A Simpler Argument For Completeness

Drawing heavily on Gustafsson's money pump

Gustafsson presents his money pump argument for completeness, drawing on a variety of plausible axioms. But I think that you can derive completeness from more plausible axioms.

The important entailment of incompleteness that I intend to argue is refuted here is the following. Believers in completeness think that sometimes A can be better than B, though B and A are both not better than C — they’re both incommensurable with C.

Here are the four things I’ll assume — the first two come from Gustafsson. Note that what I intend to show here is that if we were fully rational we would have complete preferences.

1 The Principle of Rational Decomposition If an agent, whose credences and preferences are not rationally prohibited, makes a sequence of choices which violates a requirement of rationality, then some of those choices are rationally prohibited.

2 Decision-Tree Separability The rational status of the options at a choice node does not depend on other parts of the decision tree than those that can be reached from that node

3 More Options Beneficism: If a choice is not irrational, then if after the choice you’d gain another choice, the choice still isn’t irrational.

4 Irrationality of Inferior Choices: If A>B, then if a sequence of choices starts with A and ends with B, that sequence of choices is irrational.

Each of these is very plausible.

The principle of rational decomposition follows from an even more plausible principle.

Causal irrational introduction: If prior to making some choice you have not been irrational and after making it you have been irrational, then that choice is irrational.

Causal irrational introduction is very obvious. Remember, irrationality is a type of mistake, so this boils down to saying “if before you make a decision you haven’t made a mistake but after you make the decision you have, then that decision is a mistake.”

To see how this implies the principle of rational decomposition, consider two choices. If after choice 1 you haven’t done anything irrational, but after choice 2 you have, then that would mean that choice 2 is irrational by this standard. Thus, because there has to be a sequence of choices at which point you have first been irrational, causal irrational introduction entails the principle of rational decomposition.

Decision-Tree Separability is also very plausible. When assessing a choice, you should just look at whether it’s a worthwhile choice. What happened in the past doesn’t affect that — therefore, it shouldn’t be part of your considerations.

More options beneficism is also obvious. You shouldn’t be less willing to make a choice because it gives extra options. Giving extra options is a good thing — a perfectly rational decision maker wouldn’t regard something as less choiceworthy because they now have more options.

Irrationality of inferior choices is also obvious. If a sequence of choices results in you getting something that you want less, you’ve clearly gone wrong somewhere. If you value 10 dollars more than 5 dollars and you make a series of trades that involve going from 10 dollars to five dollars, you’ve clearly made an error.

But together these entail completeness — or at least entail that if A is preferred to B and B and C are both equally preferrable (in the sense that you prefer neither), then A is more preferrable than C.

Say you start with A, trade it for C, and then for B. By Irrationality of Inferior Choices, your sequence of choices is irrational. By The Principle of Rational Decomposition, you’ve made at least one irrational choice. But it can’t be the first one, because trading something for something equally choiceworthy isn’t irrational, and the extra option doesn’t make it less choiceworthy, because of More Options Beneficism. However, it also can’t be the second, because of Decision-Tree Separability — what happened in the past doesn’t matter — trading something for something equally choiceworthy is not irrational. Thus, such a view is ruled out by these 4 obvious principles.

Objections?