Recombinations, Alien Properties and Laws of Nature


Alexander R. Pruss


March 16, 2002


           Introduction.  A recombinationist like the earlier Armstrong (1989) claims that logically possible worlds are recombinations of items found in the actual world, with some items reduplicated if need be and others deleted.  An immediate consequence of this is that if an alien property is a property that could only be defined in terms of fundamental properties that are actually uninstantiated, then it is logically impossible that an alien property be instantiated as no recombination of the items in the actual world can yield a world with an entity having such a property. 

           Recombinationism immediately implies that S5 is false.  To see this, suppose for simplicity, as I will throughout this paper, that electric charge is a fundamental property--otherwise, a different example would have to be used.  Then, let w be a possible world lacking any charged objects.  At w, then, it is true that it is logically impossible that there be a charged particle since no recombination of the entities in w yields a charged particle.  Therefore, contrary to S5, what is possible at w differs from what is possible at the actual world, since charged particles are actual and hence logically possible at the actual world.  While this argument may make one sceptical of recombinationism, the recombinationist will say that it is not surprising that if we follow out the Aristotelian intuition that possibility is to be grounded in actually existing entities, then what is possible will depend on what is actual.[1]  Henceforth I will no longer assume S5, and so logical possibilities will have to be relativized to worlds if recombinationism is true: it is logically possible at the actual world for charged particles to exist, but at a world at which there are no charged particles it is logically impossible for charged particles to exist.[2]

           However, there is a more decisive alien-properties argument against recombinationism.  I will shortly argue that the recombinationist is committed to at least one of two conceptual truths not being a conceptual truth.  The first roughly states that the laws of nature govern and explain physical events in the world rather than the other way around and thus it is logically impossible for a physically and causally possible event to affect what the laws of nature are.  This is not simply the trivial claim that laws of nature are abstract propositions and hence cannot be caused by physical events, but rather the claim is that it is impossible for a physical event to bring it about that some proposition is or is not a law of nature.  The second conceptual truth states that what is physically possible is also logically possible.

           There are in general two ways of construing a recombinationist approach to modality.  The items one recombines might be terms in those propositions or sentences that express truths about the actual world, and then one might construct worlds out of thus recombined propositions or sentences, for instance as maximal compossible collections of them.  Or one might say that a recombination of the items in the actual world is a mathematical description of how various objects in the actual cosmos are to be copied, cut, scaled, rotated, and/or translated in space-time (without overlap, if one is dealing with physical objects of a type that cannot overlap) to form a new world.  The arguments below work for both approaches.

           In fact, the argument I will give will apply not just against the 1989 Armstrong, but against any view of modality that (a) denies the possibility of alien properties while (b) allowing that it is sufficient for the possibility of a state of affairs that it should be obtainable by a recombination of the items in the actual cosmos.  A linguistic view on which possibilities are constructed out of sentences in some actual language (perhaps a mathematical language) and which unlike Sider’s sophisticated pluriverse approach[3] does not allow for alien properties will generally allow the criticism to run.  Thus, in general, we will have an argument for the possibility of alien properties—more precisely, it is left as an exercise to the reader to note that the argument shows that there is a possible world w at which it is true that it is possible for a specific alien property (electric charge, to be precise) to be instantiated even though it is not instantiated at w.

           The argument.  I will say that a proposition p is strongly physically possible (at a world w) if and only if the laws of nature operative (at w) together with the initial conditions (at w) assign a non-zero objective probability to p.  This notion is stronger than mere causal possibility, since in an indeterministic world, some zero-probability events might well be causally possible, but would not count as strongly physically possible.  We can now state the following Principle of Governance by Laws:

(PGL) It is logically necessary that if p is a strongly physically possible proposition reporting a physical event and L is a law of nature, then: Were p to hold, it would nonetheless be the case that L would (still) be a law of nature (i.e., the subjunctive conditional “were p to hold, L would be a law of nature” is true).

