Comparing axiologies

Are there ways in which it would be better if axiology were different? Here’s a suggestion that comes to mind:

1. It would be better if cowardice, sloth, dishonesty, ignorance, suffering and all the other things that are actually intrinsic evils were instead great intrinsic goods.

For surely it would be better for there to be more goods!

On the other hand, one might have this optimistic thought:

2. The actually true axiology is better than any actually false axiology.

(Theists are particularly likely to think this, since they will likely think that the true axiology is grounded in the nature of a perfect being.)

We have an evident tension between (1) and (2).

What’s going on?

One move is to say that it makes no sense to discuss the value of impossible scenarios. I am inclined to think that this isn’t quite correct. One might think it would be really good if the first eight thousand binary digits of π encoded the true moral code in English using ASCII coding, even though this is impossible (I assume). Likewise, it is impossible for a human to know all of mathematics, but it would be good to do so.

The solution I would go for is that axiology needs to be kept fixed in value comparisons. Imagine that I am living a blessed life of constant painless joy, and dissatisfied with that I find myself wishing for the scenario where joyless pain is even better than painless joy and I live a life of joyless pain. If one need not keep axiology fixed in value comparisons, that wish makes perfect sense, but I think it doesn’t—unlike the wish about π or the knowledge of mathematics.

A new solution to the non-identity problem?

Molly Gardner in a piece that just came out offers an interesting new solution to the non-identity problem, the problem of making sense of benefits and harms to people who wouldn't exist were it not for our actions of benefiting or harming. Gardner's suggestion is:

A state of affairs, A, is a benefit for an individual, S, just in case if it were true that both S existed and A did not obtain, then S would be worse off in some respect.

This is a clever solution: normally when evaluating whether an action benefits or harms someone, we simply ask how they would have done had we not done the action; but Gardner wants us further to keep fixed that the patient exists.

But clever as it is, it looks to me that it fails. First, suppose that a strong essentiality of origins thesis obtains. Then whenever we benefit or harm a future person, that person couldn't exist without our action. But that means that Gardner's conditional becomes a per impossibile conditional. And concepts of ethical importance should not be defined in terms of something as poorly understood and as controversial as counterpossible conditionals.

Suppose now that there is no strong essentiality of origins thesis. Then, plausibly, a person who was conceived through coitus could also have been conceived through IVF, at least if the same sperm and egg were involved. Now suppose that where the couple lives, IVF technology is highly experimental and works so poorly that children conceived through IVF end up having all sorts of nasty health problems. The couple is wicked and doesn't care about the health of their children, but they also haven't even heard of IVF, and so they conceive Sally the natural way. Now let's consider Gardner's conditional. What would have happened had the child existed and the couple not engaged in coitus? Well, the closest possible worlds where Sally exists and the couple did not have intercourse are worlds where the couple engaged in a poorly-functioning IVF treatment, and hence worlds where Sally has nasty health problems. So the couple benefited Sally by engaging in coitus.

The conclusion that the couple benefited Sally by coitus is, I think, true. For I believe it is always good to exist. But it is clear that Gardner doesn't want to suppose that existence is always a good. And if existence is not always a good, then we can suppose a scenario like this: Sally is going to have an on-balance bad life if she is conceived by coitus, and an on-balance worse life if she is conceived by IVF. By Gardner's criterion, the couple has benefited Sally through coitus, even though Sally's life is on-balance bad. This is surely mistaken. One might say that the couple benefited Sally by engaging in coitus rather than IVF. But since they never even considered IVF, one can't conclude that they benefited Sally simpliciter. (If Sam gives Jim a mild electric shock, he harms Jim simpliciter, but he benefits Jim by giving him a mild rather than severe shock.)

And even if we grant--as in the end we should--that existence is always good, Gardner's conditional gives us the wrong reason for thinking that the couple benefited Sally. For the benefit to Sally has nothing to do with the fact that Sally would have been worse off in the nearby worlds where she existed through IVF.

