PART III
Logical FallaciesThe mistakes and tricks in argument, so common in the markets and parliaments of every-day life, are of small interest to the logician. The logician might allow that faulty generalization, wishful thinking, etc., are sources of error in discussion. On the analogy of a manufacturing process (p. 4), he would perhaps admit that if inadequate materials or human aberration lead people to maintain false premises, they should hardly be surprised when their conclusions are unreliable. But the logician has as his only concern with fallacy the avoidance of error in the processes of argument. He is ceaselessly watchful for breakdowns in the functioning of processes designed to produce valid conclusions. As to the premises from which the conclusions are drawn, he merely assumes them true or false for the purposes of his demonstrations. The logician, as logician, does not care at all what fallacies might have entered into the formulation of the factual premises he is borrowing. As an ordinary man, he may be tremendously concerned, like the rest of us, for on the truth of some moral or political premises may hang the happiness and even the future of the race. But in logic, taken in isolation, the problem of error is a formal problem, quite like mistakes in mathematical proofs. Such mistakes are of great importance, nonetheless, and if they are not avoided, the most vigilant care in safeguarding the purity of the premises will be wasted, since the conclusion will be contaminated. The history of reason, since the invention long ago of logic, is in part the history of guaranteeing the purity of the processes of argument.
Two of the most important notions in the history of reason are logical truth and validity. The elements of these notions must be understood before there can be any profitable discussion of the processes of argument. Though they have been prominent in analysis since the days of Greece, recent decades have seen great progress in the understanding of them. The subject still bristles with difficulty for logicians and mathematicians. Yet the elements perhaps can be explained with sufficient clarity for the present purpose.
LOGICAL TRUTH There are two kinds of true sentences in discourse. One kind is empirical, which is to say, derived from experience. This is what people most often mean when they say that something is true or is a fact. When someone says, for instance, "It is almost four o'clock," the kind of truth involved is empirical: the sentence states a fact or not, depending on the time. All scientific observations and historical statements have this kind of truth or falsity. "The hyena belongs to a species somewhere between dogs and cats." "It was in the Ford Theatre that Abraham Lincoln was assassinated." "The narrator in Moby Dick calls himself Ishmael." "I was driving sixty miles an hour in a twenty-mile zone." There are sometimes great difficulties in ascertaining the truth of empirical sentences or even in understanding what it means to say of such sentences that they are true. Some of these difficulties have been explored in the section on material fallacies. But it can be seen already that where one is content with the truth of such sentences, one can employ them as premises in argument. For the present, we are concerned with the second kind of true sentences, those that are logically true.
Of logically true sentences there are simple varieties that most people are already familiar with. Consider the following sentence, "If today is Tuesday, then today is not not Tuesday," This sentence is true on Wednesday or any other day as well as on Tuesday. "If today is not Tuesday, then it is not not not Tuesday." Thus if there is no negative in the if-clause, any number of true sentences can be made out of the whole conditional by adding even numbers of nots to the then-clause. And where there is a negative in the if-clause, one can know that the whole conditional is true if there is an odd number of nots in the then-clause, no matter how large a number. These sentences are true because of the way "if-then" and "not" are used in English. "If a, then not not a" is always true; in fact, logicians say it is "trivially" true. No matter what sentence, including a false sentence, one substitutes for a, the complex sentence in this case will be true. It is trivial because nobody learns anything about the world of fact from such a sentence.
"If he's an uncle, he's a male" is also trivially true, not only because of the syntax of "if-then," but because of the meanings of "uncle" and "male." "If it is red, it is colored" is likewise true because of the meaning of "red" and "colored." Such sentences explore the semantics of a language for connections and implications between the meanings of the various terms. They may give information about the language in the sense of making evident those relations not previously clear, but they give no other information. Arithmetic in a similar way makes clear the relations among numbers. David Hume, the first to understand well the distinction between empirical and logical truth, gives the example "three times five is equal to the half of thirty." The truth of this proposition, he says, can be ascertained without assuming any knowledge of the world of fact; it is known to be true by reasoning alone -- to deny it implies a contradiction. Let us try to deny it, that is, assume that it is false:
Assuming, 3 x 5 ≠ 30/2 Then, 5 + 5 + 5 ≠ 15 Or, 15 ≠ 15 The above contradiction, saying that fifteen does not equal fifteen, is implicit in the meanings of the various numbers and in the use of the signs for division, addition, etc. To deny that three times five is the half of thirty involves denying that fifteen equals fifteen. This is a contradiction, since it is prerequisite for intelligible discourse that a term be taken in the same sense whenever it occurs in the same context -- that it be identical with itself. Such mathematical propositions, Hume says, are trueby necessity, since one cannot assert them false without contradiction, in tact, cannot even imagine them false.
There is a third variety of true sentences which logicians frequently speak of, "true by definition." There are all sorts of definitions, but it is not the truth of the definitions that logicians have in mind. A dictionary definition can be a correct report of usage, and hence regarded as true like any other report, but this would be empirical truth, which logicians are not concerned with. Moreover, most of the other kinds of definition are not reports of usage.
