(Originally appeared in Lognet 96/1)

# Ordinary Propositions That Can Neither Be Accepted Nor Rejected

By Jerome Frazee

We can neither accept nor reject some ordinary propositions without running the risk of being wrong for the wrong reason. The reason for this is that the rejecter both affirms the antecedent and denies the consequence of a conditional. Most of this article gives examples of the problem. Finally, I offer a not very satisfying solution.

If (A ⊃ B) is false then (A & -B) is true. If (A & -B) is true then A is true. Consequently if (A ⊃ B) is false then A is true. We cannot deny a conditional without affirming its antecedent. Curiously, this leads to a common type of proposition that cannot be rationally labeled “true” or “false”. The following scenario illustrates the point.

Mr. Music, the music teacher at the school, has a pupil named Hornblower who his friend, A. Slyfellow, says is actually the archangel Gabriel. Slyfellow says (1) If Hornblower will blow his horn this noon, then the dead will rise this noon. Sentence (1) is either true or false; for it and its denial cover all possibilities. So, the question is: (2) Is (1) true? Mr. Music says (1) is ridiculous and bets A. Slyfellow \$5.00 that it is false. At noon Hornblower does not blow his horn. So A. Slyfellow collects \$5.00 from Mr. Music (which, we presume, he immediately splits with Hornblower).

No one can rationally answer question (2) with Yes or No. In this story Mr. Music is set up by A. Slyfellow; he cannot win, for if he bets that sentence (1) is true then Hornblower will blow his horn and the dead will surely not rise (surely the dead will not rise!). But, it need not have been a set-up; after all, Hornblower might have died, become sick, or simply forgot. However, ordinary debate—especially between politicians—is filled with these kinds of questions. Here is an even better example:

Two legislators were debating a proposed law. Ms. Good said If we pass this proposed law tomorrow, then the people will be better off than they are now. But, Mr. Lawson answered, That is false! The next day Mr. Lawson casts the deciding vote against the proposed law, and it does not pass.

What Ms. Good said was true and what Mr. Lawson said was false, but he was wrong for the wrong reason! The antecedent was false. He voted against what could have caused his words to be true and for what caused them to be false! Nevertheless, he voted for what he held to be true: If we pass this proposed law, then the people will not be better off. Also, if he had agreed with Ms. Good, then she could have broadcast to his constituents that he would be irrational if he voted against the proposed law. With these kinds of conditionals no one can deny the conditional without running the risk of being wrong for the wrong reason, i.e., the antecedent being false instead of both the antecedent and the consequent being true. In the final analysis, Mr. Music, and everyone in a similar situation, cannot make their position known by denying what is affirmed. They must make a different statement without denying the other one! It is even more disconcerting to realize that the affirmer is also saying it the right way. There is no better way for him to say it. Either Hornblower will blow his horn and the dead will rise, or else he will not and the dead will or will not rise; i.e., it is not the case that he will blow his horn and the dead will not rise. This states the affirmer’s position exactly; he does not need to say more and he does not want to say less. If the conditional’s antecedent is true and its consequent is false, then the conditional is false; if the conditional is false, then its antecedent is true and its consequent is false; the two are equivalent. And that is the way it should be. Given that this is true, why can’t his opponent simply deny it? Instead, he must make a new statement like, Whether X happens, Y will not happen. Or better still, he can say (3) If X happens, then Y will not happen and also be careful not to deny his opponent! For if he denies him, then he cannot fix it afterward by adding (anding) something to it, because together they will either reduce to the denial or a contradiction. For the denier, the better choice is (3), for that is exactly what he wants to say. Now, the only time he will be wrong is when the antecedent is true and the consequent is true...and, this is also the way it should be. His opponent now runs the risk of being wrong for the wrong reason if he denies that (3) is true.

In English, we usually accept that the denier does not intend to affirm the antecedent (but he can’t deny it either), and so we do not hold him to the logical conclusion...unless we are meanspirited or A. Slyfellow. But, when we translate the English into symbolic logic, then it becomes clear that there is no possible way to refrain from holding the denier to the antecedent. With symbolic logic we cannot express in an aside that we are not affirming the antecedent.The problem simply does not arise when we are dealing with cause and effect...or with the future, as the following scenario will show.

