Getting More from External Relations

A speculative post. Consider two cases where it has been thought that substantive new metaphysics would be needed:

  • ‘Strong’ emergence, to allow for the apparent conceptual possibilities of wholes whose properties don’t supervene on the properties of their parts.
  • Haecceities, to ground identity and distinctness for individuals in the light of mirror worlds.

In each case, more familiar metaphysics turns out to be able to do the job for us. We just need to appeal to the right perfectly natural external relations (PNERs):

  • ‘Weak’ emergence: emergent properties are the result of perfectly-natural-external-relation(-non-spatio-temporal) (PNERNSTs) holding between the elements. The correct supervenience thesis is not supervenience of whole on parts, but supervenience of whole on parts plus the PNERs they stand in to one another. (Consider two spatiotemporally isolated spheres, and contrast with two adjacent spheres. The wholes composed by the pairs are not intrinsically similar.)
  • Anti-haecceitism: Individuals can be ‘weakly discerned’ by their place in a network of irreflexive PNERs. No two worlds differ merely by a permutation of individuals.

It’s tempting to try to extend this strategy from haecceities to quiddities, given the close similarity between these two kinds of mysterious entity. Instead of positing:

  • Quiddities, to ground identity and distinctness for properties in the light of mirror worlds;

we can appeal to PNERs between properties:

  • Anti-quidditism: Properties can be ‘weakly discerned’ by their place in a network of irreflexive PNERs. No two worlds differ merely by a permutation of properties.

The PNERs that hold between properties are not of the same kind as the PNERs that hold between individuals, of course. The former are nomic relations, such as are expressed directly in the equations that figure in the fundamental laws of nature. The latter we might call configuration relations: they’re usually taken to be spatio-temporal relations and entanglement relations and possibly more.

All sides agree that configuration relations are recombinable, at least to a limited degree: many patterns of configuration relations are possible. But necessitarians deny that nomic relations are recombinable. In fact, it is open to the necessitarian to draw the distinction between nomic relations and configuration relations in this modal way: configuration relations are whichever PNERs admit of some recombination. That in turn promises a novel-looking necessitarian account of the difference between individuals and properties: individuals are those things which only stand in configuration PNERs, while properties are those things which only stand in nomic PNERs.

What of the other cases – things that stand in no PNERs, and things that stand in both kinds of PNER? I think the former kind of thing is otiose: what possible grounds could we ever have to posit them? The latter kind of thing might conceivably be wanted, though; perhaps tropes would fit the bill, or some located entity which featured directly in the laws of nature (God? The Universe?).

Getting More from External Relations

Sleeping Beauty and the Mathematical Alarm-Clock

In the chancy Sleeping Beauty case, popularized by Elga [2000], Beauty goes to sleep on Sunday night knowing that she will be awakened on Monday morning and then put back to sleep on Monday evening. A fair coin is tossed: if the coin lands tails, Beauty is awakened again on Tuesday with her memory of the Monday awakening erased, and if the coin lands heads, she sleeps through to Wednesday. Beauty knows all this. The puzzle is to say what credence Beauty should have on Monday in the proposition that the coin lands heads (call this proposition Heads).

In the mathematical Sleeping Beauty case, uncertainty about the result of a fair coin toss is replaced by uncertainty about the truth of a mathematical proposition. On Sunday night Beauty has credence 1/2 that Fermat’s Last Theorem is true. She will be awakened on Monday if the theorem is true, and on both Monday and Tuesday (again with her memories from Monday erased) if the theorem is false. Beauty knows all this. The puzzle is to say what credence Beauty should have on Monday in the proposition that Fermat’s Last Theorem is true (call this proposition True.)

The setup of the case entails that Beauty cannot be a fully rational Bayesian agent, since such agents are required to be logically omniscient.  Is there then any sense in asking what her credences ought to be? – after all, in one obvious sense, her credence in True ought to be 1. I think so: it is common to distinguish diachronic and synchronic constraints on credences, and evaluate an agent’s performance with respect to the diachronic constraints independently of her performance with respect to certain synchronic constraints (such as logical omniscience). Epistemologists had better hope that something like this is possible: we’re not perfect Bayesians, after all, and agents with non-trivial degrees of belief in mathematical propositions need to be modelled by any candidate epistemology for mathematics.

The difference between the two cases may seem somewhat incidental. General considerations relating to topic-neutrality perhaps make it natural to assume that the two puzzles have the same solution. In any case, discussions of Sleeping Beauty have usually paid little attention to the role that chance plays in the story. For example, in a discussion of the chancy Sleeping Beauty case Chris Meacham remarks that “the chanciness of the coin toss only plays a superficial role in the argument….  the argument goes through just as well if heads and tails are replaced by two different hypotheses we have other reasons for having ½ / ½ credences in.” (Meacham [2008]). Sleeping Beauty is usually described as a puzzle about self-location; and our two cases seem to involve structurally similar self-locating uncertainty. In both the chancy case and the mathematical case, when she awakens on Monday Beauty is unsure whether it is Monday or Tuesday.

In this post I will give some reasons for suspecting that the two puzzles may in fact have different solutions. Two of the most powerful arguments for the answer 1/3 in the chancy case (both of which appear in Elga’s original article) are inapplicable in the mathematical case. Moreover, an analogy between Sleeping Beauty and confirmation in Everettian quantum mechanics gives us reason to prefer the answer 1/2 in the mathematical case.

Consider first the long-run frequency argument for the answer 1/3. Beauty knows that if the chancy case is repeated infinitely many times, the ratio of Tails-awakenings to Heads-awakenings will (with chance 1) tend to 2:1. Since any awakening is indistinguishable from any other, her credence that a randomly-chosen awakening is a Heads-awakening should be 1/3.

This argument lapses in the mathematical case. The truth-value of mathematical propositions cannot vary between awakenings, so Beauty knows that in the long-run either all the awakenings will be True-awakenings or all the awakenings will be False-awakenings.

A similar case involves different mathematical questions on each run of the experiment. This variable-question mathematical case will probably pattern with the chancy case rather than with the fixed-question mathematical case, depending on the procedure by which the questions are selected. Be that as it may, here I will only address the fixed-question mathematical case.

Consider now the Principal Principle argument for the answer 1/3. It is a mere procedural irrelevance whether the coin is tossed on Sunday night or on Monday night – so let us assume that it is tossed on Monday night, and that Beauty knows this. Now consider Beauty’s credences on learning that it is Monday. She knows that whether Heads is true depends on a future toss of a fair coin, so the Principal Principle constrains her to have credence 1/2 in Heads. But learning that it is Monday ruled out some Tails possibilities without ruling out any Heads possibilities; so conditionalization requires that Beauty’s credences prior to learning that it is Monday must be less than 1/2. If we assume that on initially awakening Monday and Tuesday deserve equal credence conditional on Tails (perhaps justifying this assumption via a restricted principle of indifference, as Elga does) then Bayes’ theorem tells us that Beauty’s credence in Heads on initially awakening on Monday should be 1/3.

This argument too lapses in the mathematical case. Since mathematical propositions have objective chances of either zero or one, the constraint that the Principal Principle places on credence in mathematical propositions amounts to the requirement of logical omniscience, which is explicitly suspended in the mathematical Sleeping Beauty case. Accordingly, nothing in the setup of the mathematical case prohibits Beauty from adopting credence 1/3 in True on being informed that it is Monday, and hence nothing prohibits her from adopting credence 1/2 in True when she initially awakens on Monday.

The breakdown in the mathematical case of these influential arguments for the answer 1/3 provides reason to doubt whether our two cases have a uniform solution. Moreover, consideration of the analogy between the mathematical Sleeping Beauty case and confirmation in Everettian Quantum Mechanics (EQM) provides reason to think that the correct answer in the mathematical case is 1/2.

The analogy between SB and confirmation in EQM was first discussed in print by Lewis [2007]. Papineau and Durà-Vilà [2009] reply to Lewis, and an extended discussion is provided by Bradley [2010]. Lewis and Bradley both claim that applying ‘thirder’ reasoning to confirmation in EQM yields the result that evidence that a quantum measurement has taken place confirms EQM over a one-world stochastic theory, even if the result of the measurement is unknown. This ‘easy evidence’ for EQM is deeply problematic, not only for Everettians but for anyone who is not prepared to rule out EQM a priori. It is highly implausible to think that we get evidence for EQM simply by finding out that a measurement has taken place.

