What if the government hadn't bailed out the banks? Some of them would almost certainly have gone bankrupt, and other companies would probably have followed.

Here we have some sort of conditional probabilities: "if A, then probably/almost certainly C". But they aren't ordinary conditional probabilities of the kind that go in the ratio formula, P(A/B) = P(AB)/P(B). I do not believe that if the government actually didn't bail out the banks (but only made everyone believe it did), then some of the banks went bankrupt. That is, my ordinary conditional probability in the bankruptcies given that there was no bailout is fairly low. Nevertheless, I believe that if the government hadn't bailed out the banks, some of them would probably have gone bankrupt. My subjunctive conditional probability in the bankruptcies given no-bailout is high.

At first glance, one might think that subjunctive conditional probability is the probability of a counterfactual: I am fairly confident that if the government hadn't bailed out the banks, then some of them would have gone bankrupt. Or one might think that a statement of subjunctive conditional probability expresses a counterfactual conditional with a probabilistic consequent: I believe that if the government hadn't bailed out the banks, then it would have been highly probable that some of them go bankrupt.

However, as Lewis argued in 1973, neither analysis is correct. Lewis instead proposed the imaging analysis of subjunctive conditional probability. On this analysis, the subjunctive probability of C given A is the probability of C after the probability of any non-A world has been moved to the closest A world(s). In a more general version due to Gaerdenfors (and endorsed by Lewis and many others), the probability of non-A worlds may be divided among A worlds in proportion to their closeness.

More precisely, assume that for any world w1 and (suitable) proposition A there is a probability distribution Q_A(w1) such that Q_A(w1)(w2) specifies the fraction of w1's probability that lands on w2 when all probability is shoveled onto A worlds. The subjunctive probability of a world w2 given A is then analysed as \sum_w1 P(w1) Q_A(w1)(w2). The subjunctive probability of any proposition C given A is the sum of the probabilities for individual C worlds given A.

This is not much of an analysis unless more is said about the family Q_X of transition probabilities. Here are two silly ways of putting flesh on the skeleton that illustrate its skeletonosity.

1. Define Q_A(w1)(w2) as P(w2/A). Then the subjunctive conditional probability of C given A equals the ordinary, indicative conditional probability P(C/A).
2. Define Q_A(w1)(w2) as P(w2). Then the subjunctive conditional probability of C given A equals the unconditional probability of C.

Robbie's worry was that the imaging analysis might clash with the idea that the subjunctive conditional probability of C given A equals the expectation of the conditional chance of C given A, i.e. \sum_x x P(ch(C/A) = x). However, consider the subjunctive conditional probability of a particular world w2 given A. On the chance account, this equals \sum_w1 P(w1) ch_w1(w2/A). On the imaging account, it equals \sum_w1 P(w1) Q_A(w1)(w2). So here is a third way of fleshing out the imaging analysis.

3. Define Q_A(w1)(w2) as ch_w1(w2/A). Then the subjunctive conditional probability of C given A equals the expectation of the conditional chance of C given A.

Robbie actually has a good reason for his worry. In this paper, he assumes the chance account and effectively shows that if the imaging analysis of subjunctive conditional probability is correct, then a certain platitude about counterfactual conditionals comes out false. Now since the chance account is actually a version of the imaging analysis, the middle if cancels out: the chance account entails the imaging analysis, and hence also the falsehood of the platitude. So what the argument refutes is not the imaging analysis, but the chance account.

At least that's what I think is going on. I have only half-heartedly worked through the details.

Update 17 Feb: The details are actually quite interesting. So here's the full argument against the chance account.

First some definitions. Let -> be a binary operator so that A->C is true at world w1 iff C holds at all the closest A worlds to w1. More precisely, given a closeness measure Q that assigns to each world w1 and proposition A a probability distribution Q_A(w1), let A->C be true at w1 iff C holds at all worlds w2 such that Q_A(w1)(w2) > 0. To have a short notation, let P^A(C) be the subjunctive conditional probability of C given A on the imaging account; that is, P^A(C) = \sum_{w2 \in C} \sum_w1 P(w1) Q_A(w1)(w2) -- again, relative to some closeness measure Q.

Fact 1. P^A(C) >= P(A->C).

