Even if its premises and conclusion cannot be refuted by any counterexam- ple or reductio, a deductive argument can also be refuted by showing that it is invalid. We know that an argument is not valid if it starts from true prem- ises and leads to a false conclusion. Often, however, we cannot point this out to refute an argument, because the truth or falsity of the conclusion is the very thing at issue. When this problem arises, a typical device is to point out that by arguing in the same way, or a similar way, we can reach a conclusion that is unsatisfactory.

Here is a simple example:

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Cary: Most of the people in this class are college students. Most college students study hard. Therefore, most of the people in this class study hard.

David: That’s just like arguing that most whales live in the sea, and most animals that live in the sea are fish, so most whales are fish.

At first sight, it might not be clear how the second argument could show anything about the first argument. What do whales have to do with stu- dents? The point, however, is simply that the two arguments share a basic form. Thus, if the second argument is not valid by virtue of that form, then the first is also not valid by virtue of that same form. The second argument is obviously not valid, since its premises are true but its conclusion is false. This shows that the first argument is not valid, at least by virtue of this shared form. Even though the first argument still might be valid on some other basis, its defenders at least owe an alternative account of its validity. Often there will be none.

Refuting an argument by showing that it is just like another argument that is obviously no good is a common device in everyday discussions. Here’s another example:

Matthew: If I had a higher salary, I could buy more things; so, if every- one had higher salaries, everyone could buy more things.

Kirsty: That’s just like arguing that, if one person stands up at a ball game, he will get a better view; so, if everyone stands up, everyone will get a better view.

At first sight, it may not be obvious whether Matthew’s style of reasoning is valid or not. Kirsty’s response shows that Matthew’s argument is invalid by providing an instance in which the same style of reasoning takes us from something true to something that is obviously false, because, if everyone stands up at a ball game, only the tallest people will be able to see better. Kirsty’s response also shows why Matthew’s argument is invalid: Just as one person’s ability to see can be affected by other people standing up, because this raises the height that is necessary to see, so one person’s ability to buy can be affected by other people having more money, if this raises prices and thereby raises the amount of money that is necessary to buy things.

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