Choices on a spectrum

My usual story about how to reconcile libertarianism with the Principle of Sufficient Reason is that when we choose, we choose on the basis of incommensurable reasons, some of which favor the choice we made and others favor other choices. Moreover, this is a kind of constrastive explanation.

This story, though it has some difficulties, is designed for choices between options that promote significantly different goods—say, whether to read a book or go for a walk or write a paper.

But a different kind of situation comes up for choices of a point on a spectrum. For instance, suppose I am deciding how much homework to assign, how hard a question to ask on an exam, or how long a walk to go for. What is going on there?

Well, here is a model that applies to a number of cases. There are two incommensurable goods one better served as one goes in one direction in the spectrum and the other better served as one goes in the other direction in the spectrum. Let’s say that we can quantify the spectrum as one from less to more with respect to some quantity Q (amount of homework, difficulty of a question or length of a walk), and good A is promoted by less of Q and incommensurable good B is promoted by more of Q. For instance, with homework, A is the student’s having time for other classes and for non-academic pursuits and B is the student’s learning more about the subject at hand. With exam difficulty, A may be avoiding frustration and B is giving a worthy challenge. With a walk, A is reducing fatigue and B is increasing health benefits. (Note that the claim that A is promoted by less Q and B is promoted by more Q may only be correct within a certain range of Q. A walk that is too long leads to injury rather than health.)

So, now, suppose we choose Q = Q₁. Why did one choose that? It is odd to say that one chose Q on account of reasons A and B that are opposed to each other—that sounds inconsistent.

Here is one suggestion. Take the choice to make Q equal to Q₁ to be the conjunction of two (implicit?) choices:

1. Make Q at most Q₁

2. Make Q at least Q₁.

Now, we can explain choice (a) in terms of (a) serving good A better than the alternative, which would be to make Q be bigger than Q₁. And we can explain (b) in terms of (b) serving good B better than the alternative of making Q be smaller.

Here is a variant suggestion. Partition the set of options into two ranges R₁, consisting of options where Q < Q₁ and R₂, where Q > Q₁. Why did I choose Q = Q₁? Well, I chose Q over all the choices in R₁ because Q better promotes B than anything in R₁, and I chose Q over all the choices in R₂ because Q better promotes A than anything in R₁.

On both approaches, the apparent inconsistency of citing opposed goods disappears because they are cited to explain different contrasts.

Note that nothing in the above explanatory stories requires any commitment to there being some sort of third good, a good of balance or compromise between A and B. There is no commitment to Q₁ being the best way to position Q.

Are free actions a counterexample to the PSR?

I’ve argued somewhat as follows in the past:

1. Necessarily, no one is responsible for a brute fact—an unexplained contingent fact.

2. Necessarily, someone is responsible for every free decision or free action.

3. So, it is impossible for a free decision or free action to be a brute fact.

But then:

4. Necessarily, a counterexample to the Principle of Sufficient Reason (PSR) is a brute fact.

5. So, no free decision or free action can be a counterexample to the PSR.

One may imagine someone, however, arguing that although a free decision or a free action cannot be a counterexample to the Principle of Sufficient Reason, a contrastive report, such as that x freely chose to do A rather than B, could be a counterexample to the Principle of Sufficient Reason. But notice that if x freely chose to do A rather than B, then x is responsible for choosing to do A rather than B. Similarly if x freely chose to do A for reason R rather than B for S, then x is responsible for doing so. Freedom implies responsibility. But no one is responsible for a brute fact, so such contrastive reports cannot be reports of a brute fact.

Objection 1: Incompatibilism is true, and on incompatibilism it is obvious that no possible explanation can be given for why x freely chose to do A for R rather than B for S. Hence the Principle of Sufficient Reason is false.

Response: Given that no one is responsible for what has no explanation, if the “no possible explanation” claim is correct, then free will is impossible. Thus, rather than showing that the PSR is false, the argument would show that if incompatibilism is true, free will is impossible. As a libertarian, I think free will is possible (and actual). But it is important to keep clear on what it is that is really endangered by the argument: it is free will and not the PSR.

Objection 2: Freedom is a necessary but not sufficient condition for responsibility.

