Many people talk as if all you need to do in order to refute a position is simply to deny it or say anything at all in reply to it. Such talk is too loose. Monty Python taught that ‘argument is not just contradiction’ or denial. Even if you go beyond denial and say something in reply, not every response is a refutation.

For example, suppose a theist argues, ‘God exists, because nothing else could explain the existence of the Universe.’ An atheist cannot refute that argument simply by saying ‘No, God does not exist’ or ‘I do not believe in God’ or ‘That’s stupid.’ The same goes for the other side. If an atheist argues, ‘Evil exists, so God does not’, a theist cannot refute that argument simply by saying ‘God does exist’ or ‘I believe in God’ or ‘That’s silly.’ These simple responses are not refutations.

In order to refute an argument, you need to give an adequate reason to doubt that argument. We saw that some arguments give reasons that justify belief in their conclusions, whereas other arguments give reasons that explain phenomena. In contrast, refutations give reasons to doubt other arguments. Thus refutation is a new purpose of arguments in addition to justification and explanation.

The reasons supplied by refutations are reasons to doubt rather than reasons to believe. In order to refute a theist’s argument that God exists, atheists do not have to show that God does not exist. All atheists need is an adequate reason to doubt that the theist’s argument provides enough reason to believe that God does exist. Similarly, theists can refute an atheist’s argument against God’s existence without giving any reason to believe that God does exist. All the theist needs is an adequate reason to doubt that the atheist’s argument shows that God does not exist. Refutation can lead to doubt and suspension of belief in both directions.

Many people who refute arguments do go on to deny those arguments’ conclusions. That additional move results in part from the discomfort of admitting, ‘I don’t know.’ Many atheists who refute arguments for God’s existence conclude that God does not exist, partly because they do not want to end up as a wishy-washy agnostic. For similar reasons, many theists who refute arguments against God’s existence jump to the conclusion that God exists. That additional claim does not, however, follow from the refutation alone. All that the refutation by itself supports is doubt, not belief.

What does it mean to doubt an argument? It means simply to doubt that the argument gives enough reason to believe its conclusion. This doubt can be directed at different parts of the argument. According to our definition of arguments, an argument includes premises and a conclusion and presents the premises as a reason for the conclusion, so a refutation has three main targets to aim at. First, refutations can give reasons to doubt one or more premises. Second, refutations can give reasons to doubt the conclusion. Third, refutations can give reasons to doubt that the premises provide adequate support for the conclusion. We will survey these forms of refutation in turn.

Does the exception prove the rule?

The first way to refute an argument is to cast doubt on its premises. This task can be accomplished either by giving some reason to believe that the premise is not true or by finding some fallacy in the strongest argument for that premise. We will focus here on one common method of refuting premises, namely providing counter-examples.

Suppose that a business owner argues, ‘Higher taxes always reduce employment, so we need to keep taxes low.’ One way to raise doubts about this argument is to give a reason to doubt or deny its premise that higher taxes always reduce employment. That’s easy. Just point to one time when taxes went up to a high level without employment going down. That one counter-example is enough to show that higher taxes do not always reduce employment.

But is this refutation strong? Not if the opponent has an easy reply. To respond, all the arguer needs is a guarding term: ‘Fine, so high taxes do not always reduce employment. Still, they usually do – almost always.’ A single counter-example cannot raise doubts about this guarded premise. The arguer can claim that this counter-example is the exception that proves the rule in the sense that its exceptional features show that the rule holds in normal cases (rather than in the original sense of this slogan, which was that the exception tests the rule).

That response is not the end of the discussion, however. As soon as the arguer admits an exception, it raises the question of whether the case under discussion is more like the rule or more like the exception. If we are trying to determine whether ‘we need to keep taxes low’ (as the conclusion claims), then we need to figure out whether our current circumstances are more like the exceptional period when taxes go up and employment does not go down, or more like the usual periods when taxes go up and employment does go down. It is not enough to give a single counter-example and then stop thinking. That further issue will not be easy to settle, but it should not be ignored.

The same goes for every counter-example. Many religious and cultural traditions espouse something like the golden rule: ‘Do unto others as you would have them do unto you’ (Matthew 7:12). It is easy to think up counter-examples to this esteemed principle. It is not wrong for judges to sentence murderers to prison, even though the judges would want not to be sentenced to prison themselves. It is not right for sadomasochists to whip their victims, even if they would like to be whipped themselves.

