Counterexamples
Chapter 3
Direct Counterexamples
The definition of soundness gives us a very direct way to expose an unsound argument. Recall our second formulation of the definition:
Soundness B: A sound argument is one for which there is no possible world in which the premises are true and the conclusion false.
If we can show that there is a possible world in which the premises are true and the conclusion false, then we will show that the argument is unsound. How do we show that there is a possible world in which the premises are true and the conclusion is false? The method that philosophers have used for thousands of years for this purpose is simple. We (carefully) imagine a possible world. In fact we already have done just this, in our exploration in Chapter 2 of unsound arguments. There is the example we were working with (2.4) and what we said about it:
Example of an unsound argument with true premises and a true conclusion
2.4 Most plants have green leaves. Therefore, many people enjoy ice cream.
Well,
that was easy. Of course it is true that most plants have green leaves, and
that many people enjoy ice cream. But the truth of the first claim is not
connected to the truth of the second claim in the right way. The fact that most
plants have green leaves does not guarantee
that many people enjoy ice cream. It is obviously possible for most plants to have green leaves even in a world in
which very few people enjoy ice cream. Imagine, for example, a world that is
like this one except that people experience ice cream the way that we
experience chalk, or bitter medicine. Plant leaves would still be green, but
not many people would enjoy ice cream. So in that world, the premise is true
and the conclusion is false. So by definition B of soundness, the argument is
not sound.
Look carefully at the sentence in the middle of the paragraph that begins "Imagine, for example, a world ." In that section, we simply describe, or construct, a possible world with the desired features. Features we desire are that the premises be true and the conclusion false. In this case, that means that it will have to be a world in which most plants do have green leaves, but not many people enjoy ice cream. Notice that we gave enough detail about the world we were describing to persuade the reader that such a world really is possible.
The description of the possible world above is called a direct counterexample.
Definition: A direct counterexample to an argument is a description of a possible world in which the premises of the argument are all true but the conclusion is false.
Let's look at another example of an unsound argument:
Most children like eating green vegetables. Moreover, when kids don't like to eat something, it's not good for them. Therefore, green vegetables are good for kids.
There is a little bit more going on in this argument. How about those premises? Do most kids really like eating green vegetables? And the second sentence looks like a pretty wild claim. You might think immediately about bitter medicine for a sick child. So these premises look pretty shaky. Do we care? No, we couldn't care less. Remember, this is not nutrition class, it is logic class. We are totally uninterested in whether the premises are true. Likewise the fact that the conclusion looks pretty plausible is of no interest to us whatsoever.
Instead, supposing that the premises were true, would the conclusion have to be true too? To see that it wouldn't, that this argument is unsound, we can give a direct counterexample. (In constructing possible worlds, we will often give them names. As a reminder, we will give them names that begin with the letter 'w'.
Imagine a world, call it world Wanda, in which kids are incredibly cheerful eaters; they like to eat almost everything. There are only a few things that kids dislike, and those happen to be things that are really lethal. So in world Wanda, if kids don't like something, it is really not good for them it would kill them. Certainly if something is good for them they like it, since they like almost everything except for those few lethal foods. But since they are such cheerful eaters, there are some other things that they also like, but that aren't very good for them. For example in world Wanda most kids like sweets. But as in our real world, sweets aren't very good for kids in Wanda. In addition, in Wanda green vegetables cause serious digestive problems in kids. But they like them anyway; green vegetables aren't among the really lethal that kid's don't like. In fact in this world kids just love to eat green vegetables, so parents have to keep the green vegetables locked up in special refrigerators. Too bad green vegetables usually make kids sick.
We have probably described this possible world in more detail than we really needed, but that's OK. The detail is useful because it helps to make clear that this world really is possible. It's very different from our world, but it could happen. That is very important. We will see shortly that sometimes what looks like a possible world turns out not to be possible after all.
Let's look at some of the features of this world, and check whether the premises and conclusion of argument 3.1 are true or false in it. The first premise of 3.1 says that most children like eating green vegetables. In Wanda that is clearly true; kids there are crazy about green vegetables.