Response: I am not sure about this. When I think about what other conditions we need to add to freedom to yield responsibility, the only one I can think of is something like knowledge of what is at stake. But it is arguable that without knowledge of what is at stake, a choice is not free. Moreover, even if one does not know what is at stake with A and B beyond what is contained in the respective reasons R and S, one will still be responsible for choosing A for R rather than B for S if one chooses freely for these reasons. One just won’t be responsible for the further aspects, beyond those captured by R and S, that one does not know.

But let’s grant for the sake of argument that other conditions need to be added to freedom to yield responsibility. If so, then the claim has to be that free but non-responsible decisions or actions or contrastive reports thereof are a counterexample to the PSR although free and responsible ones are not. In other words, one has to hold that the alleged additional conditions that need to be added to freedom to yield responsibility are what secures explicability. But given that the most plausible candidate for the other conditions is knowledge of what is at stake, this is implausible. For a free action based on mistaken or limited knowledge is no less explicable than an action based on full knowledge, once one takes into account the agent’s epistemic deficiency.

"Despite" explanations

The phenomenon of contrastive explanations has been explored by a number of authors. There is another phenomenon in the vicinity, that of explanations of despite-claims, that has not received as much attention, even though it’s also interesting. Suppose Bob hates bananas and eats a banana.

1. Why did Bob eat a banana? – Because he was hungry.

2. Why did Bob eat a banana despite hating bananas? – Because he was very hungry.

A contrastive request for explanation, say

3. Why did Bob eat a banana rather than an apple?

doesn’t so much ask for an explanation of a special contrastive proposition, but rather constrains what kind of answer is acceptable—an answer that provides a contrastive answer. Thus, saying that Bob was hungry is not an acceptable answer since it fails to be contrastive between the banana and apple options, while saying that Bob was hungry and a banana was closer at hand is an acceptable answer. However, whenever one constrains what kind of an explanation is acceptable, one runs the risk that—even without any violation of the Principle of Sufficient Reason—there is no answer. For instance, the question

4. Who killed the mayor and why?

is a request for explanation that has no answer if the mayor died from a tornado, because (4) constrains us to agentive explanation, and in this case there is no agentive explanation.

Are requests for explanations-despite like requests for contrastive or agentive explanations, requests that constrain the type of explanation that is acceptable, rather than simply modifying the proposition to be explained?

I am inclined to think that the answer is negative. Here is a preliminary analysis for what is going on when we ask:

5. Why p despite r?

First, the question carries a presupposition that the fact that r is antiexplanatory of p or that it has a tendency against p. If that presupposition is false, the question has no answer, being akin to one of the standard trick questions with false presuppositions (like “Have you stopped beating your spouse?”).

Second, what we are asking is something like this:

6. How was the antiexplanatory force of the fact that r against its being the case that p countered such that p is true?

And this seems to be a straightforward request for an explanation of an admittedly complex proposition, without any constraints being placed on what explanations are acceptable.

If I am right about this, then while a failure to have a good answer to contrastive explanation question does no damage to the Principle of Sufficient Reason (PSR), a failure to have a good answer to an explanation-despite question, when the presuppositions of the question are correct, would be a violation of the Principle. This suggests that some of the attention focused on contrastive explanation in connection with critique of the PSR should be redirected towards explanation-despite. I think the PSR can survive such attention, but the investigation is worthwhile.

Free will and the PSR

Even though I think one of the biggest challenges to the Principle of Sufficient Reason (PSR) is the feeling that something is unexplained in the case of free actions. I think this can be answered: see Section 4 here. But in this post I want to make a very small and simple point that just occurred to me.

The puzzle of free actions is not the lack of reasons. It is a surfeit of reasons. Suppose I eat a donut rather than an apple. It is easy to give a reason: the donut is more delicious. If that’s all we had, there would be no felt difficulty about the explanation. But the felt difficulty comes from the fact that while the donut is more delicious, the apple is more nutritious, and hence while I have a reason for eating the donut rather than the apple, I also have a reason for eating the apple rather than the donut.

But while a shortage of reasons would be a problem for a principle like the PSR that affirms the existence of reasons, a surfeit of reasons is not a problem for it!

So whatever one might say about the puzzle of free will, it is not problem for the PSR.

Contrastive PSR

In my Principle of Sufficient Reason (PSR) book, I defend a PSR that holds that every contingent truth has an explanation, but I do not defend a contrastive PSR. Many think this is a cop-out.

