Inconsistency is a very important notion for analyzing and evaluating argumentation in dialogues. Suppose that in the dialogue on tipping in chapter1Helen has argued for her thesis that tipping is a bad practice. But then suppose that later in the dialogue she brings forward an example, from her own experience, where she tipped a taxi driver who had gotten her to the airport on time. She was happy to give him a tip, and she admitted that in this particular case, giving him a tip was a good thing. Let’s suppose that when Helen argued for her thesis that tipping is a bad practice, she meant this statement to be an absolute generalization. In other words, she was meaning to say that all cases of tipping fall under the heading of something that is a bad practice. But then later in the dialogue, when describing this example, she admitted that giving the tip was a good thing. Something being a good thing is the opposite of its being a bad thing. Therefore, it seems that in this dialogue, Helen’s earlier statement was inconsistent with her later statement. The reason is that it is not logically possible for both statements to be true. Inconsistency can be defined as follows. Two statements are inconsistent if it’s not logically possible for both of them to be true. By ‘inconsistent’ we mean logically inconsistent and not just practically or physically inconsistent. ‘Bob is a sixty-nine- year-old man’ may be practically or physically inconsistent with ‘Bob can run a mile in under four minutes’, but the two statements are logically consistent with each other. However, ‘Bob is a sixty-nine-year-old man’ is logically inconsistent with ‘Bob is not a sixty-nine-year-old man’. In the dialogue, it seems to be a bad thing that Helen has committed herself to an inconsistency. So we have to ask: What is wrong with inconsistency when it occurs in argumentation?
Inconsistency is very common, especially in long and complex argu- ments where we tend to lose track of our previous arguments. Thus, in a way, inconsistency in argumentation is quite understandable. But the problem is that when you have an inconsistent pair of statements, it is not possible for them both to be true. Thus, in order to maintain truth, in such a case, an arguer has to give up the one statement or the other. Suppose, for example, that Bob, in the dialogue on tipping, says that Helen’s two state- ments are inconsistent. In other words, suppose that Bob points out that Helen had earlier maintained that tipping is not a good practice, but then
1. Inconsistency 45
in her later example she had maintained that tipping is a good practice. Suppose that Bob said, “Look Helen, you can’t have it both ways, either tipping is a good practice or it is not.” What Helen has to do in such a case is to make a retraction. She could do this in various ways. She could say, for example, “I didn’t really mean to say that all cases of tipping are bad. All I meant to say was that tipping is generally a bad practice, subject to exceptions.” Or she could retract her later statement. She could say, “Well I didn’t really mean to say in this example that my tipping the taxi driver was a good thing. I still maintained that all tipping is a bad practice, but in this case I just wanted to make the point that sometimes the tip can be deserved, and that the act of tipping can be good in certain respects, even though looking at it from an overall point of view, it is a bad practice.” We can see here that Helen has retracted the inconsistency by clarifying what she was trying to say. Although inconsistency can in many cases be understood and can be resolved, in general it is something that has to be addressed in argumentation once it has been discovered. The reason, once again, is that when you have an inconsistent set of statements, not all of them can be true.
A very simple case of inconsistency, called a contradiction, occurs when one statement is the direct opposite of another. The following two statements, taken together, constitute a contradiction.
Mount Lemmon is in Arizona. Mount Lemmon is not in Arizona.
In this case it is clear that the one statement is the direct opposite of the other. One is the negation of the other, indicated by the word ‘not’ appearing in the second statement. Because of this word, it is immediately evident that it is not possible for both statements to be true.
In other cases of inconsistency, one statement is not the negation of the other, yet it is clear that one asserts what the other denies. For example, consider the following set of statements, where ‘all’ is taken as an absolute universal generalization.
(a) All wolves are pack animals.
(b) Some wolves are pack animals.
(c) Some wolves are not pack animals.
(d) No wolves are pack animals.
(f)It is false that some wolves are not pack animals.
(g)All wolves are not pack animals.
It is clear enough that (a) is consistent with (b). Two statements are con- sistent if it is possible for both to be true. But is (a) consistent with (c)? No, it would seem not, for it is not logically possible for both statements to be true. If all wolves are pack animals, with no exceptions, the possibility that some are not is ruled out. (a) is the direct opposite of (d). What one asserts the other denies. Hence, (a) and (d) are inconsistent. On the other hand, (a) is consistent with (f). But (c) is not, for (c) is the direct opposite of (f). And clearly, (a) is inconsistent with (f), for it is not possible that both could be true. In contrast, (b) is consistent with (c), because it is possible that both are true. So we see that in many cases of inconsistency, there is no direct contradiction, but one can still determine that a pair of propositions is inconsistent by asking whether it is possible for both to be true.
