Deontic logic: value-rankings

The historical opening chapter of the Handbook of Deontic Logic and Normative Systems shows that, in various forms, this has been a typical way to connect ‘ought’ statements with values:

[O] It ought to be that A if and only if it is better that A than that ~A

as well as

[Cond O] It ought to be that A, on supposition that B if and only if it is better that (B & A) than that (B & ~A)

But in addition, deontic logics typically include the law that carries logical implication into the derivation of ‘ought’ statements:

[IMP] If A implies B then (It ought to be that A) implies (It ought to be that B)

(and the similar law for conditional ‘ought’ statements), important to keep the logic within the range of normal modal logics.

But do [O] and [IMP] go well together? That depends on the character of the value ranking which defines the ‘better than’ relation among propositions. Specifically, it requires that

[MON] If A implies B, and A is better than ~A, then B is better than ~B.

Problem: I will give examples below of ‘better than’ relations which do have property MON but which are intuitively unsatisfactory. But a ranking by expectation value does not have property MON.

Ranking by expectation value does not have MON: For example, a bank robber is confronted by the police. His best option (by expectation value) is to surrender. What about the option to (surrender or resist arrest)? This is America! If he resists arrest he will likely be shot. That lowers the expectation value considerably. (Reminiscent of Ross’ paradox, also about IMP.)

Solution: There is something right about [O] and [CondO], namely that value rankings have an important role to play in the understanding of ‘ought’ statements. But there is also something wrong about [O] and [CondO], namely that they presuppose that it is just, and only, value rankings that must determine the status of ‘ought’ statements.

But let me first give examples of rankings that do have MON and say why I find them unsatisfactory. In my own essay on deontic logic as a normal modal logic (1972) I gave this definition:

Ought(A) is true exactly if opting for ~A precludes the attainment of some value which it is possible to attain if one opts for A

or less informally,

Ought(A) is true in possible world h exactly if, there are worlds satisfying A which have a higher value than any worlds that satisfy ~A.

Very unsatisfactory! Today I opted not to buy a lottery ticket, thereby precluding a million dollar windfall (larger than anything I could get otherwise) and so I was wrong. I ought to have bought that ticket! As gamblers say, when you talk about prudence, “Yes, but what if you win!” Sorry, gamblers — this is not a good guide to life …

Jeff Horty offered a more sophisticated formulation in his 2019 paper (p. 78, the Evaluation Rule) as his explication of the Meinong/Chisholm analysis, which would in a normal modal logic context amount to:

Ought(A) is true in world h exactly if A is true in all worlds h’ possible relative to h, such that there is no world h” which satisfies ~A and has a value higher than h’.

Except for the relativization of possibility, this is like the preceding. Horty rightly rejects this as unsatisfactory, using the example of a forced choice between two gambles which has the same expectation value, but of which one carries no risk of loss. (One has outcomes 10 and 0, the other has only outcome 5 with certainty.) It is certainly not warranted to say that we ought always to make the gamble with the higher prize but higher risk.

There are surely other value rankings to to try, and I thought of this one:

Ought(A) is true in world h exactly if there is a one-to-one function f mapping the worlds that satisfy ~A into the worlds that satisfy A, such that for all worlds h in the domain of f the value h is less than the value of f(h).

This one too has property MON. Informally put, it means that whatever outcome you get if you opt for ~A, you realize you might have done better by choosing for A.

But imagine: gamble ~ A has with certainty one of the outcomes 5, 7, or 9 dollars, while gamble A has with certainty one of the outcomes 1, 10, 12, 14 dollars. However, to make gamble A you have to buy a ticket for $4. So your net outcomes for A are loss of $3, or win of 6, 8, 10. Clearly by the above principle you should take gamble A, for 5 < 6 < 7 <8 < 9 < 10. But is that really the right thing to do? If all the outcomes are equally likely then ~A has expectation value (21/3) and A has expectation value (21/4), which is less.

In previous posts I have discussed how Horty goes beyond this.

Now I just want to explain how the use of expectation value ranking works well with what I proposed some weeks ago in the post Deontic logic: two paradoxes” (which I gave a more precise formulation in the next post, “Deontic logic: Horty’s new examples”.) By “works well” I mean that the principle [IMP] is valid.

My proposal was that in setting up the framework for deontic logic, , we need to include both imperatives and values. So I envisage the agent as first of all recognizing the imperatives in force in his situation (‘if you have sinned, repent!’). The agent’s next step is to take account of the satisfaction regions for those imperatives (or better, maximally consistent sets of them). Then the value-ranking is applied to those satisfaction regions, and the ones that count are the ones that get highest value (the optimal regions). Next:

It ought to be that A if and only if there is an optimal region that implies A.

(This can be extended to conditional oughts in the way Horty does: go look at the alternative situation in which the agent has the condition added to his knowledge.)

When entered at this point, it does not matter whether the ranking has property MON. For what ever the ranking is, if an optimal region is part of A, and A is part of B, then an optimal region is part of B.

The story for choices, decisions and action planning is similar. It is not that the agent ought to do what is best, but rather that he has to make a best choice (the moral of Horty 2019). Suppose it is already settled that I will gamble, and I have a choice between several gambles. Now what ought to be the case (about what I do, about my future) is whatever is implied by my making a best choice. And I propose that the best choices are those which are represented by propositions (the choices themselves, not the possible outcomes of those choices) which have highest expectation value.