Fallacy

44 Trouble with Conditionals and Alternatives
Sometimes one of the premises is expressed in the form of a conditional -- "If the parties agree in principle, a compromise is always possible," "If I arrive before the instructor, I am not really late." Conditionals have the form "If a (antecedent), then b (consequent )." Now if, in the second premise, one affirms the antecedent, he may affirm the consequent as a conclusion. This form of argument, called the "modus ponens" in logic, is a perfectly valid and useful kind of inference -- "The parties in this case do agree in principle, so a compromise is now possible." "But I see that the instructor is not here yet, so I am not really late, after all."
A second form of valid inference from conditionals results from denying the consequent and then denying the antecedent as a conclusion -- "If it has been raining, the streets will be wet. But the streets are not wet. Therefore, it has not been raining."
If the conditional is true in general, that is, if the antecedent gives a sufficient condition for the consequent, then when the antecedent is true in this particular case, the consequent must be true also. And when the consequent is not in this particular case true, then the antecedent could not be true. If it is true that every time it rains the streets are wet, then if it rains now, the streets must be in fact wet. And if it is false that the streets are now in fact wet, then it can not have been raining.
There are two common mistakes in using this form of argument, corresponding to the two valid forms. The valid forms can be summarized as affirming the antecedent, or denying the consequent. The invalid forms are called traditionally "denying the an-tecedent" and "affirming the consequent." The mistake seems to consist of treating the conditionals as expressing, not only sufficient, but also necessary conditions. Suppose I can affirm that if and only if it rains the streets are wet. I can now affirm, when it has not been raining, that the streets will not be wet. Or, in the case when the streets are wet, I can affirm that it has been raining. But the conditional in the if-and-only-if interpretation is false: as a general principle, rain is not a necessary condition for the wetting of streets -- the water-wagon may have washed them down. Clearly, in view of this possibility, from the mere fact that the streets are wet, I cannot affirm that it has been raining. Rain is a sufficient condition for wet streets, but not a necessary condition. This being the case, nothing follows from "denying the antecedent" or "affirming the consequent."
Alternatives also frequently serve as premises in argument. The speaker affirms that a or b; then he affirms that not-a, and in a conclusion that, therefore, b. When any sentence is substituted for the letters, if the alternative is true, then by denying one of the sentences, the speaker obtains a valid inference to the other. "The chairman is sick, or he would surely be here. But he is not here, so he must be sick." Or, with the same alternative, "I know he is not sick, so he must be around here somewhere."
The improper use of the alternative corresponds to that of the conditional. The speaker supposes that the alternative divides the universe of discourse so that nothing is left over: a or b is equivalent to a or not-a. This kind of total division which leaves nothing over is called a "disjunction." Sometimes it is possible to substitute sentences for the letters so that a disjunction results: "It is either raining or it is not raining." "Isn't a student either present or absent?" Unless the alternative is a proper disjunction, it is a fallacy to suppose that the affirmation of one sentence entails the denial of the other. "He is sick, or he will come. But he will come; therefore, he is not sick/' He may, of course, come even if he is sick. Like the other logical fallacies, arguments based on affirming the consequent or denying the antecedent, or on affirming one of the two sentences in an alternative, are often disguised by a shift of expression. Moreover, in common speech there is the ambiguity of "if," which may mean simply "if" or may mean "if and only if." Likewise, "or" may mean "either one or the other but not both," or it may mean "possibly both, but certainly one or the other." In the course of a long argument, speakers often busy themselves producing lengthy evidence that one of the sentences in an alternative is true on the assumption that this disproves the other, or alleging evidence against the antecedent of a conditional, as if this disproves the consequent.
EXAMPLE COMMENT
A political commentator writes, "If the farmers will organize, they have a. good chance of keeping the price supports. But who ever heard of farmers really getting together on anything? They are by occupation and conviction individualists to the core." This is an enthymeme, and the readers are invited to supply the missing proposition, in this case the conclusion: ""Therefore, the farmers do not have a good chance of keeping price supports," an invalid inference. The Administration may hope to retain the farm vote by keeping the supports, even if the farmers do not organize to fight their political battles. This is an instance of the fallacy of denying the antecedent.
Peter argues with the reader, "If I don't get B in this course, there is no justice in this world." Later on, surprised at getting a B, he reminds the reader of what he had said and exclaims, "So there is justice in this world, after all!" It probably was not a logic course. This is another instance of denying the antecedent. Since the antecedent is negative, the denial of it consists in negating a negative: "I did not not-get-B in the course," or, cancelling, "I got B in the course." The denial of the negative consequent operates similarly. But the denial of the consequent does not follow. Translating into an alternative, we have, "I get B in the course, or there is no justice. But I get B." Therefore? These alternatives are not proper disjunctives; one might, for instance, get his B because of favoritism, a form of injustice in the world.
Heard on a broadcast sermon: "If a man is self-seeking in all the important crises of life, his life as a whole will become a permanent crisis of lonesomeness, bitterness and futility. No true saint is ever lonesome, even in the desert. Nor is he bitter in disappointment. He may be despised and even persecuted and all his works destroyed, but no one could call such a life a life of futility. The saint, if only by his shining example, is the most dedicated to the service of his fellow man of all the noble classes of human beings. Thus the saint can never be self-seeking, even if he retires from the world to devote himself exclusively to seeking himself, in the sense of seeking his personal salvation." Here is a typically complicated argument, exploiting all sorts of ambiguities and rhetorical flourishes, operating on complex and vaguely delimited categories (self-seeking, futility), which allow for paradoxical contrasts (not self-seeking though seeking only himself), and a multiplicity of terms, all lumped together. Yet the skeleton of the argument remains clearer than is often the case in practice. "If a, then h; but not-fe, thus not-a" -- a valid inference.
Mrs. Peter complains to her neighbor, "When it rains, it pours. This must be my rainy season, as everything is happening to me at once. Just let me tell you . . ." When has the force of if. The conclusion is given vaguely, "This must be my rainy season," but it still does not follow from affirming the consequent. Other things might cause "it" to "pour" besides the metaphorical rain.
A newspaper editorializes: "The candidate is a fool or a knave. We know that he is not very bright. He played footsie with left-wing groups one day only to turn around the next and make anti-communist speeches. I guess we exonerate him from knavery -- while noting that a well-intended dupe often does more damage than a dyed-in-the-wool villain." Again much freedom with language, yet, again, the form of the argument is clearer than is usual in such writing: "a or b; but a, therefore not-b." Invalid. It seems highly likely that many communists, if that is what is meant, are both fools and knaves. Perhaps all of them are, in some sense of these vague terms.

