Appendix

SOME LOGICAL PARADOXES

This discussion is appended for those who might have an interest in modern logic. It is not generally known that some of the apparent paradoxes of material implication also occur in the ordinary language, where they are used for special effects.

The logical relations between sentences hold even when the sentences are relevant to one another only in the largest sense, their truth values. Thus the usual interpretations in logic of the terms "or," "if," and the like, has no regard for the content of the sentences related by them, but considers only whether or not the sentences are true. Persons untrained in logic find some of the examples used by logicians very paradoxical: "Today is Tuesday, or Truman is President. But Truman is not President, so today is Tuesday." This is a perfectly valid inference, and every Tuesday since the inauguration of Eisenhower, it has yielded a true conclusion. The mutual irrelevancy of the two sentences leads to a feeling that it is not very good reasoning to decide the day of the week on the grounds that somebody is not President.

The logician is not interested in the grounds of belief in so far as they involve content other than truth value. His grounds for affirming or denying a conclusion are the relations obtaining among true or false sentences when they are combined according to exactly determined operations. An even more striking instance of this attitude results from what is called "material implication." This is a rule of interpretation which takes every complex sentence of the form "if a, then b" to be syntactically equivalent to any compound sentence of the form "not-a, or b." It is evident that in all cases in common speech, in fact in all cases whatever, where the conditional is true, the corresponding alternative is also true. "If it rains, the streets are wet" has the same truth value as "It does not rain, or the streets are wet."

Rule: If a, then b is equivalent to Not a, or b.

Yet one feels that a compound sentence expresses a weaker relationship than a complex sentence. In this case they have the same truth values, of course, but in an accidental sort of way. The conditional seems to show a relationship, to express a tighter, more essential connection between the two facts. And when one considers some of the consequences of the equivalence that Bertrand Russell and other logicians have explored, one finds them shocking to common sense. A few examples will illustrate this feeling of paradox.

An equivalence, of course, works both ways. Not a, or b is equivalent to If a, then b; "Red is not a primary color, or the earth revolves around the sun." The first sentence is false, the second true, and thus the compound sentence, as a whole, is true. This is already bad enough, though we are familiar with this use of "or," to show that one side or the other must be true. But when we translate into the conditional, according to the rule, we get "If red is a primary color, then the earth revolves around the sun." This seems to suggest that a truth about colors is grounds for the astronomical truth. But the point here is that any true sentence can be the consequence of any other true sentence, in this interpretation.

Suppose we put a true sentence on each side of the alternative. "Arsenic disagrees with me, or the moon is a heavenly body." We are also familiar with the use of "or" which allows both sides to be true: "I'll phone or drop in to see you" remains true if I do both, though, of course, I must do one or the other. Now the arsenic-and-moon example translates to "If arsenic agrees with me, then the moon is a heavenly body." (Note: "not disagrees" equals "agrees.") As this case is entirely representative, we can say that any false sentence implies any true sentence. Yet a known false sentence is strange grounds for believing a true sentence, even one surely known to be true.

So far we have seen that any sentence, true or false, is "grounds" for affirming any true sentence. What of false sentences? We already know that any false sentence implies any true sentence. It now develops that any false sentence also implies any other false sentence. To take an example, "If Carroll is a figment of the Red King's dream, then Brooklyn is the capital of Ethiopia." This is equivalent by the rule to "Carroll is not a figment of the Red King's dream, or Brooklyn is the capital of Ethiopia," perfectly true, since the first member is true.

These paradoxical results of the material interpretation of If a, then b as Not a, or b are often summarized as follows: a true sentence is implied by any sentence, true or false; a false sentence implies any sentence, true or false.

