Topics Logic

Question Are there as many true propositions as false ones? More of one than the other?

Accepted:
October 20, 2007

Comments

Each true claim can be paired with a unique false one, namely its negation (i.e., the result of prefixing the original claim with "It is not the case that ..."). And each false claim can be paired with a unique true one (again, its negation). So, there are exactly as many true claims as there are false ones.

Professor George's conclusion is probably true, but the reasoning seems to me invalid. This is so, because the two "pairing" operations produce different pairs. For example, the first operation might create the pair <"Bush is married"; "it is not the case that Bush is married">. The second operation might create the pair <"it is not the case that Bush is married"; "it is not the case that it is not the case that Bush is married">. The first operation finds one unique false claim for every true one -- but some false claims are left over (for example, "3+3=9") . The second operation finds one unique true claim for every false one -- but some true claims are left over (for example, "3+3=6"). Therefore, the argument works only if it can be shown that the two sets of "left-over" claims are equal in number of members.

One might try to avoid this problem by redefining Professor George's operations so that any claim that begins with an odd number of iterations of "it is not that case that" gets paired with the claim that removes these words. This would create the needed one-to-one mapping.

But this solution brings on a new difficulty. Some hold that any claim must have exactly one of these three properties: true, false, or meaningless; and they hold that, if a claim is true, then its negation is false, and if a claim iseither false or meaningless, then its negation (as defined by ProfessorGeorge) is true. This view is not implausible. Consider the claim "it is not the case that the number 3 is in love with Bush." One can categorize this claim as true and still deny that, when we remove the words "it is not that case that", we get a falsehood. One can say instead that "the number 3 is in love with Bush" is a meaningless claim -- neither true nor false -- because the number 3 isn't eligible for the relational predicate "is in love with". If we accept this view, then the one-to-one mapping fails because in some cases removal of the six words transforms a true proposition into a meaningless one.

One could also try to reach Professor George's conclusion by showing that there are infinitely many true and also infinitely many false claims. But I am worried that this would get us entangled with higher orders of infinity, because there are actually more true claims, and also more false claims, than there are natural numbers. We know since Cantor that the set of real numbers is more than countably infinite; and the set of true claims of the form "x is a real number" is then also more than countably infinite, and ditto for the set of false claims of the form "x is not a real number". Now, unfortunately, there are infinities larger even than that of the real numbers (called aleph-1 -- see mathworld.wolfram.com/Aleph-1.html). I suspect that one could give a good argument to the effect that the set of true claims transcends any one of these infinities, and ditto for the set of false claims. But I must confess that I am not licensed above aleph-29.

A few comments on the answers from Professors George and Pogge:

1. If two statements are logically equivalent, do we think of them as expressing the same proposition or two different propositions? If we think of logically equivalent statements as expressing the same proposition, and we use classical logic, in which "it is not the case that it is not the case that P" is logically equivalent to P (and if we ignore, for the moment, Prof. Pogge's worries about meaningless statements), then there's no problem with Prof. George's original pairing. In the example given by Prof. Pogge, "Bush is married" and "it is not the case that it is not the case that Bush is married" express the same proposition, and that proposition is paired with the proposition expressed by "it is not the case that Bush is married".

2. If we do not group together logically equivalent statements as suggested in 1 above, then as Prof. Pogge points out, Prof. George has paired each true statement with a false one, but some false ones are not used in the pairing. And he has also paired each false statement with a true one, with some true ones not used. In more technical mathematical language, we could say that we have a function from the true statements to the false ones that is one-to-one, but not onto, and we also have a function from the false statements to the true ones that is one-to-one but not onto. In this situation, the Cantor-Schroeder-Bernstein theorem says that there is a one-to-one correspondence between the true statements and the false statements. In fact, the proof of the theorem leads to exactly the solution that Prof. Pogge proposes: for statements that start with an even number of iterations of "it is not the case that," add an extra one, and for those that start with an odd number of iterations, remove one. (You can find more information about the Cantor-Schroeder-Bernstein theorem here.)

3. Suppose that, as suggested by Prof. Pogge, we allow for meaningless statements, and we say that if P is meaningless, then "it is not the case that P" is true. What about "it is not the case that it is not the case that P"? Is it equivalent to P, and hence meaningless? Or is it false, since "it is not the case that P" is true? If it is regarded as meaningless, then we do seem to have some "extra" true statements that are not paired with anything. But if it is false, then "it is not the case that P" can be paired with "it is not the case that it is not the case that P," and it will still be possible to make the pairing work.

4. Finally, what about the issues raised by Prof. Pogge about how many claims there are? I'm not sure exactly what is supposed to count as a "claim." Prof. Pogge suggests that we consider claims of the form "x is a real number." Well, certainly "2 is a real number," "23/57 is a real number," and "pi is a real number" are true claims. Prof. Pogge suggests that for every real number x, "x is a real number" is a claim. If that's right, then since there are uncountably many real numbers, there are uncountably many claims. But 2, 23/57, and pi are all "nameable" real numbers. Is there really any such thing as the claim "x is a real number" in the case of an "unnameable" number? If we require that a claim must be expressed by a finite sequence of letters, digits, and punctuation marks, then there are only countably many claims.

Question Are there as many true propositions as false ones? More of one than the other? Accepted:October 20, 2007

Question Are there as many true propositions as false ones? More of one than the other?

Accepted:
October 20, 2007