Gödel’s famous proof demonstrates that arithmetic is incomplete. A similar demonstration can be constructed with morality as the subject instead of arithmetic.
Say that, for moral standard Q and object of assessment x, Q returns a verdict of worthwhile or not worthwhile.1 I will represent this as Q(x) = w or Q(x) = ~w. Suppose that any plausible moral standard is within its own scope, so it makes sense to ask whether Q(Q) = w or Q(Q) = ~w. If Q(Q) = w, say that Q is self-assured. If Q(Q) = ~w, say that it is not.
Consider moral standard M. Suppose that M is complete. In other words, suppose that, for any x, M returns a verdict of w or ~w. There is no input x for which M does not render a verdict. Furthermore, suppose that M is sound. In other words, suppose that there is no input x for which M renders both w and ~w. M is consistent in its assessments.
On the basis of these assumptions, it is possible to reach a contradiction. Consider the pathological standard P. P renders verdicts as follows:
If M(P) = w, then:
If M(x) = w, then P(x) = ~w.
If M(x) = ~w, then P(x) = w.
If M(P) = ~w, then:
If M(x) = w, then P(x) = w.
If M(x) = ~w, then P(x) = ~w.
This formula says that P renders verdicts as follows: If M approves of P, then P and M disagree about x. If M disapproves of P, then P and M agree about x.2
Consider M(P). Suppose that M(P) = w. It follows that P disagrees with M’s every verdict. But if they disagree so profoundly, then clearly P is not a worthwhile moral standard by the lights of M, so M(P) = ~w. Suppose instead that M(P) = ~w. It follows that P agrees with M’s every verdict. Seeing as they agree, M has every reason to approve of P, so M(P) = w. We have, then, that: If M(P) = w, then M(P) = ~w. And if M(P) = ~w, then M(P) = w.
It follows from this contradiction that one of the assumptions that we started with is incorrect. Either M is incomplete, or M is unsound, or there is a plausible moral standard that is not within its own scope.
The English language is subject to incompleteness as is, of course, arithmetic. We have not stopped using English for this reason, nor have we stopped using arithmetic. This is because they remain useful despite incompleteness. I suspect that morality is the same way. My proof, then, does not show that we should throw out morality. It does show that it is not possible to construct a computer program that will render sound answers to any possible moral question, but no one was trying to do that anyways. More importantly, it casts doubt on the view that there exists a god who knows the true answer to every moral question.
1) Of course, there are many moral operators besides ‘worthwhile’ and ‘not worthwhile.’ These include ‘right,’ ‘wrong,’ ‘indifferent,’ ‘virtuous,’ and so on. I ignore these because I suspect that they complicate the analysis without changing the results. A fair way to disagree with the results is to show that they complicate the analysis and change the results.
2) Consider P(P). P is not self-assured.
Cheerful Ostrich 