Proof sketch. To compute P(A->C), we go through all worlds w1 and add up P(w1) whenever C holds at all w2 with Q_A(w1)(w2) > 0. To compute P^A(C), we go through all worlds w1 and add up \sum_w2 P(w1)Q_A(w1)(w2) over all C worlds w2. If w1 is such that C holds at all w2 with Q_A(w1)(w2) > 0, this means we simply add P(w1); so the result for P^A(C) must be at least as great as for P(A->C).

Fact 2. If the expected chance of A given C is generally >= (A->C), then the chance of C is generally >= the chance of A->C.

Proof: see Robbie's paper. The reasoning closely follows Lewis's triviality result about conditional probability. I have some reservations about the proof due to some of my opinions about chance, but set them aside for now.

Now we've seen that the expected chance account is a form of the imaging account -- namely the one for the closeness measure Q_A(w1)(w2) = ch_w1(w2/A). It thus follows from Fact 1 and Fact 2 that the chance of C is never less than the chance of A->C:

(*) ch(C) >= ch(A->C).

What does this say? By definition of the arrow, A->C says that C holds at all the closest A worlds. On the chancy closeness measure, A->w2 is true at w1 iff ch_w1(w2/A) = 1. And so the chance of A->w2 at w1 is the chance at w1 that ch_w1(w2/A) = 1. Can chance propositions themselves be chancy? Many say no. Then the chance at w1 of A->w2 is either zero or one. Assuming that propositions with chance 1 are true and propositions with chance 0 false, the chance of A->w2 at w1 is 1 if ch_w1(w2/A) = 1, and otherwise 0. So we have two cases. First, suppose the actual chance of A->w2 is 0. Then (*) says that the chance of w2 is not below 0. This is clearly unproblematic. Second, suppose the actual chance of A->w2 is 1, and hence ch(w2/A) = 1. Then (*) says that the chance of w2 must also be 1. Again, this is unproblematic. For instance, by the ratio formula, ch(w2/A) = ch(w2 & A)/ch(A), which can only be positive if w2 is an A world, in which case ch(w2/A) = ch(w2)/ch(A) = ch(w2)/ch(w2)+ch(A minus w2). If this is 1, ch(A minus w2) must be 0, and so ch(w2) must be 1. Just as (*) says.

(What if chance propositions can be chancy? What if conditional chance doesn't obey the ratio formula? What if propositions with chance 1 are sometimes false? It might be worth thinking through these variations, but the outcome is clear anyway: (*) cannot be absurd, because it is true.)

In Robbie's paper, -> is not some made-up connective, but our ordinary counterfactual conditional. On this reading, (*) is absurd. On the other hand, (*) is true if we assume that the ordinary counterfactual conditional A->C is true iff the conditional chance of C given A equals 1.

This yields the promised argument against the conditional chance analysis of subjunctive conditional probability. To bring it out more forcefully, let A=>C formalise "if A were the case then it would certainly be the case that C". Clearly A=>C is true iff the subjunctive probability of C given A is (near) 1. However, if subjunctive conditional probability equals expected conditional chance, then it follows from Fact 1 and Fact 2 that the chance of A=>C never exceeds the chance of C -- which is absurd (try A=C). Hence subjunctive conditional probability does not equal expected conditional chance.

One of the many places where this hinders progress is the introduction of centered (de se) contents. Take Lewis's suggestion that the objects of attitudes are properties. What kind of that-clause would express, say, the property of living in Berlin? On the assumption that the objects of attitudes are expressible by that-clauses, Lewis's suggestion is a non-starter.

Philosophers who are sympathetic to Lewis's proposal (including Lewis himself) sometimes put it in terms of self-ascription: I self-ascribe the property of living in Berlin. What is it for me to self-ascribe this property? Presumably it is to believe that I live in Berlin. Here we have our that-clause! On this interpretation of Lewis's proposal, the object of attitudes are expressed by that-clauses of the form "that I am F".

But this leads to trouble. In chapter 28 of his book Perspectival Thought (2007), Francois Recanati wonders:

How can I (pretend to) self-ascribe the property of being Napoleon and fighting the battle of Waterloo, if those are properties that it is impossible for me to instantiate?