But it makes sense to ask why it is that

1. The moon is round and I don’t have an odd number of fingers.

The answer is, presumably, that gravity pulled the matter of the moon into a ball and I was sufficiently careful around power tools. And yet it doesn’t make sense to ask why it is that

2. The moon is round rather than my having an odd number of fingers.

This point shows that it makes no sense to have a contrastive Principle of Sufficient Reason of the following form:

3. For all contingent truths p and contingent falsehoods q, there is an explanation of why p rather than q is true.

The only time it makes sense to ask why p rather than q is true is when q is some sort of a “relevant alternative” to p. So the contrastive Principle of Sufficient Reason would have to say something like:

4. For all contingent truths p and contingent falsehoods q, if q is a relevant alternative to p, there is an explanation of why p rather than q is true.

But now note that (4) is way messier than the standard PSR, and depends on an apparently contextual constraint in terms of a “relevant alternative” which feels ill-suited to a fundamental metaphysical principle. So, I do not think a contrastive PSR just is a plausible metaphysical principle.

Eleven varieties of contrastive explanation

In connection with free will, quantum mechanics or divine creation it is useful to talk about contrastive explanation. But there is no single generally accepted concept of contrastive explanation, and what one says about these topics varies depending on the chosen concept.

To that end, here is a collection of definitions of contrastive explanation. They all have this form:

• r contrastively explains why p rather than q if and only if r explains why (p and not q) and [insert any additional conditions].

They vary depending on the additional conditions to be inserted. Here are some options for these:

1. No additional conditions.

2. r makes p more likely than q.

3. r cannot explain q.

4. r wouldn’t explain q if q were true instead of p.

5. r wouldn’t explain q as well as it now explains p if q were true instead of p.

6. q wouldn’t be explained by r or by any proposition with r’s actual grounds if q were true instead of p.

7. q wouldn’t be explained by r or by any proposition with r’s actual grounds as well as r now explains p if q were true instead of p.

8. the conjunction of everything explanatorily prior to p makes p more likely than q.

9. r entails (p and not q).

10. r entails the truth of p.

11. r entails the falsity of q.

It is not possible to normally have contrastive explanations of indeterministic free choices or quantum events in senses 9–11, and probably sense 8, but it is possible (with an appropriately metaphysical theory of free choice or quantum events) in senses 1-7. As for the case of contingent divine creative decision, things depend on divine simplicity. Without divine simplicity, contrastive explanations are possible in senses 1–7. Interestingly, if divine simplicity is true, then it is not possible to have contrastive explanations of contingent divine creative decisions in senses 6 or 7.

In what I said above, I assumed that the explanandum cannot be a part of the explanans. If following Peter Railton one drops this condition, then contrastive explanation of all three phenomena (with or without divine simplicity) becomes possible in all the senses.

Lesson: When one talks about contrastive explanation, one needs to define one’s terms.

Acknowledgments: I am grateful to Christopher Tomaszewski for in-depth discussion that led me to recognize the important difference between 4–5 and 6–7. And the Railton point is basically due to a remark by Yunus Prasetya.

Statistically contrastive explanations of both heads and tails

Say that an explanation e of p rather than q is statistically contrastive if and only P(p|e)>P(q|e).

For instance, suppose I rolled an indeterministic die and got a six. Then I can give a statistically contrastive explanation of why I rolled more than one (p) rather than rolling one (q). The explanation (e) is that I rolled a fair six-sided die. In that case: P(p|e)=5/6 > 1/6 = P(q|e). Suppose I had rolled a one. Then e would still have been an explanation of the outcome, but not a statistically contrastive one.

One might try to generalize the above remarks to conclude to this thesis:

1. In indeterministic stochastic setups, there will always be a possible outcome that does not admit of a statistically contrastive explanation.

The intuitive argument for (1) is this. If one indeterministic stochastic outcome is p, either there is or is not a statistically contrastive explanation e of why p rather not p is the case. If there is no such statistically contrastive explanation, then the consequent of (1) is indeed true. Suppose that there is a statistically contrastive explanation e, and let q be the negation of p. Then P(p|e)>P(q|e). Thus, e is a statistically contrastive explanation of why p rather than q, but it is obvious that it cannot be a statistically contrastive explanation of why q rather than p.

The intuitive argument for (1) is logically invalid. For it only shows that e is not the statistically contrastive explanation for why q rather than p, while what needed to be shown is that there is no statistically contrastive explanation.

