Counterfactuals and Chance

Contents 1 The Problem 2 The Threshold Strategy 3 The Drop-Truth Strategy 4 Lewis's Strategy 5 Hawthorne's Objection

The Problem

According to the Lewis's semantics for counterfactuals, A>C is true if and only if, roughly, C is true at all the closest A-worlds. This demands that in order for a counterfactual to be true, the consequence should be true at all the closest worlds where the antecedent is true. This demand may be seen too strong, however, especially when we consider that under the quantum mechanical picture of the world, there is always a real, albeit very tiny, chance of occurring of some extremely bizarre event in any mundane situation. So there is a worry that Lewis's analysis renders most ordinary counterfactuals false. Here is a John Hawthorne's (2005, "Chance and Counterfactuals" in Philosophy and Phenomenological Research, vol. LXX: pp. 396-405) example:

(1) If I had dropped the plate, it would have fallen to the floor.

We all want to say that (1) is true. But our accepted physics says that there is a tiny chance of the particles comprising the plate flying off sideways. We are then led to accept:

(2) If I had dropped the plate, it might have flown off sideways.

which implies the falsity of (1), given suitable assumptions about 'would' and' might' counterfactuals. Obviously, this is an untoward consequence; Lewis's analysis seems to be committed to an error theory of everyday counterfactuals.

One might try to bite the bullet by saying that our counterfactual locution is built on the assumption that our world is deterministic; so if the actual world is indeed indeterministic, our ordinary counterfactuals are in fact false. But it would be better if our analysis can also accommodate indeterministic cases and provide a method of treating deterministic and indeterministic cases in the uniform way.

The Threshold Strategy

There are several such strategies. The first, the threshold strategy, is to replace 'all' in the analysans with 'most'. Then the analysis amount to this:

A>C is true iff C is true at an enormously high proportion of the closest A-worlds.

This analysis saves ordinary counterfactuals from being false by weakening the meaning of them. There is a cost, though. The cost is that, Hawthorne points out, we have to deny the following inference rule:

[Agglomeration] P>Q, P>R entails P>(Q&R)

Hawthorne says that Agglomeration is overwhelmingly intuitive: speech of the form "if I had dropped the cup, thus-and-so would have happened and if I had dropped the cup, such-and-such would have happened, but it is not the case that if I had dropped the cup, thus-and-so and such-and-such would have happened' strikes us as profoundly odd." (2005, pp.397-8)

One might not share this intuition with Hawthorne. At any rate, there is a more plain and immediate worry about the strategy. The worry is about the threshold: how high is high enough? To be sure, this is a vexing problem, which leads us to the idea of the second strategy.

The Drop-Truth Strategy

The second is what Jonathan Bennett calls the "drop-truth" strategy. This strategy recommends us to drop truth form our counterfactual discourses. According to this line of thought, counterfactual conditionals concern what would have been probable if A had obtained, rather than what would have been the case. So more basic and important locution of counterfactuals is the form that "if A had been the case, C would have been probable to degree n," or for short, A>(P(C)=n)'. Then the analysis has the following form, Bennett suggests:

A>(P(C)=n) iff n is the proportion of all the closest A-worlds that are C-worlds.

This strategy seems to avoid the above vexing question about threshold. (But I don't know whether the label "drop-truth" is good; anyway, can we still say that A>(P(C)=n) is true if and only if such-and-such?) And many philosophers, such as Stephen Barker, Jonathan Bennett, and Dorothy Edgington, seem to be sympathetic to this line of thought.

What about the Agglomeration rule? Consider the following counterpart of the Agglomeration rule:

[Agglomeration*] A>(P(B)=n), A>(P(C)=n) entails A>(P(B&C)=n)

This rule fails in the same reason that the Agglomeration fails in the threshold approach. One might think again that this is not a high cost. At any rate, there is a more serious problem, what Bennett calls the "infinity" problem. The problem is that there are infinitely many closest A-worlds and then the numerical ratio of all the closest A-worlds that are C-worlds may not be available. Bennett proposes the "clumping" method to solve the problem. The basic idea is to put the infinite closest A-worlds into a finite set of clumps of worlds. But the account of clumping is rough and ready.

Lewis's Strategy

Lewis's (1986, Postscript to "Counterfactual Dependence and Time's Arrow" in Philosophical Papers II, pp. 52-66) strategy for handling the counterfactuals in indeterministic situations is to rule out the bizarre possibilities from the closest A-worlds, while retaining his original analysis that A>C is true if and only if C is true at all the closest A-worlds. Lewis suggests a particular understanding of 'closest' in analysans, according to which the bizarre possibilities such as the one expressed in (2) are excluded from the closest A-worlds. The basic idea is that worlds with bizarre low probability outcomes are more distant from the actual world than worlds without such outcomes, even if the bizarre outcomes do not violate laws of nature. Here the notion of bizarreness has become crucial. What then make a certain event bizarre? The first thing to note is that mere low probability does not by itself make an event bizarre. For example, consider Lewis's example where the monkey at the typewriter produces a 950-page dissertation on the varieties of anti-realism. Clearly, this would be a bizarre event--in Lewis's term, a 'quasi-miracle'. On the other hand, what if the monkey instead types 950 pages of jumbled letters? This is not at all bizarre or miraculous. Here, however, it should be noted that the one text is exactly as improbable as the other.

