Emergence and Incremental Probability

In 'Emergence and Incremental Impact', I argued (contra Kingston and Sinnott-Armstrong) that emergent properties do nothing to undermine the basic case for individual impact: they're just another kind of threshold case, and thresholds are compatible with difference-making increments.

In that old post, I assumed counterfactual determinacy to make the case for there being some precise increment(s) that make a difference whenever a collection of increments together does.  But while revising my paper on collective harm, it occurred to me that the case becomes much more clear-cut when made in terms of probabilities.

Consider.  Kingston & Sinnott-Armstrong object (p.179):

[T]he expected disvalue approach requires that the probability of dangerous events can themselves be increased (minutely) by the addition of relatively tiny emissions. But why should we assume this? ... Emergence affects probability as it does other properties. While adding oil to an engine reduces the probability of a moving part failing, it is implausible that adding a molecule of oil reduces that probability of failure by 1/10^25.

Why is this implausible? Suppose that adding a large drop of oil containing 10^23 molecules would reduce the probability of engine failure by at least 1/100. Now consider the sequence of possible futures M[n] that consist in adding precisely n molecules of oil to the engine. By our initial supposition, the probability of engine failure in M[10^23] is at least 1/100 less than in M[0]. But then it's logically impossible to assign probabilities of engine failure to each intermediate state in the sequence without some of those values in adjacent states differing by at least 1/10^25. ...

Of course, it may well be that adding only the first molecule of oil would indeed have a much lower than average chance of making a difference. But even if so, this merely ensures that some other increments--namely, those in the threshold vicinity--have a correspondingly higher chance of making a difference. This is the familiar structure of expected-value reasoning in threshold cases. As previously argued [in my paper], if we've no idea where the thresholds lie, or no special reason to expect ourselves to be disproportionately likely to be distant from them, then the mere existence of such thresholds makes no difference to the expected value of our contribution: it remains equal to the average value of many such contributions. Nothing about emergent properties changes this basic reasoning. But it does help to emphasize a crucial dialectical point, that the important question is not whether a single increment in isolation makes a difference (it need not), but rather whether some increment in context does so (that is, given how many previous increments have already been made).