The feeling of "paradox" arises from two facts of common usage. The first has already been mentioned, that language characteristically employs conditionals to express general principles. This is to say that the conditionals are themselves general principles or are particular applications of them. Take the case "If 7 = 3 + 2, then 7 = 2 + 3." Not bad; though the hypothesis is false, the conditional does express a general principle about the meaning of "=" and "+". Apparently we don't mind reasoning on false hypotheses when a condition of relevance is preserved, that the conditional express a general principle: "If Hoover is the uncle of Washington, then Hoover is a man" is a fanciful but clear illustration of the general principle "If x is the uncle of y, then x is a man." Now if one should say, "If 7 = 3 + 2, then daffodils bloom in the fall," this condition of relevance is lacking. There is no general principle expressed here; that is, the sentence is itself no general principle about the meaning of relations like "=", nor is it a special application, like the uncle sentence, of some principle, say, about the meaning of "7" or "daffodils," or the like. The conditional is true but vacuous. Our feeling is as if, turning from the street, we opened the front door of a house and stepped through into a vacant lot. We've "entered," all right, but we have got nowhere -- like a Hollywood set.

The second fact of usage is that ordinarily we don't start with sentences already ear-marked "true" or "false." When we use conditionals, we are trying to give our grounds for affirming that some proposition is true or false. Yet there are occasions when ordinary people speak rather like logicians. These are the cases when people want to demonstrate their conviction of the certain truth of a proposition. This works very well for alternatives: "Today is Tuesday, by heaven, or my name is mud!" "I paid you back that five dollars, or I'll eat my hat." Something similar also occurs in conditionals -- we have seen how alternatives can always be translated into conditionals. "If today is not Tuesday, my name is mud!" Falstaff is very fond of this form of asseveration. "If manhood, good manhood, be not forgot upon the face of the earth, then am I a shorten herring." "An [if] I have not forgotten what the inside of a church is made of, I am a pepper-com." "If I be not ashamed of my soldiers, I am a soused gurnet."

Falstaff's semi-oaths are a reminder that these paradoxical uses are perfectly valid. When the conditionals are true, a true conclusion can be derived from them. Since arguments based on the material interpretation are valid, it is a fallacy to reject these arguments as invalid, no matter what feeling of paradox they arouse. The exploration of the forms of valid inference based on the nature of logical operations is a proper study of logic. We must conclude our discussion with a few examples of misunderstandings of the nature of the logical enterprise.

EXAMPLE COMMENT
Paul, who knows some logic, argues with Peter. "Either you should take a logic course, or if you should take a logic course, then you should not take a logic course." Peter exclaims, "And talk like Gertrude Stein! If I ever in my life heard an invalid argument, that is one." This certainly sounds like nonsense. Presumably Paul would explain that all complex sentences in the form of "a; or if a, then b" are necessarily true. Examination shows that, as one side or the other of this alternative must be true, therefore, the alternative is true. If we suppose a to be true, there is not a problem: one side is true. The whole conditional on the right side will be false, in Paul's argument, as it says that a true sentence a implies its contradictory. Now if we suppose a to be false, then the right side is true, for the conditional will simply state that a false sentence implies some other sentence, in the example its contradictory, a true sentence. But every conditional whose antecedent is a false sentence is, as we have seen, necessarily true. If Paul had said, "You should study logic, or at least if you should study logic anything is possible," the nature of the argument might have been clearer to his friend.
Paul tries again, "You should study logic or if you should study logic, then you are able to profit by it." Peter throws up his hands and turns on the radio very loud. The same tautology -- a vacuously true sentence, but nevertheless, true. If it is true that Peter should study logic, then what follows the "or" can be false. If it is false that Peter should study logic, then either he should study it or . . . , and for the dots we must supply a true sentence. As we just saw, on the hypothesis that "a" is false, "if a, then b" is necessarily true, as any false sentence implies any other sentence, true or false. True then: "If you should study logic (but you shouldn't), then you are able to profit by it" even though you wouldn't learn a thing.

Again, suppose Paul had said, "You should study logic, or by heaven, if that's false, then anything else goes in this idiotic world -- such as that you could learn something from it." In this form, for some mysterious psychological reason, the argument seems convincing.