April 5, 2009 — -- Elementary propositional logic (the logic of "and," "or," "not," and "if... then...") can sometimes give rise to vexing puzzles. Here is one rather whimsical example that occurred to me after reading about Bernard Madoff's Ponzi scheme (and others) and being struck by the word "seductive" appearing in some of the news accounts.
Being a logician, I thought of a kind of seduction that relies on logic rather than psychology to ensnare and entangle its targets.
Let me first state the puzzle in terms of a more standard sort of seduction.
Suppose a man flirts with a woman and then asks her, "Will you solemnly promise to give me right now your telephone number if I make a true statement and, conversely, not give me your number if I make a false statement?"
Maybe he can soften the statement a bit, but let's assume that this is its gist.
Feeling that this is a flattering and benign request, the woman promises to give him her number if and only if he makes a true statement.
The man then makes his statement: "You will neither give me your telephone number now nor will you sleep with me tonight."
What's the trick? Note that she can't give him her number since, if she were to do so, his statement would be made false, and so she would have broken her promise to give him her number only if he made a true statement. (This is the crux of it.) Therefore, she must not give him her number under any circumstances.
But if she also refuses to sleep with him, his statement becomes true, and this would require her to give him her number.
The only way she can keep her promise is to sleep with him so that his statement becomes false. The woman's seemingly innocuous promise ensnares her.
Fortunately or unfortunately, I suspect that the class of people for whom this seduction technique would prove effective is probably rather small. Nevertheless, it might make an interesting premise for a Star Trek episode or perhaps form part of a logicians' dating manual.
A Similar Financial Con Game
Consider now a slight variant of the above story. Suppose an investment con man is talking to a prospective client.
He flatters the client, tells him about the stock newsletter he e-mails to his clients, and asks, "Will you right now solemnly promise to give me your e-mail address if I make a true statement and, conversely, not give it to me it if I make a false statement?"
Feeling that this is a harmless request to consider his newsletter, the prospective client promises. The con man then makes the statement: "You will neither invest all your money with me today nor give me your e-mail address."
As with the seduction above, note that the prospective client can't give the con man his e-mail address since if he were to do so, the con man's statement would be made false, and so he would have broken his promise to give him his e-mail address only if the con man made a true statement. Therefore, he must not give the man his e-mail address under any circumstances.
But if he also refuses to invest all his money with the con man, the latter's statement becomes true, and this would require him to give his e-mail address. The only way the prospective client can keep his promise is to invest all his money with the con man so that the latter's statement becomes false. Again, a seemingly innocuous promise ensnares somebody.
Seduction, whether with an amorous or an acquisitive intent, often follows the same interactive pattern, the hook sometimes being logical, sometimes psychological.
One might think this example absurdly unrealistic, but consider the opaqueness of derivatives and other complicated financial instruments. There too, the commitments one unknowingly takes on are invisible and substantial. The seduction scam is child's play compared to securitized mortgages.
Nevertheless, perhaps more typical than the seduction question is the following, which is still a matter of elementary propositional logic. The question: For the following set of premises, determine whether the conclusion below (the part after "Hence") follows from them. (Don't worry about whether the premises are true or false.)
If private sector investment stays the same, then government spending will increase or more unemployment will result. If government spending will not increase, taxes can be cut. If taxes can be cut and private sector investment stays the same, then more unemployment will not result. Hence government spending will increase.
There are, of course, simple techniques for deciding if the conclusion follows from the premises in propositional logic or whether a set of such statements is consistent, but from sad experience and casual observation, I suspect that only a sprinkling of policy-makers know them.
In any case, the real economy is complex, interactive, convoluted and impossible to describe with elementary propositional logic. The logic required is more extensive, involving quantifiers -- words like "all," and "some" and "none."
An adequate description also calls on arithmetic, probability, and statistics, calculus, and differential equations, non-linear dynamics, etc. It's not too surprising that few if any people can assert with much confidence that we should take this economic course rather than that and come to this conclusion rather than another.
There are, however, some conclusions using propositional logic that can be made with the utmost confidence.
Recall the logician/economist riding an elevator who, when the doors open, is asked whether the elevator is going up or down.
He replies, "Yes."
John Allen Paulos, a professor of mathematics at Temple University, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as "Irreligion: A Mathematician Explains Why The Arguments for God Just Don't Add Up." His "Who's Counting?" column on ABCNews.com appears the first weekend of every month.