If the correct answer to the chancy Sleeping Beauty case diverges from the correct answer to the mathematical Sleeping Beauty case, then we may ask whether Everettian confirmation aligns with the chancy case or with the mathematical case. Since it is not a matter of chance whether Everettian quantum mechanics is correct, I suggest that confirmation in EQM ought to align with the mathematical case rather than with the chancy case. If Everettian confirmation scenarios are analogous to the mathematical Sleeping Beauty case, then a rejection of easy evidence for EQM generates the answer 1/2 in the mathematical case.

I conclude that there are important disanalogies between the mathematical case and the chancy case. Two influential arguments for the answer 1/3 in the chancy casehese arguments are inapplicable to the mathematical case. And the similarity between the mathematical case and confirmation in EQM provides an indirect argument for the answer 1/2 in the mathematical case.

These differences between the two Sleeping Beauty cases are of interest in their own right. But they also cast an interesting light on the chancy case. Sleeping Beauty is usually described as a puzzle about self-locating belief. But the preceding discussion suggests that the chanciness of the coin toss is also playing a crucial role in generating the answer 1/3. Sleeping Beauty may turn out to be as much a puzzle about chance as it is a puzzle about self-location.


Bradley, Darren [2011]. ‘Confirmation in a Branching World: The Everett Interpretation and Sleeping Beauty’. British Journal for Philosophy of Science 62: 323-342.

Elga, Adam [2000]. ‘Self-locating belief and the Sleeping Beauty problem’. Analysis 60: 143-47.

Lewis, Peter [2007]. ‘Quantum Sleeping Beauty’. Analysis 67: 59–65.

Meacham, Christopher [2008]. ‘Sleeping Beauty and the Dynamics of De Se Belief’, Philosophical Studies 138: 245-269.

Papineau, David and Durà-Vilà, Victor [2009]. ‘A Thirder and a Everettian: A Reply to Lewis’ ‘Quantum Sleeping Beauty’’, Analysis 69(1):78-85.

Sleeping Beauty and the Mathematical Alarm-Clock

The Everettian Doomsday Argument

If the most familiar overlapping version of Everettian quantum mechanics (EQM) is correct, then we are constantly branching into multiple people. The corresponding reconvergence is rendered effectively impossible by decoherence. This asymmetry between splitting and reconvergence generates a disparity in numbers between young and old temporal parts of people. To every newborn infant, there correspond a multitude of teenagers, and to every teenager, there corresponds a multitude of geriatrics. Thus the vast majority of temporal parts of people, spread out over the entire Everettian multiverse, are closer to the ends of the lives of the people of which they are a part of than they are to the beginnings of the lives of those people. This consequence gives rise to the Everettian Doomsday Argument.

The original Doomsday argument aims to show that the end of the human race may be sooner than we would initially have thought. Roughly, it assumes that we are a member of the human race typical with respect to birth rank – an ordering from the first human to be born to the last – and moves to the conclusion that we should expect the human race to end within a couple of hundred years, since otherwise our birth rank would be surprisingly low. The Everettian doomsday argument aims to show that death may come sooner than we would initially have thought. Roughly, it assumes that we are a member of the set of temporal parts of a branching person typical with respect to birth distance­ – an ordering from the temporal part closest to birth to the furthest – and moves to the conclusion that we should expect that we are much closer to death than to birth, since otherwise our birth distance would be surprisingly low.

I won’t assess the argument here. Those who think there is a problem with the original Doomsday argument may well find that the objection it carries over to the Everettian Doomsday argument. And some may baulk at applying the randomness assumption to the selection of observer-moments, rather than just to the selection of observers. What I think is striking is that the diverging version of EQM, which I’ve defended in print, completely undermines the Everettian Doomsday argument. If diverging EQM is correct, then there are just as many early temporal parts of a given person as there are late temporal parts. Is this is a reason to prefer divergence to overlap, or just another reason to reject the Doomsday argument?

The Everettian Doomsday Argument

Property Maximalism

Consider three questions about the metaphysics of properties:

– What is the cardinality of the set of actually instantiated perfectly natural properties?

– What is the cardinality of the set of possible perfectly natural properties?

– Are any possible perfectly natural properties uninstantiated at the actual world?

Even if the answer to the first two questions is the same, this does not fix an answer to the third question. I’ll look at the first two questions in a future post, but for today I’ll concentrate on the third.

Call a world at which all possible perfectly natural properties are instantiated a maximal world. Although on some views of modality it might be that no worlds are maximal, it does seem to follow from the common Humean view that distinct perfectly natural properties are freely recombinable that some worlds are maximal (at least: size and shape of spacetime permitting). The question I’m interested in in this post is: what kinds of considerations might lead us to think that the actual world is or is not maximal?

Another way of putting the same question is: why should we believe in alien properties? This version of the question will of course be favoured by those who think there is no reason to believe in aliens, and who wish to shift the argumentative burden onto those who think the actual world is or might be non-maximal. The implied argument in this challenge is an argument from ideological parsimony. We can explain all of the phenomena with which we can have causal interaction without positing alien properties; so why posit them?

An interesting form of this parsimony argument appeals to the nature of reference. Here’s an initial stab:

1. We can only refer to contingently-instantiated properties if we have causal transaction with those properties.

2. We have no causal transaction with alien properties.

3. We cannot refer to alien properties.

4. We cannot grasp propositions which involve the existence of any entity to which we cannot refer.

5. We should not assign any credence to a proposition that we cannot grasp.

6. We should not assign any credence to the existence of alien properties.

This argument is not a very good one. If there are alien properties, then I have (plurally) referred to them in this very sentence. So 3) will be rejected by anyone who posits alien properties.

We might try to rescue this argument by explicitly restricting it to singular reference. But then we get a rather different conclusion:

1a. We can only refer to contingently-instantiated properties if we have causal transaction with those properties.

2a. We have no causal transaction with alien properties.

3a. For any property F, if F is alien then we cannot singularly refer to F.

4a. We cannot grasp propositions which are solely about entities to which we cannot singularly refer.

5a. We should not assign any credence to a proposition that we cannot grasp.

6a. For every property F, if F is alien then we should assign no credence to the proposition that something is possibly F.

The conclusion of the second argument is much weaker than the first. And it is not sufficient to support a prohibition on positing alien properties.

Consider an analogy from cosmology. We believe there are stars beyond the visible universe. We ourselves cannot have any causal interaction with them during our lifetimes, but light from them will reach our solar system in about one billion years time. Call any star which is more than one billion light-years beyond the edge of the currently-visible universe an alien star. We can formulate the following argument about them:

1b. We can only refer to contingently-instantiated objects if we have causal transaction with those properties.

2b. We have no causal transaction with alien stars.

3b. For cannot refer to alien stars.

4b. We cannot grasp propositions which are solely about entities to which we cannot singularly refer.

5b. We should not assign any credence to a proposition that we cannot grasp.

6b. For every object F, if F is an alien star then we should assign no credence to the proposition that F exists.

This argument calls for no revisions in cosmology. Obviously we cannot know anything at all about individual alien stars, but this presents no obstacle to an inference to the best explanation to their existence.

Of course, the analogy is not perfect. We inhabit a common spacetime with the alien stars, and so even if we can enter into no causal interactions with them, we can still be causally connected, as our future light cones overlap with theirs. And we arguably share a common cause with alien stars – the big bang. But it is hard to see why these disanalogies should make a difference. These sorts of causal relationships still aren’t sufficient to ground singular reference.

If these arguments are correct, then there is no obstacle to positing alien properties as part of a metaphysical inference to the best explanation. This was the line taken by David Lewis: a version of modal realism with alien properties came out, he thought, better in the overall cost-benefit analysis than a version of modal realism without alien properties. Here is the whole of his argument from On the Plurality of Worlds:

A world to which no … properties are alien would be an especially rich world. There is no reason to think we are privileged to inhabit such a world. Therefore any acceptable account of possibility must make provision for alien possibilities. (OPW p.92)

This argument is prima facie puzzling, in the light of methodological manuoevres that Lewis makes elsewhere. In responding to an argument that modal realism leads to inductive scepticism, as there are many more worlds in which induction is reliable than worlds where it is reliable, Lewis denies that we can formulate any preferred measure over the space of possibilities. The cardinality of worlds with alien properties is infinite, and so is the cardinality of worlds without.

Thus, by Lewis’ own lights, a belief in the existence alien properties cannot be based on the thought that if there were no alien properties, our world would be objectively special in virtue of being maximal. Even if our world is maximal and recombination holds, we can still find a measure over the space of worlds which makes the maximal worlds predominate over the non-maximal ones.

