Mellor on the Chances of Effects*
ABSTRACT: In the Facts of Causation (1995), D.H. Mellor includes, as a part of his theory of causation, an account of the chance that a cause gives its effect. He proposes that this chance can be analyzed as a certain kind of conditional, a closest world conditional with a chance consequent. I show that there are problems with Mellors account, but also attempt to show how these can be remedied. This analysis highlights important issues concerning the concept of components of single case objective chance.
Mellor takes the chance he is concerned with to be objective single case chance measured by the probability calculus. It is not frequency nor credence, although it has important connections to both frequencies and credences.(1)
According to Mellor facts which have chances can have more than one chance, for example, by having them at different times. Suppose we have two unstable atoms A and B in close proximity, each of which has a low chance of decaying, and suppose that atom A, if it decays, may bombard atom B with its product, thereby driving atom B into a state in which its chance of decaying is quite high- much higher than otherwise (see figure 1). Suppose this in fact happens, and let us consider the chance E that atom B will decay at a later time tE, when an observation will be made. The chance of E changesincreases, in factat tb, the time at which B is bombarded. Thus E has two chances, at different times.
Mellor also holds that chances are contingent, but not on the fact that they are "chances of" (in our example, E), since they can exist when that fact does not. Atom B may not decay at t0, but it still had a chance of doing so prior to that time; so the chance existed but the state of affairs that it was about never did. Since chances are contingent, they must therefore be properties of other facts, facts without which they would not exist. The chance E that atom B will decay at time t0 is a property of facts about the structure and nature of that atom, together with facts concerning the nature and proximity of atom A. Mellor writes this chance as "chQ(E)" where Q is the fact or conjunctive fact of which the chance is a property.
Applying this to the case of causation, the chance of the effect is a property not of the effect but of another fact, the cause C or the conjunction of C and S, where S is the circumstances in which C causes E. According to Mellor, for every cause and effect there exists such a chance, the chance the cause gives the effect,(2) chC(E). Similarly, every effect has a chance of existing in the circumstances with C, even if that chance is zero. This chance is written as ch~C(E).
Such chances can be analysed in terms of a closest world conditional of a certain kind, Mellor argues. Taking the deterministic case, the chance that the cause gives its effect in the circumstances is 1, viz., chC(E) = 1. In such cases the causal conditional, if C then E holds. This conditional should be read as a closest world conditional in the manner of Lewis (1986), but with the consequent not being E but the chance of E:
CÞch(E) = 1
If my taking a certain dose of arsenic is sufficient in the circumstances for my death, then the chance that drinking arsenic gives me of dying in the circumstances is given by the chance I have of death in the closest world in which I drink arsenic, which is one. (The problem with interpreting the causal conditional as a closest-world conditional with consequent E is that that makes every actual cause a sufficient cause of its effect, since when ever C actually causes E the relevant closest-world is the actual world, and so if C then E is automatically true, even if C is not a sufficient cause of E.)(3)
This can be generalised to the probabilistic case. In general, Mellor says, the chance a cause gives its effect is:
(1) chC(E) = (ip)(CÞch(E) = p)
with ip for the p such that (p. 28)although this is qualified later (p. 178). Thus (1) reads the chance that a cause C gives its effect E is the p such that in the closest world where C obtains, the chance of E is p. Mellor gives the example of Don dying (E) because he falls (C). The chance chE(E) that Dons falling gives his dying is the chance that he dies in the closest world in which he falls.
The chance that E has without C is given by:
(2) ch~C(E) = (ip')(~CÞch(E) = p')
This leads Mellor to his reading of deterministic causes: C is sufficient for E just if CÞch(E) = 1 and C is necessary for E just if ~CÞch(E) = 0. These are not causal conditionals in the usual sense of if C then E and if ~C then ~E because the consequent is ch(E) rather than E. However, Mellor argues that nothing quite fits the normal sense of causal conditionals, but that these closest-world conditionals come closest. On the grounds that nothing fits the bill Mellor proceeds to put the term to use by denoting these closest-world conditionals by causal conditionals (p. 29).