Examples like these raise doubts about the golden rule, but how could its defenders respond? The obvious point about sadomasochists is that their (non-masochistic) victims do not consent to being whipped, whereas sadomasochists would like being whipped only in ways and at times to which they consent. Thus the golden rule still holds if we apply it only to the act of whipping without consent. Nobody likes to be the victim of that.

In the other counter-example, the judge would not like to be sentenced to prison even if she deserved it because she was guilty of a crime. However, the judge would presumably admit that punishing her would be fair in those circumstances. If so, then we can avoid this counter-example by reformulating the golden rule like this: ‘Do to others as it would be fair for them to do to you.’ What is wrong is then determined by what is fair instead of what you happen to like. The problem is that this reformulation of the golden rule cannot be applied to cases without determining in advance what is fair in those cases. That makes it hard to see how this rule could function as a basic principle of morality.

When a counter-example casts doubt on a premise that an argument depends on, the counter-example raises doubts about whether the argument provides adequate reason for its conclusion. After all, if the premise is false, the argument fails. That is how counter-examples to premises can refute arguments. Nonetheless, the conclusion could still be true. Moreover, the argument still might succeed if it can be reformulated in a way that avoids the counter-example and still provides a strong enough reason for the conclusion. Thus, this form of refutation, like all others, is inconclusive. It moves the discussion forward instead of ending it.

Is this absurdity made of straw?

The second way to refute an argument is to cast doubt on its conclusion. If a refutation shows that a conclusion is false, then there must be something wrong with the argument for that conclusion. At least it cannot be sound. This kind of refutation might not reveal specifically what is wrong with the argument, but it can still show that something went wrong somewhere in the argument. We know that we took a wrong turn somewhere if we end up in a ditch.

The strongest refutations of this flavour are reductio ad absurdum – they reduce the conclusion to absurdity. The clearest absurdities are outright contradictions. If someone gives reasons to believe that China has the largest number of citizens, an opponent could reply: ‘That’s absurd. Just wait a minute, and it will have more. If China had one more citizen, then it would have an even larger number of citizens, so the number that it used to have cannot be the largest number.’ It is contradictory to claim that any number is the largest number.

This reductio ad absurdum obviously rests on a misinterpretation. What the arguer meant was not that the number of citizens in China is the largest of all numbers, but only that China has a larger number of citizens than any other country. When a refutation misinterprets a claim in order to make it look absurd (although it is not really absurd when interpreted correctly), the argument attacks a straw man or a straw person. The best response to this trick is simply, ‘That’s not what I meant.’

Real cases are usually subtler. In June 2017, a member of the Israeli parliament pushed for a bill that would have required all professors to give equal time to any position that any student wanted to be discussed. The goal was to enable conservative students to require their liberal professors to consider the conservative side of controversial issues so that students would not be brainwashed towards liberalism. That goal might seem reasonable, but the law would quickly lead to absurdity.

Just imagine a course on neuroscience, whose professor emphasizes the role of the hippocampus in memory. One student says that memory might instead be lodged in the temporal pole. Another suggests that it could be the cingulated cortex. A third suggests the striatum. And so on for every part of the brain. The proposed law requires the professor to give equal time to all of these possibilities. That would be absurd for two reasons. First, there is little evidence linking memory to those other parts of the brain, so what is the professor supposed to discuss? Second, it would take every minute of every class to discuss all of these possibilities, so the course could never proceed to other topics in neuroscience. These absurdities can be cited to refute anyone who argues, ‘Every student opinion deserves equal consideration, so professors should give equal time to any position that any student wants to discuss.’

Does this refutation attack a straw man? That is not clear. On the one hand, the proponents of the law were probably thinking of positions in politics rather than neuroscience. If so, these advocates might be able to avoid absurdity by restricting the law to political issues somehow. On the other hand, it is not always clear which issues are political, so proponents of the law might have meant to include debates about politically controversial positions in history and science, such as global warming, the origins of life and the Earth, the efficacy of torture, the causes of certain wars and so on. If the law covered all of these issues as well, then any student could stop professors from discussing any of them simply by advocating an endless number of alternative views with nothing to recommend them (except the student’s desire to avoid an impending test). That threat shows that the law would effectively prevent professors from discussing any topic within its scope. Is that absurd? I think so, but maybe that’s just because I am a professor. If that result is what proponents of the law want, then they might not see it as absurd.

