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IFL: Logicbite 4
The counterexample method
Peter Smith
You are given a (one-step!) argument to assess. You suspect it is invalid. How could you show this?
In IFL §4.4, we looked at the mini-example
(1) All philosophers are logicians. So (2) All logicians are philosophers.
This argument is evidently invalid. But, to press home the point, I remarked that you might as well argue
(3) All women are human beings. So (4) All human beings are women.
I showed that the inference move in the first argument is unreliable by giving a parallel second argument which relies on the same pattern of inference but which is obviously hopeless. In other words, I offered a counterexample to the reliability of the inference.
The idea that we can challenge an inference by giving a counterexample in this sort of way is surely familiar enough before we get round to officially studying logic: the problem, as we will see, is in saying clearly just how the challenge works.
Before proceeding, a general remark. When quoting from other authors’ textbooks, I’ve occa- sionally quibbled and criticized, pointing out some less-than-ideal phrasing. I’m about to do that again, at some length. I’m not, I hope, just being captious. Rather, I want to encourage a habit in you: you need to read carefully, stay critical, even when tackling a logic book – especially including mine!
Here is Patrick J. Hurley in his A Concise Introduction to Logic^1 (compare the similar quote from Lemmon in Logicbite 2):
This section shows that the truth of a deductive argument’s inferential claim (that is, the correctness of the argument’s reasoning) is determined by the form of the argument. In other words, validity is determined by form. For these purposes, consider the following argument:
All adlers are bobkins. All bobkins are crockers. Therefore, all adlers are crockers.
Because the words “adlers,” “bobkins,” and “crockers” are nonsensical, we do not know whether any of the statements in this argument are true or false.
Really? If those three expressions are nonsensical, the putative premisses and conclusion here are nonsense too. So we do know their truth-status: they can’t be either true or false. To proceed, we need to assume that the words are not nonsense, but meaningful though unfamiliar.
(^1) The title is a little comical, as this book weighs in at over 600 pages, even before the answers to exercises start. Apparently it is the most widely used logic text in North America, and is now in its thirteenth edition. Which I find a rather depressing thought. Anyway, I’m quoting here from the seventh edition. The thirteenth edition does just a little better, though the main complaint still stands. 1
Yet, we do know that if we assume that the premises are true, it is impossible for the conclusion to be false. That is, if we assume that the adlers, whatever they might be, are included in the bobkins and the bobkins in the crockers, then we must accept the conclusion that the adlers are included in the crockers. According to the definition of validity, therefore, the argument is valid. This fact is important for understanding the nature of validity because it shows that the validity of an argument has nothing to do with its specific subject matter.
That’s badly ambiguous. Does Hurley mean (a) that the validity of an argument – any argument
- has nothing to do with its specific subject matter? Or does he just mean (b) that we’ve found a case, this case, where the validity of the argument has nothing to do with its specific subject matter? He is of course only entitled to (b). But judging from his opening words about validity being “determined by the form [as opposed to the subject matter] of the argument”, it looks as if Hurley might think he has established that (a) is true. He hasn’t.
Even though we know nothing about adlers, bobkins, and crockers, we still know that the argument is valid. The validity of the argument arises from the way the terms “adlers,” “bobkins,” and “crockers” are arranged in the statements. If we represent these terms by their first letters, we obtain the following argument form. We use a line to separate the premises from the conclusion. All A are B. All B are C.
All A are C. This is a valid argument form. Its validity rests purely upon the arrangement of the letters within the statements, and it has nothing to do with what the letters might stand for. In light of this fact, we can substitute any terms we choose in place of A, B, and C, and as long as we are consistent, we will obtain a valid argument. For example, we might substitute “daisies” for A, “flowers” for B, and “plants” for C and obtain the following valid argument:
All daisies are flowers. All flowers are plants. Therefore, all daisies are plants. Any argument, such as this, that is produced by uniformly substituting terms or statements in place of the letters in an argument form is called a substitution instance of that form.
We could quibble that Hurley hasn’t explicitly told us what he means by ‘valid argument form’: but presumabbly it is one all of whose substitution instances are valid arguments.
Hurley now continues
Let us turn now to the concept of invalidity. Consider the following argument: All adlers are bobkins. All crockers are bobkins. Therefore, all adlers are crockers.
As with the previous argument, we do not know whether the premises and conclusion of this argument are true or false. But if we assume that the premises are true, it is possible for the conclusion to be false. It might be the case, for example, that the adlers make up one part of the bobkins, that the crockers make up another part, and that the adlers and the crockers are completely separate from each other. In this case the premises would be true and the conclusion false. The argument is therefore invalid.
form. But that doesn’t by itself entail that it is an invalid argument because some substitution instances of invalid forms are actually valid. Hurley seems to be sliding from the observation that the adler-bobkin argument is an instance of the displayed invalid form to the implicit claim that the adler-bobkin argument ‘has’ that invalid form in his special double-barrelled sense. And he isn’t entitled to that as yet.
Nor is he entitled, in his final sentence, to that talk about ‘the’ form of an argument, given he has himself just remarked that an argument can be an instance of more than one form, some valid, some invalid.
I said at the outset that the ‘you might as well say’ or counterexample method for showing an argument is invalid is common and familiar. We’ve just seen one author getting into some tangles in trying to explain how the counterexample method works. And it is difficult to find authors doing much better.^2 But we now know some pitfalls to avoid: do I do any better in IFL Chapter 5?
(^2) Let me know who I have overlooked! Yes, looking ahead, lots of authors say just the right things in formal contexts when talking about counterexamples to claims of tautological entailment, for example. But what about authors talking about more informal uses of the counterexample method?
