4.3: Soundness

Logic is largely about validity. So to understand clearly what much of the rest of this book is about, you must clearly distinguish validity from some other things.

If I give you an argument by asserting to you something of the form "X. Therefore Y", I am doing two different things. First, I am asserting - the premise or premises, X. Second, I am asserting to you that from these premises the conclusion, Y follows.

To see clearly that two different things are going on here, consider that there are two ways in which I could be mistaken. It could turn out that I am wrong about the claimed truth of the premises, X. Or I could be wrong about the 'therefore'. That is, I could be wrong that the conclusion, Y, validly follows from the premises, X To claim that X is true is one thing. It is quite another thing to make a claim corresponding to the 'therefore', that the argument is valid, that is, that there is no possible case in which the premises are true and the conclusion is false.

Some further, traditional terminology helps to emphasize this distinction. If I assert that the argument, "X. Therefore Y", is valid, I assert something about the relation between the premises and the conclusion, that in all lines of the truth table in which the premises all turn out true, the conclusion turns out true also. In asserting validity, I do not assert that the premises are in fact true. But of course, I can make this further assertion. To do so is to assert that the argument is not only valid, but Sound:

An argument is Sound just in case, in addition to being valid, all its premises are true.

Logic has no special word for the case of a valid argument with false premises.

To emphasize the fact that an argument can be valid, but not sound, here is an example;

Teller is ten feet tall or Teller has never taught logic. AvB
Teller is not ten feet tall. ~A
Teller has never taught logic. B

Viewed as atomic sentences, 'Teller is ten feet tall.' and 'Teller has never taught logic.' can be assigned truth values in any combination, so that the truth table for the sentences of this argument looks exactly like the truth table of Section 4-1. The argument is perfectly valid. Any assignment of truth values to the atomic sentences in which the premises both come out true (only case 3) is an assignment in which the conclusion comes out true also. But there is something else wrong with the argument of the present example. In the real world, case 3 does not in fact apply. The argument's first premise is, in fact, false. The argument is valid, but not sound.

Exercise $$\PageIndex{1}$$

4-1. Give examples, using sentences in English, of arguments of each of the following kind. Use examples in which it is easy to tell whether the premises and the conclusion are in fact (in real life) true or false.

a) A sound argument

b) A valid but not sound argument with a true conclusion

C) A valid but not sound argument with a false conclusion

d) An argument which is not valid (an invalid argument) all the premises of which are true

e) An invalid argument with one or more false premises

4-2. Use truth tables to determine which of the following arguments are valid. Use the following procedure, showing all your work: First write out a truth table for all the sentences in the argument. Then use a '*' to mark all the lines of the truth table in which all of the argument's premises are true. Next look to see whether the conclusion is true in the *ed lines. If you find any *ed lines in which the conclusion is false, mark these lines with the word 'counterexample'. You know that the argument is valid if and only if there are no counterexamples, that is, if and only if all the cases in which all the premises are true are cases in which the conclusion is also true. Write under the truth table whether the argument is valid or invalid (i.e., not valid).

a) ~(A&B) b) ~AvB c) AvB d) AvB e) A
~A A ~BvA ~AvB Bv~C
~B B A A (A&B)v(A&~C)

4-3. Show that X is logically equivalent to Y if and only if the arguments "X. therefore Y and "Y. Therefore X" are both valid.