Abstract
Attention has been paid to the prospects of the Best System Analysis (BSA) for yielding high-level chances, including statistical mechanical and special science chances. But a foundational worry about the BSA lurks: there don’t appear to be uniquely appropriate measures of the degree to which a system exhibits the theoretical virtues of simplicity, strength, and fit, nor a uniquely appropriate way of balancing the virtues in determining a best system. I argue that there’s a set systems for our world that are tied-for-best given the limits of precision of the notions of simplicity, strength, fit, and balance. Some of these systems entail different high-level chances. I argue that the Best System analyst should conclude that (some of) the chances for our world are imprecise.
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- 1.
The present paper elaborates an idea that my collaborators and I presented in Dardashti et al. (2014). However it goes significantly beyond that article in to principal respects. First, it provides a more detailed explanation of why we can expect systems among the tied-for-best to entail differing high-level probabilities. Second, it provides an explanation of why sets of probabilities entailed by the tied-for-best systems deserve to be called imprecise chances and an explanation of how imprecise chance guides rational credence.
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- 3.
Where FD is quantum rather than classical, the uniform distribution is instead over the set of quantum states compatible with PH.
- 4.
Lewis (1983, 367–368) takes only systems formulated in perfectly natural kind terms to be candidate Best Systems. Yet, given that naturalness admits of degrees (Lewis 1983, 368), a more reasonable view is that naturalness of the predicates that a system employs is a theoretical virtue. If an axiom system achieves great simplicity, strength, and fit by employing not-too-unnatural predicates like ‘low entropy’—as the Mentaculus does—then it’s a plausible best system. As with the other theoretical virtues, there plausibly isn’t a uniquely correct metric of naturalness.
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I make no claim that the systems for our world that I’ll argue are tied-for-best are ‘very different’. All that matters is that some of them entail divergent probabilities.
- 6.
Or must be expressed in highly unnatural vocabulary (a theoretical disvirtue) to make it simple—I’ll drop this rider in what follows.
- 7.
If the tied systems all entail that the probability for entropy increase given a non-equilibrium macro-state of an isolated system is very high (as they must to be well-fitting), but don’t agree on a precise real value, we might say there’s a qualitative probabilistic version of SLT that qualifies as a genuine (BSA) law. Nothing I say about chances turns upon this.
- 8.
The possibility of imprecise chances, especially in the context of the BSA, is noted in passing by Hájek (2003a) (cf. Hájek and Smithson 2012, 39), who suggests that the Best System analyst might take QM chances to be imprecise if there are ties between systems that entail different QM probabilities. I’ve been arguing that competing axiomatizations of SM provide a strong motivation for thinking that there’s a tie. It’s plausible, given the imprecision of simplicity, strength, fit, and balance, that among the tied-for-best may be ones entailing differing QM probabilities. But I shan’t argue this here.
- 9.
A referee put the following interesting concern to me:
If there are multiple “best systems”, it seems possible that different best-systems might posit different ontologies. If so, then the set/algebra of chance propositions might differ from one best system to the next. In contrast, imprecise probabilities are typically defined on the same algebra.
To respond, we should begin by noting that, on the BSA, the question concerning the true ontology of the world is prior to that concerning what is the best system (or which are the best systems). As Lewis puts it, candidate systems “must be entirely true” (Lewis 1983, 367) in what they say about how the world is. It must also be the case that “the primitive vocabulary that appears in the axioms refer[s] only to perfectly natural properties” (Lewis 1983, 367–368). Such requirements would hardly make sense unless the true ontology were metaphysically prior to the best system (specifically: unless the true ontology determined what counted as a candidate best system, rather than the true ontology being whatever the best system implies it is). Even if we relax the requirement that the axioms of candidate systems contain only perfectly natural kind terms (see Footnote 4), and allow them to contain imperfectly natural kind terms (sensu Lewis 1983, 347), we must suppose a prior ontology of imperfectly natural kinds.