Logicians customarily treat fallacies that take advantage of the imprecisions of language as part of formal fallacies. Ambiguities, doubleness in syntax, and the like, can technically be regarded as instances of what used to be called "the four terms." This is a reference to the rule (#40, p. 145) that syllogisms cannot be validly constructed with more than three terms. Clearly, where a word is used in two several senses, there only appear to be three terms, and fallacy results just as surely as if a separate term had been frankly introduced. The following fallacies we treat rather loosely as a group of technical vices or devices that exploit the vagueness of ordinary language.
In the ordinary language a relatively small number of structures must cover a multiplicity of functions. Webster's Collegiate says that the preposition in is used to show the relationship of containment ("Peter is in the breakfast room."), of part to a whole ("Peter is the dullest boy in his class."), of condition or circumstance ("Peter is in business." "His house is in escrow -- it is also in ruins!"), of certain limitations ("Peter argues only in anger -- or in circles."), as well as relationships of many other sorts. It is also used in idioms: "in as much as," "in fact," "in particular." Can any two persons be certain that they understand the following sentences in the same sense as to the force of the prepositions (to say nothing of the vagueness of the other terms)?

In the architectural structure, man's pride, man's triumph over gravitation, man's will to power, assume a visible form. Architecture is a sort of oratory of power by means of forms,
Nietzsche

The process by which language handles different material contexts with the same linguistic forms is called "equalization." Forms logically distinct are equalized. The most notorious fallacies resulting from equalization are the fallacy of ambiguous terms and that of amphibole. Other fallacies that involve a play on vagueness or general ambiguity are those of accent, punctuation, and the like.

EXAMPLE	COMMENT
A political commentator writes, "If the farmers will organize, they have a. good chance of keeping the price supports. But who ever heard of farmers really getting together on anything? They are by occupation and conviction individualists to the core."	This is an enthymeme, and the readers are invited to supply the missing proposition, in this case the conclusion: ""Therefore, the farmers do not have a good chance of keeping price supports," an invalid inference. The Administration may hope to retain the farm vote by keeping the supports, even if the farmers do not organize to fight their political battles. This is an instance of the fallacy of denying the antecedent.
Peter argues with the reader, "If I don't get B in this course, there is no justice in this world." Later on, surprised at getting a B, he reminds the reader of what he had said and exclaims, "So there is justice in this world, after all!"	It probably was not a logic course. This is another instance of denying the antecedent. Since the antecedent is negative, the denial of it consists in negating a negative: "I did not not-get-B in the course," or, cancelling, "I got B in the course." The denial of the negative consequent operates similarly. But the denial of the consequent does not follow. Translating into an alternative, we have, "I get B in the course, or there is no justice. But I get B." Therefore? These alternatives are not proper disjunctives; one might, for instance, get his B because of favoritism, a form of injustice in the world.
Heard on a broadcast sermon: "If a man is self-seeking in all the important crises of life, his life as a whole will become a permanent crisis of lonesomeness, bitterness and futility. No true saint is ever lonesome, even in the desert. Nor is he bitter in disappointment. He may be despised and even persecuted and all his works destroyed, but no one could call such a life a life of futility. The saint, if only by his shining example, is the most dedicated to the service of his fellow man of all the noble classes of human beings. Thus the saint can never be self-seeking, even if he retires from the world to devote himself exclusively to seeking himself, in the sense of seeking his personal salvation."	Here is a typically complicated argument, exploiting all sorts of ambiguities and rhetorical flourishes, operating on complex and vaguely delimited categories (self-seeking, futility), which allow for paradoxical contrasts (not self-seeking though seeking only himself), and a multiplicity of terms, all lumped together. Yet the skeleton of the argument remains clearer than is often the case in practice. "If a, then h; but not-fe, thus not-a" -- a valid inference.
Mrs. Peter complains to her neighbor, "When it rains, it pours. This must be my rainy season, as everything is happening to me at once. Just let me tell you . . ."	When has the force of if. The conclusion is given vaguely, "This must be my rainy season," but it still does not follow from affirming the consequent. Other things might cause "it" to "pour" besides the metaphorical rain.
A newspaper editorializes: "The candidate is a fool or a knave. We know that he is not very bright. He played footsie with left-wing groups one day only to turn around the next and make anti-communist speeches. I guess we exonerate him from knavery -- while noting that a well-intended dupe often does more damage than a dyed-in-the-wool villain."	Again much freedom with language, yet, again, the form of the argument is clearer than is usual in such writing: "a or b; but a, therefore not-b." Invalid. It seems highly likely that many communists, if that is what is meant, are both fools and knaves. Perhaps all of them are, in some sense of these vague terms.