In still other cases, we are confronted with a set of statements, often more than two, that are collectively inconsistent. In many of these cases, one cannot prove that they are inconsistent so easily, and more work is involved. One has to draw out the contradiction by analyzing the argu- mentation. For example in the tipping case above, we have to show that the statement that a thing is bad is inconsistent with the statement that it is good. We can easily draw out this assumption by observing that part of the meaning of the term ‘bad’ is that something is being condemned as ‘not good’.
It is also common to have cases where there is no direct contradiction, but where a set of propositions is collectively inconsistent. Consider the following set of statements. Let’s assume that the universal generalization in the first premise is absolute.
All police chiefs are honest. John is a police chief. Taking a bribe is dishonest. John took a bribe.
If you consider all four statements together as a set, it is clear that not all of them can be true. They contain an inconsistency. Hence at least one is false. But there is no direct contradiction. There is no one statement that is the opposite of another one in the set. However, a contradiction can be derived from the set by a chain of argumentation. Consider the first pair
1. Inconsistency 47
of statements in the set. From them, the conclusion follows that John is honest, by the following argument.
PREMISE: All police chiefs are honest.
PREMISE: John is a police chief.
CONCLUSION: John is honest.
Next, consider the last two statements. Let’s assume that the first of the pair is an absolute generalization. By expressing it in this way, as below, the following argument is produced.
PREMISE: All persons who took a bribe are dishonest. PREMISE: John took a bribe.
CONCLUSION: John is dishonest.
The conclusion of the second argument is the opposite of the conclusion of the first one. The reason is that ‘dishonest’ means ‘not honest’. Thus, it has been revealed that if you consider the whole set of four statements together, you can show by a chain of argumentation that they lead to a direct contradiction. If someone engaging in argumentation in a dialogue were to state all four of these propositions as part of his argument, and the other party in the dialogue showed that they are collectively inconsistent, then the party who stated them would have to resolve the inconsistency. The obvious way to do this would be for him to give up one of the propo- sitions in the set.
To review the basic definition of inconsistency, a set of statements is said to be inconsistent if it is not logically possible for all of them to be true. Then we can say that a set of statements is consistent if it is logically possible that all of them could be true. Finally, let’s note that propositions can be consistent with each other even if they do not appear to be related to each other in any way. For example, consider the following two statements.
Genetically modified foods cause increases in allergies. Tipping is a bad practice.
In this case, the two statements are consistent, because it is possible for both to be true. The evidence for such a claim in this case is that the one statement is not relevant to the other. This implies that you can use one to
prove the opposite of the other. Generally speaking, then, if one statement is not relevant to another one, the two must be consistent with each other. However, the subject of relevance will not be taken up until chapter7.
All ‘consistency’ means is that it is possible for both statements to be true. It doesn’t mean that either of the statements actually is true. But a finding of inconsistency in a set of statements means that one of them has to be false. The basic problem with an inconsistent set of assertions in argumentation is that it is not possible for all of them to be true. Hence the allegation that an arguer has put forward a set of assertions that con- tains an inconsistency is a powerful and important kind of criticism. If an arguer in a dialogue has committed herself to a set of statements that are inconsistent, and the other arguer in the dialogue points out the con- tradiction, then the first arguer must deal with that criticism right away. The usual way to deal with it is to retract one of the statements. Although inconsistency is often quite understandable and can be fairly common in argumentation, it is something that has to be dealt with once it has been pointed out. If someone’s argument has been shown to be inconsistent, it does not necessarily mean it is entirely worthless and beyond repair. But it does mean that the argument is not acceptable as it stands. The arguer must deal with the criticism of inconsistency if her argumentation is to move forward successfully in the dialogue.
The finding of inconsistency is vitally important in legal argumen- tation. If a witness offers an account that contains an inconsistency, the cross-examining questioner in court can tear the story to shreds. Unless the witness can resolve the inconsistency, her credibility may be destroyed. The finding of inconsistency is also important in scientific argumentation where, for example, a theory may be shown on careful analysis to contain a hidden inconsistency. By finding and dealing with such inconsistencies, scientific discovery and theory formation is able to move forward.
EXERCISE 2.1
1.Determine which propositions are inconsistent with others, and which are not, among the following sets of propositions.
(a) Some peacocks are afraid of kangaroos.
(b) No peacocks are afraid of kangaroos.
(c) All peacocks are afraid of kangaroos.
(d) It is false that all peacocks are afraid of kangaroos.
(e) Some peacocks are not afraid of kangaroos.
2. Three Kinds of Arguments 49 (g) At least one peacock is not afraid of kangaroos.
(h) It is false that at least one peacock is afraid of kangaroos.
2.Prove that the following set of statements is inconsistent by finding a contradiction that can be derived from the argumentation in them.
(a) All romantic idealists love poetry.
(b) All persons who love poetry are affectionate.
(c) All affectionate persons love pets.
(d) Sam is a romantic idealist.
(e) Sam is afraid of my pet snake.
(f) No person who is afraid of my pet snake loves pets.