However, this account faces a number of problems. We will deal with three, in order of increasing importance. Firstly, Mellor takes the circumstances S in which C causes E to be those factors which are relevant to the chance that C gives E. These circumstances S then give E a chance of obtaining in the absence of C, which leads to an account of a necessary cause as one whose circumstances give its effect a zero chance of occuring in its absence. But the problem here is that circumstances actually relevant to the chance of E may include factors not relevant to the chance that C gives E.
For example, if Don has a chance, at the time he falls, of having his head blown off by a gun pointed at him by his companion, then his falling is not a necessary cause of his death. But suppose the gun is not relevant to the chance that his falling gives his dying. Then on Mellors account the gun is not part of the circumstances S in which Dons fall causes his death, and so will not be relevant to whether Dons fall is a necessary cause of his death. Clearly no reasonable reading of "necessary cause" can allow this.
This problem is resolved if we take the circumstances S to be those relevant to the chance of E. Then the gun is part of the circumstances S in which Dons fall causes his death, and so will be relevant to whether Dons fall is a necessary cause of his death, as it should be. From here on we will read S this way unless otherwise specified.
The second problem concerns equation (1). As we have seen, a fact that has a chance may have more than one chance by having them at different times. Thus the chance that C gives E, as defined in (1), is ambiguous, since E may have more than one chance in the closest C-world.
This problem can be avoided if we take the chance of E in (1) to be the chance of E at the time of C. We can write this as chtc(E). (I am not suggesting Mellor would endorse this solution.) Then the chance that Dons falling gives his dying is the chance that he dies in the closest world in which he falls at the time that he falls. We can rewrite (1) as
(1a) chC(E) = (ip)(CÞchtc(E) = p)
But a third problem is not so easily solved. This problem is that the chance that C gives E might not be the chance of E at the time of C because there are other factors which contribute to the chance of E at that time, quite independently of C. Such factors could be part of the circumstances C in which C causes E. Then (1a) does not quite give the chance that C gives E. Don has a chance, at the time he falls, of having his head blown off by a gun pointed at him by his companion (G), which contributes to the chance of E at tc, yet not to the chance that C gives E (suppose).
Thus facts can have more than one chance at a single time, or as I prefer to say, chances have components. The chance of E at tc is the sum (in the appropriate sense) of the chance that C gives E and the chance that G gives E and perhaps other factors. To return to the decay example, the chance of atom B decaying at to varies over time, but that chance also has components before the atom is bombarded: it has a chance of decaying in virtue of atom Bs own nature and structure, and an additional component of chance in virtue of the fact that A has a chance of decaying and bombarding B, thereby raising its chance. These components need to be summed in the appropriate way to give the true chance of E at that time.
The shortcoming in Mellors account is that it assumes that everything in S relevant to the chance of E is so in virtue of being relevant to the chance that C gives E. This is not necessarily so.
Is this problem merely a consequence of the fact that, in order to resolve the first problem, we defined the circumstances S to be those factors relevant to E rather than as Mellor has it, those factors relevant to the chance that C gives E? The answer is no, but before we show that, it will be instructive to consider some possible solutions to the problem, as with S taken to be those factors relevant to ch(E).
Perhaps we can remedy the problem by partitioning the circumstances. Suppose we partition S into S' and S", where S' is those factors in S relevant to chC(E), and S" is those factors in S not relevant to chC(E). Then perhaps we can take the chance that C gives E to be given by the formulation
(1b) chC(E) = (ip)(C&~S"chÞtc(E) = p)
This may work for the case of the gun, if, as I've insisted, that factor is not relevant to the chance that Don's falling gives his dying. But take a slightly different case. Suppose that Don has a serious heart condition H, so that H gives Don a chance of dying apart from his falling. But H also contributes to the chance that Dons falling gives his dying, since, on falling he may also have a heart attack and be unable to catch hold of a branch on the way down. Then since H contributes to the chance that C gives E it will be counted in S'. But H also contributes to the chance of E apart from C. So at the closest world where C&~S" the chance of E at time tc will include the component chance that H gives E apart from C, and therefore wont represent the chance that C gives E.