And even if essentiality of origins is true, the argument concerning Sally works. For it is still true that, per impossibile, had Sally existed but without her parents having intercourse, she would have existed through IVF and hence had very poor health.

The problem with Gardner's approach is this: the worlds that are relevant to the evaluation of her counterfactual may simply be irrelevant to the question of benefit or harm simpliciter.

A sufficient condition for a subjunctive conditional

Start with the idea of grades of necessity. At the bottom, say[note 1], lie ordinary empirical claims like that I am typing now, which have no necessity. Higher up lie basic structural claims about the world, such as that, say, there are four dimensions and that there is matter. Perhaps higher, or at the same level, there are nomic claims, like that opposite charges attract. Higher than that lie metaphysical necessities, like that nothing is its own cause or that water is partly composed of hydrogen atoms. Perhaps even higher than that lie definitional necessities, and higher than that the theorems of first order logic. This gives us a relation: p<q if and only if p is less necessary than q.

Let → indicate subjunctive conditionals. Thus "p→q" says that were it that p, it would be that q. Let ⊃ be the material conditional. Thus "p⊃q basically says that p is false or q is true or both. Then, the following seems plausible:

1. If ~p<(p⊃q), then p→q.

I.e., if the material conditional has more necessity than the denial of its antecedent, the corresponding subjunctive conditional holds.

Suppose it's a law of nature that dropped objects fall. Then the material conditional that if this glass is dropped, then it falls is nomic and hence more necessary than the claim that this glass is not dropped, and the subjunctive holds: were the glass dropped, it would fall.

Moreover, the subjunctives that (1) can yield hold non-trivially, if there are grades of necessity beyond metaphysical necessity (on my view, those are somewhat gerrymandered necessities), and this yields non-trivial per impossibile conditionals. Let p be the proposition that water is H₃O, and let q be the proposition that a water molecule has four atoms. Then ~p<(p⊃q), because p⊃q is a definitional truth while ~p is a merely metaphysical necessity. Hence were p to hold, q would hold: were water to be H₃O, a water molecule would have four atoms.

I wonder if the left-hand-side of (1) is necessary for the non-trivial holding of its right-hand-side.

Gentler structuralisms about mathematics

According to some standard structuralist accounts, a mathematical claim like that there are infinitely many primes, is equivalent to a claim like:

1. Necessarily, for any physical structure that satisfies the axioms A₁,...,A_n, the structure satisfies the claim that there are infinitely many primes.

There are two main motivations for structuralism. The first motivation is anti-Platonic animus. The second is worries about uniqueness: if there are abstract objects, there are many candidates for, say, the natural numbers, and it would be arbitrary if our mathematical language were to succeed in picking out on particular family of candidates.

The difficulty with this sort of structuralism is that while it may be fine for a good deal of "ordinary mathematics", such as real analysis, finite-dimensional geometry, dealing with prime numbers, etc., it is not clear that there are enough possible physical structures to model the axioms of such systems as transfinite arithmetic. And if there aren't, then antecedents in claims like (1) will be false, and hence the necessary conditional will hold trivially. One could bring in counterpossibles but that would be explaining the obscure with the obscurer.

I want to drop the requirement that the structures we're talking about are physical structures. Thus, instead of (1), we should say:

2. Necessarily, for any structure that satisfies the axioms A₁,...,A_n, the structure satisfies the claim that there are infinitely many primes.

If we do this, we no longer have a physicalist reduction. But that's fine if our motive for structuralism is worries about arbitrariness rather than worries about abstracta.

Next, restrict the theory to being about what modern mathematics typically means by its mathematical claims. If we do this, the claim becomes logically compatible with Platonism about numbers. Let us suppose that there really are numbers, and our ordinary language gets at them. Nonetheless, I submit, when a modern number theorist is saying that there are infinitely many primes, she is likely not making a claim specifically about them. Rather, she is making a claim about every system that satisfies the said axioms. If the natural numbers satisfy the axioms, then her claims have a bearing on the natural numbers, too.