In fact, (1) is false. The indeterministic stochastic situation is Alice’s flipping of a coin. There are two outcomes: heads and tails. But prior to the coin getting flipped, Bob uniformly chooses a random number r such that 0 < r < 1 and loads the coin in such a way that the chance of heads is r. Suppose that in the situation at hand r = 0.8. Let H be the heads outcome and T the tails outcome. Then here is a constrastive explanation for H rather than T:

• e₁: an unfair coin with chance 0.8 of heads was flipped.

Clearly P(H|e₁)=0.8 > 0.2 = P(T|e₁). But suppose that instead tails was obtained. We can give a constrastive explanation of that, too:

• e₂: an unfair coin with chance at least 0.2 of tails was flipped.

Given only e₂, the chance of tails is somewhere between 0.2 and 1.0, with the distribution uniform. Thus, on average, given e₂ the chance of tails will be 0.6: P(T|e₂)=0.6. And P(H|e₂)=1 − P(T|e₂)=0.4. Thus, e₂ is actually a statistically contrastive explanation of T. And note that something like this will work no matter what value r has as long as it’s strictly between 0 and 1.

It might still be arguable that given indeterministic stochastic situations, something will lack a statistically contrastive explanation. For instance, while we can give a statistically contrastive explanation of heads rather than tails, and a statistically contrastive explanation of tails rather than heads. But it does not seem that we can give a statistically contrastive explanation of why the coin was loaded exactly to degree 0.8, since that has zero probability. Of course, that’s an outcome of a different stochastic process than the coin flip one, so it doesn't support (1). And the argument needs to be more complicated than the invalid argument for (1).

Explanation modulo a fact

A notion that I find rather natural is the idea of explanation modulo a fact. This notion can do some of the work that contrastive explanation can.

1. Why did you eat a banana?

can be precisified contrastively in ways like:

2. Why did you eat a banana rather than something else?
3. Why did you eat a banana rather than doing something else with it?
4. Why did you eat a banana rather than not eating anything?

But one can also accomplish the same thing this with requests for explanation modulo a fact. Doing the work of 2 and 3 is easy:

5. Given that you ate, why did you eat a banana?
6. Given that you did something with a banana, why did you eat it?

Doing the work of 4 is a bit harder, but maybe this works:

7. Given that you ate a banana if you ate anything, why did you eat a banana?

When we ask for explanation why p given q, we are asking for something that isn't just an explanation why q, and that gives us a further explanation why p when added to the fact taht q. The question presupposes there is such an additional truth. If the fact that q is the one and only complete explanation why p, then the question has a false presupposition.

It seems, however, that any request for explanation modulo a fact can be rephrased as a request for contrastive explanation. For instead of asking:

8. Given q, why p?

one can ask:

9. Why p and q rather than not-p and q?

This makes explanation modulo a fact not very useful, it seems. However, one can also formulate contrastive questions as requests for explanation modulo a fact. Thus the contrastive question:

10. Why p₁ rather than p₂, p₃, ... or p_n?

can be replaced with:

11. Given p₁, p₂, p₃, ... or p_n, why p₁?

So which is more fundamental? Explanation-modulo or contrastive explanation? I don't know. The literature prefers contrastive explanation, but that may be a historical accident.

In any case, I think all this makes an important fact about contrastive explanation clearer, a fact that I learned from conversation with Dan Johnson: When you make a request for a contrastive explanation, you are also indicating a refusal to accept certain kinds of answers. In the explanation-modulo case, these are answers that merely go through the given fact. But then the defender of the Principle of Sufficient Reason should feel no embarrassment about being unable to give answers to all requests for contrastive explanation—after all, that every fact has an explanation does not make for any kind of guarantee that the explanation will be of the sort we want. Once we start ruling out certain kinds of explanations, we might well be left with none.

Leibnizian explanations

Say that p is a Leibnizian explanation of why q rather than r provided that p explains why q and not r and it is not possible for p to explain r.

I am inclined to think that a Leibnizian explanation of why q rather than r is a contrastive explanation of why q rather than r. But does the converse hold? Are contrastive explanations always Leibnizian?

The answer may depend on what we do about background assumptions in explanations—whether we count them as part of the explanation. I ask why you are wearing a watch on your right wrist rather than your left. You say:

1. I didn't want to be like everyone else.

But in a country where watches are normally worn on the right wrist, this could explain why the watch was worn on the left. So if we count (1) as a bona fide case of contrastive explanation, not all contrastive explanations are Leibnizian.

