T1.4: 5 of 7

Now we can apply some of these concepts in the informal proofs we first presented in the last chapter. For instance, it's easy to see that

If a set of sentences includes one which is logically false, then that set is inconsistent.

How do we prove this? Well, we argue as before (using the sigma '\' as name for a set of sentences; the symbol here will appear as a slash if one is without the Logic font):

Suppose that \ is any set of sentences which includes a logically false sentence A.

By the definition of logical falsity, it is not possible that A be true.

So, it is not possible that all members of \ be true together (because there is no way to make even the one, A, true).

Finally, then, by the definition of consistency, \ is not consistent: that is, \ is inconsistent.

Whew. The simple idea behind this proof gets a bit obscured behind all the verbiage. There's a better way. We'll look at a new example using this "indirect" method; then let you improve the above proof in the exercises.

So, Let's look to the indirect way to give these informal proofs.