The spectrum of values

Let’s do some rough and ready thinking about the spectrum of possible values of objects on a classical theistic view.

1. God has his value essentially.

2. Necessarily, God is more valuable than everything other than God.

3. Necessarily, everything that exists exists by participation in the good God and hence has positive value.

The spectrum of possible values thus has an upper bound: God’s value. Moreover, it follows from (2) that God is infinitely many times more valuable than anything else. For consider some object x other than God, and imagine a world (perhaps a multiverse) where x is duplicated some number n of times. By (2), God will be more valuable than the duplicates of x, and hence God is more than n times valuable than x. Since n is arbitrary, it follows that God is infinitely many times more valuable than x.

Thus, our spectrum of values has God at the top, then an infinite gap, and below that possible creatures.

What does the lower end of the spectrum of possible values of objects look like? Well, by (3), all the values are positive. So the lower end of the spectrum lies above zero. I suspect that it asymptotically approaches zero. For consider an object x and now imagine an object y which has exactly one essential causal power, that of producing x with a probability of 1/2. Intuitively, y has something like half the value of x. So it is plausible that the lower end of the spectrum of possible values approaches zero but does not reach it.

But now suppose that y has only an infinitesimal probability of producing x (imagine y has an internal spinner and it produces x whenever that spinner lands exactly at ninety degrees). Then x seems like it would be infinitely more valuable than y. If this is right, then for every value in the spectrum, there is a value that is infinitely many times smaller than it.

The spectrum of values has a top (God) but no bottom. For any value on the spectrum of values, there is a value infinitely many times smaller than it. And for any value on the spectrum of values other than God, there is a value infinitely many times greater than it.

There is thus a very natural sense in which everything is relatively infinite in value: everything is infinitely many times more valuable than something else. But only God is absolutely infinite in value: God is infinitely many times more valuable than everything else.

Incommensurability complicates things, though.

The choice between qualitative probabilities and generalized quantitative probabilities is illusory

There are two approaches to generalizing probabilities beyond the classical real-valued numerical approach.

1. Switch the values of the probability function from real numbers to values in some other ordered algebraic entity (e.g., the hyperreals, the surreals, an arbitrary totally ordered monoid).

2. Switch to qualitative probabilities, where instead of assigning values to events, one just compares the events probabilistically (this one is at least as likely than that one).

In the literature (including stuff I wrote myself!), the qualitative probability approach is treated as more general. But in fact, it’s not, at least not if we assume that the “at least as likely as” relation is transitive and reflexive, i.e., is a partial preorder. For suppose that we have a collection F of events and a partial preorder ⪅ on them. Say that A ≈ B iff A ⪅ B and B ⪅ A. Let V be the set of equivalence classes of events under the relation ≈, and define the partial order ≤ on V by [A]≤[B] iff A ⪅ B, where [A] is the equivalence class of A. (All this makes sense if ⪅ is a partial preorder.) Define P(A)=[A].

And that’s it! You now have a probability function P on F whose values are a partially ordered set V. So, what you do with qualitative comparisons you can do with values. Now that I’ve said it, it’s obvious to me and I’m kicking myself for not noticing it earlier. Perhaps some people in the field have noticed it and found it so obvious that it’s not worth saying.

And rather than thinking of there being a debate between probability functions with values and qualitative probabilities, we can now just stick to the probability function approach, and say that there is a serious debate as to what kind of a structure the values have: are they a real closed field, a totally ordered field, a totally ordered monoid (and if so, does its operation correspond to addition or to multiplication in the classical case), a mere partially ordered set, etc.?

The point here applies in other contexts, such as qualitative utilities, moral value comparisons, etc. Using the apparatus of set theory, we can replace a comparison relation by a value-assignment, which makes it more convenient to apply all the apparatus of ordered algebraic entities of different sorts.

