Shantila's Inside Logic #6+

Validity and Soundness

We say that in valid reasoning, the conclusion must be true if all the premises are true.

And we say that reasoning is sound if both (a) it is valid, and (b) all the premises are true. Notice that it follows that the conclusion of a sound arguments must be true.

It normally is not the business of logic to investigate the truth or falsity of premises, but the distinction between validity and soundness is easy to grasp.

To see clearly the difference between validity and soundness, consider the following example.

This is valid reasoning, which we can represent as follows (using S: Shaq is married to Madonna. R: Shaq is married to a rock star).

S>R, S } R

The sequent clearly is valid, conforming to our rule MP -- the conclusion R would have to be true if both premises were true. And of course we now can easily construct a proof using MP to show that it is valid.

But notice that this reasoning about Shaq and Madonna is not sound, because not all of its premises are true. Although the first premise S>R is true (since Madonna is a rock star), the second premise S is false (since Shaq is not married to her).

How do I know? Well, that is a fair question. I guess I don't really know for sure that Shaq is not married to Madonna. Indeed stranger things this have turned out true (and if they are SECRETLY married -- then no wonder we don't know about it). All the same I do have some reasons for believing that S is false (for example, I saw something recently about Madonna's family --and Shaq wasn't in it). Almost certainly I am correct in thinking they are not secretly married.

In any case, the truth or falsity of S does not depend on what I happen to know or believe. The sentence S is either true or false independent of my beliefs. And if S is false, the reasoning is not sound (even though valid). Now, if things are strange enough here so that in fact S is true (so the reasoning is both valid and sound), then I happened to pick a bad example to illustrate the difference between validity and soundness! All the same, we can grasp the difference.

There are many statements that are true that nobody knows to be true. For example, there is truth about whether there ever was water on the planet Mars --either there was water there at some time or there never was any water there at all-- but until recently nobody knew the answer. (The recent Mars exploration did find evidence of water two years ago.)

* Practice

6.5 Give simple examples in English of (a) a valid argument that you know for sure is not sound, and (b) a valid argument that you know for sure is sound.

6.6 Symbolize the reasoning in both of the following examples and construct a proof to show that the reasoning is valid. Use the following symbols. S: Squirrels have been mentioned in this text. L: This is a logic text.

(a) If squirrels have been mentioned in this text, this is not a logic text. This is a logic text. So squirrels have not been mentioned in this text.

(b) If squirrels have been mentioned in this text, this is not a logic text. Squirrels have been mentioned in this text. So this is not a logic text.

6.7 Both of the examples in the preceding exercise are valid. Now state whether or not they are sound; and if you do not have enough information to determine soundness, please explain.

6.8 Go back and prove all of our sequents 1 - 17. (Really! This is why it is called "practice.")