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«1 Introduction This paper examines the relation between three concepts: rational degrees of belief (or credences), counterfactuals and chances. All ...»

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Chances, Credences and Counterfactuals

Richard Bradley

LSE

April 24, 2016

Abstract

1 Introduction

This paper examines the relation between three concepts: rational degrees of

belief (or credences), counterfactuals and chances. All three notions are hotly

debated in philosophy and I will necessarily have to take lot for granted if any

progress is to be made on the question of how they are related. In particular

I will assume that the degrees of belief of a rational agent can be represented by a probability measure on a Boolean algebra of propositions, su¢ ciently rich as to contain both counterfactual propositions and propositions about chances.

Then my task becomes the somewhat more modest one of saying how degrees of belief in counterfactual propositions and in chance propositions are related to each other and to degrees of belief under suppositions. My hope is that doing so will shed light on all three concepts by establishing the constraints that any interpretation of them should satisfy.

The focus of this investigation will be a particular claim about how coun- terfactuals and chances are related that I will call Skyrms’Thesis in honour of its original proponent, Brian Skyrms (in Skyrms (1980) and Skyrms (1981)).

Skyrms’Thesis says, very roughly, that rational degrees of belief in a counter- factuals go by the expected conditional chances of their consequents, given the truth of their antecedents. For instance, suppose that I have in my hand a coin that might be fair or two-headed or two-tailed and that you believe each possibility to be equally likely. Then, according to Skyrms’Thesis, the degree to which you should believe that if I were to toss the coin it would land heads is given by the expected conditional chance of it landing heads given that it is tossed. Its conditional chance of landing heads equals 0.5 or 1 or 0, according to whether it is fair or two-headed or two-tailed. So its expected chance of landing heads is ( 1 1 ) + (1 1 ) + (0 3 ) = 0:5. Hence it is probable to degree one-half 1 23 3 that if the coin were tossed it would land heads.

Skyrms’Thesis is a special case of what has come to be called the Ramsey Test hypothesis, a principle that relates credence in conditionals (counterfactual 1 or otherwise) to credence under a supposition. I will discuss the Ramsey Test hypothesis and the notion of a supposition in more detail in section 4. A second claim— that I will term the Principal Suppositional Principle— relating credence under suppositions to conditional chances, is required to derived Skyrms’Thesis.

It is examined in section 5 along with one further claim about the independence of credences in chances from suppositions of a particular kind. To get to the Principal Suppositional Principle some background is required. To this end, in section 3, I recall Lewis’treatment of the relation between chances and credences, suggesting a formulation of the relationship between them that allows for elimination of the problematic notion of admissibility, so central to Lewis’ theory. This proposal depends however on understanding chances in a particular kind of way, and motivating this interpretation will be my …rst task.

Some terminology that will be used throughout. I will take the degrees of belief of a rational agent to be given by a probability function P de…ned on a Boolean algebra of propositions. Propositions will be denoted by italicised capitals and the conjunction of any two propositions X and Y by either their concatenation XY or by the pair (X; Y ). The operations of negation and disjunction will be denoted by : and _ respectively and the logical contradiction and tautology by ? and respectively. The counterfactual ‘ A were the case if then X would be’will be denoted by A ! X. Similarly the proposition that the chance that X is true (at certain point in time t) is equal to x will be denoted by Cht (X) = x and the proposition that the conditional chance that X, given that A, at t, is equal to x will be denoted by Cht (XjA) = x.For simplicity the time index will often be dropped.

2 Chance as Ideal Probability We face uncertainty about a good many things. In a sense all uncertainty stems from lack of information. But there is clearly a big di¤erence between the uncertainty I might have about the time that the bus from Tel-Aviv to Jerusalem departs in the morning, which derives from a simple failure to consult the timetable, and uncertainty that is structural or irreducible in some way. In indeterministic systems such as those described by quantum mechanics, uncertainty is deeply structural because there is simply no information to be obtained that will settle the question of where particles are located (prior to their measurement). In other cases such information may exist in principle but in practice is impossible to obtain. Our uncertainty about the rainfall in, say, Kinshasa on the 1st of January 2050, is hardly less severe for all our knowledge of the deterministic meteorological system governing the weather in Zaire, for this system is chaotic and accurate predictions about distant events in such systems is impossible. So too even for homely, deterministic and non-chaotic systems like those governing coin tosses or the development of cancers or political insurrections. The uncertainty we face regarding such events deserve the label ‘ structural’because they re‡ not the idiosyncratic state of knowledge of ect a particular individual but physical constraints on all of us on the accessibility 2 of certain kinds of information. With regard to such uncertainty we are all in the same boat.





These observations give strong support to Lewis’ widely-shared view that there are two sorts of probabilities— the subjective degrees of belief of a Bayesian agent and the objective chances of events— and that an understanding of uncertainty and how to manage it, rests on the relationship between them. There is much about this view that I think is correct. In particular, I will defend a version of what Lewis called the Principal Principle relating objective and subjective probability: informally, that a rational agent should set her degree of belief in any proposition to what she expects its objective chance to be. But contrary to the mainstream interpretation of Lewis’view, I will argue that the objectivity of chances does not stem from the fact that chances are physical properties of the world. Chances are probabilistic judgements whose objectivity resides in their being expert or ‘ best-possible’ judgements given the physical facts. They are thus neither simply frequencies, nor the propensities that purportedly explain them. Rather they are the judgements of an ideal reasoner who is fully informed of all the propensity and/or frequency facts, but not of the truth of the events that are the bearers of chances.