As an application, in a recent post I said that given two very plausible axioms on probability functions, one cannot have a regular rotationally-invariant probability function on the measurable subsets of the circle. Using the above remarks, that point immediately extends to qualitative probabilities that satisfy the following two axioms:

1. If A and B are disjoint, A and C are disjoint, and B and C are equally likely, then A ∪ B and A ∪ C are equally likely, and

2. Ω − A and Ω − B are equally likely iff A and B are equally likely,

with rotational invariance being understood as saying that A is at least as likely as B if and only if ρA is at least as likely as B if and only if A is at least as likely as ρB for every rotation ρ, and with regularity understood as saying that every non-empty event is more likely than the empty event.

Pluralism in public life

Consider this formulation of the central problem of a pluralist democracy:

1. How to have a democracy where there is a broad plurality of sets of values?

Assuming realism about the correct set of values, this is roughly equivalent to:

2. How to have a democracy where most people are wrong in different ways about the values?

But when we think about (1) and (2), we are led to thinking about the problem in different ways. Formulation (1) leads us to think the problem is with the state, which should somehow accommodate itself to the plurality of values. Formulation (2) points us, however, to the idea that the problem is with the people (including perhaps ourselves) who have the wrong set of values.

My own view is that there is partial but incomplete realism about values. Specifically, there is such a thing as the correct set of values. But there is a legitimate plurality of rankings between the values, though even there not everything goes—some rankings violate human nature. As a result, the problem is both with us, in that most of us have the wrong set of values and have some prioritizations that violate human nature, and with the state which needs to accommodate a legitimate plurality of prioritizations.

Determinates vs. values

Spot has a mass of 10kg, while Felix has a mass of 8kg. The standard Platonic way to model the facts expressed by this is to say that Spot and Felix both have the determinable property of mass and they also have the determinate properties mass-of-10kg and mass-of-8kg, respectively. But there is another Platonic way to model these facts. Rephrase the beginning statement as: "Spot masses 10kg while Felix masses 8kg." The natural First Order Logic rendering of the English is now: Masses(spot, 10kg) and Masses(felix, 8kg). In other words, there is a relation between Spot and Felix, on the one hand, and the two respective values of 10kg and 8kg, on the other.

The determinate property approach multiplies properties: for each possible mass value, it requires a property of having mass of that value. The value approach, on the other hand, introduces a new class of entities, mass-values. So far, it looks like Ockham's razor favors the standard determinate property approach, since we don't want to multiply classes of entities.

However, the determinate property approach has some further ideology. It requires a determinable-determinate relation, which holds between having mass and having mass m. The mass-value approach doesn't require that. We can define having mass in terms of quantification: to have mass is to mass something (∃x Masses(spot, x)). Moreover, the value approach might be able to greatly reduce the number of values it posits. For instance, mass, length and charge values could all simply be real numbers in a natural unit system like Planck units. If one thinks that the Platonist needs mathematical objects like numbers anyway, the additional commitment to values comes for free. Further, the determinate property approach requires positing either a privileged bijection relation (or set of bijection relations) between mass values and non-negative real numbers or enough mathematical-type relations between mass determinates (e.g., a relation of one mass determinate being the sum of two or more mass determinates) to make sense of the mathematics in laws of physics like F=Gmm'/r².

There is also a potential major epistemological bonus for the value approach if the values are real numbers. Standing in a mass relation to a particular real number will be causally relevant. Thus, real numbers lose the inertness, the lack of connection to concrete beings like us, that is at the heart of the epistemological problems for mathematical Platonism.

All that said, I'm not enough of a Platonist to like the story. Is there a non-Platonic version of the story? Maybe. Here's one wacky possibility after all: Values are non-spatiotemporal contingent and concrete beings. They may even be numbers, contingent and concrete nonetheless.

