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Not everyone who smokes gets cancer, but we still say things like "smoking causes cancer." How should we understand causal statements like this?

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A good question, but as you

A good question, but as you no doubt guessed, one that people have thought about. The short answer is that we'll say that X causes Y if X raises the probability of Y, even if it doesn't raise the probability to 100%

Let's be a bit more concrete. Think about clinical trials of a medication. Suppose we think that some new compound lowers blood pressure. We might test this by selecting a set of test subjects with hypertension, and then randomly assigning some of them to the treatment group and others to the control group. Ideally the test would be a double-blind test. That is, neither the people administering the treatment nor the people being treated would know if they were getting the actual medicine or a mere placebo. We'd measure everyone's blood pressure before the trial, and then after. And then we'd compare. Normally we wouldn't expect that everyone who received the actual treatment would have lower blood pressure at the end, and we also wouldn't expect that no one who got the placebo would end up with lower blood pressure. What we would be looking for, roughly, is a statistically significant difference between the two groups, with the people in the treatment group having a lower average blood pressure after the treatment than before, and by a larger margin than the control group. I haven't stated that carefully or precisely, but you get the idea.

So the effect is statistical. If the trial is successful, we would say that administering the medication makes it likely that the patient's blood pressure will be lowered, and makes is more likely than if the patient were left untreated.

What I've said here leaves out important nuances and qualifications, but I hope it gets the idea across. We don't think that X must invariably bring about Y to say that X causes Y, but we do think that for X to cause Y, X has to make Y more likely.

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