I was thinking about “would-counterfactuals” the other day and wondering how they’re meant to be understood on a libertarian account of free will that holds to contrary choice as a necessary condition for a free choice. I thought I had come up with some way of giving meaning to statements of the form “Agent S would do action A if put in circumstance C”. However, I realised that I had failed. Nevertheless, it was interesting thinking about metaphysics and mathematics so I thought I’d share it.

### Definitions

Before I do, though, maybe I should define my terms and set the scene. Under libertarianism there are two principles. They are:

- Principle of Contrary Choice (PCC): S (an agent) freely does A (an action) when put in C (a circumstance) only if S could’ve done not-A in C. In layman’s terms, a choice is free only if you could’ve done otherwise.
- Principle of Self-Determination (PSD): S freely does A only if S isn’t externally caused to do A.

Notice how these are stated as necessary conditions. For the sake of this post, we will use the term “PSD-libertarian” to refer to someone who thinks PSD is the necessary and sufficient condition for free will, and we’ll use the term “PCC-libertarian” to refer to someone who thinks at least PSD&PCC are necessary conditions for free will.

Next we need to talk about *possible worlds*. Roughly speaking, a possible world is a complete description of how reality *could have* been. To say “unicorns exist in some possible world” is to say that it is possible (metaphysically speaking) that unicorns could have existed. We mustn’t confuse possible worlds with alternate universes or multiverse. The other possible worlds don’t exist, they’re descriptions of how the actual world could have been. So, for example, there’s a possible world in which I don’t exist. There’s a possible world in which big foot exists. And so on. Now, naturally, the actual world is a possible world (it’s certainly true that what actually is the case is possibly the case). Given this concept of possible worlds we can define the following concepts for some proposition, P (let Ω be the set of all possible worlds, let □P denote the statement “P is necessarily true” and let ◊P denote “P is possibly true”):

- P is necessarily true if and only if P is true in every possible world (□P ⟺ (∀W∊Ω)P∊W).
- P is possibly true if and only if P is true in some possible world (◊P ⟺ (∃W∊Ω)P∊W).
- P is contingent, relative to some world W, if and only if P is true in W and not true in some other possible world.

From these definitions, we can see that

- P is impossible if and only if P is not true in any possible world (¬◊P).
- P is necessary if and only if not-P is impossible (□P ⟺ ¬◊¬P).

It should be noted that since necessary truths are true in every possible world, we only need contingent truths to distinguish between the different possible worlds. In fact, we’ll only consider the tenseless contingent truths of each possible world as the distinguishing feature. Thus we have:

(∀W1, W2∊Ω) W1 = W2 ⟺ (P∊W1 ⟺ P∊W2)

Where P is a contingent truth. We can now rephrase the PCC in terms of possible world semantics:

PCC: An action A of some agent S in some circumstance C is free in some possible world W1 only if there is another possible world W2 in which S is put in C but does some other action not-A.

PSD doesn’t change all that much. Now that we have all this out the way we can get to the issue at hand: would-counterfactuals.

### Understanding would-counterfactuals

How are we to understand the statement “S *would* freely do A in some circumstance C”? For the PSD-libertarian the answer seems clear:

S would freely do A in C if and only if in every possible world in which S is put in C and PSD isn’t violated, S does A

For the PSD-libertarian it doesn’t matter that PCC doesn’t hold. So we don’t need multiple possible worlds in which S is put in C but does something different. Thus we can safely assume that there aren’t any possible worlds in which S does something different when put in C, since this doesn’t affect the freeness of S doing A. The PCC-libertarian isn’t so lucky, it seems. For if in every possible world S is put in C, S does A, then that can’t be said to be a free action. Therefore we can’t say that S would *freely* do A. A PCC-libertarian can understand the statements “S might freely do A in some circumstance C” and “S would not freely do A in C” as follows:

S might freely do A in C if and only if there exists a possible world in which S is put in C and PSD isn’t violated and S does A

S would not freely do A in C if and only if there does not exist a possible worldin S does A when put in C and PSD isn’t violated

Fair enough, but this doesn’t solve the problem of *would*-counterfactuals. Admittedly, I haven’t read any professional treatments of this so I don’t know if this problem has been solved, but I gave it a try.