Praise and relativism

Hartry Field agrees with Putnam that values are non-factual. Of course, there is a fact of the matter about whether x values F, but there is no fact of the matter about whether x's valuing F is correct. This includes epistemic values. Field thinks this is not a problem. One simply relativizes epistemology to an "evidential system". Then, making use of a non-relativistic concept of truth, one defines the reliability of an evidential system. Finally:

if there is any "highest epistemological praise" it will be something like "is justified relative to some highly reliable evidential system" (or "is justified relative to all highly reliable evidential systems", or some such thing). This isn't really an adequate formulation of what "the highest epistemological praise" (if there is such a thing) would be, for (among other things) reliability is not the only feature we want our evidential systems to have; but it gives the general flavor. (Journal of Philosophy 79 (1982), p. 564)

Field is cautious about whether there is any such thing as the "highest epistemological praise". His caution could have two sources: he could be cautious about whether there is such a thing as "high epistemological praise" or about whether there is such a thing as the "highest epistemological praise". I shall take the latter to be his worry. Thus, on my reading, Field thinks there is such a thing as high epistemological praise, and to give it is to say something of "the general flavor" of the claim that a belief is "justified relative to some highly reliable evidential system (jrtshres)".

But now let me raise this question. What makes saying that a belief is jrtshres be a case of praise, while saying that it is justified relative to some evidential system (jrtses—note that every belief has this property) or that it was acquired during a full moon (adafm) are, presumably, not praise?

To answer this question we need to figure out the sense of the word "phrase". I see two prima facie plausible answers. On the first, to praise something is to attribute to it a property that is valued (individually or socially)—this is the relativistic notion of praise. On the second, to praise something is to attribute to it a property that is in fact valuable—this is the objective notion of praise.

Let's start with the second. This clearly has difficulties. Thus, it is easy to imagine (and I remember a claim that there is a code of honor among Russian thieves according to which this is so) a criminal subculture where to say that something was earned through honest work got is not praise, even though it is the attribution of a property that is in actual fact valuable. Similarly, it seems to be genuine praise if I say, misunderstanding the aim of checkers: "Great! You've just managed to get yourself into a position where you have no valid move." Nonetheless, there may be a sense of "objectively correct praise" on which to praise something is to attribute to it a property that one believes to be objectively valuable. But then by engaging in epistemic praise, we are presupposing something incompatible with Field's relativism about epistemic values—we're taking a belief's being jrshrtes to be objectively valuable.

On the other hand, here is a difficulty for the relativistic notion of "praise" as a reading of what Field is claiming. It seems that on a relativistic notion of praise, what is going to be the highest epistemological praise is not that a belief is jrtshres, but that it is justified according to one's own evidential system (on the individual relativist reading—the social case needs a modification in the argument). The evidential systems in Field's paper embody different individuals' evidential values, and so if one praises by attributing properties that one values, then one will be praising compliance with one's own evidential system.

I suppose Field could object that it is possible to see one thing as valuable for one's own beliefs and another as valuable for another's. Perhaps one sees epistemic caution as good in one's own case but values incaution in others, being glad that others explore crazy hypotheses, as that gives one a richer fund of ideas to work with. This example, by itself, is no good, though. Instrumentally valuing something that others do, on account of its benefits to oneself, is not really praise (unless one has an overinflated ego and one equates oneself with God or the universe or something like that). It is not, for instance, praise for the conman to say, once the con is done: "You have made me rich", though the conman values being rich. It would, on the other hand, be more like praise for the conman to say to someone: "You have made yourself rich." As long as we see others as being relevantly like ourselves, it does not seem that we can coherently praise in another what we do not value in ourselves.

Moreover, let's simplify and assume that what it is to value something is to have a certain kind of preference for it. A more sophisticated theory of subjective value will need a more sophisticated version of this argument, but I suspect the basic point will still be possible to make. Then on the relativistic reading, the force of the praise comes down to something basically like one's preference for jrtshes beliefs. But the following statement seems to me to be performatively inconsistent: "I praise you for F, because I prefer F." The relativism in the second clause undercuts the praise in the first. Epistemic praise, however, can be made both of oneself and of another. If made of another, one can hold back on the "because I happen to prefer jrtshes beliefs" clause. But if we praise ourselves in a clear-headed way, then we cannot hold back on it, and we indeed are being performatively inconsistent.

Of course, if to value something is to believe that it is objecitively valuable, the performative inconsistency disappears. But Field cannot take this route.

If all this is correct, we get a more general result: If relativism about values is correct, praise is insincere, manipulative or in some way inconsistent.