## On Brown on the composition of value

A few thoughts on Brown (2014) and Brown (2020) and the composition of value.

Some propositions (or properties, but let's run with propositions) have value. They are reasons to act one way rather than another. We may ask how this kind of value distributes over the space of propositions.

Since logically equivalent propositions plausibly have the same value, we can picture the propositions as regions in logical space – sets of possible worlds. Now how is the value of a region related to the value of other regions – to its subregions, for example? This is the question Campbell Brown raises in Brown (2014) and Brown (2020).

Brown actually favours a somewhat different way of thinking about the algebra of propositions. He construes the conjunction of two propositions A and B as the mereological sum of A and B. The conjunction has both A and B as parts. More generally, A is part of B iff B entails A and A is not the tautology (the set of all worlds). The tautology and the contradiction generally mess up the mereological picture, so let's pretend that they don't exist. (From now on, "proposition" means "non-trivial proposition".)

Brown's mereological representation can be confusing if you are used to picturing propositions as sets of worlds. In Brown's representation, A is a part of B iff B is a subset of A. You might expect it to be the other way round. Here's a trick that helps me get the right intuition: When Brown says that A is 'part' of B, he means that A is part of what's required for B to be the case. If B is a subset of A then A is part of what's required for B to be the case.

OK. Now let's ask how the value of a proposition is related to the value of other propositions. Is the value of a whole the sum of the value of its parts?

On a naive reading, this means that if a proposition A is the conjunction of some propositions B and C, then the value of A is the sum of the value of B and C. But every proposition A is the conjunction of A and A, and we probably don't want to say that the value of any proposition is twice the value of itself.

On a more charitable interpretation, the additivity conjecture ("the value of a whole is the sum of the value of its parts") says that if a proposition A is the conjunction of mereologically distinct propositions B and C, then the value of A is the sum of the value of B and C. (Brown gives a different interpretation in the 2020 paper, but I think it's equivalent to this one, which I find more intuitive.) Two propositions are mereologically distinct (in Brown's mereology) if there is no proposition they both entail; equivalently, if there is no world that they both rule out.

This is a little easier to understand if we express it in terms of mereological atoms. The additivity conjecture says that the value of any proposition A is the sum of the value of its atomic parts. An atomic part of A is a proposition entailed by A that does not entail any other proposition. It's a superset of A that contains all worlds but one. The conjecture therefore amounts to the following:

1. For each world w, the proposition that this world is not the case has a certain value. Call this value the "negative value" of w.
2. The value of any proposition A is the sum of the negative value of all worlds not in A.

In the 2014 paper, Brown rejects this idea (there called "Strong Additivism"). In the 2020 paper, he argues that it isn't a substantive assumption. The argument is that all that really matters, and all we have a good grip on, is the value of "wholes", i.e. of complete worlds. And for any assignment of value to worlds one can construct an assignment of value to atoms (i.e., a "negative value" assignment to worlds) relative to which the additivity conjecture is satisfied.

Why is the value of wholes all that really matters? Because what we should do is a matter of overall reason, not a matter of individual reasons. Also, as Brown points out in the 2020 paper, it can be hard to figure out the value of an unspecific proposition:

Consider, for example, the proposition that either you win the lottery or you are killed. Do you want this to be true? That probably depends on which of the propositions of which it is a part would also be true. Would it be true because you win the lottery, or because you are killed (or both)? These more specific propositions are what really matter to you.

Brown's view, then, is that there isn't any interesting value structure in the space of propositions. The only genuine bearers of value are the individual worlds. We can, if we want, assign values to other propositions, but such an assignment would always be arbitrary, and it wouldn't track anything real.

I disagree. I think there is an interesting value structure in the space of propositions. In fact, there are two such structures.

The first is what I'll call Jeffrey's structure.

To begin, consider two logically incompatible propositions A and B with equal value. (Two individual worlds, if you want.) What's the value of their disjunction? Here, unlike in the lottery-or-death case, there is an intuitively clear answer. The disjunction has the same value as its disjuncts. If you want A to be true, and you want B to be true to the exact same extent, then you also want A ∨ B to be true, to that extent.

