Abstract
This paper is a response to Lewis’ ‘Humean Supervenience Debugged’ (1994). Lewis was in the business of defending Humean Supervenience, and the project seemed successful until the case of chance. Lewis thus originally named chance the ‘big bad bug’ for Humean Supervenience until the aforementioned paper in which he claims victory. Here I argue that he was unsuccessful and that Humean Supervenience remains bugged by chance. I will show how this bug remains due to a misdiagnosis of where the problem lies with regard to undermining. First, I define Humean Supervenience and chance, and state the bug in its original form, then secondly I describe Lewis’ attempt to remove the bug. Thirdly, I explain why the bug persists, despite Lewis’ efforts, and show the real source of the undermining problem to be due to the circularity of Humean Supervenient style accounts of chance. Finally I describe the situation this leaves chance in, and show how the incompatibility of chance and Humean Supervenience is evidence for the nonexistence of chance. I conclude that it is the circularity of the formation of Humean Supervenient laws of chance which continue to bug Humean Supervenience, leaving it untenable and resulting in little chance for chance.
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Notes
As I am referring repeatedly to this text, future references will be given by page number only.
I will be assuming for the sake of argument that extremal chances will mean the occurrence or nonoccurence of an event, such that 0 chance is to not occur, and having chance 1 is to occur. This has been worried over by those such as Mellor (2006). For example it may not be that having 0 chance means nonoccurence in the infinite case—in a finite sequence of events each of nonzero (say ½) chance of occurrence, the chance of them all occurring is greater than zero, but in an infinite sequence of such events it turns out that ½ (for instance) to the power of infinity is zero. But this may not mean that infinite sequences cannot occur. This will not matter—see note 5.
This is a later and more important formulation called ‘the Principal Principle Reformulated’ (Lewis (1980) p97) which I believe to be a clearer version to work with than the equivalent original formulation: C(A/XE) = x where C is the initial reasonable credence, A is the thing in question, X is the proposition that the chance of A is x, and E is the admissible evidence.
Lewis (1980) p98
For the notion of admissibility to have meaning and not be redundant there must be some values that turn out to entail the occurrence/nonoccurrence of A (and the obvious candidates for these values are 1 and 0). Without these extremal values having such significance there will never be inadmissible applications of the PP, as if values 0 and 1 do not entail non-occurrence and occurrence then the principles never give information about what happens in the world and as such will always be admissible. This is very problematic, and so I will be treating chances of values 0 and 1 to have their standard entailment of nonoccurrence and occurrence.
This admissibility will be discussed further later in this paper, to the effect that HtwTw may turn out to be regarded as inadmissible evidence relative to A, otherwise it will provide no information about A at all and will thus be admissible at the cost of being useless and uninformative.
Lewis (1986) pxiv
Lewis understands “it is because the chancemaking pattern lies partly in the future that we have some chance of getting a future that would undermine present chances. This problem would go away if we could assume that the chancemaking pattern lay entirely in the past” (p483) but Lewis rejects this method as it would make Tw necessary which he argues is false/problematic.
Lewis notes: “Different alternative futures would determine different present chances… It’s not that if this future came about, the truth about the present would change retrospectively. Rather it would never have been what it actually is, and would always have been different.” (p482–3)
This diagram makes clear a parallel circularity problem to that of the paradox of time travel: If I were to travel back in time, I could do something back in the time I travelled to which would then result in a different future to that of the one I had travelled back from. The classic example of killing my grandfather makes the paradox vivid—if now, in 2014, I travelled back to 1959 (the year before my mother was born) and killed my grandfather before he and my grandmother had conceived my mother then my mother would not be born in order to give birth to me, and thus there would be no year 2014 in which I exist in order to travel back to 1959 to kill my grandfather. So similarly here is a circularity that entails a form of undermining.
Thau notes: “It sounds a bit odd to say that the theory of chance [Tw] might rule out certain futures [F*] which are nonetheless probabilistically possible, but only a bit odd.” (p495) This is along the same lines of Lewis’ dismissal of undermining being merely (and not worse than) peculiar. Tw rules out F* by being incompatible with it, yet gives it a positive chance. But to be given chance 0 may not be to be ruled out (see note 2). However the contradiction remains as the LHS will still equal 0 (despite whether this means nonoccurence) and the RHS will not equal 0. Furthermore F* will need to be ruled out as not occurring if Tw is true, as that will mean F* cannot occur (because if Tw is true then F* didn’t occur), and chance 0 will equal being ruled out.
A contradiction can also arise completely on the same side (LHS) of the PP. The LHS can be argued to equal 0 (F* being incompatible with Tw) yet also Tw combined with Htw is meant to select the conditional giving F* a positive chance and hence equals more than and not equal to 0. This neatly summarises the circularity in Humean Supervenient laws of chance I am arguing for.
He states that the PP will still be our working definition of chance, and that the NP will approximate the PP to a near enough degree, so that even though the NP is strictly the true principle the PP will approximate this closely enough to still be classed useful and a priori.
