Abstract
I articulate a Global Best-System Account (GBSA) of laws of nature along broadly Mill–Ramsey–Lewis lines. The guiding idea is that the job of laws is to capture real patterns across time—where a pattern is real if it allows to compress information about matters of particular fact. The GBSA’s key ingredient is a definition of ‘best system’ in terms of a ranking method that meets a number of desiderata: it is rigorously defined; it outputs the ranking based on the candidate systems’ epistemic virtues; it is objective, in that the output does not depend on one’s choice of basic concepts; it avoids two trivialization results (by Lewis and Arrow-Okasha); and it does not appeal to naturalness, or other metaphysical notions in that vicinity. Although the GBSA is independent of the controversial thesis of Humean Supervenience, it satisfies the weaker thesis that laws of nature supervene on the total physical state of the world. Moreover, the resulting notion of lawhood is hyperintensional, unlike the neighboring notion of nomological necessity. Finally, GBSA-laws do not track naturalness, contra the Lewisian orthodoxy.
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Notes
Universal generalizations are sentences of the form \(\forall x\phi \). One might complain that real-world laws of nature are typically formulated as equations, and so do not have the form \(\forall x\phi \). The objection is answered by pointing out that ‘form’ refers to the logical form of a sentence, as opposed to its surface syntactical form. For example, \(F=ma\) is a generalization in the intended sense insofar as its logical form is (roughly) ‘every object with mass is such that its force equals its mass times its acceleration’. I will be assuming throughout that the logical form of a sentence is revealed by a suitable regimentation. For a defense of the view that laws of nature are generalizations see (Friend, 2016).
A set of sentences \(\Gamma \) is said to strictly imply ‘p’ if, necessarily, if \(\Gamma \) then p (where ‘if...then...’ is a material conditional), cf. Lewis, (Lewis et al., 1959).
“Each of the notions involved in the BSA [...] require more elucidation than Lewis (or anyone else) has given them” Loewer (2007), p. 319.
A binary relation R is said to be complete if for any x, y, either Rxy or Ryx; it is said to be transitive if for any x, y, z, if Rxy and Ryz then Rxz.
Lewis understands strength measure-theoretically: the stronger a system, the smaller the coarse-grained proposition (set of worlds) it expresses, cf. Loewer (1996, p. 109) (2007, p. 319). Note that, since there are infinitely many measures that can be defined on logical space, Lewis-strength presupposes the existence of a privileged measure that plays the required nomological role, cf. Hicks (2018 , p. 986).
An alternative formulation of the BSA which also appeals to natural properties is defended in Bhogal and Perry (2017).
See van Fraassen (1989), Loewer (2007) for a rejection of naturalness in the particular case of lawhood. See Hofweber (2009) for a skeptical challenge to the intelligibility of naturalness and other special purpose metaphysical primitives. A different kind of skepticism, which questions the epistemic virtuousness of naturalness, rather than its intelligibility, is offered in Dasgupta (2018), and addressed in Sider (forthcoming).
By ‘cognate notions’ I mean the ones that are in the tight circle of interdefinability to which naturalness belongs, such as duplication and intrinsicality.
The trade-off of fit and simplicity is front and center of the Minimum Description Length (MDL) method of inductive inference (Grünwald, 2007; Rissanen, 1978). MDL is based on the information-theoretic notion of Kolmogorov-Chaitin complexity, which is a measure of the orderliness of a set of data in terms of the shortest program that will output it (Chaitin, 1966; Kolmogorov, 1965). Alternative formal approaches to inductive inference include Bayesian methods, such as the Solomonoff (1964) theory of inductive inference, the Akaike (1973) information criterion, as well as the method of minimizing retractions in worst-case scenarios of Kelly (2011). For a philosophical discussion of those approaches see Sober (2015), Woodward (2014). For a comparison between those approaches and the MDL method see Grünwald (2007, Sect. 17).
For an application of the real-pattern approach to mereological questions see Petersen (2019).
The idea of reformulating the BSA by separating a system from the auxiliary information is also defended in Hall (2015). Hall’s proposal and mine nevertheless differ in a number of respects. In particular, Hall does not provide a method for assessing the epistemic virtues (which I provide in Sect. 5). Also, he draws the line between system and auxiliary information by telling us that the former but not the latter is nomological in nature. As he correctly points out, however, BSA-theorists of Humean convictions cannot help themselves to the notion of ‘nomological’ on pain of circularity. My way of drawing the distinction is, on the other hand, purely syntactical, and so consistent with a Humean attitude.
paraphrase is the intended condition provided that the systems share a common logic. If that is not the case, the paraphrase will need to be extended to the logical vocabulary—a nontrivial task which I am not going to undertake here.