Daniel Nolan raises similar worries in his "Selfless Desires" (2006): can't I desire that there be no sentient life, or that my parents never met? But then the content of my desire is not adequately expressed by any clause of the form "that I am F".

In either case, the problem is that self-ascription turns a perfectly harmless property into something impossible. It doesn't help to say that people can believe and desire the impossible. Even if that is true, a desire that there be no sentient life is surely not a desire of something impossible.

When I say "I wish I was never born" or "I wish there was no sentient life", I express a desire that is satisfied at worlds where I was never born. On Lewis's proposal, the content of my desire is a property that applies to various things in worlds where I do not exist. This is a perfectly consistent property, and there is no reason why it couldn't be the content of a desire.

It would be better to avoid talk of self-ascription. If the content of beliefs and desires are properties, then they just aren't things expressed by that-clauses -- not even by that-clauses of the form "that I am F".

At t1, X = 10.
At t2, X = 20.

Or suppose you have a population of n objects with various velocities. Your statistics textbook will tell you that the variance of the velocity in the population is defined as

Var(X) = Exp[X - Exp(X)]^2,

where Exp(Phi) is the average Phi value weighted by its frequency. What does 'X' stand for in this equation? On the right-hand side, it seems to denote a number, since you subtract another number from it. But it clearly doesn't stand for any particular number, such as the velocity of the third object in the population. On the other hand, it also isn't a bound variable: it represents the same magnitude on the right-hand side that it represents on the left-hand side (where, incidentally, it cannot just stand for a number, since it makes no sense to speak of the variance of a number).

As a final example, consider the description of dynamic systems in control theory. The probability that the system moves from a certain state to another at stage k is standardly written

P(x_{k+1} / x_k).

The two terms 'x_{k+1}' and 'x_k' are supposed to denote states. But if we just call these two states s1 and s2, and if the system can't be in two different states at once, then P(s1/s2) would have to be either zero or one, depending on whether s1 = s2. So 'x_{k+1}' doesn't just denote the state s1, but somehow says that the system is in s1 at stage k+1, which is thus compatible with 'x_k'.

Publications in statistics, computer science, engineering, and physics are full of examples like this. Nevertheless, my first reaction is always to blame the authors for sloppy writing. I think to myself that 'P(x_{k+1} / x_k)' is really short-hand for

P(the state at k+1 is x_{k+1} / the state at k is x_{k}),

and 'Var(X) = Exp[X - Exp(X)]^2' is short-hand for something like

Var(X) = Exp_i[Val_i(X) - Exp_i(Val_i(X))]^2

where 'i' ranges over the individuals in the population and 'Val_i(X)' denotes the value of the maginture X (i.e. velocity) for individual i.

However, the original expressions are completely in order if we interpret the problematic terms as intensional variables. An intensional variable denotes different things relevant to different points of evaluation. It has two semantic values: an extension and an intension. Predicates and functors can apply to either of these values. Thus in 'Var(X)', 'Var' applies to the intension of 'X'. The minus sign, on the other hand, applies to the extension of 'X'. 'Exp' is a modal operator that quantifies over points of evaluation. Similarly for 'x_{k+1}'. Here 'x' is an intensional function symbol. Its extension is a function from times, given by the subscripts, to states. Its intension is the functional concept "the system's state at --". 'P' in 'P(x_{k+1} / x_k)' is a binary intensional functor. The formal semantics of such expressions has been worked out in much detail by people like Church, Montague, Cocchiarella, Garsons and Fitting, and there is little doubt about its consistency and coherence.

So maybe I should stop accusing everyone else of sloppiness, and rather start accusing us philosophers of technical narrow-mindedness. It looks like we have been indoctrinated with the idea of semantic innocence -- the idea that individual variables must be "directly referring" -- oblivious to the fact that everyone else happily violates our doctrine. Shouldn't we rather join the party and give up on our innocence?

Good: Upon hearing that the probability of rain is x, you come to believe to degree x that it will rain.

Bad: Upon hearing that the probability of rain is x, you become certain that it will rain if x > 0.5, otherwise certain that it won't rain.

The Bad process seems bad, not just because it may lead to bad decisions. It seems epistemically bad to respond to a "70% probability of rain" announcement by becoming absolutely certain that it will rain. The resulting attitude would be unjustified and irrational.

