(X ⇒^—Y) is good if, and only if is (X ⇒^—X) →^—Y on the scroll of all wisdom.

The equivalence between a lie X ⇒^—Y and the act of lying ^—X →^—Y is demonstrated in Table 1.

The problem is determining the true consequences of X(^—X), if all GURU ever does is either tell the truth or lie about X(^—X). Here GURU has to question itself. If it suspects ^—X to be a lie, and it knows that it only makes lies that have good consequences, it checks whether the lie X ⇒ ^—X would be good. Substituting this for X gives the definition (X ⇒^—Y ) ≡ (X ⇒ ^—X ) →^—Y.

Such a definition wouldn’t lead us anywhere, because it would be circular, but, since GURU’s memory contains all wisdom, (X ⇒^—Y ) can simply be defined in terms of (X ⇒ ^—X) →^—Y being on the scroll of all wisdom, which is not per se circular.

Based on these rules, GURU is now capable of telling good from bad, and even telling whether judgments about what is good or bad are themselves good or bad. But does this say anything about whether GURU is good?

 

When Gödel Meets GURU

In fact, the rules from the previous sections were taken from Smullyan [ 8] and are a subset of the formal system he presents as Craig’s machine. The reader interested in its formal details is referred to his work for proof that this system is in fact Gödelian. I built the GURU story around Craig’s machine only to provide an interpretation for it, and to show how reasoning could work on an ethical level, but a machine or a metamathematician is not interested in anything else but these four rules in their purely symbolic form.

The Gödel sentence G such that G ≡ ¬ PG is a correct sentence of a formal system that states “you cannot prove me within my system.” Note that some of the details were left out here. PG could in terms of GURU’s “machine-language” be an operation finding out whether G is on the scroll of all wisdom. One would further have to define the concepts of ≡ and ¬ in terms of GURU’s machine-language and would have to know some more details about how GURU really operates. For Craig’s machine, Smullyan shows that (m ⇒^—m) →^—m is such a sentence.

In terms of GURU, the Gödelian sentence (m ⇒^—m) →^—m would have an interpretation like, “What are the consequences of an act, where the consequences of doing this act are the same as the consequences of murder, and the consequences of not doing this act are that I suspect you of lying when you say that murder is bad?”

Although this question is a skeptical way of asking about the most basic fact, “Is murder good or bad?”, it is not possible for GURU to answer it.

 

Divide The Dollar!

In the previous sections, the first two ethical questions were considered. They were handled together because they are similar, but the third question, in contrast, is widely unrelated to the other ones. A fresh approach will be taken for this question.

The divide-the-dollar situation is one of the philosophical classics. Two people are given a dollar, but only under the condition that they find an agreement on how to divide it. This problem is representative of the collectivist/individualist-debate, yet very simple. It is in fact so simple that it is frequently found in elementary school math-

textbooks, in wordings like “This year Grandma gave Mary and John one dollar for Christmas. How much money does each of them get?” Part of the naive approach an elementary school pupil takes when confronted with this problem is the assumption that each of the two bargainers ought to get the same amount of money.

Such a standard can only work if all the individuals involved in the bargaining-process accept it without further questioning. Someone who doesn’t accept it could demand 99 cents, arguing the following way: “If I reject every possible bargain, except for the one that leaves me with 99 cents, then my opponent has only two options left: either accepting or rejecting this one-and-only offer. Accepting the offer leaves the opponent with one cent, otherwise neither of us will get anything. This is why the opponent has to accept.”

From a collectivistic standpoint, this seems completely nonsensical, because how can one individual justify this in such a way that this ethical justification wouldn’t apply if the other individual made the same argument? (This principle frequently appears in the ethical literature as reciprocity.) Note that the elementary-school approach presented above is already a collectivistic one, because it is subject to a global ethical standard. From an individualistic standpoint, on the other hand, the collectivist’s commitment to the principle of reciprocity itself seems nonsensical. What motivation drives a bargainer to accept the 50-cents-deal, if 99 cents would also be possible. Given this concept, one can now go on to generalize the idea, so it applies not only to thought-experiments.

 

About Idealized Rational Individuals

From a strong individualistic view, consider the following statement:

Given any specific situation an individual might find himself in, it will benefit the most, in the long run, by making such a decision that maximizes its own personal benefit.

 

An example where the above statement does not hold was already given. Two individualists will not come to any agreement in the divide-the-dollar situation, which is why they won’t get any money. As far as ethics is concerned, we are done at this point, and are forced to draw back to a weaker version of the individualistic view, like “there are situations, where an individual….”

The bargaining-game, as described by John F. Nash [ 7], provides some more insight, not by proving any ethical views, but by saying something about how common these divide-the-dollar situations really are. It doesn’t necessarily take two “hardcore-individualists” confronted with a situation as theoretical and as made-up as the divide-the-dollar problem, to find a counter-example for the individualistic view. It completely suffices to let two idealized rational individuals take part in a two-person game, in which they are supposed to find an agreement on trading some of the goods they possess, without the use of money (which would only be a special case of the situation considered).

 

Nash’s Approach To The Bargaining Problem Nash uses the example shown in Table 2 to illustrate a bargaining situation. The table shows goods, some of which belong to Bill and some of which belong to Jack, and their utilities to Bill and Jack. (Nash uses the notion of utility to formalize the concept behind the value of a good. The fact that goods can be of differing value for different people could be seen as the major driving force behind economy.) Bill and