But perhaps we should take the background assumptions to be tacitly a part of the explanans in (1). Thus, maybe the real explanans is:

2. I didn't want to be like everyone else, and everyone else was wearing watches on their left wrist.

The problem with this move, however, is that the game can go on. Suppose that it's Lent and we're in a community where everybody tries to frustrate their minor preferences during Lent. Then (2) could explain why one was wearing the watch on the left wrist. So it's still not a Leibnizian explanation. But maybe, once again, we should take the explanans as tacitly including information from the background, like:

3. I didn't want to be like everyone else, and everyone else was wearing watches on their left wrist, and I had no reason to frustrate my minor preferences.

But the game doesn't stop here. Imagine a world where aliens frustrate people's minor preferences when and only when the people have no reason to frustrate them. Then (3) could explain wearing a watch on the left wrist, because the aliens would ensure that it's there. We can say that there was more tacit stuff in the explanans that rules this out. But can we ever stop? Do we really want to insist that explanations are always implicitly infinite?

So it looks like it's hard to defend the claim that contrastive explanations are Leibnizian. But perhaps we can defend the claim that contrastive explanations are weakly Leibnizian, where p is a weakly Leibnizian explanation of why q rather than r provided p explains q and not r, but p does not explain r in close worlds where it is true that r. I like the context-sensitivity of the "in close worlds". But if one doesn't like it, one could instead go for:

4. p is a weakly Leibnizian explanation of why q rather than r if and only if p explains why q and not r, and were r to hold, it would be false that p explains r.

It is now fairly plausible that contrastive explanations are weakly Leibnizian. Is it plausible that weakly Leibnizian explanations are contrastive? I think so.

The Leibniz Principle for explanation

Wes Salmon thinks the following "Leibniz Principle" is incompatible with the explanation of indeterministic phenomena:

if, on one occasion, the fact that circumstances of type C obtained is taken as a correct explanation of the fact that an event of type E occurred, then on another occasion, the fact that circumstances of type C obtained cannot correctly explain the fact that an event of type E' (incompatible with E) occurred.

Salmon thinks that the Leibniz Principle is incompatible with explanations in indeterministic cases and hence false.

I don't know if the Leibniz Principle is false. But I do have an argument that it is compatible with explanations in indeterministic cases.

Consider an electron in a mixed (3/5)|up>+(4/5)|down> state. The electron then undergoes a process whereby it is measured whether it is in an up or down state, thereby requiring collapse. It has probability 9/25 of collapsing into |up> and 16/25 of collapsing into |down>. In fact it collapses into |up>. This is pretty much the hardest kind of real-life case for explanations of indeterministic cases, since it is the less likely outcome that happens. But it is also one where an explanation can be given that satisfies the Leibniz Principle.

Consider now the following circumstances:

1. The electron is in a state such that the squared modulus of its probability amplitude for |up> is at least 9/25, and it was collapsed into |up> or |down>.

Then C explains why the electron collapsed into |up>. Salmon should agree with me on this point, since given what he thinks about indeterministic explanation, he will agree that

2. The electron is in a state such that the squared modulus of its probability amplitude for |up> is 9/25 and the squared modulus of its probability amplitude for |down> is 16/25, and it was collapsed into |up> or |down>.

But the information about the |down> component is not relevant to explaining that the electron collapsed into |up>, and the information that the squred modulus of the probability amplitude for |up> is at most 9/25 does not help explain why the electron collapsed into |up>. When we remove this irrelevant information (and Salmon famously said that irrelevant information is fatal to explanations) from (2), we get (1).

Now, if we accepted (2) as an explanation of the electron's collapsing to |up>, we would also have to accept it in another case as an explanation of the electron's collapsing to |down> (an even better one, since that is a likelier result), contrary to the Leibniz Principle. This is the sort of reason for which Salmon rejects the Leibniz Principle.

But (1) has no such unfortunate result. For while (1) does explain why the electron collapsed into |up>, it cannot explain why an electron collapsed into |down>.

One could also weaken the Leibniz Principle and take it to be a constraint on contrastive explanation (cf. this paper). If so, then the above would show that we can satisfy at least one desideratum for contrastive explanation in indeterministic cases (for a different approach, see this paper).