A controversial figure stumbled into civilization calling himself Prof. Hardy and claiming to be the only survivor of a scientific expedition to the outback several years previously. He said that the expedition had found a peculiar lake with an unusual species of fish in it. All the fish were hermaphrodites and swam in schools; each school had exactly one father fish at any given time, and at a certain time in the year every fish in that school was pregnant by either itself or its school’s father fish, but not by both. Sometime after the professor died Dr. Wiseman realized that there were no Hardy fish. However he was unable to convince his colleagues of this. The debate became so heated that an expedition was sent to the outback in search of Hardy’s lake and fish. They finally found a peculiar lake that fit Hardy’s description, and it also had schools of fish in it. It was decided to forgo an exhaustive investigation of the fish for the time being and simply inquire of the natives about the old professor. At this point the leader of the expedition declared flatly, If this is Hardy’s lake, then these are Hardy’s fish! (No one holds that either Hardy’s fish caused Hardy’s lake or vice versa. The fish could have been caused by the lake in a sense, but in that same sense, they could have been caused by the ocean.)

Unlike Mr. Music, Dr. Wiseman knows that the consequent is false. However, whether he agrees or disagrees he runs the risk of being wrong for the wrong reason: if he agrees, then the lake may be Hardy’s; if he disagrees, then the lake may not be Hardy’s. Moreover, when a person denies a conditional, he may be wrong for the wrong reason because he denies the consequent. For Example, A. Slyfellow could have said, If it is not the case that the dead will rise this noon, then it is not the case that Hornblower will blow his horn this noon. This, of course, says the same thing as before, but now Mr. Music would be wrong because he denied the consequent when he denied the conditional. (A. Slyfellow’s original statement had the form (X ⊃ Y) and, consequently, this one has the form (-Y ⊃ -X) so it should still be answered by (X ⊃ -Y) and not by (-Y ⊃ -(-X)), which amounts to (-Y ⊃ X). Actually, it could be answered (now and before) by (Y ⊃ -X): If the dead will rise this noon, then it is not the case that Hornblower will blow his horn this noon.) Basically, the problem does not exist because both sides fail to recognize that both can be right. If the best the two sides can do is say, If X, then Y, and you may be right and, opposingly, If X, then not-Y, and you may be right, then something is missing. The missing part is the fact that the two sides are diametrically opposed to each other on the only part they are really arguing about. Actually, no one would say this, for neither side would think that they were both right if X were false. Usually, they would think the question remained unresolved.

Basically, we are looking for a way to express our denial of our opponent’s position without affirming the antecedent or denying the consequent. But in logic there is no way to do this, for what can we deny? Once we have denied the conditional, there is no way to doctor it into what we want to say by “anding” something to it. In English we take it for granted that we are only alleging something true about a situation when the antecedent is true. And this is true when the alleger speaks; for when the antecedent is false, he says nothing: If X, then Y; and if not-X, then Y or not-Y. But, this is not true for the denier! He alleges something is true even when the antecedent is false. This is particularly obvious when translating from English to symbolic logic or when speaking a man-made language based on logic like Loglan. In a man-made language we can remedy the situation by simply creating a new word which by definition expresses both denial and the denier’s own allegation: i.e., (X ⊃ -Y). Of course there can be no actual negation of the original statement. However, if an English-speaker expects his argument to be held up to the scrutiny of symbolic logic, he should be careful what he says; for the backbone of a very large percentage of arguments is the conditional. When an arguer denies a conditional, he says something very exact; but usually he has the feeling that the if is still hovering over the antecedent: If the antecedent is false, then all bets are off! he thinks.

The best way to answer (X ⊃ Y) is with (X ⊃ -Y). But if that doesn’t seem strong enough, one might respond with, That is false if X!, or equivalently, If X, that is false! These reduce to: (X ⊃ -(X ⊃ Y)) ≡ (X ⊃ -Y). This puts the if back over the antecedent while expressing strong denial.