Of course, the thought that maximal worlds are special is a very natural one. It looks strange to say that the world could have lacked some properties, but that it could not have instantiated any extra properties. But it isn’t easy to make this vague thought more precise.

If we believed that the actual world contains only finitely many perfectly natural properties, then that might provide a route to denying property maximality. I’ll talk about this in a future post.

Property Maximalism

Truth-makers and Truth-causers

Truth-maker theory attempts to view the relation between truth and being as more than merely global supervenience. The idea is that we can take a particular truth – for example the true proposition that there is some cheese – and ask what element of actual being takes particular responsibility for it. In simple cases the question is easy to answer; in our example it is the world’s cheese. In more complicated cases – for example the (presumably) true proposition that energy is conserved – the question is much harder to answer.

The question is posed in various ways: as the question of what grounds what (Schaffer?), as the question of what makes what true (eg Armstrong?), and as the question of what determines what (?). The disparate terminology conceals some key common ground. Truth-maker theorists tend to agree that truthmakers necessitate the truth of their associated propositions, and they tend to agree that every true proposition has a truthmaker.

This agreement is enough to guarantee the supervenience of truth on being, as endorsed by Bigelow, Lewis, and many others since. But by itself it is not enough to satisfy the Australian school of truth-maker theorists, because it is compatible with the only kind of supervenience being the monistic kind, where the actual world is the truthmaker for every contingent true proposition. As I understand it, this is Schaffer’s view; but it seems to be regarded as not doing proper justice to the motivations behind truthmaker theory.

Graham Oddie at the Philosophical Society in Oxford yesterday suggested an alternative, but still revisionary, treatment of truthmakers. By combining the view with a theory of propositions according to which they amount to ascriptions of genuine properties, such that there are no negative propositions, Oddie reduces the scale of the task faced by the truthmaker theorist. His suggestion is that truthmakers correspond to commensurate actual sufficers; that is, roughly, that they element of being that is proportionate – ‘commensurate’ – to the truth they make.

One way of being a commensurate actual suffice is to be a minimal actual sufficer; that is, some event such that, necessarily, if it obtains then the proposition is true, but such that no part of it is such that, necessarily, if it obtains then the proposition is true. Restall has a nice example which shows that minimal actual sufficers can’t be the whole story – no fusion of electrons can be a minimal actual sufficer for the proposition that there is a denumerable infinity of electrons. (Oddie had a suggestion for dealing with this which I won’t discuss here.)

The thing that struck me about the position developed was how similar the notion of truth-making which emerges is to the notions of truth-causing and truth-explaining. In the recent literatures on causation and on explanation, the idea that a cause should be proportionate to the effect and the idea that an explanans should be proportionate to the explanandum are commonplace. These notions of proportionality are often cashed out using counterfactuals, for example by Yablo and by Menzies and List.

The difference is that Oddie’s account uses an indicative conditional to characterize sufficiency – if the truthmaker obtains then the proposition is true – and something like a minimality condition to characterize proportionality. In contrast, Menzies and List use subjunctive conditionals both to characterize sufficiency and to characterize proportionality – if the cause were to occur, the effect would occur, and if the cause were not to occur, the effect would not occur. (To get this to work, they drop the strong centering assumption from a Lewisian counterfactual semantics. But it seems possible that there are ways to combine their view with an account of counterfactuals which preserves strong centring, for example by using counterfactuals prefaced by a generic operator in the analysis of sufficiency and proportionality, rather than just the bare counterfactuals.)

Applying these subjunctive conditionals to ascribing responsibility for truth gives interesting results. Let’s try saying that an event is responsible for the truth of a proposition where a) if the event were to obtain, the proposition would be true and b) if the event were not to obtain, the proposition would not have been true. A non-minimal sufficer will not comply with the second of these conditions: the closest worlds where the non-minimal suffice does not obtain is one where another sufficer obtains instead, so non-minimal sufficers will not be proportionate.

If we adopt this account of proportionality both for causation and for ascribing responsibility for truth, the resulting picture is a picture of truth-causing.

Consider the example ‘there is some cheese’, and see how the conditionals work out in this case. For any cheesy event, if that event were to obtain then the proposition would be true. So any cheesy event satisfies the first condition. But it is hard to see what event could satisfy the second condition. A truth-causer for ‘there is some cheese’ would have to be such that, if it were to not obtain, then ‘there is some cheese’ would be false. But even the totality of the world’s cheese seems not to meet this condition. Had the totality of the world’s cheese not existed, presumably some other cheese might have existed instead.

So we get the surprising result that the cause of the truth of ‘there is some cheese’ is neither any particular piece of cheese, nor the totality of the world’s cheese. Are there any other candidates for the cause of this truth? The totality of the world’s cows are neither sufficient nor proportionate; the totality of cows could have existed without there being any cheese (if there were nobody around to milk them), and there could have been cheese even without the totality of cows (if there were other cows around instead). So it looks like nothing actual is the cause of the truth of ‘there is some cheese’.

Contrast this result with singular propositions. The cause of the truth of ‘Oddie exists’ is Oddie. Were there to be an Oddie-ish event, ‘Oddie exists’ would be true; were there not to be an Oddie-ish event, ‘Oddie exists’ would not be true. So singular propositions do have truth-causers – the referent of the singular term.

So maximalism fails for truth-causers: some truths (those that are general rather than particular) are true without there being any particular actual thing or things which causes their truth. So Oddie should resist assimilating commensurate actual sufficers to commensurate causes, and using the account of commensurateness appropriate to causation.

It remains to be seen whether any non-counterfactual account of proportionality can be given which allows us to retain maximality for truth-makers. Oddie’s proposal is that proportionality amounts to a disjunction of minimality and recursively-defined smallness; perhaps it can be made to work, but it doesn’t look as elegant as the counterfactual account of proportionality available in the case of causation.

Truth-makers and Truth-causers

Reply to ‘Quiddistic Knowledge’ – Part 2

In my last post I replied on behalf of the ‘modal necessitarian’ to Schaffer’s arguments in the first half of this paper. In this post I’ll have some things to say about the second half of the paper: Schaffer’s response on behalf of the quidditist to the anti-quidditist’s epistemological argument.

Quidditism, as Schaffer uses the term, is the acceptance of merely quiddistic differences between possible worlds; that is, it is the view that there are at least two distinct worlds ‘that differ solely by swaps of which properties confer which powers’. Anti-quidditism is the denial of quidditism.

Why be a quidditist? Schaffer offers an argument from properties (that our best theory of the nature of properties allows them to be repeated across worlds) and an argument from duplication (that the thesis that qualitative duplication is intrinsic depends on quidditism). I will not have much to say about these arguments here.

Why be an anti-quidditist? The motivation Schaffer considers is an epistemological one. Here is the argument laid out in full:

22.1) If there are worlds that differ solely over which property confers which power, then there is a world w distinct from actuality solely over which property confers which power;
22.2) If there is a world w distinct from actuality solely over which property confers which power, then we cannot discriminate between actuality and w;
22.3) If we cannot discriminate between actuality and w, then we do not know whether actuality or w obtains;
22.4) If we do not know whether actuality or w obtains, then (since actuality and w differ over which properties exist) we do not know which properties exist;
22.5) Therefore: If there are worlds that differ solely over which property confers which power, then we do not know which properties exist.
23) We do know which properties exist;
24) Therefore: there aren’t worlds that differ solely over which property confers which power.

Schaffer’s response to this argument, in a nutshell, is to assimilate it to general sceptical worries, for example to external-world scepticism. It’s certainly easy to generate a an analogous argument:

32.2) If there is a world w that contains people in exactly our phenomenal situation who have no hands, then we cannot discriminate between actuality and w;
32.3) If we cannot discriminate between actuality and w, then we do not know whether actuality or w obtains;
32.4) If we do not know whether actuality or w obtains, then (since actuality and w differ over whether we have hands) we do not know that we have hands;
32.5) Therefore: If there is a world w that contains people in exactly our phenomenal situation who have no hands, then we do not know that we have hands.
33) We do know that we have hands;
34) Therefore: there are no worlds that contain people in exactly our phenomenal situation who have no hands.

Although the metaphysical possibility of zombie scenarios may be controversial, the metaphysical possibility of radical sensory error seems pretty secure. So 34 is unacceptable; so either 33 or one of the 32.x premises has to go.