The second premise of 3.1 tells us that when kids don't like to eat something, it's not good for them. Is that true in Wanda? Yes, it is completely true. There are only a few things there that kids don't like, and they are all lethal. So obviously it is true in Wanda that if kids don't like to eat something, it's not good for them. Indeed, if they don't like to eat something, it is lethal.
So both premises of 3.1 are true in Wanda. What about the conclusion? Is it true or false (in Wanda) that green vegetables are good for kids. Notice that in our actual world, green vegetables are good for kids. But that doesn't matter a whit. Because in Wanda it is clear that green vegetables cause severe digestive problems in kids problems so severe that parents must keep them locked up. So in Wanda the conclusion that green vegetables are good for kids is false.
We have succeeded in imagining a possible world in which the premises of 3.1 are true and the conclusion is false. We have given a direct counterexample to argument 3.1, and have thereby shown that it is unsound. What that means is that even if the premises were true, the conclusion does not follow. Even if kids did like to eat green vegetables, and even if (contrary to fact) when they don't like something it is always bad for them, it still wouldn't follow that green vegetables are good for kids. There would still be room in the space of possibility for green vegetables to be bad for kids.
Another example
Let's look at another argument:
3.2 Dogs like to swim, but cats don't. Therefore, people who like to swim with their pets should have dogs as pets.
Is this sound? Does it follow from the fact (if it is a fact) that dogs like to swim but cats don't, that people who like to swim with their pets should have dogs? Here is a direct counterexample that shows that it does not follow.
World Walter is very much like our world, but with some important differences. In Walter, all dogs love to swim. However, they love to swim so much that whenever they are in the water they take off very fast and swim as far as they can go. Usually they drown as a result, because they swim away until they are exhausted. So people who own dogs must be very careful to keep their dogs away from the water. Cats, on the other hand, don't like to swim (just like our world), so they don't make very good swimming pets either. However in Walter miniature pigs are the most popular pet. They are very friendly, intelligent and devoted. More importantly, they are accomplished swimmers gentle, considerate, and enthusiastic. Almost everybody in Walter who wants to swim with a pet has a pet pig just for that reason.
In giving a direct counterexample to show that 3.2 is unsound, we need to construct a possible world in which the premises are true and the conclusion false, that is, in which dogs do like to swim, cats don't, but nevertheless people who want to swim with their pets should not buy dogs. It is clear that Walter has all these features. So argument 3.2 is unsound. Its conclusion does not follow from its premises. We will examine later how the argument might be improved in response to this counterexample.
Setting up a counterexample
3.3 Britney Spears likes pistachio ice cream. John Ashcroft doesn't. Therefore Britney Spears likes more kinds of ice cream than John Ashcroft.
Hopefully you are starting to get the hang of this and can see already that this argument doesn't look good. Of course we don't care about what kind of ice cream Britney Spears or John Ashcroft like. We care only whether the conclusion follows from the premises.
Can we give a direct counterexample to this argument? Let's start by setting up the counterexample. By "setting up the counterexample" we mean describing for ourselves what a counterexample would look like, without actually giving one.
We know that a direct counterexample will be a possible world in which the premises are true and the conclusion false. So we want our counterexample to be a story a description of a possible world in which Britney Spears really does like pistachio ice cream and John Ashcroft doesn't, but in which John Ashcroft likes more kinds of ice cream than Britney Spears does. We have just set up the counterexample. Having done so, actually constructing one is now easy.
Imagine a world, Wendell, in which Britney Spears likes only one kind of ice cream: pistachio. She even turns up her cute little nose at vanilla. No matter what she is offered, if it's not pistachio she hates it. However, she loves pistachio. John Aschroft, on the other hand, is just the opposite. He loves every possible kind of ice cream except pistachio. Ginger and jalapeρo ice cream? He loves it. Asphalt ice cream? Bring it on. He loves everything except pistachio. Obviously in world Wendell, John Ashcroft likes more kinds of ice cream than Britney Spears does.
In world Wendell the premises of 3.3 are clearly true and the conclusion is clearly false. So 3.3 is unsound. It does not follow from the facts (if they were facts) that Britney Spears likes pistachio and John Ashcroft doesn't, that she likes more kinds of ice cream than he does.
