bad ones. This strongly discharges any of the real force of the point that was trying to be made.
The claim of the irrationality of induction was not true, but in a sense, it doesn’t make any material difference even if it is true; we still need to distinguish the better inductions from the worse ones.
4. Justifying induction with God
Some theists suggest that they have an answer to this problem which is not available to an atheist. The idea is that through his revelation to us, God has communicated that he will maintain the uniformity of nature. Given this metaphysical guarantee of uniformity, inductive inferences can be deductively justified. When we reason from the set of all observed a‘s being F to all a‘s being F, we are projecting a uniformity from the observed into the unobserved. Yet we were unable to justify making this projection. The theist’s answer is that God guarantees the projection.
We may initially suspect foul play here. After all, how do we know that God will keep his word? It does not seem to be a logical truth that because God has promised to do X, that he will do X. It is logically possible for anyone to promise something and not do it. Thus, it seems like we have just another inductive inference. We are saying that because God has always kept his promise up till now, he will continue to do so in the future. The best we can get out of this is an inductive justification for induction, which is just as question begging as the atheist version of appealing to the past success of induction. I think this objection is decisive. However, let’s suspend this objection for the time being. Even if somehow we could get around this, maybe by saying that it is a necessary truth that God will not break his promise or something, I say that even then we have an insurmountable problem.
5. Why that doesn’t help
The problem now is that while God may have plausibly promised to maintain uniformity of nature, he has not revealed to us precisely which inductive inferences are the right ones; i.e. the ones which are tracking the uniformity he maintains, as opposed to those which are not. God’s maintaining the uniformity of nature does not guarantee that inductive inferences are suddenly truth-preserving. Even if it were true, it did not stop the turkey making the unsuccessful inference that he would get fed tomorrow on Christmas eve, and it did not stop those people who boarded that plane which ended up crashing. Even if God has maintained uniformity of nature, and even if he has revealed that he has done so to us in such a way that we can be certain about it, we are still totally in the dark about which inductive inferences we can successfully make.
So let’s suppose we live in a world where God maintains the uniformity of nature, and that he has told us that he does so. When faced with a prospective inductive inference, and trying to decide whether it is more rational (like the plane ride) or irrational (like the card colour) to make the inference, what could we appeal to in order to help us make the distinction? We cannot appeal to God’s word, as nowhere in the bible is there a comprehensive list of potential inductive inferences which would be guaranteed to be successful if made (which would be tantamount to a full description of the laws of nature). Priests were not able to consult the bible to determine which inductive inferences to make when the plague was sweeping through medieval Europe. They continued to be unaware of what actions of theirs were risky (and would lead to death) and which ones were safe (and would lead to them surviving). The only way to make the distinction between good inductive inferences and less good ones is by looking at the evidence for them out there in the world. Knowing that God has guaranteed some regularity or other is no help if you don’t know which regularity he has guaranteed.
The problem is that we are unable to determine, based only on a limited sample size, whether any inductive generalisation we make is actually catching on to a uniformity of nature, or whether it was just latching on to a coincidence. When Europeans reasoned from the fact that all observed swans were white to the conclusion that all swans were white, they thought that they had discovered a uniformity of nature; namely the colour of swans. They didn’t know that in Australia there were black swans. And this sort of worry is going to be present in each and every inductive inference we can make, even if we postulate that we live in a world where God maintains the uniformity of nature and has revealed that to us. The problem is primarily epistemological; how can we know which inductive inference is truth-preserving? The apologist’s answer is metaphysical; God guarantees that some inductive inferences are truth-preserving (i.e. the ones which track his uniformities). For the apologist’s claim to be of any help, it would have to be God revealing to us not just that he will maintain the uniformity of nature, but which purported set of observations are generalisable (i.e. which ones connect to a genuine uniformity). Unless you know that God has made the whiteness of swans a uniformity of nature, you cannot know if your induction from all the observed cases to all cases is truth-preserving. And God does not reveal to us which inductive inferences are correct (otherwise Christians would be have a full theory of physics).
In short, even if we go all the way down the road laid out by the apologist, they still have all the same issues that atheists (or just people of any persuasion who disagree with the theist’s argument laid out here) do. They have no option but to use the very same evidential tools that atheists (etc) do to make the distinction between the more rational and less rational inductive inferences.
The apologist’s claim was that inductive inferences were question begging. I showed that this is not the case (and that in fact it could not be the case). Then I went on to see what would be at stake if the apologist had scored a point. We saw that still the apologist would need to distinguish better and worse inductive inferences, just like the atheist, and would have no other option but to use evidence to make this case. Then we looked at the idea that God guarantees that there would be some uniformity of nature. We saw that this claim does not make any material difference to the status of inductive inferences, and so cannot be seen to be a justification of induction in any real sense.