# Does the scientific method rest on the fallacy of affirming the consequent?

0. Introduction

There have been some rather strange suggestions from certain apologists recently about the nature of the scientific method, such as here and here. Prime among the criticisms is the claim that the scientific method rests on a fallacy called ‘affirming the consequent’. However, this is a strange claim for various reasons. Firstly, the criticism doesn’t engage with how philosophers of science actually talk about the scientific method. From around 1960, with the work of Thomas Kuhn, attempts at summing up the scientific method in a simple inferential procedure have been largely abandoned. It is now widely taken in the philosophy of science that there is no one simple pattern of reasoning that completely captures the scientific method – a phenomenon known as the ‘demarcation problem‘. So there is no simple logical model of inference which completely covers the everything in the scientific method. But this means that there is no simple model of fallacious inference which completely characterises the scientific method either. In short, the scientific method is too complex to be reduced to a simple informal fallacy.

However, if we pretend that this Kuhnian sea-change had not taken place, then we would most naturally associate the scientific method with the notion of inductive inference, with evidence being given in support of hypotheses (or theories).  However, induction is not guilty of the fallacy of affirming the consequent, as I shall show here.

After this wander through induction, I will try to explain what the motivations are for the apologetical critique and how that misses the mark by failing to appreciate that scientific advances are often made through falsification rather than verification.

1. Affirming the consequent

The fallacy of affirming the consequent is any argument of the following form:

1. If p, then q
2. q
3. Therefore, p

The inference from the premises to the conclusion is invalid, because it could be that the premises are true and the conclusion is false. For example, if p is false and q is true, then the premises are true and the conclusion is false. If you want a proof of this, let me know and I will provide it in the comments.

The reason it is a fallacy to use affirming the consequent is just that the argument is deductively invalid. The lesson is this: if you have a true conditional, then you cannot derive the truth-value of the antecedent from the truth of the consequent.

2. Science affirming the consequent

The idea that the scientific method commits the fallacy above can be explained very easily. We might think that theories makes predictions. This could be thought of like a conditional, where the theory is the antecedent and the prediction is the consequent; if the theory is true, then something else should be true as well. So, take a scientific hypothesis (such as ‘evolution is true’, or whatever), and a prediction that the theory makes (‘there will be bones of ancient creatures buried in the ground’, etc). Here we have the conditional:

If evolution is true, then there will be bones of ancient creatures in the ground.

Now we make a measurement, let’s say by digging in the ground to see if there are any bones there, and let’s say we find some bones. So the consequent of our conditional is true. The claim by the apologists is that when a scientist uses this measurement as support for the hypothesis, they are committing the fallacy of affirming the consequent, as follows:

1. If evolution is true, there will be bones of ancient creatures in the ground
2. There are bones of ancient creatures in the ground.
3. Therefore, evolution is true.

This is the sort of reasoning that is being alleged to be constitutive of the scientific method, and, as it is stated here, it is an example of affirming the consequent.

The problem with this line of thinking is not that it isn’t fallacious (it is clearly fallacious), but it’s that it is not what goes on in science.

3. Induction

In 1620, Francis Bacon published a work of philosophy called the ‘Novum Organon‘ (or ‘new tool’), in which he proposed a different type of methodology for science than the classical Aristotelian model that came before (Aristotle’s collected scientific and logical works had been collected together under the title ‘Organon‘). One way of characterising the Aristotelian method was that one does science by applying deductive syllogistic logic to ‘first principles’ (which are synthetic truths about the world). An example of this sort of first principle in Aristotelian physics might be that all things seek their natural place. It is of the nature of earth to seek to be down, and air to seek to be up, etc. This is, supposedly, why rocks fall to the ground, and why bubbles raise to the surface of water.

Part of Bacon’s dissatisfaction with this idea is that it provides no good way of discovering what the first principles themselves are; it just tells us what to do once you have them. Aristotle’s own ideas about how one discovers first principles are not entirely clear, but it seems that he thinks it is some kind of rational reflection on the nature of things which gets us this knowledge. Regardless, Bacon’s new method was intended to improve on just that, and is explicitly designed as a method for finding out what the features of the world actually are, of discovering these synthetic truths about the world. His precise version of it is a bit idiosyncratic, but essentially he advocated the method of induction.

Without going into the details of Bacon’s method, the idea is that he was making careful observations about the phenomenon he wanted to investigate, say the nature of heat, trying to find something that was common to all the examples of heat. After enough investigation the observation of a common element begins to be reasonably considered as not just a coincidence but as constitutive of the phenomenon under question. (He famously carried out just such an investigation into the nature of heat and concluded that it was ‘the expansive motion of parts’, which is actually pretty close to the modern understanding of it.)