So Lewis says, "what makes a quasi-miracle is not improbability per se, but rather the remarkable way in which the chance outcomes seem to conspire to produce a pattern. A quasi-miracle would be such a remarkable coincidence that it would be quite unlike the going-on we take to be typical of our world." (1986, p. 60).

Here the question arises what exactly mean by the expression "a remarkable coincidence that it would be quite unlike the going-on we take to be typical of our world"? Lewis does not provide a detailed discussion about this. Hawthorne points out that human psychology will inevitably be the basis of any articulate distinction between remarkableness and unremarkableness that can do the job here. In his paper (2005), Hawthorne aims to show that any development of Lewis's view will run into problems. So let us assume that "remarkableness" is a primitive notion that we understand well enough.

Lewis goes on to say that if a world contains a quasi miracle, though it is entirely lawful, that detracts from its similarity to the actual world: "Like a big genuine miracle, it makes a tremendous difference form our world."

Let us go back to our examples (1) and (2). Given the above notion of quasi-miracle, Lewis can give the right verdict that (1) it true. The reason is that although there is a chance that the plate flies off sideways, worlds in which such a quasi-miracle occurs is more distant form the actual world than typical worlds in which a quasi-miracle does not occur; so quasi-miraculous worlds are excluded from the closest antecedent worlds; and, thus, the plate falls to the floor in all the closest worlds in which I drop the plate; (1) turns out true.

What about then (2)? It seems that we are inclined to think that (2) is true, given such a real, albeit very tiny, possibility that the plate flows off sideways. But this contradicts Lewis's definition of might-counterfactuals, i.e., A>mC =df. ~(A>~C), given the truth of (1). So Lewis distinguishes two readings of (2). One is a not-would-not reading of might-counterfactuals which is faithful to his original definition of might-counterfactuals, on which (2) comes out as:

(2-1) It is not the case that if I had dropped the plate, it would not flown off sideways.

On this reading, (2) is false, and so there is no problem. On the other hand, there is another reading of (2), a would-be-possible reading, on which (2) comes out as:

(2-2) If I had dropped the plate, its flying off sideways would be possible.

On this reading (2) is true. The two readings differ in that (2-1) means that some of the closest A-worlds where I dropped the plate are worlds are worlds where a quasi miracle happens--that is, the plate flies off sideways; while (2-2) means that all of closest A-worlds are worlds where it is possible for a quasi miracle to happen. So (1) conflicts with (2-1), but not with (2-2). This way Lewis explains why we intuit (2) is true without discarding his original definition of might-counterfactuals.

Hawthorne's Objection

Lewis has told a story according to which (2-2) is perfectly compatible with (1). Hawthorne shows that this leads us to unacceptable consequences. (Hawthorne (2005); Also see Ryan Wasserman (2006), "The Future Similarity Objection Revisited", in Synthese 150: 57-67.) Suppose that a coin flipper, Joe, is poised to toss a fair coin a million times. You kill Joe. Consider then the following counterfactual:

(3) If you had not killed Joe, he might have tossed all heads.

This counterfactual seems a natural enough claim. But Lewis cannot give the positive verdict here. After all, Joe's tossing a million heads would be a remarkable and surprising outcome which requires a quasi miracle. So Lewis is committed to the truth of the following counterfactual:

(4) If you had not killed Joe, he would not have tossed all heads.

Consider now some unremarkable pattern of sequence S (for instance, let S begin: HTTHTHHTH). Joe's tossing S would have been extremely unlikely, but it would not have at all surprising. Thus, Joe's tossing S would not involve any quasi-miracle. Hence, Lewis has the truth of the following counterfactuals:

(5) If you had not killed Joe, he might have tossed S.

And the following counterfactual is incontrovertibly true:

(6) If you had not killed Joe, the chances of his tossing all heads would have been the same as the chances of his tossing S.

Lewis is then committed to (4), (5) and (6), which, it seems, cannot all be true. That is, Lewis is committed to the following awkward speech: "If you had not killed Joe, he would have tossed his coin a million times. In fact, he might have tossed the sequence S. And his tossing S would have been exactly as likely as his tossing all heads. But if you had not killed Joe, he still would not have tossed all heads."

Things get worse. Joe's tossing all tails, like Joe's tossing all heads, would count as a quasi-miracle. Hence we have the truth of the following:

(7) If you had not killed Joe, he would not have tossed all tails.

So, from (4), (7) and [Agglomeration], we can obtain the following:

(8) If you had not killed Joe, he would not have tossed either all heads or all tails.

Now, the following is incontrovertibly true:

(9) If you had not killed Joe, his tossing either all heads or all tails would have been twice as likely as his tossing S.

Then the claims (5), (8) and (9), when taken together, produce an unacceptable consequence. That is, Lewis is committed to the following awkward speech: "If you had not killed Joe, he would have tossed his coin a million times. In fact, he might have tossed the sequence S. And his tossing S would have been less likely that his tossing either all heads or all tails. But if you had not killed Joe, he still would not have tossed either all heads or all tails."

These considerations seem to show that Lewis's proposal to invoke quasi-miracles to deal with the counterfactuals in inderministic situations fails.