Nevertheless it’s true that, given an algebra of propositions concerning the instantiation of perfectly and imperfectly natural properties throughout space-time, members of the set of tied-for-best systems may entail probability distributions that are defined merely on (strict) sub-algebras of this algebra. (Or—and this is a possibility to which the same referee alerted me—it may be that a single system entails two or more distinct chance functions, defined on disjoint sub-algebras of this algebra, in which case it may nevertheless be that the union of these sub-algebras is still a strict sub-algebra of the whole algebra.) This will be so if entailing a probability distribution over the whole algebra requires a complexity that is not offset by the resulting increase in strength. If this is the case, then I think the correct thing to say is that the (imprecise) chance associated with each element in the algebra is the set of probabilities that members of the tied-for-best systems assign to that element. If none of the systems assign a probability to that element, its chance is undefined. If only one does (or if more than one do, but they each assign the same probability), then its chance is precise. If systems assign it differing probabilities, then its chance is imprecise.
- 10.
Joyce (2005, 171) argues that imprecise credences “are the proper response to unspecific evidence”. I believe they’re also the proper response to specific evidence about imprecise chances. Elga (2010) and White (2010) argue that imprecise credences are irrational. Joyce (2010) and Hart and Titelbaum (2015) provide responses that, in my view, are compelling.
- 11.
After the Latin cadentia from which the word ‘chance’ derives.
- 12.
A name inspired by the fact that imprecise probabilities are sometimes described as ‘mushy’.
- 13.
I think MushyP is intuitive. But an ‘epistemic utility’ justification of it might be attempted, analogous to Pettigrew’s (2012, 2013) attempts to give an epistemic utility justification of the Principal Principle. Doing this would require taking \(\boldsymbol{ch}\) to constitute a ‘vindicated’ set of probability functions, and showing that, according to some reasonable measure of distance between two sets of probability functions, obeying MushyP minimizes ‘inaccuracy’ (understood as the distance between \(\boldsymbol{ch}\) and \(\boldsymbol{cr}\)). Carr (2016) has proposed a reasonable inaccuracy measure that she has shown to vindicate White’s Chance Grounding Thesis (CGT): that “one’s spread of credence should cover the range of possible chance hypotheses left open by your evidence” (White 2010, 174). Carr’s argument involves taking the set of vindicated functions to be those that aren’t excluded from being the chance function by one’s total evidence. If we instead take the set of vindicated functions to be \(\boldsymbol{ch}\), Carr’s demonstration appears to go through just as well. Where the elements of \(\boldsymbol{ch}\) are interpreted as the probability functions associated with the tied-for-best systems, there’s a natural justification for treating these as a set of ‘vindicated’ functions because they’re constrained (by the requirement of fit) to be close to the actual pattern of outcomes.
- 14.
An alternative approach would be to attempt to extend epistemic utility arguments (Hicks 2016) for calibrating one’s credences to the probability function associated with the best system to the case where there’s no unique best system and associated probability function.
- 15.
Plausibly it would also maximize epistemic utility in the sense of minimizing the distance of one’s credences from the frequency function freq(X | Y ) which, for all propositions X and Y (within the domain), yields the relative frequency of X given Y (cf. Hicks 2016), or the ‘truth’ function which, for all propositions X yields P(X | ⊤) = 1 if X is true and P(X | ⊤) = 0 if X is false (cf. Joyce 1998).
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Acknowledgements
For helpful discussion, I’d like to thank Seamus Bradley, Radin Dardashti, Matthais Frisch, Karim Thébault, and audiences at EPSA15, the universities of Leeds and Cambridge, LSE, and the Institute of Philosophy.
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Fenton-Glynn, L. (2017). Imprecise Best System Chances. In: Massimi, M., Romeijn, JW., Schurz, G. (eds) EPSA15 Selected Papers. European Studies in Philosophy of Science, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-53730-6_24
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