Impossible duties and consequentialism

Intuitively, sometimes you’re obligated to do something you can’t do. For instance, you promised to visit a friend at 5 pm, and at 4:45 pm you are hiking a one-hour drive away. Or you did something bad, and now you owe the victim a sincere apology, but you’re a vicious person and not psychologically capable of rendering an apology that is sincere.

Consequentialist theories, however, have to limit their consideration to actions you can do, since otherwise everything we do is wrong. For whatever we do, there is an impossible action with even better consequences. You spend a day volunteering at a homeless shelter. That may sound good, but the consequences would have been better if instead you magically cured all cancer.

Thus, it seems:

1. If consequentialism is true, you are only ever obligated to do something possible.

2. Sometimes, you are obligated to do the impossible.

3. So, consequentialism is false.

That said, I am not completely convinced of (2).

How impossible can we get?

I've been thinking about a framework for really impossible worlds. The first framework I think of is this. A world w is a mapping (basically, a function, except that the propositions don't form a set) from propositions to truth values. Thus, if w₀ is the actual world, w₀(<the sky is blue>)=T and w₀(<2+2=5>)=F. But there will be a world w with all sorts of weird truth assignments, for instance where the conjunction is true but the conjuncts are false, or where p is false but its negation is also false.

But I then wondered if this captures the full range of alethic impossibilities. What about impossibilities like this: Worlds at which <2+2=4> has no truth value? Worlds at which every proposition is both true and false? To handle such options it's tempting to loosen the requirement that w is a mapping to the requirement that it be a relation. Thus, some propositions might not be w-related to any truth value and some propositions might be w-related to multiple truth values. But we can get weirder than that! What about worlds w at which the truth value of <2+2=4> is Sherlock Holmes? Nonsense, you say? But no more nonsense than something being both true and false. So perhaps w should be a relation not just between propositions and truth values, but propositions and any objects at all, possible or not. But even that doesn't exhaust the options of truth assignments. For what about a world where truth is assigned to every cat and to no proposition, or where instead of <2+2=4> having truth, truth has <2+2=4>? So perhaps worlds are just relations between objects, impossible or possible?

Of course, it feels like we've lost grip on meaningfulness somewhere in the last paragraph. But it's not clear where. My suggestion now is that none of the complications are needed. In fact, even the initial framework where a world is a truth assignment may be needlessly complicated. Let's take instead the simpler framework that a world is a collection of propositions.

Thus, p is true at w if and only if p is a member of w. And p is false at w if and only if ~p is a member of w.

But what about the bizarre options? On this framework, for any world w, either <2+2=4> is a member of w and hence true at w or it's not. What about the possibility that it is both true and non-true at w? I think the framework can handle all the bizarre possibilities provided that we understand them as world-internal. What is true at w is a question external to w, a question to be settled by the classical logic that is actually correct. Either p is true at w or it's not, and it can't be both true and non-true. But, nonetheless, although while it can't be that p is true at w and not true at w, it can be that p is true at w and p is false at w (just suppose both p and ~p are members of w). So that p is false at w does not imply the denial of the claim that p is true at w.

All the bizarreness, however, is to be found in world-internal claims. Let's say that p is (not) true in w provided that the proposition <p is (not) true> is true at w (in the external sense), i.e., <p is true> is a member of w. Likewise, say that p is (not) false in w provided that <p is (not) false> is true at w. And so on: in general, S(p) in w provided that <S(p)> is true at w. Then while truth-at w is relatively tame, truth- and falsity-in w can be utterly wild. We can have p true in w and p not true in w. We can have a world w in which <2+2=4> has the truth value Benjamin Franklin and is also false and true. There will be a world in which ~(2+2=4) but it is nonetheless true that 2+2=4. And so on. It's all a matter of getting the scope of the world-relativizing operator right.

A very impossible world?

In a criticism of the Pearce-Pruss account of omnipotence, Scott Hill considers an interesting impossible situation:

1. Every necessary truth is false.

While the criticism of the Pearce-Pruss account is interesting, I am more interested in a claim that Hill makes that illustrates an interesting fallacy in reasoning about impossible worlds. Hill takes it that a world at which (1) holds is a world very alien from ours, a world at which there are "infinitely many" "false necessary truths".
But that's a fallacious inference from:

2. (∀p(Lp→(p is false))) is true at w

(where Lp says that p is necessary) to

3. ∀p(Lp→(p is false at w)).

Indeed, there is an impossible world w with the property that (1) is true at w and there is no necessary truth p such that p is false at w. Think of a world as an assignment of truth values to propositions. A possible world is an assignment that can be jointly satisfied—i.e., it is possible that the truth values are as assigned. An impossible world is an assignment that cannot be jointly satisfied. Well, let w₀ be the actual world. Then for every proposition p other than (1), let w assign to p the same truth value as it has according to w₀. And then let w assign truth to (1).

Deep Thoughts XXXIII

Everything is possible except the impossible.