SOUNDNESS

Now suppose we agree that a given argument is valid. Should we then accept its conclu­sion? Let’s look at an example from the Zen Masters of Logic, Monty Python. In their movie Monty Python and the Holy Grail (1975), Sir Bedevere manages to elicit the fol­lowing argument from a group of willing bystanders:

If she’s made of wood, she’s a witch.

If she weighs the same as a duck, she’s made of wood.

Therefore, if she weighs the same as a duck, she’s a witch.

Although I haven’t given here the reasoning by which he establishes each premise, it is, of course, awful, full of the most flagrant fallacies. Only someone as utterly stupid as the peasants in Sir Bedevere⁵ s audience would accept the conclusion. But that’s only because the premises are totally ridiculous. If, however, we put that to one side and, for the sake of argument, accept the above two premises, then we have to accept the conclusion. That is, denying the conclusion would be incompatible with accepting the premises, so the argument stated above is valid! (We’ll prove it so in chapter 7.)

What this example shows is that in order to be convinced by an argument, we need to be persuaded of more than just the validity of the reasoning. If we are going to accept the argument’s conclusion, we also need to be convinced of the truth of the argument’s prem­ises. Now it is often said that all that is necessary for a good argument, in addition to the validity of its reasoning, is that its premises be true. Such an argument is defined as sound:

An argument is sound if and only if it is valid and all its premises are true.

Otherwise it is unsound.

A few notorious examples, however, suffice to show that the mere truth of the premises is not enough, in addition to the argument’s validity, to make an argument a good one. There is, for instance, a whole range of arguments in which the statement being argued for as the conclusion is already presupposed as one of the premises.

(1) A religion is a system of beliefs based on some commitment to the supernatural.

Since (2) Scientology is a religion, (3) we know that its credo centres on such a com- mitment.Therefore, having such a commitment, (4) it must be classified as a religion.

Here the conclusion (4) is derived from the definition (1) and the statement (3). But (2) is given as a premise for (3) (as shown by the premise indicator “since”). This means that (4) is derived from (2) as a premise, but the two statements say the same thing! In case the controversial nature of that subject throws you off, here is another example of the same form:

(1) An irrational number cannot be expressed as the ratio of two integers. Since (2) π is an irrational, (3) we know that it cannot be expressed as the ratio of two integers. Therefore, not being so expressible, (4) it must be an irrational number.

Such arguments are called circular arguments. By our definition of validity, they are necessarily valid, because denying the conclusion is incompatible with accepting all the premises.^[8] Even if all the premises are true, including the statement in question, clearly this type of argument should not persuade anyone, and therefore should not be regarded as a good argument, even though it is sound by the above definition.

Another objection to soundness as a criterion for a good argument is that a valid argu­ment may perhaps have true premises, but if they are not known to be true, it will still not be persuasive. Indeed, outside of purely formal contexts, we often do not know for certain whether many of the statements we use as premises are infallibly true. But this will not prevent us from judging whether, given appropriate standards of evidence, we should accept the conclusion on the basis of the premises. But this is just validity.

Thus the concept of soundness does not appear to be an adequate notion for evaluating the strength of arguments.

Finally, notice that in logic, as opposed to uncritical speech, we NEVER speak of an argument or the reasoning in it as being true or false; in logic the only thing that can be true or false is a statement or proposition. Likewise we NEVER speak of a statement being valid or invalid, reserving those terms strictly for arguments, inferences, or forms of reasoning.

SUMMARY ______________________________________________________________

• An argument or inference is valid if and only if denying its conclusion is incom­patible with accepting all its premises. Otherwise it is invalid.

• This definition is preferred to ones defining a valid argument as one for which it is impossible that its premises are all true and its conclusion false, (1) because it can be successfully applied to a much wider class of arguments than can that definition, and (2) because (unlike that definition) it can be applied whether or not the premises of a given argument are actually all true.

• An argument is sound if and only if it is valid and all its premises are true. Oth­erwise it is unsound.

• However, soundness so defined is inadequate to define a good argument, since (1) circular arguments can be sound but should persuade no one, and (2) whether all an argument’s premises are true is something we often do not know for certain, yet we still make judgements about whether it should persuade us of its conclusion.

3. Is the following argument valid?

Every sound argument is a valid argument.

Every sound argument has true premises.

.∙. Every valid argument has true premises.

4. A famous example of a circular argument is that of the “Cartesian Circle.” Although scholarly opinion differs on whether Descartes was actually guilty of it, he is alleged to have argued that the fact that clear and distinct ideas have true contents (C) is guaranteed by the fact that an Omnibenevolent God exists (G); but we know that an Omnibenevolent God exists because this is the content of one of our clear and distinct ideas. Thus we have C because G; but G, because C. Show that this argument is valid according to the definition Ofvaliditygiven in the text.

5. Consider the statement “A square has four sides.” Determine whether it is (i) logi­cally inconsistent with, (H) incompatible with (given the meanings of ‘square’ and ‘triangle’), or (Hi) neither incompatible nor logically inconsistent with, each of the following statements:

(a) A square has three sides. (b) A square does not have four sides.

(c) A triangle has three sides. (d) No four-sided figure is square.

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