An argument is said to be valid or to have a valid form if each deduction in it can be justified with one of the rules of inference listed in the previous section. The form of an argument might be valid, but still the conclusion may be false if some of the premises are false. So to show that an argument is good we have to be able to do two things: show that the argument is valid (i.e. that every step can be justified) and that the argument is sound which means that all the premises are true. If you start off with a false premise, you can prove anything!

Consider, for example the following “proof” that \(2=1\).

Suppose that \(a\) and \(b\) are two real numbers such that \(a=b\).

	by hypothesis, \(a\) and \(b\) are equal, so
\(a^2 = ab\)
	subtracting \(b^2\) from both sides
\(a^2 - b^2 = ab - b^2\)
	factoring both sides
\((a+b)(a-b) = b(a-b)\)
	canceling \((a-b)\) from both sides
\(a+b = b\)

Now let \(a\) and \(b\) both have a particular value, \(a=b=1\), and we see that \(1+1=1\), i.e. \(2=1\).

This argument is not sound (thank goodness!) because one of the premises — actually the bad premise appears as one of the justifications of a step — is false. You can argue with perfect logic to achieve complete nonsense if you include false premises.

So, how can you tell if an argument has a valid form? Use a truth table. As an example, we'll verify that the rule of inference known as “destructive dilemma” is valid using a truth table. This argument form contains 4 predicate variables so the truth table will have 16 rows. There is a column for each of the variables, the premises of the argument and its conclusion.

Now, mark the lines in which all of the premises of this argument form are true. You should note that in every single situation in which all the premises are true the conclusion is also true. That's what makes “destructive dilemma” — and all of its friends — a rule of inference. Whenever all the premises are true so is the conclusion. You should also notice that there are several rows in which the conclusion is true but some one of the premises isn't. That's okay too, isn't it reasonable that the conclusion of an argument can be true, but at the same time the particulars of the argument are unconvincing?

As we've noted earlier, an argument by deductive reasoning can go wrong in only certain well-understood ways. Basically, either the form of the argument is invalid, or at least one of the premises is false. Avoiding false premises in your arguments can be trickier than it sounds — many statements that sound appealing or intuitively clear are actually counter-factual. The other side of the coin, being sure that the form of your argument is valid, seems easy enough — just be sure to only use the rules of inference as found in Table 2.6.2. Unfortunately most arguments that you either read or write will be in prose, rather than appearing as a formal list of deductions. When dealing with that setting — using natural rather than formalized language — making errors in form is quite common.

Two invalid forms are usually singled out for criticism, the converse error and the inverse error. In some sense these two apparently different ways to screw up are really the same thing. Just as a conditional statement and its contrapositive are known to be equivalent, so too are the other related statements — the converse and the inverse — equivalent. The converse error consists of mistaking the implication in a modus ponens form for its converse.

The converse error:

Consider, for a moment the following argument.

If a rhinoceros sees something on fire, it will stomp on it. A rhinoceros stomped on my duck. Therefore, the rhino must have thought that my duck was on fire.

It is true that rhinoceroses have an instinctive desire to extinguish fires. Also, we can well imagine that if someone made this ridiculous argument that their duck must actually have been crushed by a rhino. But, is the conclusion that the duck was on fire justified? Not really, what the first part of the argument asserts is that “(on fire) implies (rhino stomping)” but couldn't a rhino stomp on something for other reasons? Perhaps the rhino was just ill-tempered. Perhaps the duck was just horrifically unlucky.

The closer the conditional is to being a biconditional, the more reasonable sounding is an argument exhibiting the converse error. Indeed, if the argument actually contains a biconditional, the “converse error” is not an error at all.

The following is a perfectly valid argument, that (sadly) has a false premise.

You will get an A in your Foundations class if and only if you read Dr. Fields' book. You read Dr. Fields' book. Therefore, you will get an A in Foundations.

Suppose that we try changing the major premise of that last argument to something more believable.

If you read Dr. Fields' book, you will pass your Foundations class. You did not read Dr. Fields' book. Therefore, you will not pass Foundations.

This last argument exhibits the so-called inverse error. It is by no means meant as a guarantee, but nevertheless, it seems reasonable that if someone reads this book they will pass a course on this material. The second premise is also easy to envision as true, although the “you” that it refers to obviously isn't you, because you are reading this book! But even if we accept the premises as true, the conclusion doesn't follow. A person might have read some other book that addressed the requisite material in an exemplary way.

Notice that the names for these two errors are derived from the change that would have to be made to convert them to modus ponens. For example, the inverse error is depicted formally by:

If we replaced the conditional in this argument form by its inverse (\({\lnot}A \implies {\lnot}B\)) then the revised argument would be modus ponens. Similarly, if we replace the conditional in an argument that suffers from the converse error by its converse, we'll have modus ponens.