Web sites on which refer to or discuss this paper:
Judea Pearl Cognitive
Systems Laboratory
University of Wisconsin Course discussion groups (Ellery Eells and
Dan Hausman) 1,
2,
University
of Bristol, Philosophy of physics course
Robin Cowan
and Mario J. Rizzo
Iain Martel
Greg Ray
DISCUSSION
TWO NOTES ON THE PROBABILISTIC APPROACH TO CAUSALITY *
GERMUND HESSLOW
University of Lund
From Philosophy of Science 43 (1976) pp. 290-292.
1. According to the Humean theory of causality the individual event A_{t}(the event of kind A occurring at time t) caused B_{t’ }(t’ is assumed to be later than t) if and only if events of kind A are always followed by events of kind B. Many everyday examples of causal connections however refute this view, since it is seldom that the cause is always followed by the effect. Mumps, e.g., can cause sterility, but the probability of sterility given this illness is minimal.
For those who wish to retain the basic idea of the Humean approach there seems to be two ways of solving the problem. The first, which I will call the deterministic approach, can be briefly described as follows. An event A can be construed as a set of more elementary events A_{1}, A_{2}, ...A_{n}, and the statement that A occurs as a conjunction A_{1} occurs, A_{2} occurs, etc. We then introduce the following definitions:
The second approach, the probabilistic, requires of the cause that it should raise the probability of the effect. Suppes ([1], p 12) gives the following definition:
2. The basic idea in Suppes’ theory is of course that a cause raises the probability of its effect, and it is difficult to see how the theory could be modified without upholding this thesis. It is possible however that examples could be found of causes that lower the probability of their effects. Such a situation could come about if a cause could lower the probability of other more efficient causes. It has been claimed, e.g., that contraceptive pills (C) can cause thrombosis (T), and that consequently there are cases where C_{t }caused T_{t’}. But pregnancy can also cause thrombosis, and C lowers the probability of pregnancy. I do not know the values of P(T) and P(T/C) but it seems possible that P(T/C) < P(T), and in a population which lacked other contraceptives this would appear a likely situation. Be that as it may, the point remains: it is entirely possible that a cause should lower the probability of its effect.
Obviously this example is quite consistent with the deterministic theory.
3. Another obvious objection to D3 is that it allows as causes events that did not occur. Suppes deals with this objection m the following manner: We can "say that A_{t} is a potential prima facie cause of B_{t’}. When both events occur the potential becomes actual" ([1], p 40). (This is related to drawing a distinction that Suppes’ theory lacks, namely between causality as a relation between kinds of events and as a relation between individual events.) Instead of saying that A_{t} causes B_{t’ }we should say that A_{t} can or may cause B_{t’ }and that A_{t} did cause B_{t’ }only when both events occurred.
The relationship between potential and actual (or between generic and individual) causal relations is not so simple however. From the fact that A_{t} may cause B_{t’} (e. g. smoking causes cancer) and A_{t }and B_{t’ }both occurred (John smoked and got cancer) it does not follow that A_{t}caused B_{t’} (John’s smoking caused his getting cancer).
Let us, returning for a moment to the terminology of the deterministic theory, assume that there are two kinds of events A and B that are complete causes of E and that A consists of A_{1} and A_{2}. It would
then be reasonable to say that A_{1} may cause E (and normally A_{1} will raise the probability of E). According to Suppes we should then say that A_{1t} caused E_{t’}if A_{1t} and E_{t’ }occurred. But A_{1t}caused E_{t’}only if A_{2t}occurred. If this was not the case but rather A_{1t}, -A_{2t}, B_{t}and E_{t’}occurred, then B_{t}but not A_{1t} caused E_{t’}.
Suppose that some people have a certain property A, such that those who smoke and have A always get cancer, while smoking has no effect on those who lack A, but those belonging to the latter group can cancer in other ways. If John, who smoked and got cancer, was an A-person, the smoking caused the cancer, but if he was not an A-person, his cancer must have been caused by something else.
The fact that A_{t} may cause B_{t’} and that both events occurred does not automatically warrant the conclusion that A_{t}, caused B_{t’}. The truth of this statement is dependent on the presence of those factors which together with A_{t} constitute a sufficient condition for B_{t’}. Thus the statement that A_{t}, caused B_{t’} presupposes the presence of a sufficient condition, i.e. it presupposes the kind of determinism that follows from D1 and D2.
4. If a physician has two patients with the same illness and receiving the same treatment, but only one of them recovers, he can be sure to ask in what way they differed. I.e. he assumes that there is a factor which was present in one patient but not in the other that explains the different outcomes. This is the kind of determinism which is reflected in D1 and D2, and it precisely this that Suppes finds objectionable and which is his main reason for adopting a probabilistic approach. In this example it is of no consequence if the physician actually believes that there is an explanation or if he merely acts as if there were, but in the example in the foregoing section this is not so. There the truth of a causal statement is dependent on the presence of the other factors.
The fact that determinism is doubtful or extravagant as a metaphysical thesis is not really relevant to the analysis of causation. What is relevant is if a deterministic assumption is so embedded in ordinary discourse as to affect the language of causality.
REFERENCES
[1] Suppes, P. A Probabilistic Theory of Causality. Amsterdam: North-Holland, 1970.