Deontic logic: Horty’s gambles (2)

In the second part of his 2019 paper Horty argues that there is a need to integrate epistemic logic with deontic logic, for “ought” statements often have a sense in which their truth-value depends in part on the agent’s state of knowledge.

I agree entirely with his conclusion. But is the focus on knowledge not too strict? Subjectively it is hard to distinguish knowledge from certainty — and apart from that, when we don’t have certainty, we are still subject to the same norms. So I would like to suggest that rational opinion, in the form of the agent’s actual subjective probability, is what matters.

Here I will examine Horty’s additional examples of gambling situations with that in mind. I realize that this is not sufficient to demonstrate my contention, but it will show clearly how the intuitive examples look different through the eyes of this less traditional epistemology.

Horty’s figure 4 depicts the following situation: I pay 5 units to be offered one of two gambles X1, X2 on a coin toss. My options will be to bett Heads, to bet Tails, or Not To Gamble. But I will not know which gamble it is! You, the bookmaker will independently flip a coin to determine that, and not tell me the outcome. In the diagram shown here, X1 is the gamble on the left and X2 the gamble on the right.

On Horty’s initial analysis, if in actual fact I am offered X1 then I should bet Heads, since that has the best outcome. But as he says, rightly, I could not be faulted for not doing that, since I did not know whether I was being offered X1 or X2.

Even if the conclusion is the same, the situation looks different if the agent acts on the basis of the expectation values of the options available to him. The alternatives depicted in the diagram are equi-probable (we assume the coins are fair). So for the agent, who has paid 5 units, his net expectation value for betting Heads (in this situation where it is equally probable that he is betting in X1 or in X2) is the average of gaining 5 and losing 5. The expectation value is 0. Similarly for the option of betting Tails, and similarly for the option of Not Gambling: each has net expectation value 0. So in this situation it just is not true that the agent ought to take up any of these options — it is indifferent what he does.

Horty offers a second example, where the correct judgment is that I ought not to gamble, to show that his initial analysis failed to entail that. Here is the diagram, to be interpreted in the same way as above — the difference is in the value of the separate possible outcomes.

Reasoning by expectation value, the agent concludes that indeed she ought not to gamble. For by not gambling the payoff is 5 with certainty, while the expectation value of Betting Heads, or of Betting Tails, is 2.5.

So on this analysis as well we reach the right conclusion: the agent ought not to gamble.

Entirely in agreement with Horty is the conclusion that these situations are adequately represented only if we bring epistemology into play. What the agent ought to do is not to be equated with what it would objectively, in a God’s eye, be best for her to do. It is rather what she ought to do, given her cognitive/epistemic/doxastic situation in the world. But she cannot make rational gambling decisions in general if her knowledge (or certainty) is all she is allowed to take into account.

It would be instructive to think also about the case in which it is known that the coin has a bias, say that on each toss (inlcuding the hidden first toss) it will be three times as likely as not to land heads up. Knowledge will not be different, but betting behavior should.