Here is one reason to think that she's saying that. Mathematical practice is centered on getting what generality you can. What mathematician would want to limit a claim to being about the natural numbers, when she could, at no additional cost, be making a claim about every system that satisfies the Peano axioms?

Now, if we go for this gentler structuralism, and allow abstract entities, we can easily generate structures that satisfy all sorts of axioms. For instance, consider plural existential propositions. These are propositions of the form of the proposition that the Fs exist, where "the Fs" directly plurally refers to a particular plurality. We can define a membership relation: x is a member of p if and only if x is said by p to exist. Add an "empty proposition", which can be any other proposition (say, that cats hate dogs) and say that nothing is its member. Then plural existential propositions, plus the empty proposition, with this membership relations should satisfy the axioms of a plausible set theory with ur-elements. If all one wants is Peano axioms, we can take them to be satisfied by the sequence of propositions that there are no cats, that there is a unique cat, that there are distinct cats x and y and every cat is x or is y, that there are distinct cats x and y and z and every cat is x or is y or is z, and so on.

I am not completely convinced that this sociological thesis about modern mathematics is correct. Maybe I can retreat to the claim that this is what modern mathematics ought to claim.

Non-triviality of conditionals

Here's a rough start of a theory of non-triviality of conditionals.

A material conditional "if p then q" is trivially true provided that (a) the only reason that it is true is that p is false or (b) the only reason that it is true is that q is true or (c) the only reasons that it is true are that p and q are true.

A subjunctive conditional "p □→ q" is trivially true provided that (a) the only reason that it is true is that p and q are both true or (b) the only reason that it is true is that p is impossible or (c) the only reason that it is true is that q is necessary or (d) the only reasons that it is true are that p is impossible and q is necessary.

For instance, "If it is now snowing in Anchorage, then it is now snowing in the Sahara" understood as a material conditional is trivially true, because the falsity of the antecedent (I just checked!) is the only reason for the conditional to be true. The contrapositive "If it not now snowing in the Sahara, then it is not now snowing in Anchorage" is trivially true, since it is true only because of the truth of the consequent. On the other hand, "If I am going to meet the Queen for dinner tonight, I will wear a suit" is non-trivially true. It is true not just because its antecedent is false--there is another explanation.

Likewise, "Were horses reptiles, then Fermat's Last Theorem would be false" and "Were Fermat's Last Theorem false, horses would be mammals" are "Were I writing this, it would not be snowing in Anchorage" are trivially true, in virtue of impossibility of antecedent, necessity of consequent and truth of antecedent and consequent, respectively. But "Were horses reptiles, either donkeys would be reptiles or there would no mules" is non-trivally true--there is another explanation of its truth besides the impossibility of antecedent, namely that reptiles can't breed with mammals and mules are the offspring of horses and donkeys.

The reliability of intuitions

Often, analytic philosophers give some case to elicit intuitions. Intuitions elicited by certain kinds of cases count for less. Here is one dimension of this: Intuitions elicited by worlds different from ours are, other things being equal, reliable in inverse proportion to how different the worlds are from ours. Here is the extreme case: intuitions elicited by cases of impossible worlds.

For an example of such an intuition, consider the argument against divine command theory that even if God commanded torture of the innocent, torture of the innocent would be wrong. Now, the obvious response is that it's impossible for God, in light of his goodness, to command torture of the innocent. But, the opponent of divine command theory continues, if God were per impossibile to command it, it would still be wrong, but according to divine command theory, it would be right. (E.g., Wes Morriston has given an argument like that.)

I've criticized this particular argument elsewhere (not that I think divine command theory is right).[note 1] But here is a point that is worth making. This argument elicits our intuition by a case taking place in an impossible world. But impossible worlds are very different from ours. So we have good reason to put only little weight on intuitions elicited by per impossibile cases.

This does not mean that we should put no weight on them.