One lesson from this example is that absurdity is sometimes in the eye of the beholder. Not so in the case of outright contradiction, but often in real cases. Does that mean that reductios cannot refute any real arguments? No, but it does reveal that those refutations will work only for limited audiences. This refutation cannot work against extremists who hold that professors should not be able to discuss any controversial issues. Nonetheless, it can still work for moderates who think that professors should be able to discuss the main alternative positions on a controversial issue without spending equal time on every possibility that any student might like to bring up for whatever reason. This case reinforces my earlier point that arguments will never satisfy anyone whose standards are too high, such as those who seek certainty, but they can still be very useful for people with reasonable goals, such as justifying their conclusion to reasonable moderates with open minds.

What is ‘just like arguing …’?

The third way to refute an argument is to give reasons to doubt that its premises provide adequate support for its conclusion. This variety of refutation targets defects in the relationship between premises and conclusion rather than in the premises or conclusion themselves.

We saw examples in our discussion of fallacies. Equivocation occurs when a word has a different meaning in the conclusion than it had in a premise. Ad hominem arguments and appeals to authority use premises about believers to support conclusions about their beliefs. And arguments beg the question when their premises are not independent of their conclusions – that is, when premises and conclusion are too closely related.

The relation between premises and conclusion can also be defective in other arguments that do not fit the patterns of standard fallacies. How can we tell whether that relation is defective? The most direct method is to look closely at the argument itself and assess it for validity (if it is deductive) or for strength (if it is inductive). Recall that inductive strength is the conditional probability of the conclusion given the premises. That probability is often hard to calculate or even estimate, so this method has its limits.

Another method is less direct, but sometimes easier to apply. Try to construct a parallel argument that mirrors the form of the argument being assessed and has obviously true premises and an obviously false conclusion. If opponents admit that the premises are true and the conclusion is false, then this parallel argument can reveal something defective in the relation between the premises and conclusion in the original argument being assessed. In other words, when someone presents an argument, critics respond, ‘That’s just like arguing in this parallel way’ where the parallel argument has an obvious defect. The original argument can then be defended only by showing that it does not share the same defect.

Martin Luther King deployed this strategy in his ‘Letter from Birmingham Jail’. He had been jailed for marching in favour of racial equality and civil rights. His jailers and critics argued that he should not have marched, because this protest would inspire his opponents to violently attack him and other marchers. King replied, ‘In your statement you asserted that our actions, even though peaceful, must be condemned because they precipitate violence. But can this assertion be logically made? Isn’t this like condemning the robbed man because his possession of money precipitated the evil act of robbery?’ In this case, King’s critics argued, ‘The marchers precipitate violence, so they must be condemned.’ He replied, in our terms, ‘That’s just like arguing that the robbery victim’s possession of money precipitated robbery, so the robbery victim must be condemned.’

Pretty powerful reply, right? But what is going on? King does not deny the truth of the premise that the marchers precipitate violence. They do. King also does not argue that the conclusion is false. That could not be shown by switching the subject to robbery. Indeed, King’s reply might seem irrelevant. How could talking about robbery show anything about marches? The key lies in the form of the arguments. Because they share a similar form, if one is defective in its form, so is the other. The parallel argument about robbery is supposed to move from a true premise that the robbery victim’s acquisition of money precipitated robbery to a false conclusion that this victim should be condemned. That movement shows that there must be some defect in the relation between premises and conclusion in the argument about robbery. If the argument about marches has the same form and the same relation between its premises and its conclusion, then the relation between premises and conclusion in the argument about marches must also be defective.

This reply does not attempt to show that the conclusion of the argument about marches is false. It still might be true that the marchers ought to be condemned. All King has shown is that this one argument is not enough to support that conclusion. He casts doubt on one argument without arguing for the opposite. Moreover, he casts only some doubt. He does not prove beyond any question that the argument fails. His critics still have several moves available.

First, King’s critics can accept the conclusion that the robbery victim should be condemned. If that conclusion is true, then the parallel argument is not obviously defective, so this refutation fails to reveal a defect in the original argument. But this reply seems implausible in this case.

Second, King’s critics can deny the premise that the robbery victim’s possession of money precipitated the robbery. If the robbed person hid his money, as most people do, then the robber would not know whether he had money, so he would have robbed this victim even if he had had no money with him. Since possessing money is not necessary for him to be robbed, his possession of money might not be what causes or precipitates the robbery. This reply is perhaps more plausible, but still problematic.