Compelling counterexamples
Think about this argument and the counterexample to it that follows.
3.4 Bicycles cause much less air pollution than cars do, so people should ride bicycles to work.
Here's a counterexample:
Imagine a world, Woody, in which cars are much less polluting than they are in this world, but not as polluting as bicycles. In Woody most people are very lazy and don't like to exercise very much. They would be really unhappy biking to work every day, and it doesn't cause a lot of extra pollution for them to drive. Clearly in Woody it's better for people to drive to work, and that's what they should do.
This is a pretty lame counterexample; it doesn't show very well that 3.4 is unsound. Let's try to figure out what is wrong with it. Let's start by setting up our counterexample. We would want a world in which bicycles do cause much less air pollution than cars do but in which people should nevertheless not ride bikes to work. Remember that our counterexample is aimed at an audience. Somebody has actually proposed argument 3.4. Let's suppose her name is Abby. Abby thinks that the conclusion follows from the premises. We need to present her with a scenario a possible world in which it will be really clear to Abby that her premises are true, but her conclusion false. This is because she is the audience for the counterexample; she is the person we are trying to persuade.
So in order to be able to persuade Abby, we want it to be clear even to her that her premises are true in Woody, but her conclusion is false there. How did we do? First, is it really, really true in Woody that bicycles cause much less air pollution than cars do? No, it is not as clear as we would like. The opening sentence of the counterexample says that cars are less polluting than they are in this world. That is actually undermining the premise making it seem less true than it is in our world. Later on the counterexample says that it doesn't cause a lot of extra pollution for people to drive. Is it looking really true that bikes cause much less pollution than cars? No, just the opposite. It may be true, but it is looking barely true. The counterexample has undermined, or attacked, the premise.
Second, will it be clear to Abby that the conclusion is false in Woody? Remember that Abby is a person who believes that people in this world should bike to work to cut down on air pollution. Is she likely to be sympathetic to the citizens of Woody who are so lazy that they would rather cause a little pollution than bike? Probably not. If Abby thinks that we should bike to work, she is probably going to think that people in Woody should bike to work too. In short, Abby is not likely to find this counterexample very compelling.
Definition: A direct counterexample is compelling to the extent that it is obvious that in the constructed world the premises of the argument are true and the conclusion false. The more obvious that is, the more compelling the counterexample.
Woody is not a very compelling counterexample. Abby would not be convinced. Could we do better? Let's try again.
Imagine a world, Wilda, in which for purposes of energy efficiency people have clustered workplaces close to each other. At the same time, they have created living communities that are quite separate. This allows them to optimize their living space. However most homes are dozens of miles from where people work. As a result it would take people so long to ride bikes to work that they wouldn't have time to get anything done, or to spend time with their families. The only feasible way for people to get to work is to drive, or take public transportation. Even though cars pollute much more than bikes, it would obviously be foolish for people to ride their bikes to work.
This counterexample is far more compelling. There is no attempt to undermine the premise it is really true in Wilda that cars pollute a lot more than bikes. But even Abby would probably agree that in Wilda people shouldn't ride bikes to work.
Dialogue and Audience
When somebody makes an argument for a claim, he has the oppourtunity to begin a dialogue. A dialogue is a discussion between two or more parties in which the parties generally respond to, and build upon, others' contributions. Not all arguments are dialogues. Here is a classic argument between kids:
Jake: You took my Lego toy.
Preethi: No I didn't!
Jake: Yes you did!
Preethi: Didn't!
Jake: Did too!
Preethi: Stop hitting me
Jake: Give it back!
Preethi: I'm going to tell Mom.
This is not a dialogue, since Jake and Preethi are not responding and building upon each other's claims. Indeed, although Jake and Preethi are having an argument, nobody is making an argument, since no reasons are being given. Unfortunately this model is all too representative for what passes, certainly on television and radio, as an argument. On many shows that claim to explore issues of public policy there is as little sign of dialogue as in the fight between Jake and Preethi. One of the objectives of this text is to develop a model of argument in which arguments become dialogues.