In mathematics, for instance, where often one symbol is simply substituted for another by a process of definition, it would be meaningless to ask whether or not the definition is true. In general argument, persons often introduce a special definition of a term which may or may not conform very closely to ordinary usage -- that is not the point of the definition. In such cases the definition is intended as a decision to use the term in a particular way for the duration of the argument. This is done, among other seasons, to control the ambiguity of ordinary usage. Clearly it is also meaningless here to ask if the definition is true. How can a decision be true or false? It can be kept to or overlooked. (It is a grievous fallacy in argument to take advantage of this convention, to define a term in one way and then slip over into conventional meanings -- see p. 158.)
What logicians mean when they speak of "true by definition" is the status of a sentence which is implied as a logical consequence of a definition. Suppose in the course of an argument I define "extravagant" as "spending more than you earn." Then I demonstrate that my friend has consistently for years done just that. It follows that my friend is extravagant as defined. The sentence, "My friend is extravagant," granting the other premise about his spending, is true by definition. Like the other kinds of logical truth, definitional truth is also innocent of any factual substance -- the only information it gives is that if you define a term in a certain way, then, the conditions of the definition being met, you can use the term.
These, then, are the three most important kinds of logically true sentences, true by definition, true by the meanings of words. tru,e by the use of syntactical elements. They give no information about the world, but only about the use of language in reasonable discourse. It is this lack of material content that is referred to when it is said that such truth is tautological or trivial.
Not all logical truths are as obvious as "Mars is inhabitable or not." Many sentences in books of logic are true by analysis, as logicians say, but some of them provide the greatest difficulty in that analysis. In less rigorous fields, students are sometimes warned against tautologies. A student theme replete with them may be inane -- of course, this is what teachers rightly protest -- but at least it contains some certainly true sentences, since all tautologies are true.
VALIDITY The reason for the importance of the notion of validity is connected with what has been said about logical truth. If an argument is valid and if the premises are true, then the conclusion must also be true. This "new" truth is also essentially empty of any new information about the world -- it merely reformulates what was already expressed in the premises. It is a mere consequence of truths already stated -- and sometimes, as in arithmetic, the truth of the premises themselves is merely logical, consisting of definitions, tautologies, and the like. The conclusion, however, may embody important novelty, and the mathematician feels himself possessed of a new truth indeed. The student of geometry who masters the Pythagorean theorem is often struck with the fact that this very complicated and unexpected result is a consequence of a chain of reasoning reaching back to a handful of postulates and axioms. If there were men of godlike intellect, as the French philosopher Laplace pointed out, they would immediately see all the possible consequences of whatever consistent set of postulates one could offer them. But men are not gods and thus are capable of surprise.
Moreover, people often wish to test the consequences of their premises without being in a position to say whether or not they are true. The "postulates" may not be obviously inconsistent, but if they are inconsistent, this will become evident as their consequences are explored. It is indifferent to logic whether or not the premises are well founded in experience. One can assume them true and explore the results. Similarly, in ordinary things, one need not wait for certain proof of some propositions before finding out what other propositions would be true, granting them. For instance, some astronomers have asserted that lichens probably grow on the planet Mars. Now lichens are a fungus living symbiotically with an alga. With this fact as a premise, it is easy to construct an argument proving that the cooperative phenomenon of symbiosis occurs on Mars, if one assumes other premises, including "There are lichens on Mars" to be true. Sometimes propositions are actually known to be false, but it would be interesting to see what would be the case, if only they were true. No language has succeeded in becoming an international language, neither Esperanto, nor Io, nor Basic English, but it is certainly legitimate to speculate on the consequences of the adoption of one or another of them. The results of such speculations may actually suggest the design of a practical course of action.
The process we have been describing is possible only because some propositions can be shown to entail other propositions. If one understands the notion of entailment in the sense that conclusions can be entailed by premises merely on the assumption that the premises are true, then he is getting close to understanding what is meant by speaking of valid arguments.
It is the validity of the argument that guarantees the production of true conclusions when they are entailed by true premises. Thus when a conclusion is derived from premises, we cannot know that it is true because of this fact, unless we know that the premises are true. What we can know in many well understood forms of proof -- such as the syllogistic form, which will be examined in some detail -- is whether or not the argument is valid in itself. If the conclusion necessarily follows, then the argument is valid; if it does not, then this is invalid. "If this is an omen, then it is also a portent. But it is an omen. Therefore, it is also a portent." There may be no such things as omens or portents, but the argument is valid.
We will not attempt to define "validity" in the abstract. Let us rather adopt the following rule: An argument is valid if and only if to assert the premises true and the conclusion false involves a contradiction. The notion of validity, it is seen, depends upon the notion of truth. Yet there is a sharp distinction between validity and truth.