Skeptics, of course, reject 33. Different non-skeptical theories of knowledge might give different diagnoses about which of the 32. premises is the culprit. Dogmatists deny 32.2 or 32.3. Contextualists typically reject 33 in skeptical contexts, and 32.3 in non-skeptical contexts. Anti-closure theorists deny 32.3 or 32.4. Contrastivists say that 32.4 has a true reading and a false reading, depending on how we fill out the contrast class. What all these responses agree on is that 34) should not be rejected as a result of the skeptical argument. But that is the analogous response to the response the anti-quidditist recommends we make to their epistemological argument concerning quiddities.

Schaffer accordingly concludes that the epistemological argument against quidditism is a failure. An exactly analogous line of reasoning would lead us to external-world skepticism; and any strategy for defending external-world realism can equally be applied to defend quidditism against the quiddity-skeptic.

I want to question this response.

We should be alert to the risk of equivocation over ‘quiddistic knowledge’. Certainly, we can have knowledge that there are quiddities, at least according to many of the theories Schaffer outlines. We can discriminate between worlds with quiddities and worlds without – because metaphysics might just crash in worlds without. And it obviously does not crash in ours. Or at least, that seems to be something like the Lewisian line of thought.

But knowing that there are quiddities (derived, perhaps from knowledge that there have to be quiddities) is not the same thing about having knowledge about individual quiddities – knowledge that some particular property has some particular quiddity. Lewis takes it that the former sort of knowledge is accessible by a priori reasoning, but argues at length against the possibility of the latter sort.

The key point is that the epistemological objection to anti-quidditism needs only the latter sort of ignorance. An epistemic situation vis-a-vis quiddities such that we could know that some quiddity or other was present, but never know which, is an unacceptable one; or at least, many people have been unwilling to follow Lewis in biting this bullet.

Schaffer shows every sign of wanting to capture the stronger, which-is-which sort of knowledge of quiddities. The problem is that several of the anti-skeptical strategies he outlines start to falter at this stage. The reason is that it seems impossible to specify any reliable method for gaining which-is-which knowledge of quiddities.

This is most obvious in the case of reliabilist anti-closure theories. They get round the skeptical challenge to external-world knowledge by positing that sense perception is in fact a reliable way of coming to know what the external world is like. But there is no analogous story for our which-is-which knowledge of quiddities. Scientific evidence is evidence only for the Ramsey sentence (let us assume); and sensory evidence is continuous with scientific evidence. Without some Godelian extra-sensory perception of quiddities, it seems that we cannot maintain that we have a reliable route to which-is-which knowledge of quiddities.

Schaffer mentions, with apparent approval, direct realist theories according to which we become acquainted with quiddities directly in the act of sense perception. This is indeed, I think, the most promising line for a quidditist to take. It was Russell’s view: hence his thesis that we know all and only the quiddities of the properties which we encounter in sense perception. The other quiddities we can know only though description – ‘the quiddity of electronhood’. Perhaps some will find a stable resting place here. But this view invokes an unattractively sharp theory/observation divide, and still renders the vast majority of quiddities unknowable in the desired sense.

In sum: even granting that there are quiddities, the quidditist has given no positive proposal about how we acquire knowledge of the quiddities. Herein lies a significant difference with the case of external-world scepticism – it is part of our current best theory that we have a faculty of sensory experience, which in fact usually provides a reliable method for coming to acquire knowledge that we have hands. We thus have a naturalistic story about how we acquire our knowledge that we have hands. Nothing of the sort is available to the quidditist – or at least, all such stories are highly controversial and Schaffer shows no signs of willing to embrace them.

The project of defending a type of knowledge from the sceptic presupposes that we already have well-grounded knowledge of it in non-sceptical contexts. But that is not the case when it comes to quiddities. Nowhere in our everyday or scientific epistemic projects do we actually acquire knowledge that some particular quiddity is instantiated by some particular property. We can get at them by definite descriptions such as ‘the suchness of red’ – but it is not clear that we are talking about anything other than qualitative character of experiences of red.

My preferred way of putting the epistemological argument against quidditism is that of John Hawthorne. Schaffer quotes Hawthorne: “We don’t need quidditative extras in order to make sense of the world… Why posit from the armchair distinctions that are never needed by science?” This way of putting pmatters makes sure that the burden of proof is in the right place. It is the quidditist, with their infinitely-many extra merely quiddistically different possible worlds, that needs to convince us that their extra ontology should be quantified over in our best total theory. (Call a world a world, or dress it up as primitive ideology – it comes to the same thing, in that it adds to total theoretical complexity.)

It seems that worlds differences between which make no phenomenal difference are not needed in our system-of-the-world. There are two ways that the quidditist could respond. The first is to give a non-trivial epistemological story about how we discriminate between different quiddistic possibilities. This is likely to involve direct perception of some sort, I suspect. The second is to insist that objects must have quiddities, and that we must be able to know that they do, but that we do not need to be able to tell which quiddity is had by which property. That leaves us back with the view of David Lewis and, roughly, of Immanuel Kant. If this is where we end up, we should take another look at our motivations for believing in quiddities in the first place. Would metaphysics really crash without them, as Lewis and Schaffer imply?

There is reason to think that quidditism and haecceitism ought to be given a uniform treatment. Those like Adams, who are full-blown haecceitists and full-blown quidditists, have a consistently-motivated, if implausible, position. An anti-haecceitist anti-quidditist has a remarkably austere ontology – one which could perhaps be exhaustively specified by a space of possible mathematical structures and an ‘is physically realized’ primitive. But the positions in-between – the anti-haecceitist quidditists such as Schaffer and Lewis, and the haecceitist anti-quidditists such as, perhaps, Hawthorne – seem to me to be unstable positions.

One lesson is about how metaphysicians skeptical (in the ‘unconvinced’ sense) should approach epistemological arguments against their targets. Don’t tacitly tively grant the possibility of knowledge about your subject, and then seek to undermine it by indiscriminability arguments. We have plenty of lines of defence against those arguments and – while none of these lines of defence ought to work out to deliver knowledge of entities that do not in fact exist, it may not always be obvious that they fail. We are dealing with some potentially faulty theories of knowledge. Are we so sure that every one of the pictures of knowledge Schaffer adduces are coherent and fully tenable positions? If not, why think that some theory’s ability or inability to refute a particular sceptical argument isn’t due to a problem with the theory?

The subtleties of the way different theories of knowledge cope differently with our epistemic position vis-a-vis particular metaphysical posits are liable to be a distraction. If there really is methodological reason to avoid positing quiddities, we shouldn’t need to check how this pans out with each and every theory, and particularly not with the more controversial ones.

I think what are often called ‘epistemological’ arguments in metaphysics are often better construed as methodological arguments. Very often, we find employed conditionals of the form ‘if theory T were right, there would be a distinct range of facts unknowable by us.’ the conclusion is drawn that this is a regrettable thing to suppose. It is better to argue thus: ‘T undermines its own motivation, for the facts it invokes to explain the phenomena are in fact incapable of doing the job.’ This form of methodological argument for eliminativism, I think, can help us dispense with quiddities, haecceities, absolute simultaneity, absolute actuality, God, and much else besides.

Reply to ‘Quiddistic Knowledge’ – Part 2

Reply to ‘Quiddistic Knowledge’ – Part 1

[Warning – very long post!]


‘Quiddistic Knowledge’ by Jonathan Schaffer (Schaffer [2005]) is a great paper. It is a model of philosophical clarity and vigour. Nonetheless, I think that nearly all of its arguments can be resisted, and that both its main conclusions are false. Here are those conclusions:

1) Necessitarianism about laws has no good motivation, is subject to devastating objections, and ‘collapses into an incoherent mess’ because the (anyway bad) motivations for it pull in different directions.

2) Scepticism about quiddities is no more plausible than scepticism about the external world.

In this post I’ll explain why I think 1) is false; 2) will have to wait for next week.

My strategy for rejecting 1) is to describe a particular necessitarian picture, and to show how it accommodates all of the genuine motivations for necessitarianism while avoiding all of Schaffer’s objections. I do not dispute that other less plausible and less well-motivated versions of necessitarianism are indeed subject to his objections, or that the various versions have not always been clearly distinguished.

The version of necessitarianism that I will defend is, in Schaffer’s terminology, modal necessitarianism. It is the thesis that the actual laws of nature hold with metaphysical necessity; that is, that they are the laws of every metaphysically possible world. Bird [2004] calls this thesis ‘strong necessitarianism’; and it is indeed a stronger – and a more interesting – thesis than the other forms of necessitarianism that have been defended in the literature.