This is enough to cast doubt on the additivity conjecture. If all worlds have equal value, then all propositions should have equal value, by the argument I just gave. But if the value of a proposition is the sum of the value of its atoms, and all atoms have equal value, then "larger" propositions (with more atoms) have greater value than "smaller" propositions (with fewer atoms).

We can go further. What makes the lottery-or-death case puzzling is that one disjunct is good and the other is bad. As Brown points out, whether we want this disjunction to be true "depends on which of the [disjuncts] would also be true". In the worst case, we would be killed. In the best case, we would win the lottery. We can say, however, that the value of the disjunction lies somewhere in between the value of its disjuncts.

To say more, we need more information about which disjunct would be true. We don't need perfect information. A probability measure will do. If, for example, winning the lottery has value 10 and getting killed has value -100, and there's a 99% chance that we'll get killed given that the disjunction is true, then the value of the disjunction is plausibly the weighted average of the value of the disjuncts, weighted by these chances.

With the help of a probability measure, we can systematically and non-arbitrarily fill in the value of every proposition. This has all been worked out in Jeffrey (1983).

In Jeffrey's structure, the value of complete worlds has primacy. The structure is "wholistic", in the sense of Brown (2014). It is also "context-insensitive": the value of a proposition is not relative to a world at which the proposition is true.

In (philosophical) decision theory, people often take a value function over worlds as their starting point. As Pettit (1991) points out, this is not how we normally think of value or desirability or reasons. When we judge that one outcome is better than another, we consider different features of these outcomes. We look at their pros and cons. There aren't just overall reasons. There are also individual reasons.

These pros and cons or individual reasons are not adequately represented anywhere in Jeffrey's structure. That's why we need a second kind of value structure, which I'll call the intrinsic value structure.

Intrinsic value, unlike Jeffrey value, is independent of probability. If a proposition is intrinsically desirable, then it is desirable not because it is likely to bring about other desirable things, or because it is evidence for other desirable things. It is desirable in itself.

Most propositions arguably have no intrinsic value. It's not that their intrinsic value is zero. They don't have an intrinsic value at all. If, for example, your only intrinsic goals are to be rich and to be famous, then the proposition that you are rich has intrinsic value for you, and so does the proposition that you are famous, but I don't think the proposition that you are rich or famous has any intrinsic value, nor does the proposition that you have porridge for breakfast. These may have derivative value, but to determine their value we need information about how they are related to the things you ultimately care about. They only have Jeffrey value.

We do, however, need to say how individual reasons aggregate into overall reason – how the value of a world is determined by its membership or non-membership in the propositions that have basic intrinsic value. So we have opportunity for a little structure. This is the structure I was talking about in my previous blog post. I'd like to say that it is an "additive" structure. But not in Brown's mereological sense. (In fact, additivity in my sense is incompatible with additivity in Brown's sense.)

In the 2020 paper, Brown mentions that Oddie apparently defended a similar view. Brown complains that it would be arbitrary to say that the value of a world is determined by such-and-such intrinsically valuable propositions. Why not choose other propositions as the basis, with other values, if that would determine the same assignment of value to all worlds?

I'm not sure how to respond. If the aim is to systematize our value judgements, then I guess I simply disagree that all we have to go by are judgements of overall value for entire worlds.

Brown, Campbell. 2014. “The Composition of Reasons.” Synthese 191 (5): 779–800. doi.org/10.1007/s11229-013-0299-8.
Brown, Campbell. 2020. “The Significance of Value Additivity.” Erkenntnis. doi.org/10.1007/s10670-020-00315-3.
Jeffrey, Richard. 1983. The Logic of Decision. Second. Chicago: University of Chicago Press.
Pettit, Philip. 1991. “Decision Theory and Folk Psychology.” In Foundations of Decision Theory: Issues and Advances, edited by M. Bacharach and S. Hurely, 147–75. Cambridge (MA): Blackwell. 