All the cases we have been discussing can be shown to be consistent with this rule. The most important case occurs when the premises are in fact true. Here a new truth emerges, for the conclusion is necessarily true, by the rule. Most every-day arguments try to be of this sort. Another case is that of dubious premises assumed to be true for the sake of testing the validity of the argument. "If all the people in New York have purple eyebrows, then they do not believe in Baal. But all the people in New York do believe in Baal. Therefore, they do not have purple eyebrows." The premises are presumably false; it is unlikely that all the people in New York believe in Baal, or that, if they do, this is related to their not having purple eyebrows. Yet the conclusion follows from these premises. Moreover, this conclusion seems in fact true, though foolish. The truth of the conclusion must be independent of these premises, since they are false. This, then, is a case of a valid argument leading to a true conclusion from false premises. How is it seen that the argument is valid? By the rule, because the rule states only that the conclusion could not be false if the premises were true, and we can see that this is so in the example. What we do to see it is assume the premises true for the sake of the argument (as people say), and then observe if the conclusion must follow. This may be hard to do in an absurd case like the above; for this reason it is customary to show the shape of the argument by the use of a notation. A simple notation will show that the argument about the purple eyebrows has the shape: "If a, then not b, but b; therefore, not a." All arguments in this shape are valid. Let the student experiment with this shape 1) using true premises, and 2) using false premises leading to conclusions in fact false.
It is a fallacy, in a way the most fundamental fallacy, to confuse validity with truth. Persons are often convinced of the truth of a proposition because they can see that it follows rigorously from certain premises, while failing to notice that the premises are by no means certainly true. The Nazis convinced themselves that the Germans should rule the world by asserting (a) the master race should rule the world, and (b) the Germans were the master race.
It also happens that an argument seems sound to some people because the conclusion is acceptable and they are told that it derives from premises which they see to be true. I may reason (a) if I can afford to build my own home, I shall not have to go on paying rent, (b) but I can't afford to build my own home, (c) so I shall have to go on paying rent. The conclusion may be a true proposition and so may each of the premises, but the argument is still invalid, since the conclusion could be false, though the premises are true (see pp. 142 ff.). When this happens, the argument does, in a way, contain a statement that is false. This is the statement, implicit in the order of the propositions, or partially avowed in some such expression as "therefore" or "hence" or (as in the example) "so" -- the statement that the conclusion follows. In valid arguments this implicit statement about the argument is true, in invalid arguments, false. Sometimes one explicitly: "it follows that," "Q.E.D.," "you have to admit that," and the like.
Let us turn now to a simple traditional example which applies what we are learning about the difference between valid and invalid argument. (The reason for using the letters "M," "S," and "P" will become apparent.)
so All M is P
All S is M
All S is Pso All P is M
All S is M
All S is PThe scheme on the left can be seen intuitively to be valid. That on the right seems to be a mistaken version of it. No matter what values we substitute for the letters, just as the formula on the left will result in a valid argument, mistaken formulas such as that on the right will result in an invalid argument.
Valid: "All good students like to argue; all our freshmen are good students; so all our freshmen like to argue."
Invalid: "All good students like to argue; all freshmen like to argue; so all freshmen are good students."
Or another invalid case: "All loyal Americans are anti-communist; the American Only League are anti-communist; so the American Only League are loyal Americans."
Suppose the premises perfectly true, the conclusion does not follow -- the League might be fascist or in some other way both anti-communist and still not loyally American. Even if true also, the conclusion still does not follow.
The relationship between the propositions of a syllogism can be diagrammed by the use of circles to represent the classes involved. In the case of valid arguments there is no way to draw the circles to avoid showing the conclusion true. Take the simple form already given:
so All M is P
All S is M
All S is P
The argument is revealed to be invalid if there is any place permitted by the statements made about the classes in the premises so that one can escape the position required by the conclusion. Let P represent the class of loyal Americans, M the class of anti-communists, and S the America Only League. Then the invalid example just cited outlines and diagrams as follows:
so All P is M
All S is M
All S is P
The process is to take the premises as giving directions for the construction of circles. In a valid argument they compel the circles to show that the conclusion is inescapable. Circle diagrams are a sort of game in which one seeks to embarrass the contention that the conclusion follows. If one wins, then the argument is invalid. Let the reader invent arguments constructed according to the above valid and invalid examples. He will gain some insight into the notion that validity is a matter of structure, of form, not simply of the truth or falsity of the premises.
There are large numbers of valid forms of argument. In the discussion that follows we confine ourselves to the traditional forms. We choose to do this, as most of the formal fallacies have been discussed in the literature of the syllogism (a form of the syllogism is given at the left on p. 141). Moreover, as the use of the circles shows, the validity of the syllogism can be seen intuitively and thus does not require the elaboration of difficult formal systems.
Now there are theoretically as many formal fallacies as there are ways to argue invalidly, that is, to make mistakes in arguing from premises to conclusions. But our concern here is with commonly occurring and well recognized errors. Some of the logical fallacies were known to the ancients; philosophers in the Middle Ages were often adroit at detecting them. They persist, nevertheless, to our day, and every educated person should know the name and face of them.