Tw may be compatible with many mosaics, as the mapping from the arrangement of particular matters of fact to the laws need not be one-one, but could be many-one. So it could be that not only will the actual mosaic be somehow stored or compatible with Tw but also any mosaic that doesn’t contain in it an undermining event which would produce a different T*w. See note 18.
This would make Tw by itself inadmissible evidence, as to be admissible requires that it doesn’t say anything about how things actually turn out. If Tw were to be inadmissible in this way, then both the PP and the NP would never apply and would be regarded useless or incorrect. Lewis to some extent recognizes this aspect of HS and the inadmissibility of HtwTw relative to undermining futures (in that HtwTw rules out F* by being incompatible with it) and concedes to allow for degrees of admissibility so that HtwTw can qualify as being an acceptable degree of admissible. It turns out though, that due to HS, Tw and HtwTw are always admissible by saying nothing at all. Interestingly then the PP and NP are useless either way: if HtwTw is admissible by saying nothing then they are completely uninformative, and if HtwTw is inadmissible then they cannot ever apply.
Thanks to John Horden for pointing out that the fact that the future is chancy and needs to be formed in order to determine the laws Tw may not pose a problem for Lewis who holds the difficult combination of eternalism with realism about chance. Under eternalism, the future will be as formed as the past, and so the complete subvenient base will already somehow be there ready to determine Tw, even when the future is chancy and hasn’t happened yet. There seems to be a tension between this eternalist block universe view and its compatibility with nonextremal objective (nonepistemic) chance, but I shall consider it here as a possible escape for Lewis. Lewis would say the whole mosaic is out there, extending into the unforeseen future, regardless of what the chances are. So Tw would be determinately true even though it has some present chance of being false (by the future turning out differently having a positive chance). This is because being determinately true at a time and being nomologically determined to be true at that time (having chance 1) are not the same, and as such Tw needn’t have chance 1 to be formed. Yet for the purposes of this paper, my target is not Lewis specifically, but rather any advocate of Humean chance. So unless every member of this target also endorses eternalism, there remains a problem of Tw being undefined due to being formed in part by an unformed chancy future.
Thanks to Paul Noordhof who noted there needn’t be a one-one mapping from arrangement-theory. It could be that multiple/variable realisation is true, such that there could be many different arrangements that all determine the same theory Tw. This implies that ‘if A1 then Tw, or if A2 then Tw’ but not ‘if Tw then A1’. Tw doesn’t require that one particular arrangement occurred over others supporting (roughly speaking) the same frequencies. To illustrate, suppose that all future coin tosses have a 50 % chance of landing heads; and that this is just because throughout history (past, present and future), half of the coin tosses land heads. Suppose there are 100 coin tosses in the whole of history and that there have been 50 coin tosses so far, and 25 have landed heads. So there are 50 coin tosses yet to come, and 25 will land heads—otherwise the relevant chances would be different. But clearly there are many different ways in which there can be 25 heads out of 50 tosses. There could first be 25 heads, and then later 25 tails, or vice versa, or alternating heads and tails, and so on. The order doesn’t affect the frequency, and by hypothesis the frequency determines the relevant chances. And even if actual chances have a (more sophisticated) best-system analysis of the sort advocated by Lewis, it still seems that, analogously, many different (but relevantly similar) patterns in the Humean mosaic would determine the very same chances that we have in the actual world. This gives Tw a certain degree of independence, loosening my circularity claim somewhat. However given that there still are arrangements (presumably the majority of the possible arrangements) that will be undermining for Tw it still holds that Tw has a chance of being incorrect. The chance of Tw is now just the chance that the future wont be undermining. It is not the case that every nonactual future would determine the truth of a different theory of chance, but there are still some (again presumably most) that would. Given that they have a positive chance, and what gives them this chance therefore has a chance of being incorrect, the circularity from the undermining problem remains, as if there’s a chance of undermining in a theory then that theory has a chance of being incorrect.
Perhaps ‘govern’ is inaccurate, as a strict Humean/Lewisian does not regard laws as really governing events but rather merely reflecting them. Under HS in a chancy world w the chance of A at t in w will be x simply because the frequency of A-type events (in the circumstances of the type which hold at t in w) are those that would make the Best System Analysis of Laws give chance x to A-type events as part of the best (simple, informative and most fitting) description of the patterns of events which hold in the world w. So really the laws just reflect and describe the frequencies. Nevertheless, since the frequencies have an overall chance of coming about, the laws themselves have a chance to correctly reflect those frequencies or not, and so we must question how we understand these chances that laws have, and I propose it is either circular or regressive.
I argue that undermining is much more severe than mere peculiarity, and occurs due to the circularity in all contingent Humean Supervenient accounts of chance, which the NP can’t resolve.
I would like to thank Stephen Barker, Paul Noordhof, Barry Lee, John Horden and Dave Ingram for useful discussion and comments relating to earlier drafts of this paper, and to the audiences of the Graduate conference at the University of Stockholm and the Mind and Reason research group at the University of York at which I presented versions of this paper.
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Finn, S. Humean Supervenience Rebugged. Philosophia 42, 959–970 (2014). https://doi.org/10.1007/s11406-014-9545-7
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DOI: https://doi.org/10.1007/s11406-014-9545-7