I will appeal to one additional premise in my defence of modal necessitarianism. It is the premise that quantum indeterminism is a part of the actual laws of nature; that is, that the actual laws are indeterministic in roughly the sort of way invoked by quantum mechanics. This assumption, while obviously not in any sense a priori, seems very likely to be true, and it is certainly not in any way in tension with modal necessitarianism. Schaffer will not, of course, be willing to grant this premise: he considers the deterministic Bohmian mechanics to be an ’empirically open possibility’. I agree that the case for quantum indeterminism is not totally conclusive, and that the appearance of indeterminism could arise from a deterministic Bohmian conspiracy, but I do think that the case for indeterminism is very strong. I am accordingly happy to make my conclusions conditional on this premise.  (It should, however, be noted that I only use this premise in the course of responding to one particular argument offered by Schaffer. My other responses are independent of the question of indeterminism.)

So, on to the arguments.

Arguments for modal necessitarianism

Schaffer first seeks to undermine two arguments for necessitarianism – the argument from natural necessity and the argument from sustaining counterfactuals. He correctly notes that – if valid – these arguments count in favour only of modal necessitarianism, the version I seek to defend. So what does Schaffer think is wrong with these two arguments?

The argument from natural necessity:

1) If the relation between properties and their powers is contingent, then like charges might not repel;
2) Like charges must repel;
3) Therefore: the relation between properties and their powers is not contingent.

Schaffer diagnoses this argument as equivocating on the modal strengths of the ‘might not’ and the ‘must’ which appear in premises 1 and 2 respectively. He claims that ‘the “must” of natural necessity in 2) is a restricted necessity, and the “might” in 1) is unrestricted. Hence they are compatible.’

This objection initially seems successful. The response of the modal necessitarian will be to deny that the ‘must’ in 2) is restricted; but that is precisely what is at issue in the debate over necessitarianism, so this denial is dialectically unavailable for use in a suasive argument against contingentism.

The right course, therefore, for the modal necessitarian is not simply to deny that the ‘must’ in 2) is restricted. Rather, the modal necessitarian should deny this on the grounds that a reading of 2) as a restricted necessity makes it inexplicable why we should be interested in claims like 2). If it is genuinely possible, in the unrestricted sense, for like charges to repel, why should we care whether it is impossible in some restricted sense?

Schaffer, presumably, will seek to justify the treatment of 2) as involving a restricted necessity on the grounds that the restriction involved is an interesting and a relevant one. Even if like charges might not repel, that they repel in all the worlds which share our laws is interesting and informative if a restriction to worlds which share our laws is an interesting and relevant restriction.

This justification depends on an adequate contingentist treatment of laws of nature. This, I submit, we do not have. I am essentially convinced by van Fraassen [1989] that all extant contingentist theories of laws fall prey either to the ‘inference problem’ (given knowledge of the laws, why is it rational to draw inferences from them about unknown events) or to the ‘identification problem’ (how do we acquire knowledge of the laws anyway?). Humean theories that have the laws supervene on total history face the identification problem; Humean theories that have the laws supervene on past history face the inference problem; and arguably non-Humean accounts such as those due to Armstrong [1984] face both problems.

I cannot consider possible contingentist lines of response to the inference and identification problems here. My conclusion is conditional: as long as there is no satisfactory contingentist account of the knowability and rational relevance of the laws of nature, there can be no contingentist explanation of why we should invoke a form of necessity restricted to worlds in which the actual laws in the semantics for assertions like 2). And if the best semantics for assertions like 2) does not involv e a modality restricted to worlds in which the actual laws hold, then there is no equivocation between the modalities in 1) and 2), and the argument from natural necessity stands.

The argument from sustaining counterfactuals:

4) If the relation between properties and their powers is contingent, then there is nothing that guarantees that charges repel in any other possible world;
5) In the nearest possible world, like charges repel;
6) Therefore: the relation between properties and their powers is not contingent.

Schaffer’s response to this argument is to deny that the consequent of 4) is in contradiction with 5). He argues that although there is no guarantee that charges repel in some arbitrary possible world, there can be a guarantee that charges repel in the nearest possible world if fixity of laws is partly constitutive of nearness.

The reason this response fails is that it makes it inexplicable why we should be interested in the counterfactual construction thus constituted. It is certainly open to Schaffer to postulate, along with Lewis, that fixity of laws is an important ingredient in closeness for some logical construction counterfactual*. But unless we can explain why the counterfactual* construction is of interest and use to us, we have no reason to think the counterfactual construction that we use in ordinary discourse is the counterfactual* construction. And since, in light of the inference problem and the identification problem, it seems that the contingentist cannot explain why fixity of the laws is an interesting condition, and hence that the contingentist cannot explain why we should be interested in a counterfactual construction which incorporates it into closeness.

Until a contingentist account of laws can be given which solves the inference and identification problems, the fixity-of-laws condition is unmotivated as an element of counterfactual closeness. The contingentist can give no explanation of why the nearest worlds are always worlds which share the laws of the actual world, and consequently can give no explanation of why laws of nature play a role in sustaining counterfactuals.

In contrast, the modal necessitarian has a straightforward explanation of why the nearest worlds are always worlds with the same laws as the actual world; it is that all worlds have the same laws as the actual world. The argument from sustaining counterfactuals is thus sustained.

Arguments against modal necessitarianism

Taking himself to have punctured the two arguments just described, Schaffer moves on to providing arguments against necessitarianism. He gives five such arguments: from modality, from counterfactuals, from propositions, from conceivability, and from recombination. There is a common form to these arguments: in each case, Schaffer argues that the best philosophical theory of the topic in question relies essentially on contingentism. There is also a common form to my replies: in each case, I will argue that the contingentist theory Schaffer provides is inferior, or at least not clearly superior, to the necessitarian alternative.

The argument from modality:

The contingentist analyses natural necessity as a restricted form of necessity. According to Schaffer, only this theory ‘can assimilate natural necessity to the general pattern of restricted necessities found across the historical, epistemic, deontic and conventional necessities.’

The necessitarian response to this argument is straightforward – it is not at all clear that natural necessity should be assimilated to this general pattern. There are obvious differences between natural necessity and the restricted forms of necessity Schaffer mentions;

(Note that the analyses of epistemic and deontic necessity are by no means universally accepted. According to the strategy that analyses epistemic necessity as a restricted necessity, all metaphysically necessary truths are epistemically necessary; but we lack knowledge of many metaphysically necessary truths of mathematics and of logic. And according to the strategy that analyses deontic necessary as a restricted necessity, ought implies can – it will never be the case that we ought to do something that is metaphysically impossible. Such problems cast significant doubt on the analyses of these forms of necessity as restricted.)

And there is independent reason, as argued above, to think that the conception of natural necessity as a restricted necessity is problematic; it is that it renders it inexplicable that we should be interested in the particular restriction which corresponds to natural necessity.

Kit Fine [2002] has pushed an alternative argument for the conclusion that natural necessity is not best seen as a restricted necessity; this argument maintains that the laws of nature themselves should be ascribed natural necessity, but that the ‘restriction strategy’ renders the natural necessity of the laws simply as truth relative to themselves. Since every truth is true relative to itself, Fine argues that this renders the natural necessity of the laws ‘cheap and trivial’. I will not attempt to adjudicate on how successful this argument is; but, if it is found convincing, it provides a further reason not to conceive of natural necessity as a restricted necessity.

The modal necessitarian picture can still accommodate restricted modalities, of course; it is just that the restrictions are placed on a space of possible worlds which only includes worlds in which the actual laws hold.

I conclude that the strategy which analyses natural necessity as a restriction of metaphysical necessity is not clearly superior to the necessitarian alternative, which simply identifies natural necessity with metaphysical necessity. This undermines Schaffer’s argument from modality.

The argument from counterfactuals:

Schaffer argues that the best semantic theory of counterfactuals requires that we recognise possible worlds containing miracles: small violations of the actual laws. Miracles of this sort are selected for by the Lewisian criteria for nearness of possible worlds set out in Lewis [1979]; they allow the closest antecedent world(s) to match the actual world exactly up to some time, and then to smoothly diverge in such a way as to make the antecedent true. Without worlds containing miracles, if the laws are deterministic then the closest world in which an antecedent which contradicts actuality holds would be one in which the initial state of the universe is different. Miracles therefore allow determinism and our intuitive judgments about counterfactuals to co-exist: we do not have to accept that, if the laws are deterministic, then had I scratched my nose just now the state of the universe at the big bang would have been different.