Why is dialogue important? Dialogue focuses on finding common ground. This does not mean that it tries to play down differences. Rather it uses what common ground it can find as a way of identifying and focusing on real differences.
Remember that an argument involves two parties, an arguer and a target. An arguer proposes an argument, trying to persuade the target. So the target is the audience for the original argument. Now the target responds with a counterexample. The target is trying to persuade the arguer that the original argument was unsound. Therefore the arguer is the audience for the counterexample. A compelling counterexample is one for which it is obvious that the premises of the argument are true and the conclusion false. Obvious to whom? In most cases we want it to be obvious to the audience in this case to the original arguer. The target is trying to persuade the arguer of the unsoundness of the argument. So it is the arguer who must see, and come to agree, that the premises are true and the conclusion false in the possible world of the counterexample.
So the audience will have an impact on the choice of counterexample. Here is an example.
3.5 Improving education would benefit everybody in important ways, since everybody is better off with a good education. Therefore the government should spend more on education.
On the surface this looks like an argument that almost everybody could get behind. However, a compelling counterexample will show that it ignores what might have to be sacrificed in order to improve education. Here the question of audience becomes important in framing a compelling counterexample. To set up the counterexample, we want to construct a world in which it is clear (to the arguer) that although increasing funding would definitely benefit everybody (not just some remember not to undermine the premise), and everybody really would be better off with a good education, nevertheless it is obvious (to the arguer) that the government should not spend more on education.
Suppose the arguer is Dave the Democrat. Here is a counterexample for Dave:
Imagine a world, Will, in which education has been made a huge priority so much so that a major part of the national budget has gone toward education. This is because everybody recognizes how important education is to everybody. So most people in world Will have gone to college and graduate school, and in wonderful schools too. Nevertheless, they would be even better off if they could get more education (remember not to undermine the premise). However the government budget has been so drained in support of education that the only funds that could be diverted to improving it even more would have to come from health care support. So although the people would benefit from being better educated, actually paying to give them a better education would cause a health crisis, plague, death and destruction. So even thought the education would be beneficial, the people can't afford for the government to spend any more money on it.
But if the arguer were Rita the Republican, the following counterexample might work better:
Imagine a world, Willa, in which education has been made a huge priority so much so that a major part of the national budget has gone toward education. This is because everybody recognizes how important education is to everybody. So most people in world Willa have gone to college and graduate school, and in wonderful schools too. Nevertheless, they would be even better off if they could get more education (remember not to undermine the premise). However the national defense budget has been so drained in support of education that the only funds that could be diverted to improving it even more would have to come from pay for the military and weapons development. As it is now, many soldiers don't even have adequate combat boots. So although the people would benefit from being better educated, actually paying to give them a better education would cause a crisis in the military, rendering it incapable of protecting us against foreign invasion. So even thought the education would be beneficial, the people can't afford for the government to spend any more money on it.
In proposing a counterexample we appeal to a set of beliefs the beliefs of the audience for our counterexample. In actual practice, when we are talking with somebody, we will have a good sense of who the audience is, and can frame counterexamples accordingly. But sometimes, for example when we are writing, or speaking in public, or doing homework exercises, we will not have a very good idea of what our audience believes. In those cases we will need to appeal to as wide an audience as possible in order for our counterexamples to be compelling. We would need to pick counterexamples that did not presuppose unusual beliefs, but only beliefs that most people have in common.
Embracing the premise
Sometimes it is possible to construct a counterexample in which you make the premises true in a sneaky way.
3.6 Dogs are really cute, so it is wrong to hit a dog.
This is obviously not a great argument. What do you think of this counterexample?
Imagine a world, Waverly, in which Marcia has an incredibly cute
stuffed dog that her therapist gave her as a way of working out her anger.
Whenever she gets angry at her parents she is supposed to hit the dog as hard
as she can. (Notice
in giving the counterexample we don't have to say that the premises are true
and the conclusion false. In this case the point is clear enough.)
