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.

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.

A theory about counterpossibles

I suspect that non-trivial per impossibile counterfactuals, true subjunctive conditionals of the form "p→q", where p is impossible and the conditional is not simply said to be true on account of the falsity of p, are in a way like poetry: They tell us things that are hard to express in more ordinary language and that, moreover, have a deeper resonance with us, and are more plausible, when put poetically.

But we can, I think, give a sufficient condition for the truth of a counterpossible: if the material conditional "if p, then q" is true in virtue of a fact explanatorily prior to or independent of not-p, then p→q holds. This condition seems to me to also hold in the case of ordinary counterfactuals. Thus, the laws of nature are explanatorily prior to or independent of ordinary non-nomic facts. Thus, if it is a law of nature that if something is a raven, then it is black, we can say that if there were a raven in this room, it would be black, because the conditional "if something is a raven, then it is black"[note 1] is explanatorily prior to or independent of the absence of ravens from this room.

In particular, when the consequent of the material conditional is true and explanatorily prior to or independent of the antecedent, the subjunctive conditional holds trivially. For instance: "Were God not to have commanded respect to parents, there would (still) be a duty to respect parents." Here, the corresponding material conditional holds in virtue of the consequent's holding, and the consequent is (or so the asserter of the conditional claims) independent of or explanatorily prior to God's commanding respect to parents.

I don't know if the condition I have given is necessary for a conditional's truth. But at least sometimes, I think, we use a per impossibile counterfactual precisely to express a claim about explanatory priority or independence.

Here is a seemingly different sufficient condition for the subjunctive conditional p→q. If the material conditional "if p, then q" is more strongly necessary than not-p, then p→q holds. The idea of grades of necessity is perhaps best introduced by example: nomic necessity is stronger than practical necessity; metaphysical necessity is stronger than nomic necessity; narrowly logical (or conceptual?) necessity is stronger than metaphysical necessity.

We can combine the two conditions. Suppose that the material conditional "if p, then q" follows with a necessity of grade n₁ from some fact F, and this fact F is (a) explanatorily prior to or independent of not-p, and (b) the truth of not-p is not necessary with a necessity of grade n₁, then the subjunctive p→q holds. I don't know if this is a necessary condition for a subjunctive to hold. Maybe it is.