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.

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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.

In this series of posts, I will survey Hume’s moral philosophy. To begin with, I will look at his discussion of conventions. Conventions are central to Hume’s theory of justice, as well as his understanding of political obligation, international relations, and promise making.

In the second Enquiry, Hume illustrates the notion of a convention with a few examples:

…two men pull the oars of a boat by common convention for common interest, without any promise or contract: thus gold and silver are made the measures of exchange; thus speech and words and language are fixed by human convention and agreement.   

Enquiry Concerning the Principles of Morals, Appendix III, Paragraph 8

Hume’s examples have five important features: mutual benefit, multiple solutions, different preferences, unplanned agreement, and reciprocal performance.

Mutual Benefit

In Hume’s examples, conventions are beneficial for all parties. The men in the boat will not get where they are going unless their rowing is synchronized, and a common language and currency serve the interests of anyone who wants to communicate and trade. When Hume refers to “common interest,” this sort of mutual benefit is what he has in mind.

Multiple Solutions

In Hume’s examples, there is more than one way to solve the same problem. The men in the rowboat can row either at a fast pace or a slow pace. So long as their rowing is synchronized, they will reach their destination. Similarly, though there are many languages, each serves the same purpose, to facilitate communication. Lastly, though there are many currencies, each serves the same purpose, to facilitate trade.

Different Preferences

It is often the case that different people prefer different solutions to the same problem. The rowboat example illustrates this point nicely. Suppose that one man is in a hurry, but the other is not. Most likely, they will prefer to row at a different speeds. Whether they settle on a fast pace or a slow pace, one man will be a little disappointed. However, he will be even more disappointed if they are unable to settle on either pace. If they cannot agree to coordinate their efforts, they will not reach their destination. Although it is often the case that different people prefer different solutions to the same problem, everyone prefers some solution over no solution at all.

Unplanned Agreement

As Hume says, the men in the rowboat reach agreement without any explicit promise or contract. To go further, their agreement does not need to be spoken or even planned. For example, imagine that each man varies the pace at which he rows until, by chance, they become synchronized. Once they are synchronized, it is in each man’s interest to maintain synchronization by continuing to row at the same pace.

Reciprocal Performance

In the rowboat example, neither man has an incentive to row unless he expects that the other man will row as well. It is a waste of energy to row alone. Hume makes this point by saying:

Whatever is advantageous to two or more persons, if all perform their part; but what loses all advantage if only one perform, can arise from no other principle [besides common interest]. There would otherwise be no motive for any one of them to enter into that scheme of conduct.

Enquiry Concerning the Principles of Morals, Appendix III, Paragraph 8

The currency example and the language example follow the same pattern. There is no incentive for me to use dollars as currency unless I expect that other people will too, and there is no incentive for me to learn a language unless other people speak it as well. In general, in Hume’s examples, there is no incentive for one person to perform his part unless that person expects that everyone else, or almost everyone else, will perform their parts too.

Conclusion

A convention is an agreement, often unspoken and unplanned, between two or more people to coordinate their actions for the benefit of everyone involved. Usually, there will be more than one way of coordinating, and usually there will be disagreement over which way to choose. Everyone will agree, however, that it is better to be coordinated than to be uncoordinated.