In other words, starting with a limited number of observations of a trend, we move to the tentative conclusion that the trend is in fact indicative of a law. So the general pattern of reasoning would be that we move like this:

1. All observed a‘s are G
2. Therefore, all a‘s are G

The qualification of ‘all observed’ in premise 1 does most of the work in this argument. Obviously, just observing one a to be G would not count as much support for the conclusion. Technically, it would be ‘all observed’ a‘s, but it wouldn’t provide much reason to think that the conclusion is true. In order for the inductive inference to have any force, one must try to seek out a’s and carefully test them appropriately to see if they are always G’s. One must do an investigation.

So if we make a careful and concerted effort to investigate all a’s we can, and each a we come across happens to be G, then as the  cases increase, we will become increasingly confident that the next a will be G (because we are becoming increasingly confident that all a‘s are G). This is inductive inference.

With an inductive argument of this form, it has to be remembered that the conclusion does not follow from the premises with deductive certainty. Rather than establish the conclusion as a matter of logical consequence from the truth of the premises, an inductive argument makes a weaker claim; namely that the truth of the premises supports the truth of the conclusion; the truth of the premises provides a justification for thinking that the conclusion is true, but not a logically watertight one. Even the best inductive argument will always be one in which the truth of the premises is logically compatible with the falsity of the conclusion. The best one can hope for is that an inductive argument provides very strong support for its conclusion.

3. Induction affirming the consequent?

It is this inductive type of argument which the apologetical critique above is trying to address, it seems to me. They are saying that this type of scientific argument is really of the following form:

1. If all a‘s are G, then all observed a’s will be G               (If p, then q)
2. All observed a’s are G                                                         (q)
3. Therefore, all a‘s are G.                                                      (Therefore, p)

Notice that the 2nd premise and the conclusion (2 and 3) is precisely the inductive argument from above; we have just added an additional premise (1), the conditional premise, onto the inductive argument. This fundamentally changes the form of the argument. Now the argument has the form of the deductively invalid argument ‘affirming the consequent’.

There are three problems with this as a critique of scientific inferences. Firstly, we have added a premise to an already deductively invalid argument, and shown that the result is deductively invalid, which is kind of obvious. Secondly, it characterises scientific inferences as a type of deductive inference, when there is good reason for thinking that they are not (at least if scientific inferences are supposed to discover synthetic truths about the world). Lastly, the addition of the first premise seems patently irrational, and obviously a perversion of normal inductive arguments. Let’s expand on each of these three problems:

Firstly

All the apologetical critique has demonstrated is that one can make a fallacious deductive argument by adding premises to an inductive argument. However, inductive arguments are already deductively invalid. There is a fallacy called the inductive fallacy. It consists of taking an inductive inference to be deductively valid. So if you thought that all observed swans being white logically entailed that all swans are white, then you have committed the inductive fallacy, because you would have mistaken the relation between the premise and the conclusion to be one of deductive validity, when it is merely that of inferential support. All observed swans being white does provide some reason to think that they are all white, but the fallacy is in thinking that it alone is sufficient to establish with certainty that they are all white.

The addition of the first premise does nothing to undermine an inductive inference. It doesn’t make it more fallacious than it was in the first place. In a sense, this analysis commits the essence of the inductive fallacy, in that it says that scientific inferences are deductive when they are not; the claim that scientific inferences are guilty of affirming the consequent is itself an instance of the inductive fallacy.

Secondly

We could, if we wanted to, add premises to an inductive argument to make it deductively valid, as follows:

1. If all observed a’s are G,  then all a‘s are G        (If p, then q)
2. All observed a’s are G                                             (p)
3. Therefore, all a‘s are G.                                          (q)

Now the addition of the first premise has made the argument deductively valid, as it is just an instance of modus ponens.

The apologists were reconstructing scientific inferences as fallacious deductive arguments. Yet, even if we patched up the argument, as above with a deductively valid version of the inference, we still face a problem. This is that now we have a deductive argument, just like with Aristotle’s methodology. The very same reasons would remain for rejecting it, namely that as a methodology it provides no new synthetic truths; it only tells you what follows from purported first principles, not what the first principles are. We would be back to Aristotle’s dubious idea of introspecting to discover them. Thus, it isn’t desirable in principle to reconstruct an inductive argument as a deductive argument – even if the result is deductively valid. This means that the claim, that scientific reasoning is a failed attempt at being deductively valid, is implausible; even if scientific reasoning succeeded in being deductively valid that would be no help. The lesson is that they are a different type of inference, not to be judged by wether they are deductively valid or not.