The idea of an expert probability for an agent goes back to Haim Gaifman (1988), who characterised it as a probability assignment the agent was committed to tracking in the sense of taking as a constraint on her attitudes, the principle that her degrees of belief in some proposition X should equal the expert’ probability for X. In a sense, Truth is an expert of this kind, requiring s

an agent who aims at the truth to obey the principle that:

–  –  –

Similarly, as Hall (2004) and Joyce (2007) suggest, the Principal Principle can be read as saying that Chance is an expert probability, requiring those who seek objectivity to align their credences with the objective chances in the sense of

satisfying for all events X:

–  –  –

when Ch is a probability measure of the true chances of events.

This proposal is, on the face of it neutral about what chances are, and indeed di¤erent authors have …lled it in di¤erent ways. Hacking (1965), for instance, suggested that relative frequencies were expert probabilities and hence that rational degree of belief in any repeatable event X should equal its relative frequency in the appropriate reference class of events. More often though, Lewis’ principle has been read through the prism of a propensity interpretation s of chances. Neither, it seems to me, o¤er the possibility of a su¢ ciently general interpretation because of some well-known limitations of each.

The main limitation of frequentism is that it does not allow for single case chances. Yet there are many non-repeatable events for which talk of objective probability of its occurrence seems perfectly sensible. In debate about the e¤ects 3 of climate change, for instance, there is much discussion of the chances of human extinction and other catastrophic events. Is all such talk purely subjective?

The propensity interpretation does not su¤er from this limitation; indeed several variants of it are explicitly designed to deal with single-case chances.

The main problem here is that propensities are not probabilities at all in the strict sense (as …rst observed by Paul Humphreys (1985)). A propensity is a disposition of a set-up to produce certain kinds of outcomes: of spins of roulettes wheels to cause to ball to land on even numbers, of weather systems to produce snow, of levels of sugar consumption to result in diabetes. Such talk is causal in nature and, indeed, the propensity interpretation provides a natural home for accounts of probabilistic causation. But causation by its nature is typically one-directional, while probabilities are always two-directional. The chance of a window shattering if a stone is thrown at it might sensibly be viewed as a physical propensity of a set-up involving ‡ ying stones and windows, but the chance of the stone being thrown, given that the window shattered, cannot.

But the two chances are equally well-de…ned.

To elaborate, consider a simple example in which a fair coin will be tossed …ve times. What is meant by fair depends, of course, on the interpretation that is given to chances. According to the propensity theorist, the coin will have a certain disposition to land heads whose magnitude will depend on the set-up: the properties of the coin, the manner in which it is tossed and various environmental factors. A coin is fair therefore when the set-up is such as to ensure that the coin is equally disposed to land heads as to land tails. The frequency theorist on the other hand will say that the coin is fair because on half the tosses in the relevant reference class of tosses of this coin it lands heads, and on half it lands tails.

Let’ consult our intuitions on some basic cases. What is the chance of the s coin landing heads on the …rst toss? One-half is the only reasonable answer in view of the fairness of the coin. And the chance of it landing heads on the last (…fth) toss, given that it has landed heads on the …rst four tosses? Again the answer is one-half, absent any grounds for thinking that the tossing of the coin has undermined the conditions for its fairness. Finally what is the chance of it landing heads on the last toss given that it has landed heads on the …rst four tosses and that it will land heads in only four out of the …ve tosses? The answer it seems to me, must be zero. For we cannot accommodate this information about the proportion of heads landings without drawing this conclusion.

Let Hi be event of the ith coin toss landing heads and Ch be a chance function on the Boolean algebra based on the events fH1 ; :::; H5 g. Let E be the event of four out of …ve tosses landing heads, i.e. E = H1 H2 H3 H4 :H5 _ ::: _ :H1 H2 H3 H4 H5. Now the description of the set-up plus our answers to the

three questions constrain Ch as follows:

–  –  –

Now the propensity view cannot give an interpretation of the (fragment of) a chance function I have just implicitly de…ned on the algebra of coin landing events. For the fact that the coin has landed heads on the …rst four tosses and that it will land heads in only four out of the …ve tosses does not give us much grounds for thinking that the coin is not fair. One should expect that frequencies in small classes of events will diverge from propensities and so such divergences are slim evidence for a change in the dispositional facts. So on the propensity view, the chance of a fair coin landing heads on the …fth toss is still one half, no matter what the frequency facts are. The root problem here, it seems to me, is that the conditional probability of the last toss landing heads, conditional on four out of …ve tosses landing heads is not really a propensity at all. It is a judgement that is made in the light of knowledge of the relative frequencies, knowledge that in this case overrides anything that we know about the physical propensities.

Finite frequentism also cannot give an interpretation of these chances, though for somewhat di¤erent reasons. From the frequentist’ point of view, it makes no s sense to talk of the chance of a fair coin landing heads given that the frequency of heads landing is greater than one-half. If the coin has landed heads four out of …ve times then it is not a fair coin (by the frequentist de…nition of fair). So on the interpretation of chances as frequencies, constraint 3 is meaningless. It does not matter whether the relevant frequencies come from a …nite reference class or an in…nite one. Whatever the relevant reference class, the frequency of heads must be one-half if the coin is fair. But in the reference class picked out by the condition that H1 H2 H3 H4 the frequency of heads is not one-half.



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