I think the following passage, from John Earman, John Roberts and Sheldon Smith’s paper “Ceteris Paribus Lost” (Erkenntnis 57: 281–301, 2002) is somewhat mistaken.
But the second problem with CP laws, their untestability, is decisive in our view. In order for a hypothesis to be testable, it must lead us to some prediction. The prediction may be statistical in character, and in general it will depend on a set of auxiliary hypotheses. Even when these important qualifications have been added, CP law statements still fail to make any testable predictions. Consider the putative law that CP, all Fs are Gs. The information that x is an F, together with any auxiliary hypotheses you like, fails to entail that x is a G, or even to entail that with probability p, x is a G. For, even given this information, other things could fail to be equal, and we are not even given a way of estimating the probability that they so fail. Two qualifications have to be made. First, our claim is true only if the auxiliary hypotheses don’t entail the prediction all by themselves, in which case the CP law is inessential to the prediction and doesn’t get tested by a check of that prediction. Second, our claim is true only if none of the auxiliary hypotheses is the hypothesis that “other things are equal”, or “there are no interferences”. What if the auxiliaries do include the claim that other things are equal? Then either this auxiliary can be stated in a form that allows us to check whether it is true, or it can’t. If it can, then the original CP law can be turned into a strict law by substituting the testable auxiliary for the CP clause. If it can’t, then the prediction relies on an auxiliary hypothesis that cannot be tested itself. But it is generally, and rightly, presumed that auxiliary hypotheses must be testable in principle if they are to be used in an honest test. Hence, we can’t rely on a putative CP law to make any predictions about what will be observed, or about the probability that something will be observed. If we can’t do that, then it seems that we can’t subject the putative CP law to any kind of empirical test.
They are arguing against the claim that there are any ceteris paribus laws with ineliminable CP clauses. And they claim, plausibly, that if the claim that other things are equal is statable in physical terms then it quite well can be eliminated. So far so good.
The problem is that they leave out a very easy way in which the CP law could be testable without the CP clause being statable in physical terms. Assume that there is a p such that p entails that other things are equal, but is not entailed by it. Now nothing in what has been said about CP clauses rules out this possibility. Indeed, if CP clauses are infinite disjunctions of physical state descriptions, that situation will be easily possible. Then if p is one of the auxiliary hypothesis, the CP law will be testable even though the CP clause cannot be (finitely) stated.
I don’t have any particular fondness for CP laws, but this argument that they are untestable seems to have a hole here. (We’re bracketing questions about the philosophical importance of testability here by the way. I also have no idea whether this point has been made elsewhere in the literature – this is a diary entry not a quasi-publication.)