I agree that this argument spells trouble for the modal necessitarian who wants to reconcile deterministic laws with the denial that, were things to differ from actuality in any way, the initial state of the universe would have differed. However, even holding fixed our intuitive judgments about counterfactuals, this is not a natural position for the modal necessitarian to adopt. Rather, the best form of modal necessitarianism will hold that the actual laws involve quantum indeterminism, and hence that the laws of all possible worlds are indeterministic.

The claim that quantum indeterminism holds at all possible worlds allows the modal necessitarian to account for counterfactuals without worlds involving miracles. There are two ways this can be done. Either the modal necessitarian can replace miracles with highly unlikely but still lawful quasi-miraculous quantum fluctuations, and preserve the rest of the Lewisian semantics for counterfactuals unchanged; or the modal necessitarian can adopt a modified semantics which makes no appeal to miracles at all. While I will make no attempt to provide such an alternative semantics here (I venture an initial proposal in my doctoral thesis), and the obstacles which must be overcome in providing one are significant, I am optimistic that this can be done. The main advantage of such an account would be that it would not render true counterfactuals like ‘if I had scratched my nose just now, a highly unlikely quantum event would have had occurred’. But, more modestly, the modal necessitarian can resist the argument from counterfactuals simply by adopting the modified Lewisian semantics which replaces miracles by quantum quasi-miracles.

Of course, relying on quantum indeterminism makes the acceptability of modal necessitarianism hostage to empirical fortune. But I am happy to accept the risk that future developments will reveal that the world is deterministic, since I take that revelation to be extremely unlikely.

The thought that miracles are not needed for the best account of counterfactuals is an unfamiliar one. That is because work on counterfactuals tends to take for granted that, whether or not it they are an open epistemic possibility, deterministic laws are a metaphysical possibility, and we must allow for the truth of our ordinary counterfactual judgements to be preserved in deterministic worlds. That is, work on counterfactuals tends to presuppose contingentism. But this is just a sociological observation; it does not amount to any kind of argument for contingentism.

If modal necessitarianism is correct, then the best semantics for counterfactuals will not involve worlds containing miracles. And if quantum indeterminism is a part of the actual laws, then the best necessitarian semantics for counterfactuals is at least as good as the best contingentist semantics. So Schaffer’s argument from counterfactuals has no force against the modal necessitarian who accepts quantum indeterminism.

The argument from propositions:

Schaffer’s argument from propositions is straightforward. He assumes that propositions can be identified as sets of worlds such that p is true at w, and that there are contentful propositions involving actual properties under alien laws. An example which is meant to underwrite this latter assumption is that ‘a misinformed scientist might believe that like charges attract’.

Here is the way Schaffer formalizes the argument:

7) If the relation between properties and their powers is necessary, then there is no contentful proposition that like charges attract;
8) There is a contentful proposition that like charges attract;
9) Therefore: the relation between properties and their powers is not necessary.

Schaffer suggests that the necessitarian will reject 8). I agree that rejecting 8) is an option for the necessitarian, though I do not think it is the only option – 7) could also be rejected. But I take particular issue with Schaffer’s claim that the only way for a necessitarian to reject 8) is to say that the supposed proposition that like charges attract is the proposition that like schmarges schmattract, where schmarge and charge are distinct properties governed by distinct laws. As Schaffer correctly points out, this response calls for nomic or causal necessitarianism – the modal necessitarian will deny that there is any such property as schmarge or any such possible behaviour as schmattraction. And, as Schaffer correctly argues, the response lacks independent motivation; he also argues that charge and schmarge cannot be epistemic duplicates.

The way that the modal necessitarian should respond to the argument from propositions is by drawing an analogy with mathematical necessities. Consider the analogous argument:

7a) If the relation between numbers and number-theoretic truths is necessary, then there is no contentful proposition that 1+1=3;
8a) There is a contentful proposition that 1+1=3;
9a) Therefore: the relation between numbers and number-theoretic truths is not necessary.

Nobody will be willing to accept this argument. But reasons for rejecting it may differ.

Some will say that there is indeed a contentful proposition that 1+1=3 – it is just that it is necessarily false. Saying this requires a theory of propositions more fine-grained than the propositions-are-sets-of-worlds theory. (I am assuming that the null proposition, true at no world, is not ‘contentful’ in Schaffer’s sense.) Such people will reject 7a). But if 7a) is rejected then there seems no reason to uphold 7); and the argument from propositions fails.

To give it any chance of success, we must bolster the argument from propositions with the assumption that the best theory of propositions is indeed the propositions-are-sets-of-worlds theory. But even given this assumption, the argument can be resisted by the modal necessitarian. For given that theory of propositions, the modal necessitarian will deny 8), for the same reason that proponents of the propositions-as-sets-of-worlds theory must deny 8a).

Schaffer’s motivation for 8) is equally motivation for 8a). A misinformed mathematician might believe that Fermat’s Last Theorem is false. But according to the propositions-as-sets-of-worlds theory, interpreting this mathematician as believing a contentful proposition is impossible. If Schaffer’s argument from propositions were correct, by parity of reasoning we would have to reject the necessity of the truths of mathematics. Since the truths of mathematics are undeniably necessary, the modal necessitarian can accordingly resist the argument from propositions.

The argument from conceivability:

Schaffer’s argument from conceivability maintains that the link between conceivability and possibility is an indispensable part of modal epistemology, and that the modal necessitarian ‘is committed to a complete collapse of any conceivability-possibility link’. He formulates the ‘argument from conceivability’ as follows:

10) If the relation between properties and their powers is necessary, then it is inconceivable that like charges attract;
11) It is conceivable that like charges attract;
12) Therefore: the relation between properties and their powers is not necessary.

Schaffer expects the response from the necessitarian that 11) is false; that when we take ourselves to be conceiving that like charges attract, we are in fact conceiving that like schmarges schmattract. He correctly maintains that this response is not available to the modal necessitarian, but only to the causal or nomic necessitarian, and quite rightly criticizes it as lacking independent motivation.

It is clear, though, that the modal necessitarian ought to respond to the argument from conceivability by rejecting 10). Whether it is conceivable that like charges attract depends on us, and on our conceptual apparatus. Whether the relation between properties and their powers is necessary depends not at all on us, or on our conceptual apparatus, but on the properties and powers themselves. 10) is prima facie highly implausible.

Schaffer does in fact go on to provide additional support for 10). In a footnote he argues that ‘conceivability sees to be our main guide to knowledge of what is possible. This suggests that it is preferable to restrict conceivability rather than reject it outright, on pain of modal skepticism.’ This motivation for the conceivability-possibility link may be persuasive to the contingentist (who faces notorious difficulties when it comes to modal epistemology, and who might thus be driven to embrace a problematic epistemology rather than give up on the project altogether) but it is totally unpersuasive for the modal necessitarian.

According to the modal necessitarian, we gain knowledge of what is (unrestrictedly) possible by engaging in empirical scientific enquiry. No special epistemology is required for modal truths; and the problematic conceivability-possibility link need play no special role. The argument from conceivability is accordingly totally impotent against modal necessitarianism.

The argument from recombination

Schaffer’s final argument against necessitarianism runs as follows:

13) If the relation between properties and their powers is necessary, then some combinations of charge and acceleration would be impossible;
14) All combinations of charge and acceleration are possible;
15) Therefore: the relation between properties and their powers is not necessary.

As should be obvious, the modal necessitarian will deny 14), and maintain that not all combinations of charge and acceleration are possible. This need not be in conflict with the principle of recombination (as Schaffer states it, that if x and y are distinct existences, then there is a possible world with just x, a possible world with just y, and a possible world with x and y). That is because, for the modal necessitarian, charge and acceleration are not distinct existences.

Schaffer anticipates this necessitarian response, and argues that it ‘preserves the letter of recombination, but dashes its spirit.’  The argument for this conclusion involves the supposition that the laws are deterministic. For reasons discussed above, the necessitarian need not allow this supposition; the best form of necessitarianism has it that the laws are necessarily quantum-indeterministic. However, even granting this supposition, Schaffer’s rebuttal of the necessitarian response fails.

The rebuttal starts with the observation that ‘every actual existence is a correlate of a common cause: the Big Bang’ and argues that, if necessitarianism and determinism are true, that entails that ‘zero recombination of actual existences is allowed. The world has become an indivisible Parmenidean unity, the essential outpouring of the initial singularity. This is not a minor restriction on recombination, but rather an unprecedented rejection of any recombination of actual elements.’