What strongly physically possible things happen cannot affect what the laws of nature are, because it is the laws that govern and explain lawlike events and not the other way around.  Laws are cited in explanations of events, whereas events are not cited in explanations, but only in epistemic justifications, of laws.[4]

           To see the non-triviality of PGL, note that someone who takes an indeterministic, frequentist Humean view of laws of nature will have to deny PGL.  For suppose that it is a law of nature that coins are indeterministic and come up heads with probability 1/2.  Nonetheless, assuming the number of flips made in the history of the universe is finite, there is a non-zero probability that they will all come out heads.  But were they all to come out heads, on frequentist Humean principles it would no longer be a law of nature that they have probability 1/2.  Rather than being an argument against PGL, however, this can be seen as an argument against the Humean, in fact a special case of the general objection that Humean laws are not explanatory. 

           Wayne Davis in private correspondence came up with the following objection to the above argument:  The Humean could properly claim that a world in which all the coin tosses come up heads is just not a physically possible world, since after all the law of nature that the coin has probability 1/2 of coming up heads is falsified in that world.  However, plausibly, if a physical state of affairs has a non-zero probability of occurring conditionally on the initial conditions of the universe and on the actual laws of nature, then the state of affairs is physically possible.  Hence, if there is a non-zero probability, given the laws of nature and initial conditions, that the coin would always come up heads, then this state of affairs is physically possible.  The probabilities in non-deterministic laws of nature are supposed to be non-epistemic, and thus a non-zero probability implies a real possibility—indeed, a real physical possibility since we are talking of the laws of nature, contrary in this case to the Humean objection.

           The second much less controversial conceptual truth I will need is the Principle of the Possibility of Physical Possibilities (PPPP):

(PPPP)     Logically necessarily, if p is strongly physically possible, then p is logically possible.

PPPP also follows from the fact that the objective probability of any event that is logically impossible is zero.

           We are now in a position to see how recombinationism violates the conjunction of PGL and PPPP.  First, consider an indeterministic world w­0 which starts out at a time t0 containing a bunch of photons interacting randomly according to the laws of nature that hold in our actual world, but with w0 not containing any charged particles at t0.  However, according to current physics, two photons can collide and produce two charged electrons, and I shall suppose this in fact happens at some time after t0 in w0.  For simplicity, I will use the term “electron” as short for the compound term “charged electron”, and without prejudging whether, as is highly plausible, it is a necessary truth that all electrons are charged.  I now assume w0 has such initial conditions that it was strongly physically possible at w0 that no charged particle production process ever happened.  Presumably if w0’s universe is sufficiently quickly expanding this can be ensured.  Observe that w0 can be constructed by an appropriate recombination of the items inhabiting the actual world and therefore is logically possible (at our world) by the recombinationist’s lights.

           Now, let p be the proposition that no charged particles are ever formed.  Then, p is strongly physically possible at w0 by the above assumptions.  Consider now the law of nature holding at w0 according to which two photons can collide and produce two charged electrons and according to which this has a non-zero probability given the initial conditions of w0.  Let w1 be any world with the same initial conditions as w0 but in which p holds.  Then, assuming electric charge is a fundamental property, the existence of a charged particle is logically impossible at w1, according to recombinationism.  Therefore, by PPPP, it is not strongly physically possible.  But if L were a law of nature at w1, then it would be strongly physically possible that a charged particle should come to exist, given the initial conditions.  Thus, L is not a law of nature at w1.[5]  Hence, in no world with the same initial conditions as w0 and in which p holds is it the case that L is a law of nature.

           It now follows that according to recombinationism it is true at w0 that were p to be true, L would not be a law of nature.  Since p is strongly physically possible at w0, this is a violation of PGL.

           Therefore, given the possibility of the above constructions, the recombinationist is committed to the denial either of PPPP or of PGL or of both.  Clearly, PPPP is the less controversial of the two.  A Humean might not mind having to deny PGL.  But Humeanism leads to problems with the apparent explanatory prowess of nomological explanations.  If one thinks of the laws of nature as in some way above the objects and events of our universe, one will find the denial of PGL highly problematic.