### My Analysis

So, for my analysis we need a metric, d:ΩxΩ→ℝ (a function maps a pair of possible worlds to a real number), so that we can consider (Ω, d) a metric space. As soon as we have this we can talk about “closeness” of possible worlds. Let W,Y,Z∊Ω be three possible worlds. Then for d to be a metric, the following properties need to hold:

- d(W, Y) = 0 ⟺ W = Y (separation conditions)
- d(W, Y) = d(Y, W) (symmetry condition)
- d(W, Y) ≤ d(W, Z) + d(Z, Y) (triangle inequality)

It seems obvious to me that d can be easily defined as follows:

d(W, Y) = the number of propositions that are true in W but not in Y

Clearly the the separation conditions hold. What about the symmetry condition? Let’s say that A⊆W is the set of all propositions true in W but not in Y and B⊆Y is the set of all propositions true in Y but not in W. Note that d(W, Y) = |A| and d(Y, W) = |B|. Now clearly |A| ≤ |B| since if P∊A then ¬P∊B. Similarly |B| ≤ |A|, thus

d(W, Y) = |A| = |B| = d(Y, W)

which is what we needed. What about the triangle inequality? If all any of the worlds are equal to one another, then it is trivially true. Consider, then, the case when we’re dealing with three different possible worlds. Consider a proposition P∊W such that ¬P∊Y. Now, assuming the principle of excluded middle, we know that P∊Z or ¬P∊Z. Either way we can see that d(W, Z) + d(Z, Y) is always going to be at least as big as d(W, Y). Thus the triangle inequality holds. Thus d is a metric.

It might be tempting to go straight from this metric to a definition of would-counterfactuals relative to some possible world:

Relative to some possible world W, S would freely do A in C if and only if in the closest possible world, Y, in which S is put in C, S does A

Here the term “closest” can be given meaning by the metric. So the closest possible world in which S is put in C can be understood as:

Y, such that d(W, Y) = min{d(W, Z) | “S is put in C”∊Z}

There is, however, a problem in this: what if two possible worlds Y1 and Y2 both include the proposition “S is put in C”, with S doing something different in each, and d(W, Y1) = d(W, Y2). Then there doesn’t seem to be a unique minimum. In fact, we’re right back where we started. We might as well just use this metric to define might-counterfactuals more specifically.

But all is not lost. If we allow for the axiom of choice, then there is a function, Γ:P(Ω)→Ω, where P(Ω) is the set of all subsets of Ω, excluding the empty set. We can speculatively use this function in our definition of would-counterfactuals. First we need a set, which we’ll define as follows:

<S,C,W> := {Y∊Ω | d(W, Y) = min{d(W, Z) | “S is put in C”∊Z}}

We have, then, that Γ(<S,C,W>)∊Ω. With this we can define would-counterfactuals as:

Relative to some possible world W, S would freely do A in C if and only if in the possible world Y=Γ(<S,C,W>), S does A

This will certainly only give us a single action which S *would* do. I feel like this is sufficient for our needs.

### The Problems

Now we come to the problems with this analysis of would-counterfactuals for PCC-libertarians. Firstly, it should be noted that there was one thing we didn’t check about our metric, namely that it always maps a pair of possible worlds to a finite number. It seems possible that two possible worlds (W and Y) differ in an infinite number of propositions, in which case d(W, Y) does not map to a real number, meaning the function isn’t left total, meaning it isn’t a function! There seems to be three ways around this:

- Insist that all possible worlds are made up of only a finite number of propositions
- Define d:RxR→ℝ where R⊆Ω and (W,Y)∊RxR if there are only a finite number of propositions P∊W such that ¬P∊Y
- Define d:ΩxΩ→ℝ∪{∞}, where min(A) = ∞ ⟹ A = {∞}

None of these is very nice to be honest. Secondly, we can’t just take any function Γ:P(Ω)→Ω. This is because will-counterfactuals are a subset of would-counterfactuals. What I mean by this is the following:

S *will* freely do A in C if only if relative to the actual world, S *would* freely do A in C

“will” refers to what S will do in the actual world when put in some circumstance C. We can solve this problem by picking Γ such that whenever it is given a set with the actual world (call it AW) in it, it must map to the actual world:

AW∊G ⟹ Γ(G) = AW

While these problems aren’t fatal, they do make this analysis of would-counterfactuals seem even more unintuitive than when it began. But such is life I guess. The third problem, however, *is* fatal. Did you notice how we didn’t actually give an analysis of would-counterfactuals at all? We gave an analysis of what we might call “relative would-counterfactuals”. Would-counterfactuals seem to be true regardless of which world they’re “seen from”. It seems that only the PSD-libertarian can make sense of absolute would-counterfactuals after all.

O well 🙂

Hey. So I think one way to solve your metric problem would be to consider the different kinds of infinity. Assuming that the class T of all propositions that are true in W but not in Y is a set, we have that |T|=aleph_n for some nonnegative integer n, where aleph_n is the (n+1)-th infinite number (provided W \neq Y). Then perhaps you could define d(W,Y)=n+1, and maybe the axioms of a metric space would still hold.

Another possibility is the following definition of the distance between two possible worlds, which you might be able to use: http://youtu.be/kTZmLuhJgag