In a previous post, I introduced David Lewis’ approach to measuring similarities between possible worlds. He argued that facts pertaining to laws of nature make a bigger difference when comparing worlds than particular facts do. Thus, discrepancies between their laws of nature count for more in terms of dissimilarity than discrepancies between their past or future regions.
Lewis’ way of comparing worlds enabled him to evade Bennett (1974) and Fine’s (1975) objection that “miracle worlds” falsify his theory of how counterfactuals are made true. Lewis argued that the objection only works if such worlds contain many exceptions to their laws of nature; and these exceptions make them too unlike the actual world to serve as a counterexample to his theory.
We saw Lewis’ point born out in previous posts when we considered the example of W3, a world where several miracles were needed to erase the many effects of Tom igniting his pile of gunpowder. But what if W3 could be modified to ensure that Tom’s action only had one effect? Such a world would only need a small, localized miracle to erase that effect; it would be indistinguishable from the actual world in all other respects. Is this kind of “small miracle” world even possible?
Michael Tooley (2014) thinks it is possible. He invites us to imagine a world (W4) in which Tom is psychokinetic. In this world, Tom has some three unusual abilities. First, his mental acts do not require any changes in the physical world in order for them to occur. Second, he can bring about physical events directly so that no intervening or mediating process is needed. And third, his mental acts are so precise that they produce no effects other than the physical events which he wills.
Now, let’s suppose that W4 is logically possible. Can we generate a valid counterexample to Lewis’ theory? I think so. Suppose that W4 contains a fission bomb so powerful that it can incinerate all life on earth through a nuclear holocaust. Imagine further that Tom uses psychokinesis to split a single uranium atom inside the bomb. When Tom uses his mind to split the atom, no events other than his mental act are enlisted to perform it: neurons do not activate; nerves do not fire; and limbs do not move.
Moreover, Tom’s psychokinetic powers are so precise that only the splitting is produced by his mental act; there are absolutely no side-effects of his act in terms of forming memories or nudging other parts of the bomb. Finally, at the exact moment when Tom splits the atom, a small, localized miracle returns the atom to its initial state and stays the fission process. So instead of a nuclear holocaust, the future of W4 is an exact match with the future in the original world where Tom does not split the atom psychokinetically.
What does this counterexample tell us about Lewis’ theory? It tells us that W4 only needs a small, localized miracle to satisfy Lewis’ criteria of similarity. If so, then the following counterfactual S^ will be true:
S^: If Tom had psychokinetically split the uranium atom, a holocaust would not have occurred.
But we know that S^ is false! Therefore, Lewis’ criteria generate the wrong truth-value for this counterfactual.
 It is worth noting that if laws play an essential role in Lewis’ analysis of counterfactuals and if counterfactuals play an essential role in his analysis of causation, then his reduction of causation presupposes laws rather than explains them. This comes as no surprise to Lewis because he introduced his theory of counterfactuals (in part) to make up for what was lacking in the nomological/regularity analysis of causation. For more on this point, see Swinburne (1997) “The Irreducibility of Causation” Dialectica. 51 p.82, footnote 7.
 One might criticize Lewis for being ad hoc when he gives more consideration to nomic facts than non-nomic facts when measuring similarities and differences between worlds. To avoid this charge, Lewis needs to provide independent reasons for comparing worlds in his preferred way, other than simply avoiding counterexamples!
 Lewis could reject Tooley’s example on the grounds that psychokinesis is logically impossible. But Tooley has at least two counters here: first, psychokinesis might be incompatible with the actual laws of nature but different laws seem to be logically possible, so psychokinesis should not be ruled out prematurely. Second, branching processes that terminate after a short time might do the trick instead of non-branching processes like psychokinesis. For more information on this second option, see section 5.2.5 of Tooley (2014).