           Objection 1.  One non-Humean way to deny PGL would be to propose an Aristotelian view of laws of nature on which the laws of nature supervene on the powers and dispositions of substances.  Then, the laws would not be above the substances in any obvious way.  However, a modification of the above argument against the recombinationist can be run on the Aristotelian view.  For suppose (as we can surely ensure) w0 is such that only one actual collision between photons that produces electrons occurs, and it occurs at some time t0 which is after the start of the universe’s existence in w0 (assuming it has a start).  Then, on the Aristotelian view, the photons in w0 have the capacity of colliding and producing electrons.  Since the actuation of this capacity is explanatory of the existence of electrons produced from such collisions and since the capacity is a dispositional property pre-existing its actuation, the existence of the capacity must be independent of whether in fact the capacity is actuated.  Therefore, whether the photons have this capacity cannot depend on whether in fact electrons are formed from photon-photon collisions.  Thus, just as there is a world w0 in which the photons eventually do collide to produce charged electrons, there must be a world w1 where they do not in fact collide and produce charged electrons, but have a capacity of doing so, even though there are in fact no charged particles in w1.  But if recombinationism is true, then it is true at w1 that it is logically impossible for charged particles to exist, and so the photons have a capacity for doing something logically impossible, which is absurd, and contrary to the Aristotelian analog of PPPP. 

           What if we take the way out of this by denying that at w1 the photons have a capacity for producing electrons?  Then at w0 the photons have even before t0 the dispositional property of producing electrons under appropriate conditions, but they always lack this property at w1.  Thus, the recombinationist who takes this way out is committed to the idea that an event at t­0­ determines what dispositional properties photons had before t­0­, which those who find backwards causation absurd will reject.  Thus, we see that an Aristotelian view of laws of nature will not help the recombinationist.[6]

           Objection 2.  But perhaps in fact the laws of nature are necessarily deterministic?  In response to this, we can redeploy the argument used against the recombinationist who took refuge in an Aristotelian view of laws of nature.  Given determinism, we want to say that photons have the dispositional property that were two to collide under appropriate circumstances, they would produce two electrons.  But once again it follows that it is true at w0 that were the two photons not to have collided at t0, then photons would never have had this dispositional property prior to t0 (since they lack it in w1 according to the recombinationist as obviously nothing has a dispositional property to do anything not logically possible and it is the case at w1 that the existence of electrons is not logically possible) and the laws of nature would always have been different.  It is no longer true that it is strongly physically possible at w0 that the two photons did not collide at t0 since determinism entails that only actual occurrences are strongly possible, but the conclusion is absurd enough.  For instance, although it is in fact true at w0 that were two photons to have appropriately collided at some time t–1<t0 then they would have produced two electrons, this is not true at w1 since according to the recombinationist at w1 it is (presumably) logically possible for them to appropriately collide at w1 while it is logically impossible for electrons to exist, and there are no true counterfactuals with impossible consequent and possible antecedent.[7]

           Objection 3.  Our basic argument consisted in constructing a world w1 at which it is the case that although it is physically possible that electrons exist, it is not logically possible.  The recombinationist might object, perhaps for reasons similar to Kripke’s (1980) rejection of the possibility of a unicorn’s existence, that there can exist no propositions involving alien properties—perhaps propositions are not necessary beings.  Thus, at w1 the proposition that electrons exist does not exist, and a fortiori the proposition that it is logically impossible that electrons exist does not exist either.  Hence, it is not the case at w1 that it is logically impossible that electrons exist, nor is it the case at w1 that it is logically possible that electrons exist.  However, this view falls afoul of the fact that as constructed it was the case at w1 that it was physically possible that electrons exist.  But plausibly if a proposition is true at a world, then it exists in that world, and hence the proposition that it is physically possible that electrons exist exists at w1, and hence so does the subproposition that electrons exist.  If the recombinationist, however, responds that something can be the case at a world without the proposition reporting its being the case existing at that world[8], then she has no objection to the argument.  For once she admits that the proposition that it is physically possible that electrons exist is true at w1 without existing there, she should likewise admit that the proposition that it is logically impossible that electrons exist is true at w1 without existing there.

           A different response to this objection would be eschew reference to electrons in our propositions, and simply to speak of alien properties in general.  Then, by the argument given, it would still be the case given recombinationism that at w1 it is physically possible that a property alien to w1 is instantiated (i.e., that there is an instantiated property F which cannot be expressed in terms of the properties instantiated at w1)—namely, should two photons collide—while simultaneously it is the case at w1 that it is logically impossible that a property alien to w1 be instantiated, since no rearrangement of the items in w1 will produce a state of affairs with any property alien to w1.