This argument fails because it neglects that the necessitarian may allow that the actual initial conditions of the universe are contingent, even if the actual laws are necessary. Schaffer recognises this option in a footnote, but dismisses it as a route to reclaiming recombination: ‘Perhaps sometimes this is possible. But, I suspect, it will still drastically limit recombination of actual elements, far beyond what intuition permits.’ He also complains that it renders recombination an a posteriori matter. I find it difficult to be worried by this line of thought. The necessitarian will typically be unmoved by appeals to intuition about what is possible; their modal epistemology is scientific and a posteriori, not intuition-based and a priori. And in any case, Schaffer has not provided any argument that the restriction on recombination which determinism and modal necessitarianism jointly produce is as extensive as he suspects it is. And finally, for a necessitarian who rejects determinism, the argument has no force. I conclude that the argument from recombination is unpersuasive.


I have replied to Schaffer’s objections to the two arguments for modal necessitarianism that he discusses, and have shown how his arguments against necessitarianism lack any force against a modal necessitarian who take quantum-mechanical indeterminism to be a part of the actual laws. In the process, I have highlighted the advantages of the necessitarian modal epistemology over the contingentist modal epistemology. I conclude that modal necessitarianism remains a tenable and attractive account of the modal status of laws of nature.



Armstrong, D. [1984]. What is a law of nature?

Bird, A. [2004]. ‘Strong necessitarianism: the nomological identity of possible worlds’

Fine, K. [2002]. ‘The Varieties of Necessity’

Lewis, D. [1979]. ‘Counterfactual dependence and time’s arrow’

Schaffer, J. [2005]. ‘Quiddistic Knowledge’.

van Fraassen, B. [1989]. Laws and Symmetry.

Reply to ‘Quiddistic Knowledge’ – Part 1

Indeterminacy of world number in EQM

One of the most puzzling features of the decoherent multiverse of Everettian quantum mechanics is that the number of branches it contains is typically indeterminate. David Wallace describes the situation as follows:

There is no sense in which [chaotic] phenomena lead to a naturally discrete branching process: … while a branching structure can be discerned in such systems it has no natural “grain”. To be sure, by choosing a certain discretisation of (configuration-)space and time, a discrete branching structure will emerge, but a finer or coarser choice would also give branching. And there is no “finest” choice of branching structure: as we fine-grain our decoherent history space, we will eventually reach a point where interference between branches ceases to be negligible, but there is no precise point where this occurs. As such, the question “how many branches are there?” does not, ultimately, make sense.
(Wallace, in Saunders et. al, 2010, Many Worlds?, p.67-68)


Advocates of EQM use a variety of terminology to express this feature of the theory: ascriptions of branch number are ‘interest-relative’ (Saunders), are ‘arbitrary conventions’ (Saunders), are subject to ‘some indeterminacy’ (Wallace), are ‘not well-defined’ (Greaves) or ‘presuppose a piece of structure which is not present in the theory’ (Greaves). The question ‘how many branches?’ is said to be ‘a non-question’ (Wallace), with ‘no good answer’ (Saunders) or simply with ‘no answer’ at all (Wallace). Branch number ‘has no categorical physical significance; it is not part of what is really there.’ (Saunders)

This treatment of an apparently meaningful question as lacking any good answer is reminiscent of Carnapian, Wittgensteinian and neo-verificationist rejections of metaphysics. One of the best responses to these critics of metaphysics is the appeal to classical logic. If mountains have no boundaries, but it is not the case that everything overlaps with every mountain, it follows that there are no mountains. And if there is no positive number N such that N is the number of branches, it follows that there are no branches. Of course, critics of metaphysics do not deny that there are mountains; and Wallace, Greaves and Saunders do not deny that there is branching. Our puzzle, then, is how to make sense of how it can be that there is branching despite there being no positive N such that N is the number of branches.

The key to resolving the puzzle, I think, is to grant that there is something defective about the question ‘how many branches?’ but to deny that this requires it to be nonsensical or to lack a true answer. We can instead say that the number of branches is indeterminate: that there is some N such that N is the number of branches, but there is no N such that N is determinately the number of branches. The question ‘how many branches?’ is defective because it has no determinate answer, even though (determinately) it is meaningful and thus (determinately) does have an answer. This sort of response is hinted at by Wallace’s talk of ‘indeterminacy’ and by Greaves’ talk of ‘vagueness’, but it has not to my knowledge anywhere been made explicit; it also appears to be in tension with phrases like ‘non-question’ that Wallace and Greaves use elsewhere. I therefore propose it not as an interpretation of Greaves, Wallace and Saunders but as a friendly amendment to their strategy, designed to make decoherence-based EQM more palatable to the mainstream metaphysician.

The bivalent indeterminacy explanation of the defectiveness of questions about branch number is only satisfying if we can give an appropriate account of the determinacy operator. The contemporary literature on vagueness provides various frameworks for thinking about determinacy and indeterminacy that allow us to retain classical logic. I will discuss only the most prominent approaches, each of which makes use of the notion of admissible precisifications. A precisification is, as the name suggests, a way of making precise a vague expression. The characterization of admissibility varies from theory to theory.

Epistemicist approaches to vagueness say that some precisification is picked out by patterns in our community’s global linguistic usage as objectively the correct precisification, but that it is typically unknowable (for a distinctive sort of reason) which is the correct one. According to epistemicism, admissibility is epistemic in nature, and indeterminacy is a kind of ineliminable ignorance, the ineliminability of which derives from our lack of knowledge of the semantic values of the words we use.

Supervaluationist approaches  in contrast hold that admissibility is semantic in nature: admissible interpretations are those whose correctness is not ruled out by the various conventions we have (explicitly or implicitly) laid down to govern our language. For supervaluationists, indeterminacy derives from semantic indecision. Fine, Lewis, and McGee & McLaughlin give influential (though differing) versions of supervaluationism. I prefer McGee & McLaughlin’s version, since (like epistemicism) it allows for the preservation of bivalence.

Let’s see how these two views apply to the indeterminacy of branch number present in decoherence-only versions of EQM. We treat ‘the number of branches’ as a vague expression, and consider various ways of making it precise. Each of these ways corresponds to some particular coarse-graining of the decoherence-based decomposition of the quantum state into components. The coarse-graining we choose is subject to constraints: if it is too fine-grained, then interactions between the components will become non-trivial and decoherence will be lost. If we make it too coarse-grained, there will cease to be any branching at all. And no coarse-graining produces non-integral values for branch number. We can therefore place two extremely minimal constraints on admissible precisifications for ‘the number of branches’ in any episode of branching: every admissible precisification must correspond to some natural number of branches, and this natural number must be greater than 1.

The constraints on precisifications described in the previous paragraph ensure the right results in the case of questions about world number. According to both epistemicism and supervaluationism, it makes perfect sense to say that there is some N such that N is the number of branches, but it is indeterminate which N is the number of branches. Despite the indeterminacy of branch number, it remains determinately true that there is branching.

As it happens, I expect that most Everettians, if they accept my account of branch number indeterminacy in terms of a precisification-based theory of vagueness, would prefer a supervaluationist reading of the indeterminacy operator. Epistemicism is widely considered implausible even in classical cases of vagueness – it is simply hard to believe in all those unknowable facts of the matter – and I see no particular reason why epistemicism should be any more or less plausible in the case of branch number than it is in classical cases of vagueness.

The claim that branch number is vague could be resisted in a number of ways. For example, van Fraassen suggests as a constraint on the usefulness of vague predicates that there be clear cases and clear counter-cases. While there are clear counter-cases for ‘is the number of branches’ (for example zero) there are no clear cases – no numbers N such that determinately N is the number of branches. This seems to me an overly restrictive conception of vagueness, however. For example, consider the predicate ‘is the smallest large number’. We might think that zero is a clear counter-case: it is as far from large as anything could be. But there are no clear cases – no number is such that it is clear that it is the smallest large number. Van Fraassen’s suggested characterization of vagueness accordingly looks inadequate. Vagueness should instead be thought of as the phenomenon of borderline cases; that characterization (which is favoured by Williamson) allows branch number to count as indeterminate.