Well, in some sense a stuffed dog is a dog. But that's sort of a sneaky way to make the premise true. In chapter zz, when we talk about strong counterexamples, we will give a clear foundation to the sense that this is a sleazy counterexample, that there is something not right with it.
In the meantime, it will be enough to point out that we want to appeal to the audiences beliefs. Obviously the arguer wasn't thinking of stuffed dogs. We want to make the strongest case possible, not to sneak through on a technicality. We want even the arguer to agree that what she had in mind as premise is true in Waverly, but the conclusion is not.
We can put this more generally by saying that in giving a counterexample you should not attack, or undermine the premise. Rather you should embrace the premise. To embrace the premise is to try to agree with it as fully and cheerfully as possible. Remember that in logic class you have no argument with the premise. Your attitude is full of generosity toward the premise. Let the premise be completely, fully true in just the way that the arguer meant. You want to show that even then, the conclusion doesn't follow. The better the premises look, the more you will have accomplished in showing that the conclusion still doesn't follow.
Imagining and possibility
Earlier in the chapter we mentioned that in imagining a possible world you need to imagine carefully, to be sure that it really is possible. Although the practice of imagining a scenario in order to show that it is possible dates back at least to the time of Plato (450 BC 385 BC), has got a good bit of attention from philosophers in recent years.
One of the problems is that it seems sometimes that we might imagine things that turn out to be impossible. For example, recently the "Four-color theorem" was proved by using exhaustive computer searches. This theorem shows that every map can be colored with only four colors in such a way that no bordering areas have the same color. But some people think that they can imagine maps so complicated that you would need five colors. You might imagine countries that get smaller and smaller, swirling in a spiral, or something like that. And then imagine working and working to color it with four colors and finally hitting your forehead with the heel of your hand and saying angrily, "That's it! It can't be done."
But have you really imagined a map so complicated that it requires five colors? Many claim that you haven't. You haven't imagined it because there is no such thing possible. So it is not possible that you really imagined it. It may have seemed to you that you imagined it, but you were mistaken.
The lesson is that just because you think you have imagined something, it doesn't follow that you have succeeded. You might be mistaken about what you can imagine. To imagine something successfully, you may have to be careful enough in the detail to be sure that you really are imagining it. Here is an example of how you could go wrong:
3.7 Anybody who is off the Dean's list wants to get money from his parents. But nobody is both on the Dean's list and unhappy in college. Moreover, people who are self-sufficient don't want money from their parents. Mark, unfortunately, is not happy. Therefore Mark is not self-sufficient.
And here is the counterexample (?):
Imagine a world, world Willow, in which Mark is having trouble with relationships. He is so self-sufficient that he never really needs anything from others. This, of course, makes it hard for him to establish a deep relationship with anybody and so he is unhappy. This is affecting his school work, but he is so self-sufficient that he is managing without needing any money from his parents.
Does this counterexample work? Let's set it up. For convenience, we'll number the premises.
Therefore,
To construct a counterexample we would need a world in which all of the premises (1) (4) are true, but in which Mark is self-sufficient. Have we succeeded? Superficially it may appear that we have. But a closer look will show that we have not. Since Mark is unhappy in Willow, by premise 2 his school work is not only affected, as the counterexample says, but he is not on the Dean's list. For if he were, then he would be both unhappy and on the Dean's list, if premise 2 is to be true in Willow, that can't happen. So Mark is off the Dean's list. Hence if premise 1 is to be true in Willow, Mark must want money from his parents. But then by premise 3, he can't be self-sufficient, since if he were he wouldn't want money from his parents.
But in setting up the counterexample we saw that we needed a world in which Mark is self-sufficient. And evidently despite our best efforts, he can't be self-sufficient in Willow.
Try as we might, we can't really imagine a world in which these premises are true and the conclusion false. As soon as we attend carefully to the details of the world, we see that no matter how we twist and squirm, if the conclusion is false if Mark is self-sufficient then at least one of the premises will have to be false too.
These examples show that in describing a possible world, there must be enough detail to assure us that the world described really is possible. That is part of what it is for a counterexample to be compelling. Remember that a compelling direct counterexample is one in which it is very obvious that the premises of the argument are true and the conclusion is false. If there is sufficient detail to make it obvious that the premises are really true in the world and the conclusion false, then there will also be enough detail to assure us that the world really is possible.
