Section 1.5: Argument Forms

Reading

Read sections 1.5 in your text.

Instructor's Commentary

At first, you may have difficulty with the counterexample method. If you do, don’t feel you’re all alone. Many students do not grasp this method after reading about it for the first time. However, if you come to a good understanding of the counterexample method, you will also have a good understanding of two of the major concepts in this course: validity and invalidity. You should also find it easier to understand why other tests for validity and invalidity really work. So, let us review the basis for the counterexample method and some common difficulties with that method.

The counterexample method is based on the definition of validity. Because invalidity is the opposite of validity, you can also say that the method is based on invalidity. Let’s look at it from the perspective of validity. The definition of validity can be stated in the following way:

You might say that a valid argument is one in which no matter what you use for each of the terms of an argument, you can never get an argument where the premises are all true and the conclusion is false. This surmise leads us to the best perspective on the counterexample method. Since this method only identifies invalid argument forms, it makes sense to look at the method with the definition of invalidity in mind:

This definition should not be surprising if you followed the earlier definition of validity. Since the definition of validity claims that there is no substitution instance for each term such that all the premises are true and the conclusion is false, it follows that an invalid argument would have substitution instances for the terms such that all the premises are true and the conclusion is false.

By producing a counterexample to refute an argument, you are showing that the reasoning in the original argument is faulty. You are demonstrating that anyone who uses that argument form is not entitled to claim that the truth of the conclusion follows from the premises. So, no matter how true the premises of the original argument are, the conclusion does not follow from those true premises if the argument form is invalid.

Let’s work through a problem. Take problem 2 of part I of exercise 1.5, which reads:

The first thing you want to do is find out which statement is the conclusion and which are the premises, and then arrange them in the right order:

Next, you want to get it into an argument form, so you must determine what the terms of the argument are. In problem 2 we find the following terms:

Thus, the form of the argument would look like this:

To show that this argument form is invalid, we should first find an obviously false conclusion which has the same form as the conclusion in the original argument. Why should we start with the conclusion? If we start with the conclusion, we will already have two of the three terms that we need. So, all we’ll have to do is think of a third term that makes both premises obviously true. For the above argument, an obviously false conclusion would be “Some dogs are not mammals.” If our audience knows some ordinary zoological facts, this should be obvious. Now, we should try to find a third term that makes both premises obviously true. Do not look at both premises at the same time, because doing so will confuse you. Try to find a term that makes one of the premises obviously true, and then substitute the third term into the other premise to see whether it makes the other premise obviously true. As a third term, try cats. We will find that both of the premises are obviously true (”No cats are dogs” and “Some mammals are not cats”). Thus, our counterexample would be:

Most audiences would find this counterexample sufficient to dismiss the reasoning of the original argument as invalid.

The difficulty with the counterexample method is coming up with a convincing counterexample. Producing a good counterexample requires two things: (1) an awareness of your audience’s experiences and (2) a little imagination on your part. Remember, it is important that you choose a conclusion that is obviously false and premises that are obviously true. Why is it important? What may be obvious to one audience may not be to another. Let me give you an example.

Suppose a Zulu woman from Africa who had never seen the big city and all of its confusion gave you an argument that you knew was invalid. Suppose also that you were interested in explaining her faulty reasoning to her. You do not want to give her a counterexample that contained terms about computers. Most likely, she cannot tell whether the statements of your syllogism are true or false. The same would be true if she tried to use objects of her culture that were unfamiliar to you to show that your reasoning was faulty. The problem cuts both ways. Since a counterexample requires you to give premises that are obviously true and a conclusion that is obviously false, you must give an analogy that your audience understands. Counterexamples that rely on a knowledge of computers will probably not wash in rural Africa. Even within your own culture, you can run into trouble. Try using technical plumber‘s language in a conference filled with theoretical physicists. Theoretical physicists may be bright people, but they—like most other people—call plumbers when they have leaky pipes. Thus, a counterexample requires a sensitivity to other people’s perspectives.

This next point is about imagination, and it is connected to the first point about your audience. You must have a sympathetic imagination. Put yourself in the shoes of your audience. With what are they going to identify? Don’t use language that is foreign to your audience. If you engage in public speaking at all, the counterexample method is one of the most effective tools you can develop. Watch any candidate who is running for political office. Most likely, a candidate will criticize an opponent’s program by showing, through a counterexample argument, that the argument supporting that program is invalid. But, as all good citizens (and logicians) know, one must make sure that the counterexample argument has the same form as the original argument.

With practice you will find that it gets easier to produce counterexample arguments. So, try the problems assigned in the exercise below. If you need more practice, try the other problems in exercise 1.5.

Exercises

Do exercise 1.5, pages 63–64, as follows:

• Part I, problems 4 and 7.
• Part II, problems 1 and 4.