Suppose instead we partition S into S* and S**, where S* is those factors in S relevant to the chance E has in the absence of C, ch~C(E), and S** is those factors in S not relevant to ch~C(E). Then we take the chance that C gives E to be given by
(1c) chC(E) = (ip)(C&~S*Þchtc(E) = p)
The idea here is that since H will be in S*, we screen out the contribution that H makes to the chance of E apart from C, so that it is not included in our reckoning of the chance that C gives E. This it does, but unfortunately the solution will not work because in excluding factor H, we are excluding the contribution that H makes to the chance that C gives E, so that the chance of E at time tc at the closest world where C&~S* is still not the true chC(E).
This seems to indicate that in gereral partitioning circumstances will not work as a strategy to save (1), because of the possibility of factors relevant both to chC(E) and to ch~C(E).
Before we consider a different kind of solution, we should return to a question noted above, namely, is the problem merely a consequence of the fact that, in order to resolve the first problem, we defined the circumstances S to be those factors at tc relevant to E rather than as Mellor has it, those factors relevant to the chance that C gives E? Will this not make a difference to the partitioning solutions just canvassed? The answer is that it does make a difference, but not one that will solve the problem. Suppose with Mellor we take S to be the factors which are relevant to C causing E. Then in the closest C-world the chance of E at tc willinclude the contribution H makes to ch(E) apart from C, and so again will not represent the chance that C gives E. Partitioning S will again not help for the same reasons as before.
I propose that a different kind of approach can resolve all these difficulties. Suppose we can sum, in some sense, resultant probabilities. Suppose objective single-case chance is (sometimes) resolvable into components, by analogy with force. The force acting on the moon is the sum (in the appropriate sense) of the gravitational force due to the sun and the gravitational force due to the earth. So, suppose that component chances at a time are probabilities which can be summed (in the appropriate sense) to give the chance, or the total chance at that time.
To apply this to our problem, we write
CÞ (chtc(E) = p1)
~CÞ (chtc(E) = p2)
chC(E) = p3
We have seen that p3 ¹ p1, and futher that p1 includes as components chC(E) and the chance that H gives E apart from C, and p2 includes as a component the chance H gives E apart from C, but not the contribution that H makes to ch(E) in virtue of C.
This schema suggests that what we need to do to find the component chance that C gives E is to subtract the probability p2 from p1, in whatever is the appropriate way to sum the components of chance. Writing this subtraction operator as -,
chC(E) = p3 = p1 p2
Put another way, ch(E) at C-worlds has as components chC(E) and ch~C(E).
This solves our problem with factors such as H, because subtracting the component ch~C(E) = p2 from chtc(E) = p1 removes the contribution that H makes apart from C but not the contribution that H makes to the chance that C gives E. In other words, the chance that Dons falling gives his dying is the chance Don has of dying at the time he fell minus the chance Don has of dying if he doesnt fall; since the chance Don has of dying at the time he fell includes the contribution his heart condition makes, both in virtue of C and apart from C, and the chance Don has of dying if he doesnt fall includes the contribution his heart condition makes apart from C but not the contribution his heart condition makes in virtue of C.
So Mellors approach can be defended against this difficulty, if we are allowed the concept of component chances which can be summed in the appropriate way.
Notes* I would like to thank Hugh Mellor and Anderw Oakley for helpful discussions on the material in this paper.
(1) Mellor, 1995, ch. 3,4.(2) Mellor believes, like many philosophers, that causes need not determine their effects. (Dowe, 1993; 1996; 1997; forthcoming; Suppes, 1970).
(3) Mellor, 1995, p. 19.
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