Another failure of imagination
Look at this sound argument:
3.8 It is always wrong to hurt a child. Therefore Marcia, who was torturing her 3-year old sister, was wrong.
There had better not be a counterexample to this argument! What would we say if somebody offered the following?
Imagine a world, Wilfred, in which the word 'torturing' means 'playing happy games with', and imagine that Marcia was playing really happy games with her sister, Joanne. Joanne loved the games and was not hurt at all. In Wilfred Marcia was torturing her sister (that is, playing happy games with her), but she wasn't hurting her. So even though in this world it is always wrong for somebody to hurt a child, Marcia wasn't wrong.
Does that counterexample look like it is cheating? Yes, it is. Let's set up a counterexample for this argument. We would need a world in which it is always wrong to hurt a child, and in which Marcia really was torturing her 3-year old sister, but in which she was not wrong. Is Wilfred such a world? Well, Wilfred is a world in if somebody said 'Marcia is torturing her sister' people would agree. This is because they would understand that sentence to be talking about playing happy games. But is Wilfred really a world in which Marcia is torturing her sister? No. When we talk about what is happening in other possible worlds, we use the language of this world. Thus the meanings of the words we are using are the meanings they have in our actual world. If in Wilfred some of our words have a different meaning from what they have in the actual world, it is still the meanings in the actual world that we use in deciding what is true and what is false in Wilfred. In short, in imagining other worlds, we keep the meanings of our words fixed so that we can use them to imagine all sorts of changes in the facts of those worlds.
Sometimes this distinction between the meanings of words and the facts of the world can get fuzzy. Here is another purported counterexample to argument 3.8.
Imagine a world, Wende, in which people mature very quickly. Indeed, in Wende the average life-span is 10 years. Most people marry and have children when they are 2. In Wende, Marcia's 3-year old sister, Joanne, had a husband and a baby. Unfortunately, Joanne was just about to step on her baby and Marcia had to push her very hard to protect the baby. This hurt Joanne, but since in Wende Joanne wasn't a child, but an adult, it was not wrong.
Is this cheating, or is it a good counterexample? Is it part of the meaning of the word 'child' that a 3-year old is a child? Or is that just one of the deep facts about our world? Most philosophers believe that there is no clear answer to this question that the lines between differences of meaning and differences of fact are not sharp.
What then do we say about the counterexample? We say that a compelling counterexample will embrace the premise, not undermine it. This is therefore not a very compelling counterexample, since it calls into question whether the premise that it is always wrong to hurt a child is really true in Wende. Perhaps 3-year olds are not children in Wende and it really is always wrong to hurt a child. Or perhaps 3-year olds are still children, but just not called 'children', and sometimes it is OK to hurt a child. Since it is not completely clear which of these ways of looking at it is the right one, it is not completely clear that the premises of 3.8 are true in Wende. So the counterexample is not terribly compelling. Nor is it clearly irrelevant.
In short, argument 3.8 looks pretty sound, but maybe this counterexample points to a small problem with it. Later, in chapter zz, we will see how even a frail counterexample like this one points us to a missing premise that could be added in order to make 3.8 rock solid. In the meantime, here by way of summary are the rules we have arrived at.
Rules for imagining
Rule 1: Imagine carefully.
Don't clutter your counterexample, but be sure to give enough detail to persuade yourself and your audience that the world you describe really is possible.
Rule 2: Keep the meanings of the words in the argument fixed.
The premises need to be true, and the conclusion false, relative to their meanings in the actual world. Don't try to play with the meanings of the words.
Rule 3: Embrace the premises.
Don't try to make the premises true in a half-hearted or slimy way. Try to be thoroughly cheerful about the premises; make them as completely true as possible.
Imagine a world in which people are only children until they get to be 2-years old, at which point they are adults.
<arg with ce that looks possible but isn't. Rely mostly on complicated form, not meaning shifts or math. Then 2 rules for imagining. 1. Detail; 2. words keep meaning
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