Can we explain the comparative badness of the Bad process solely by appeal to the overriding epistemic goal of truth? It seems not.

Let's compare the objective reliability of the two methods. Roughly speaking, the reliability of a belief-generating process measures its tendency to generate true beliefs. How do we apply this to the Good process which typically generates only partial beliefs? We don't want to count an 0.3 belief that it will rain as true iff it will rain. A natural move is to measure reliability in terms of the distance between the degree of belief and the truth value of the relevant proposition (1 for true, 0 for false). So a reliable process would tend to generate high degrees of belief in true propositions, and low degrees in false propositions. Call the distance between degree of belief and actual truth value the belief's inaccuracy.

Using this measure, it turns out that under normal circumstances, the Bad process is more reliable than the Good process. To illustrate, consider 100 occasions on which the weather forecast announces a 70 percent probability of rain. Let's also assume that in 70 of these cases, the announcement is followed by rain, and in 30 it isn't. (It is a pretty good weather forecast.) The Bad method would make you certain that it rains on each occasion, so your distance from the truth is 0 in 70 cases and 1 in 30. Average inaccuracy: 0.3. The Good method would give you a belief of degree 0.7 that it will rain, so your distance from the truth is 0.3 in 70 cases and 0.7 in 30 cases. Average inaccuracy: 0.42. On average, the Bad process brings you closer to the truth.

Note that as a counterexample to reliabilism, this is quite different from thought-experiments involving far-fetched scenarios where the reliability of a process comes apart from (what we take to be) its actual reliability. Even under perfectly normal and common conditions, the Bad process is more reliable than the Good one.

The above reasoning also shows that the Bad process has lower expected inaccuracy that the Good method from the point of view of an agent who uses the Good method. If you follow the Good method and hear the "70% chance of rain" announcement, you can calculate that the expected inaccuracy of the Bad method is 0.3, while the expected inaccuracy of your own method is 0.42. (If you follow the Bad method, you'll judge the Bad method to have 0 expected inaccuracy, compared to 0.3 for the Good method.)

This is odd. Here we have two methods; we know that one of them is more reliable, that it is more likely to bring us closer to the truth; but still we think it would be irrational to use it. We judge that, from a purely epistemic perspective, one ought to use the other, less reliable method.

A while ago, I claimed that the appearance of truth as the unifying epistemic virtue might be an illusion based on the fact that any belief-forming method whatsoever will automatically be regarded as truth-conducive by agents who use it. It looks like we have a counterexample to this as well. At least it is not true that any belief-forming method automatically appears more accuracy-conducive than its rivals to agents who use it.

Four quick comments.

First, the present puzzle is closely related to the puzzle discussed in Allan Gibbard's "Rational Credence and the Value of Truth" (2008), who should therefore get all the credit. Gibbard assumes that it is irrational to have degrees of belief which, by their own lights, are less accurate than certain other degrees of belief; and he argues that this requirement of rationality cannot be explained solely on the assumption that rational belief "aims at truth", although it can be explained on the assumption that rational belief aims at successful action.

Second, it is tempting to argue against the Bad method by considering its long-run accuracy. If you follow the Bad method and otherwise update by conditioning, you may easily end up with more inaccurate beliefs than if you follow the Good method. But if truth is your goal, why not combine a deviant response to weather forecasts with a deviant update process? It is easy to show that some such combinations have higher reliability, and higher expected long-run accuracy than the sensible combination of the Good method with conditioning (see Gibbard's follow-up note "Aiming at Truth over Time" (2008), page 6).

Third, another problem with the Bad method is that it doesn't generalise well. Suppose the weather forecast says that there's a 30 percent chance of rain, a 40 percent chance of sunshine, and a 30 percent chance of neither rain nor sunshine. And suppose you apply the Bad method not just to the rain statement, but also to the two others. You'd end up being a) certain that it won't rain, b) certain that the sun won't shine, and c) certain that it will either rain or the sun will shine. But arguably, this is an impossible state of mind; the attitude ascriptions (a)-(c) are inconsistent. So if one could show that the Bad method, even restricted to rain forecasts, may (easily) lead to impossibilities like this, that would solve the puzzle. I don't think this can be shown. But another possibility is that the compensatory methods you have to use in addition to the Bad method in order to restore consistency will have a cost in expected accuracy. And then maybe any consistent package of methods containing the Bad process would end up less accuracy-conducive than the reasonable package containing the Good process. That would be nice.

