# Contrastive probabilistic explanation

I want to propose something I’m not totally convinced is correct, but that I think is worth considering. In general we have the question about contrastive indeterministic explanation: an antecedent A can give rise to two different consequences B and C, it actually gives rise to B, and we want to know why it gave rise to B rather than C.

There are two cases that encode this, each prima facie in different ways (though they may be ultima facie reducible to the same case, more on this later): libertarian free choice and quantum indeterminism. Let’s take them in turn.

In a free choice we are impressed by reasons R for choosing between B and C. In the event we choose B, we want to know what explains why we chose B rather than C. The answer comes in being more precise about the content of R: it includes reasons R1 for choosing B over C and R2 for choosing C over B, and it’s in virtue of this that we are choosing between B and C in the first place (see section 4 in Divine Creative Freedom by Alexander Pruss). When we choose B then R1 explains why we chose it over C, and when we choose C then R2 explains why we chose it over B. Thus, the explanation is contrastive in virtue of the reasons themselves being contrastive. We’ll return to this shortly.

In an event of quantum indeterminism we have some quantum event — radioactive decay, say — that happens with a certain probability. Let A be the circumstance involving an atom at t1 which will decay with some probability, B be the circumstance involving it having decayed at t2, and C the circumstance involving it not having decayed at t2. In B, how would we explain why it had decayed rather than not?

The first Aristotelian step is to give an account of probabilistic causation, and the second is to elucidate the explanation this affords us. With regards to the first, something like what Feser has proposed here seems plausible, namely that the probabilistic behaviour the atom exhibits is grounded in its substantial form. This explains why the atom in the same antecedent state can result in two different consequent states, in a similar way to how the form of a material thing explains its inertia (see Nature and Inertia by Thomas McLaughlin for a fantastic discussion of this). It also plausibly explains why B is realised when it is realised. But it does not seem to explain why B was realised rather than C.

And here comes my proposal: there is no contrastive fact over and above the plain fact that B occurred and C did not. The difference between the two cases is a relation, and a relation is wholly grounded in the relata themselves (see Aquinas on the Ontological Status of Relations by Mark Henninger). Thus to explain why I am taller than you, it is sufficient to explain why I am my height, why you are your height, and note that the former is greater than the latter. There is no additional fact to explain. Similarly to explain B, and note that B excludes C, is sufficient to explain why B rather than C. If the situation were slightly different such that we had two identical atoms at t1 that at t2 realised B and C respectively, then to explain B for the first and to explain C for the second just is is to explain the outcome of the difference, since this consists precisely in the two outcomes being realised.

But wait! Why was there some irreducible contrastive fact to explain in the free choice case? Because in this case the content of the choice itself was contrastive. It was not that the relation between the choices had to be explained contrastively, but rather that in order to explain every aspect of the choice we also had to explain the contrastive aspects.

# A cosmological argument from simple existential facts

There are loads of different cosmological arguments out there and hopefully someday I’ll be able to write blog posts about some of them. Right now, however, I want to share an interesting version I came up with, thanks to an argument from Alexander Pruss: define a “simple existential fact” to be a true proposition reporting simply the existence or non-existence of a specific being[1,2]. So U doesn’t exist, where U is any specific unicorn and Roland exists, where Roland is me, are simple existential facts of the actual world. Now, let K be the conjunction of all simple existential facts of the actual world that don’t involve necessary beings. So, K only reports the existence or non-existence of beings which possibly fail to exist.

Now assume that possibly, K has an explanation. From this it follows that a necessary being exists. We now show this: Let α denote the actual world[3]. Since possibly, K has an explanation, there is some possible world w, in which K is true and there is some fact q that explains K. Since K involves existential facts, q must involve the causal activity of some being that exists in w[4], call it N. Now either N exists in α or it doesn’t. Assume it doesn’t. Then N doesn’t exist is a conjunct of K. Since K is true at w, it follows that N doesn’t exist in w. But this is a contradiction. Therefore, N exists in α. Is N necessary? Well, assume N isn’t necessary. Then N exists is a conjunct of K, and in w, N explains its own existence. But surely, nothing contingent can explain its own existence. So N is a necessarily existent being.

What’s really nice about this argument is that it doesn’t assume S5 or even the Brouwer axiom. Furthermore, even though I think the Principle of Sufficient Reason is true, to say that K has an explanation doesn’t commit one to the full-blown PSR. In fact, to say that possibly, K has an explanation doesn’t even entail that K actually has an explanation: while N could explain K, N needn’t actually explain K.

### Notes

1. “Specific” has a specific meaning here: rigidly designated. A rigid designator always refers to the same entity and is different from a definite description which changes depending on the current state of affairs. For example, “the president of the United States” is a definite description, whereas “Richard Nixon” is a rigid designator (of course, if there were more than one Richard Nixon we could be talking about, then some more specificity would be needed).
2. I wonder if I couldn’t simplify this even further by saying “a specific being or class of beings” where class could be something like “unicorns”?
3. That is, α is the rigid designator for this possible world.
4. This was discussed very briefly here. It doesn’t seem possible to explain the existence of contingent beings “conceptually”.

# Explanations

Recently[1] I’ve been doing some reading on (amoung other things) Leibnizian Cosmological Arguments and the Principle of Sufficient Reason. One thing that’s involved in these arguments is the idea of an “explanation”. We generally have a firm grasp or intuition of whether something is an explanation for some fact or not. Consider the following statements involving explanations:

1. John sent his children to school A rather than school B because school A has good sports facilities
2. The kettle is boiling because John turned it on and it was working properly
3. The kettle is boiling because the heat of the flame is being conducted via the copper bottom of the kettle to the water, increasing the kinetic energy of the water molecules, such that they vibrate so violently that they break the surface tension of the water and are thrown off in the form of steam[2]
4. We die because the process of natural selection has selected organisms that die over the course of the earth’s evolutionary history

In each of the above cases, the explanation is italicised and the explanandum (the fact to be explained) comes before the “because”. Cases (1) and (2) involve personal or libertarian explanations, case (3) involves a scientific explanation and case (4) an evolutionary explanation (we’ll see later why such evolutionary explanations, including case (4), often fail to explain their explanandum).

Now, as far as I can tell, no-one has been able to give a complete reductive analysis[3] of explanations. Nonetheless, we know a number of things about explanations; two of which I wish to share here.

### 1. Explanations can be non-entailing

An explanation entails its explanadum if it is impossible for the facts in the explanation to be true and not the facts in the explanadum. Consider example (2) above: it is not possible for the facts John has turned the kettle on and the kettle is working properly to be true without the kettle is boiling following. Simply put, the facts in the explanation entail the facts in the explanadum. We can see another example of this in (3).