Wallace prefers to say that it is not vague but arbitrary how many worlds there are. However, this seems to have highly implausible consequences. Arbitrariness is a property of choices, or of decisions: if a choice is arbitrary, then it is up to us how we make it. But it is not, in any coherent sense, up to us how many worlds there are. (It may in some sense be up to us what we mean by ‘world’: but given any choice of what is to count as a world, it is not up to us how many such objects there are.) Of course, it makes sense to say that for practical purposes we must pick some coarse-graining to analyse a particular physical interaction, and to say that the choice we actually make in any given situation is arbitrary; but this is perfectly compatible with (and indeed can potentially be explained by) the thesis that world number is indeterminate. Knowing that something is indeterminate but – for pragmatic reasons – picking a single precisification to work with invariably involves making an arbitrary choice.

The claim that world number is indeterminate an unsettling one. Quineans might well worry how we give adequate identity conditions to entities indeterminate in number; and prima facie such entities might give rise to a version of Evans’  argument against vague identity. There are many reasons of this general sort to doubt that vagueness is possible in fundamental reality; the Everettian, though, has a multi-purpose response. It is that, in EQM, worlds (like people and minds) are not fundamental; rather, they are emergent phenomena. Saunders and Zurek have emphasized that there need be no vagueness at the more fundamental level, the level of the quantum state. And Wallace  argues forcefully that approximation and indeterminacy are characteristic features of all emergence in the non-fundamental sciences. According to this line of thought, EQM is altogether unexceptional in using vague terms in its explanations.

Indeterminacy of world number in EQM

Directions of explanation and of analysis

The other day U.Melbourne hosted an interesting talk by John Maier. In it he argued that a variety of ideas within philosophy of mind and action can be analysed in terms of the single primitive ‘X is an option for Y’. I’m just going to pick one strand out of the talk because it set me thinking about the nature of analysis. As an example, consider Maier’s suggested analysis of ability:

(1) S has the ability to A iff S normally, in virtue of her intrinsic properties, has A as an option.

Now, this biconditional looks true to me – indeed, necessarily true. Having an ability intuitively does entail there being something about you that is intimately related to the possible doing of A. When the ability to do some action A is lacking, it is (at least ‘normally’) the case that the intrinsic state of the agent does not meet a certain A-enabling condition.

True, perhaps, even necessarily true; but it doesn’t look like an analysis should. Several questioners expressed the feeling that ‘has as an option’ ought not to be analytically fundamental, but it wasn’t easy to pin down exactly why. Here’s my suggestion for why (1) is not a good candidate for an analysis of ‘ability’, regardless of its truth: it’s that the analysandum seems to be explanatorily prior to the analysans. That is, it seems that the having of an ability to A explains why S normally has A among her options, rather than that the presence of A amongst S’s options explains why S has the ability.

Maier wasn’t sure about this intuition about explanation, but he suggested that even if it is correct then that need not be a reason to abandon the analysis. That is, he thought that the direction of explanation over a bi-conditional need not match the order of analysis.

There is something disarming about this response. It reflects an unexpectedly modest conception of analysis. But there is also something unsettling about it. If explanatory considerations are orthogonal, what reasons could tell in favour of the claim that a given biconditional is or is not an analysis?

One response would be to point to the hierarchical nature of analysis: since the relation of analysis is non-symmetric and transitive, successful analyses will form a tree, with the most primitive notions at the base. Thus there might turn out to be lots of necessarily true biconditionals with clauses ascribing options to some subject S on one side and simple action-theoretic claims on the other, and few true biconditionals connecting (say) claims about capacities with simple action-theoretic claims. Recognizing that this situation obtains would give us reason to take having as an option as analytically basic. That, at least, is how I can best reconstruct Maier’s picture.

The problem with this response seems to be that it undermines the motivation for seeking analyses. Call an analysis where the obtaining of the analysans does not explain why the analysandum obtains a ‘mere analysis’. If some biconditional is a mere analysis, then what makes it more interesting than a necessarily true biconditional which is not an analysis at all?

A more promising defence of Maier’s biconditional as an analysis might be to grant that the direction of explanation and the direction of analysis coincide in all cases, but to argue that the normal availability of the option to S in virtue of one’s intrinsic properties does explain the ability to S. We might say that since to have the ability to S just is to normally have S as an option in virtue of one’s intrinsic properties, of course the latter explains the former – it implies it by subsumption under a general rule, which is (according to the D-N model) a form of explanation. This response will work in every case, as far as I can see.

What’s going on here is that, granting the truth of the analysis, the analysans obtaining is explaining how the analysandum obtains. But (and this is to point to one of the limitations of the D-N model) the analysans is not explaining why the analysandum obtains. So here is my half-baked proposed analysis of analysis*:

X is an analysis of Y iff 1) (necessarily) Y iff X and 2) X explains why Y.

* This reminds me of the questions following David Chalmers’ first Locke lecture earlier this year, where an obstreperous Carnapian seemed to think he’d refuted Chalmers totally by posing the question: ‘Ah, but what’s your analysis of analysis?’

– call an analysis where the obtaining of the analysans does not explain the obtaining of the analysandum a ‘mere analysis’
Directions of explanation and of analysis

‘Interpretations’ of quantum mechanics

I’m going to alternate properly new posts with short extracts from my thesis. So here’s the first such extract, on why Everettian quantum mechanics is not just one ‘interpretation’ among many:

Often EQM is presented as an interpretation of quantum mechanics, for purposes of comparison with other ‘interpretations’; examples usually given are pilot-wave theory, the ‘Copenhagen interpretation’, and dynamical collapse theories such as that of Ghirardi, Rimini and Weber (Ghirardi, Rimini, and Weber 1986). I think that that this way of conceiving the foundational issues, where the equations of quantum mechanics are common to the different approaches and they are distinguished only by a quasi-metaphysical layer placed on top of the equations, is badly misleading.

The main reason for this is that that most ‘interpretations’ impose extra dynamical structure of their own onto the basic skeleton of the quantum mechanical formalism. For example, in addition to the unitary evolution of the quantum state, (non-relativistic) pilot-wave theory postulates point-like particles with definite trajectories, and a ‘guidance equation’ which tells the particles how to move based on the structure of the state.  As such, it is strictly speaking not an interpretation of quantum mechanics at all; it is an autonomous theory which happens to share much of the theoretical structure of quantum mechanics. (In recognition of this point, the name ‘Bohmian mechanics’ is often adopted by enthusiasts of pilot-wave theory.)

The Copenhagen interpretation, as preached by Bohr (Bohr 1934), (Bohr 1958), (Bohr 1963) is different from these explicit modifications of quantum mechanics; it takes the equations of quantum mechanics as they stand but accounts for their link with our experience of the macroscopic world in an non-standard way. In this respect it has much in common with EQM. The difference is that EQM is naturally thought of as a realist theory; it interprets the quantum state as a description of the physical properties of a system. The Copenhagen interpretation (at least as it is ordinarily understood[1]) is an instrumentalist theory; it tells us how to use quantum mechanics to predict behaviour in the macroscopic world (antecedently understood in terms of the theories and concepts of classical physics), but rejects any of talk of ‘correspondence’ between the quantum-mechanical description and any macroscopic reality. It is thus natural to group EQM and the Copenhagen interpretation together, treating the former as a form of scientific realism about quantum mechanics and the latter as a form of anti-realism[2] about quantum mechanics; we can then contrast them both with those theories which explicitly modify quantum mechanics.

The reasons for preferring EQM to Copenhagen are essentially instances of our more general reasons for preferring scientific realism to instrumentalism. EQM provides us with a picture of fundamental reality, albeit a strange one, while the Copenhagen interpretation rejects any such demands. According to the Copenhagen interpretation, questions about the fundamental nature of microscopic reality without reference to experimental context are simply misguided – there can be no informative answer to such questions.

EQM and the Copenhagen interpretation do not just differ in the metaphysical picture they give us. EQM can form of a coherent and unified theory of nature in a way in which the Copenhagen interpretation cannot. EQM is a theory of closed systems, and thus can be applied to the entire universe, unlike the Copenhagen interpretation which restricts quantum mechanics to describing the behaviour of quantum systems embedded in classical environments. This is reflected in the way that quantum cosmology invariably proceeds, implicitly or explicitly, with EQM as a background assumption.

[1] Bohr himself can be interpreted rather differently, as a realist who took classical mechanics to be fundamental. See (Saunders 2005).

[2] By ‘anti-realism’ here I mean something stronger than van Fraassen’s constructive empiricism (Van Fraassen 1980), which takes physical theories about unobservable entities literally but advises a restrained epistemic stance towards them. The kind of anti-realism about the microscopic exemplified by the Copenhagen interpretation is semantic rather than epistemic in nature.

‘Interpretations’ of quantum mechanics