42 False Conversion of PropositionsThe example about the Communists and the A.S.C., if we may labor it, also illustrates another important fallacy. We remarked above (p. 147) that "Only Communists are members of the A.S.C." can be translated into "All members of the A.S.C. are Communists," presumably a false sentence. If it were true, the speaker would certainly have said it, and with emphasis, rather than using the trick of leaving it to his audience to supply the "only." But the point we wish now to make is that frequently unsound syllogisms succeed in escaping detection because the audience is prone to "convert," that is turn around, one of the premises in a way to make tenable its relation to the other premise and, hence, to the conclusion. It is sometimes difficult, in following or advancing an argument, to bear in mind just how true propositions can be validly converted and still make the same assertion. Perhaps some of the time speakers take advantage of the ambiguous English usage discussed in the last section ("Children are cruel") in order to avoid thinking out just how they are entitled to convert some proposition they know to be true, without intending to commit the fallacy of suppressed quantification. In this way they will at least avoid the fallacy of false conversion.
This is the fallacy of getting the turning-about process wrong, so that, though one starts with a true proposition, he ends up with one that does not follow. In a simple case, it is easy to see the kind of mistake one should avoid. Only a very small child, learning that dogs are furry animals, will regard all furry animals as dogs, and say, "Doggie!" when he sees a cat or a pony. "All dogs are furry animals" converts into "Some furry animals are dogs." All S is P converts to Some P is S, or Only P is S.
Particular affirmative propositions also have their valid and invalid conversions. These, however, convert directly, as one can see intuitively. Some S is P converts to Some P is S.
This is as good as we can get with the universal propositions in respect to "some." But notice, one cannot here say "only." "Some great sinners became great saints." With an adjustment of the tenses of the verbs, this converts to "Some of those who became saints had been great sinners," but not (alas!) to "Only those who became saints had been great sinners."
The negative universal propositions also convert directly. No S is P converts to No P is S.
But one is not apt to fail to notice this. "No lips that have touched wine shall ever touch mine," seems perfectly clear no matter whose lips we start with: "No lips of mine . . ."
The remaining case, the particular negative, is ticklish. "Some students are not interested in politics" does not assert that some people other than students are interested (though, of course, this is true). It does not even assert that anybody else is also not interested in politics, that is, that the class of those not interested in politics has members other than students. Nor does it assert that some students are interested; it is the temptation to make this last supposition that is dangerous here.
The temptation is to fill in the dotted lines. Here the temptation will occur in the simple case (for everybody knows from other sources that some students are in fact interested). In the case of more difficult formulations, one will perhaps be more careful. "Some stars cannot forever maintain their stability. They explode into novae." One would hesitate to convert here and say that some stars can remain stable, for it is easy to see that this statement involves a long-range prediction in a very technical matter, indeed. This kind of proposition also converts directly, but (as the diagram shows), one must be careful of the not, keeping it always with the same term. "Some S is not P converts to "Some Not-P is S." Thus there is a second source of error here. "Some disloyal Americans are not communists." False conversion: "Some communists are not disloyal Americans." This remains to be discovered, if true, from other evidence. It does not follow from the obvious truth of the first proposition. All that follows is that some people other than communists are disloyal Americans.
The fallacy of false conversion consists in making an illicit conversion of any of these four kinds of propositions. Let us gather in one place the permissible conversions:
1. All S is P converts to Some P is S or only P is S
2. Some S is P converts to Some P is S
3. No S is P converts to No P is S
4. Some S is not P converts to Some not-P is S
Even where the resulting proposition is known to be true, the conversion is not allowed. If one does know this, he knows it from some other information, and it is this information which he should codify, so to speak, into his propositions. Otherwise what he says, though true, leads to invalid argument.
It can now be seen how some kind of implicit false conversion lies behind many instances of invalid syllogisms. In a school-boy example of the undistributed middle (see #40) and many similar cases, this is very conspicuous: "All asses have ears. You have ears. Therefore, you are an ass." One would not propose this minor premise to a (humorless) superbrain, for the only plausibility comes from the fact that a false conversion of the major premise is assumed: "Only asses have ears" or "All those with ears are asses" (instead of "Some of those with ears are asses"). And in the instance of the Communists belonging to the A.S.C., the speaker leaves it to his audience to make the false conversion and thus justify the inference.
EXAMPLE COMMENT A religious pamphlet claims "As all religious men rely on a power beyond human understanding, so no one who relies on such a power can be wholly devoid of religious feeling." As in most occurrences of logical fallacies, this example disguises itself by a shift in expression. If the writer had said, ". . . so no man who relies on such a power can be a non-religious man," he would have provided a case of false conversion of a universal affirmative proposition, though still not a simple case of it. "No P is not-S" is a consequence of "All P is S," right enough, but not of "All S is P" ("All religious men rely . . ."). The actual false conversion is thus a step left out. Persons who indulge in compulsive superstitious practices are probably not "religious men" in the sense intended, though they surely can be said to rely on a power beyond, well beyond, human understanding! Employee Peter exclaims, "Some bosses just can't take advice. I tried to tell Mr. Henry that the new man isn't worth beans, and now the fellow's lost three old customers in a week. I guess most men who are humble enough to listen to advice just don't become bosses." Assuming that "are humble enough to listen to advice," means "can take advice," then Peter has falsely converted, unless the last sentence is offered independently of the first. This is quite possible in querulous material, which is notoriously innocent of relevance and coherency.