Third, King’s critics can point out differences between the supposedly parallel arguments. The robbery victim did not know that he would be robbed, but King did know that his opponents would attack violently. The robbery victim presumably hid his possessions to avoid robbery, whereas King marched in the open and hid nothing. He wanted publicity.

King cannot deny these differences between the supposedly parallel arguments, but he could deny that these differences make a difference. One way to test what makes a difference is to add premises to each argument. King’s critics could reply, ‘Fine, we spoke too quickly. But our main point still holds: the marchers knowingly and publicly precipitate violence, so they must be condemned.’ To refute this revised argument, King would need to say, ‘That’s just like arguing that the robbery victim’s possession of money knowingly and publicly precipitated robbery, so the robbery victim must be condemned.’ The problem is that this new premise is clearly false, so this new argument does not move from true premises to a false conclusion. As a result, it cannot reveal anything defective in the relation between this premise and this conclusion.

As always, the discussion can continue. The point here is only that an attempt to refute an argument by saying ‘That’s just like arguing …’ works only if the supposedly parallel argument has true premises and a false conclusion and only if the argument really is parallel. All of that needs to be shown in order for the refutation to work. It is not enough to say, ‘That’s just like arguing …’ unless it really is like arguing …

When this method of refutation is applied properly, it can be used to uncover many kinds of fallacies. Here are a few examples with varying degrees of strength:

THE FALLACY OF COMPOSITION

Argument: If one person doubles her income, then she will be better off.

Therefore, if all people double their incomes, then they will all be better off.

Refutation: That’s like arguing that if I stand up at a concert, then I will see better; so if the entire audience stands up at a concert, then they will all see better.

Lesson: What holds for parts might not hold for the whole.

THE FALLACY OF DIVISION

Argument: North Korea is an aggressive country, and you are from North Korea, so you must be aggressive.

Refutation: That’s like arguing that North Korea is a mountainous country, and you are from North Korea, so you must be mountainous.

Lesson: What holds for the whole might not hold for parts.

FALSE DICHOTOMY

Argument: You are either with us or against us, and you are not yet fully committed to our cause, so you must be our enemy.

Refutation: That’s like arguing that you are either with Fiji or against Fiji, and you are not yet fully committed to Fiji, so you must be an enemy of Fiji.

Lesson: People can be neutral – neither for nor against.

FALSE EQUIVALENCE

Argument: There is some argument for adopting this policy, but there is also some argument against it and in favour of an alternative; so both sides are reasonable, and it is unreasonable to favour one over the other.

Refutation: That’s like arguing that there is some argument for jumping off this building (how thrilling!), and there is also some argument against jumping off (how deadly!); so both choices are reasonable, and it is unreasonable to favour one over the other.

Lesson: Not all arguments and reasons are equivalent. Some are better than others. (The same point holds when there are experts on both sides.)

APPEAL TO IGNORANCE

Argument: You can’t prove that there were weapons of mass destruction in Iraq, so there must have been none.

Refutation: That’s like arguing that you can’t prove that there are tiny spiders in this room, so there must not be any tiny spiders in this room.

Lesson: There might have been lots that we did not see, because they are hard to find, even when they are there.

FALSE CAUSE (OR POST HOC ERGO PROPTER HOC)

Argument: Our economy improved right after he became President, so he helped our country a lot.

Refutation: That’s like arguing that our economy improved right after my daughter was born, so she helped our country a lot.

Lesson: The timing might be a coincidence. More generally, correlation does not imply causation.

None of these refutations is conclusive. In each case, defenders of the argument could claim that (a) the premise in the refutation is false; (b) the conclusion in the refutation is true; or (c) the argument in the refutation is not really parallel to the original argument, because they differ in some relevant respect.

Such attempts at refutation still shift the burden of proof to the defender of the argument, so even inconclusive refutations can make progress. They do not end the discussion, but that is not their purpose. Their goal is to rule out simple mistakes, and they can do that. When arguers succeed in defending their arguments against refutations by parallel reasoning, they usually need to complicate their arguments and add qualifications. The refutation shows that the original argument without the qualifications oversimplified the issues. The revised argument reveals complexities and subtleties that the original overlooked. Refutation can thereby improve discussions without ending them.