Topics Logic

Question Does Goedel's incompletness therom demonstrate that logic cannot be shown to be consistent and complete because we cannot prove a system of logic without relying on logic or begging the question? In other words; does it reveal a fallacy of "pretended neutrality"?

Accepted:
September 23, 2010

Comments

No, I do not think the incompleteness theorem has any such consequence.

First of all, although the incompleteness theorem does apply to some formal systems some people would classify as systems of logic (e.g., second-order logic), its primary application, as usually understood, is to formal systems of arithmetic or, more generally, or mathematics. What people usually mean by "logic" is first-order classical logic, and the incompleteness theorems do not tell us anything about its consistency or validity. (There is a nice proof of the undecidability of first-order logic that rests upon the completeness theorem.)

There are two versions of the incompleteness theorem. The first shows that no (sufficiently strong) formal system is ever complete (if consistent), in the sense that you can either prove or disprove every statement formulable in that system. The second shows that you cannot prove such a system to be consistent without relying upon assumptions that are, in a very specific sense, logically stronger than the assumptions made within that system (its axioms). Pure first-order logic simply does not fall within the ambit of such theorems, though second-order logic does (in fact, fairly weak fragments of second-order logic do).

The second incompleteness theorem is sometimes said to show that no system can "really" be proven to be consistent at all, but this is clearly false. There are some very nice proofs of the consistency of various systems that depend upon precisely the same sorts of assumptions one would be perfectly happy to use, say, in proving the fundamental theorem of algebra. So either you aren't prepared to accept the principles used in the proof of the fundamental theorem of algebra or else you are, and then you can prove the consistency of certain systems subject to the incompleteness theorem. I think I am right here using the fundamental theorem of algebra as an example, but someone please correct me if that example is wrong!

The more basic thing to say is this. First, you cannot prove anything without relying upon logic. So, as Michael Dummett once remarked, if there were a "logical skeptic" who questioned all the laws of logic, then there would clearly be no arguing with him! Second, you cannot prove anything without relying upon assumptions that are at least as strong, logically, as the conclusion you wish to prove. This is again fairly obvious: If the proof is correct, then the assumptions used in it logically imply the conclusion, so, of course, they are together logically as strong as the conclusion. So "begging the question" can't simply mean: relying upon assumptions logically as strong as the conclusion for which you are arguing. Otherwise, there would be no sound arguments! And the only thing Gödel's theorem shows is that any proof that a (suffuciently strong) system S is consistent must rely upon assumptions logically stronger than those in S. It does not show that no such proof can be a proper proof, or be properly convincing. It does not even show, though for more complicated reasons, that no such proof should convince anyone antecedently skeptical about the truth of the assumptions made in S, nor even (for reasons connected with the "begging the question" stuff), that it should not convince anyone skeptical of S's consistency. This is in part connected with the fact that the sense of "logically stronger" in Gödel's theorem is very delicate and quite different from the one connected with "begging the question", but that is a much more complicated issue.