Introduction to Logic
by
Stefan Waner and Steven R. Costenoble

This Section: 6: Arguments and Proofs

5. Rules of Inference Section 6 Exercises Main Logic Page "Real World" Page

6. Arguments and Proofs

We have already had a taste of proofs in Section 5. In this section, we make more precise what we were doing there, and get some practice in coming up with proofs. To begin with, let us take another look at one of the proofs we went through in Section 5.

In Example 6 of the previous section, we proved the argument:

Precisely, an argument is a list of premises followed by a conclusion. (We allow the possibility of an empty list of premises, that is, no premises at all, as in Example 5 of Section 5.) The argument

is valid if the statement (P1P2 . . . Pr)C is a tautology. In other words, validity means that if all the premises are true, then the conclusion must be true. Notice, however, that validity does not guarantee the truth of the premises; a valid argument can easily have false premises.

Now in Example 6 of the previous section we also gave the following proof:

In general, a proof of an argument is a list of statements, each of which is obtained (using statements earlier in the list) by using one of the rules of inference T1, T2, S, C or P, with the last statement of the list being the conclusion of the argument.


According to the definition, all we need to check to see whether the argument

is valid is whether [(aq)(bq)][(ab)q] is a tautology, and this we can check by looking at its truth table. Who needs a proof?

Answer

Quite correct. The approach using a truth table does have the advantage that it's mechanical (so that one could write a computer program to do it, for instance). There are however, several disadvantages:

1. Truth tables can get awfully large. For instance, the truth table for [(aq)(bq)] [(ab)q] involves eight rows and nine columns. It gets worse quickly, since each extra variable doubles the number of rows. For example, the truth table for

would require 32 rows. The proof, on the other hand, is easy and requires only three lines in addition to the premises.

2. Checking the validity of an argument mechanically is almost completely unenlightening; it tells you nothing about what is going on in the argument. Thus, from the purely intellectual point of view, a proof is far more interesting than a (possibly huge) truth table. By the end of this chapter, you should be able to look at an argument in words or symbols and decide, using your understanding of the rules of inference, whether or not it is valid. Thinking in terms of truth tables is no help towards this goal.

2.While truth tables suffice to check the validity of statements in the propositional calculus, they do not work for the predicate calculus, and hence they do not work for real mathematical arguments. One of our ulterior motives is to show you what mathematicians actually do: they create proofs.

Question

I'm convinced that proofs may be a good thing, but I'm still a little skeptical. What does a proof actually have to do with the validity of an argument?

Answer

On the one hand, a proof establishes the validity of an argument. The reason is that, in a proof, every line must be true if the preceding lines are true. In particular, the truth of the first lines, the premises, implies the truth of the last line, the conclusion. Hence a proof does show that an argument is valid. Much less obvious, but reassuring, is the fact that every valid argument in propositional calculus has a proof. In other words, an argument is valid if and only if there is a proof of it.


The only way to learn how to find proofs is by practice. This is best accomplished by doing lots of examples. We'll try to give some tips as we go along.











So far, all the arguments we have seen happened to be valid. But who says that all arguments are valid?



The following example is reminiscent of the kind of question that often appears in aptitude tests (such as the LSAT).


5. Rules of Inference Section 6 Exercises Main Logic Page "Real World" Page

We would welcome comments and suggestions for improving this resource. Mail us at:
Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: July, 1996
Copyright © 1996 StefanWaner and Steven R. Costenoble