(A lot of people think that there is nothing inconsistent about the attitude ascriptions (a)-(c), even though there is something inconsistent about the ascribed attitudes. We would then have to ask whether the rationality requirement of having consistent attitudes can be explained solely by appeal to the goal of truth or accuracy; see e.g. Jim Joyce's "Accuracy and coherence: Prospects for an alethic epistemology of partial belief" (2009) for the latest moves in this game. The short answer is that it can't be done. But as I said, I don't find this very problematic, because I'm sympathetic to Ramsey's view that the requirements of probabilistic coherence are analytic and therefore not in need of epistemic defense.)

Fourth, I've measured inaccuracy simply as the distance between belief and truth value. But there are other measures. If we use squared distance, the problem disappears. There may even be good reasons for using this measure, apart from solving the present problem (Joyce mentions some at the end of the paper just cited). But none of these reasons seem to be based on truth as the overriding epistemic goal. If all that matters for epistemic rationality is closeness to the truth, it is not clear why closeness should be measured by squared distance rather than absolute distance.

On the other hand, it is also not clear why closeness should be measured by absolute distance. So I might have to qualify the claim in the other blog post: on one disambiguation of "accuracy-conduciveness" there are clear counterexamples to the hypothesis that epistemic goodness is a matter of accuracy-conduciveness. On this reading, accuracy (or truth) therefore doesn't appear to be the overriding epistemic goal. On another disambiguation, there is such an appearance, but it is an illusion based on the fact that any belief-forming method whatsoever automatically appears accuracy-conducive by agents who use it.

The question is simply how, on Williamson's account, we can have knowledge of substantial metaphysical necessities, e.g. of the fact that gold necessarily has atomic number 79. Williamson explains that when we counterfactually imagine gold having atomic number 78 (knowing that it has number 79), we will "generate a contradiction", because we hold "such constitutive facts [as atomic number] fixed" (p.164). But the distinction between constitutive and not-constitutive facts can hardly be analysed as the distinction between whatever we happen to hold fixed and the rest, given Williamson's commitment to strong mind-independence of metaphysical modality. So what justifies our holding fixed the atomic number?

A natural thought is that we hold fixed such-and-such facts in counterfactual reasoning because we know that they are necessary. But according to Williamson, it is the other way round. Our knowledge of necessity is based on the attitude of "holding fixed" which we have towards certain propositions. So that attitude had better not be knowledge of necessity.

Someone suggested the attitude might be knowledge of constitutivity: we know that it is a "constitutive fact" that gold has atomic number 79. But then how do we know that? How do we know that it is not constitutive of this rock that it got stuck at the bush, so that what we imagine when we imagine it tumbling further is really a different rock, otherwise much like this one? And anyway, how would knowledge of contitutive facts help unless we also know that those facts are necessary and therefore can be held fixed in counterfactual reasoning?

It was also suggested that perhaps the relevant attitude towards the propositions we "hold fixed" is not knowledge at all. But whatever it is, the attitude is supposed to match mind-independent facts about necessity. And the match should not be an accident, otherwise counterfactual reasoning could not give us knowledge.

A third idea was that Williamson may not want to explain modal knowledge at all, but merely oppose philosophical exceptionalism. However, it seems that the relevant "holding fixed" attitude doesn't play much of a role in the evaluation of everyday counterfactuals. When I consider what would have happened if the rock had tumbled further, I obviously do hold a lot of things fixed, e.g. the depth of the valley. But this is not the right kind of holding fixed: when I entertain the possibility that the valley had been more shallow, I do not "generate a contradiction". So I do not hold the depth of the valley fixed in the unconditional sense in which, according to Williamson, I hold fixed the atomic number of gold -- in the sense that I hold p fixed even when I evaluate the counterfactual assumption that not-p. As far as ordinary evaluation of counterfactuals is concerned, it seems like we could do all that without ever holding anything absolutely fixed.

So what is Williamson's take on this peculiar, philosophical, knowledge-like attitude that provides us with knowledge of metaphysical necessities?