           Objection 4.  The recombinationist could insist that recombinationism is only true at the actual world.  Thus, it is true at the actual world that it is impossible for a property alien to the actual world to be instantiated, but the same cannot be said any longer if some other world is substituted in place of the actual world.  Thus, recombinationism would only be a contingent truth.  This, however, would shift the difficulty to the epistemic sphere: If recombinationism were at most contingently true, how could we know it to be actually true?  Could we know that although there are worlds relative to which alien properties are possible, nonetheless none are possible relative to our world?  (Cf. Lewis, 1986, p. 159.)

           Conclusions.  Therefore, the recombinationist would need to abandon the idea of laws or dispositional properties as being explanatorily prior to the physical events we think of as governed by them.  This is too high a price to pay, especially given the available alternative views of possibility.  Indeed, even a Platonist view of possibility, unsatisfactory as it may be in some respects (see, e.g., Pruss, forthcoming), might be preferable to recombinationism in every way, especially since the recombinationist cannot have a principled dislike of Platonic entities as she is willing to countenance recombinations which are entities as Platonic as can be.  And, the very same arguments we have give above, show that granting two plausible assumptions about laws of nature (PGL and PPPP), and given some plausible assumptions about what recombined worlds are possible (assumptions that need to be strong enough to let us construct w0 and w1), then one cannot categorically deny the possibility of alien properties—for there are possible worlds (e.g., w1) at which it is the case that an alien property (viz., electric charge) could be instantiated.[9]


Armstrong, D. M.  A Combinatorial Theory of Possibility.  Cambridge University Press, 1989.

Kripke, Saul.  Naming and Necessity.  Harvard University Press, 1980.

Lewis, David.  On the Plurality of Worlds.  Oxford / New York: Blackwell, 1986.

Pruss, Alexander R.  “The actual and the possible”, in: Richard M. Gale (ed.), Blackwell Guide to Metaphysics, Oxford: Blackwell, forthcoming.

Sider, Ted.  “The Ersatz Pluriverse”, submitted.

[1]Note, though, that the way in which possibility is grounded in actual entities by the recombinationist is not really faithful to the spirit of the Aristotelian dictum, since recombinations are themselves mathematical or logical abstractions, even if they are recombinations of actual objects.  The Aristotelian would like possibilities to be grounded in the powers of existent things (see, e.g., Pruss [forthcoming] for an approach along these lines, where, interestingly, S5 seems to continue to hold). 

[2]We must be careful not to confuse the locution “it is logically impossible at w that p” which is short for “it is the case at w that it is logically impossible that p” with the locution “it is logically impossible (that p at w)”, the latter (assuming symbols naming propositions and worlds refer rigidly) being true if and only if in fact p is false at w.

[3] Ted Sider (submitted).

[4]Nicholas Rescher in his recent book Nature and Understanding (Clarendon: Oxford, 2000, Section 4.4) disagrees with this as stated, but argues that both laws and events can be systemically explained by one another in a non-viciously circular manner through the fact that together they form an appropriately unified system.  But even if this is so, and it does seem implausible that two contingent propositions would explain one another reciprocally, the kinds of explanation given in both directions are sufficiently different that my argument should still go through.

[5]We can see this directly in Armstrong’s view of laws of nature as relations between universals.  The law L consists in a certain relation between Photonicity and Electronicity.  But there are no uninstantiated universals for Armstrong, and so Electronicity does not exist at w1, and hence neither does L.  This observation is, I think, due to Troy Cross (in discussion).

[6]Not that the Aristotelian needs to be a recombinationist!  See note 1, above.

[7]I am assuming here that it is true at w0 that were the electrons not to have collided at t0, then they would neither have collided in the future.  It might be possible to cleverly set up a situation where this is false: although the collision is the only one in the history of w0, were it not have happened, another would have happened.  However, no doubt one can by appropriately rearranging the particles in w0 ensure this would not be so.   

[8] This view is not as absurd as it sounds.  For instance, a Kripkean who thinks that no proposition making a de re reference to an individual can exist at a world at which that individual does not exist is still free to say: There is a world w at which it is the case that Socrates never existed, even though there is no world at which both the proposition that Socrates never existed is true and at which it is the case that Socrates never existed.  For perhaps when we say that such-and-such is the case at w, we are able to say this in the actual world so that the proposition in question is available to us—the proposition that Socrates never existing being available to us only because it is actually false! 

[9] I am grateful to Troy Cross, Wayne Davis, Gregory Fitch and Richard Gale for discussions on these topics and/or comments on earlier drafts.