Thirdly

Our original inductive argument went from the premise about what had been observed to what had not been observed. The whole point of inductive arguments is to expand our knowledge of the world, and so this movement from the observed to the unobserved is crucial. It is essentially of the form:

The observed a‘s are G ⇒ All a‘s are G

However, the first premise of the affirming the consequent reconstruction gets this direction of travel the wrong way round. They have it as:

All a‘s are G ⇒ the observed a‘s are G

If we keep clearly in mind that the objective of the scientific inference is to expand our knowledge, the idea of starting with the set (all a‘s) and moving to the subset (the observed a‘s) is weird. How could it expand our knowledge to do so? It is an inward move. This conditional though has been added to an inductive inference by our apologetical friends as a way of forming the ‘affirming the consequent’ fallacy out of an inductive inference.  But given that it gets the direction of travel exactly backwards, why on Earth would anyone ever accept this as a legitimate characterisation of a scientific pattern of reasoning?

This last concern highlights the cynicism inherent in the affirming the consequent critique. It isn’t a way of honestly critiquing a problem in science, but just an instance of gerrymandering an inductive inference, i.e. the change has been made just for the purposes of making the inference look bad, rather than as a way of highlighting a genuine issue. There is no independent reason for adding it on.

4. Or is it?

It might be claimed that I am pushing this objection too far. After all, there is reason to add the conditional premise on to the inductive inference. This is because theories make predictions. If a theory is true, then the world will have certain properties. And we do find examples of experiments being done in which the positive test result is used as a way of confirming the theory. And if this is right, then it looks like a conditional, and we are saying that the antecedent is true because the consequent is. So are we not back at the original motivation for the affirming the consequent critique?

Well, no. We are not. Here’s why. Let’s take an example. The textbook example. In Einstein’s general relativity, one of the many differences with classical Newtonian physics is that gravity curves spacetime. That means that there would be observable differences between the two theories. One such situation is when the light from a star which should be hidden behind the sun is bent round in such a way as to be visable from Earth:

We already knew enough about the positions of the stars to be able to predict where a given start would be on the Newtonian picture, and the details of the Einsteinian theory provided ways to calculate where the star would be on that model. So, the Newtonian model said the star would be in position X, and the Ensteinian theory said it would be in position Y.

These experiments were actually done, and the result was that the stars were measured to be where Einstein’s theory predicted, and not where Newton’s theory predicted.

Is this an example of the affirming the consequent fallacy? It might look like it. After all, it may well look like we were making this sort of argument:

1. If general relativity is correct, then the star will be at X  (if p, then q)
2. The star is at X                                                                            (q)
3. Therefore, general relativity is correct.                                (p)

However, the real development was not that general relativity was confirmed when these measurements were made, but that Newtonian physics was falsified. Corresponding to the above argument we have a different one:

1. If Newtonian physics is correct, then the star will be at Y (If p, then q)
2. The star is not at Y                                                                       (~q)
3. Therefore, Newtonian physics is not correct.                        (~p)

The first argument is a logically invalid deductive argument; it is affirming the consequent. But the second argument is just modus tollens (If p, then q; ~q; therefore, ~p), and that is deductively valid.

What we learned with the measurement of light bending round the sun was not that general relativity was true as such, but that Newtonian physics, and any theory relevantly similar to it, was false. General relativity may still be false, for all the experiment showed us, but it showed us that whatever theory it is that does correctly describe the physics of our universe is going to be more along the lines of general relativity than Newtonian physics. We learned something about the world, even if we did not confirm with complete certainty that relativity was true. And this is what scientific progress is like.

5. Conclusion

It would be affirming the consequent if someone thought that the positive measurement deductively entailed that general relativity was true. If any scientist has gone that far, then they are mistaken. It doesn’t mean that the scientific method itself is mistaken however.

## 4 thoughts on “Does the scientific method rest on the fallacy of affirming the consequent?”

1. Martin Plant says:

Nice clear explanation, thanks.
I think you’ve stated modus tollens incorrectly just before your conclusion.

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2. K says:

Thank you, Alex Malpass

Respects–KMc

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3. Juan Lebron says:

Alex, once again….very enjoyable read!

“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.”
– (Attributed to) Albert Einstein

I love that quote because it shows that the person (the discoverer* of the theory, Einstein) that could have been tempted the most to think “I FOUND TRUTH” didnt do that because he was perfectly aware of how science actually works. On the other hand, most people who didn’t discover it (that should therefore be more humble in their certainty about the theory) think of it as an immutable fact about nature.

I guess, that’s why I love this blog…because I think it is a wonderful (fairly digestible) way of educating people. It gives me hope.

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