I will be assuming both that a class S of systems has finite cardinality, and that each system in S has finite length. I consider a generalization to the infinitary case in Sect. 6.6. I will also be assuming that (1) commas separating distinct formulas do not count towards the length of their concatenation; (2) subscripts do not count towards the length, either, in such a way that \(\ell (\{t_{k}\})=1\); (3) predicate symbols have unitary length, so that \(\ell (\{\textrm{Blue}(x,t)\})=6\).
Thanks to Zachary Kofi for drawing my attention to this option.
It is straightforward to show that the local valuation validates Binary Independence. Suppose that S satisfies paraphrase, and \(\delta _{\mathcal {L}}\) is the assignment of values to systems in S. Also, suppose that \( S\subset S^{*}\), \( S^{*}\) satisfies paraphrase, and \(\delta ^{*}_{\mathcal {L}}\) is the assignment of values to systems in \( S^{*}\). If \(\Phi _{\mathcal {L}_{1}},\Psi _{\mathcal {L}_{2}}\in S\), then \(\delta _{\mathcal {L}}(\Phi _{\mathcal {L}_{1}})=\delta ^{*}_{\mathcal {L}}(\Phi _{\mathcal {L}_{1}})\) and \(\delta _{\mathcal {L}}(\Psi _{\mathcal {L}_{2}})=\delta ^{*}_{\mathcal {L}}(\Psi _{\mathcal {L}_{2}})\). Therefore, \(\Phi _{\mathcal {L}_{1}}\sqsubseteq ^{S}_{\mathcal {L}}\Psi _{\mathcal {L}_{2}}\) only if \(\Phi _{\mathcal {L}_{1}}\sqsubseteq ^{S^{*}}_{\mathcal {L}}\Psi _{\mathcal {L}_{2}}\).
The appeal to counterconventional reasoning is meant to be purely illustrative, and plays no role in the definition of a global valuation. On counterconventionals see Einheuser (2006).
In order to see that, suppose that S satisfies paraphrase, and \(\theta \) is the valuation of systems in S. Also, suppose that S is expanded to \( S^{*}= S\cup \{\textrm{X}_{\mathcal {L}^{*}}\}\), \( S^{*}\) satisfies paraphrase, \(\theta ^{*}\) is the valuation of systems in \( S^{*}\), and \(\mathcal {L}^{*}\) is not the language of any system in S. If, say, \(\theta (\Phi _{\mathcal {L}_{1}})=\theta (\Psi _{\mathcal {L}_{2}})\) and \(\ell (\mathbb {T}^{\Phi _{\mathcal {L}_{1}}}_{\mathcal {L}^{*}})>\ell (\mathbb {T}^{\Psi _{\mathcal {L}_{2}}}_{\mathcal {L}^{*}})\), then \(\Phi _{\mathcal {L}_{1}}\sqsubseteq ^{S}\Phi _{\mathcal {L}_{2}}\), but not \(\Phi _{\mathcal {L}_{1}}\sqsubseteq ^{S^{*}} \Phi _{\mathcal {L}_{2}}\).
Indeed, entangled systems of quantum mechanics on its standard interpretation are nonseparable insofar as the state of a system fails to supervene on the intrinsic properties of its parts. However, see Loewer (1996), as well as Bhogal and Perry (2017) for attempts at reconciling separability with quantum mechanics.
Here is a sketch of the proof that the GBSA satisfies physical statism. A difference in GBSA-laws requires a difference as to which systems are best in each eligible class. And a difference as to which systems are best in each eligible class requires a difference as to which classes are eligible. By the definition of eligible class (Sect. 4), a difference as to which classes are eligible requires a difference as to which propositions are made true by the total physical state of the world. Finally, a difference as to which propositions are made true by the total physical state of the world requires a difference in the total physical state of the world (provided that truth supervenes on being). Putting all together, by the transitivity of supervenience it follows that a difference in GBSA-laws requires a difference in the total physical state of the world.
See Loewer (2020) for a sketch of a BSA that is independent of the truth of humean supervenience.
Eddon and Meacham (2015, Sect. 4) have floated a similar idea. However, since their notion of lawhood fails to meet egalitarianism, the notion of naturalness they characterize is bound to be nonobjective.
For example,
- \(\mathbb {T}^{\mathcal {L}_{2}}_{\mathcal {L}_{3}}\):
-
\(=\{\mathbb {T}(x=_{t} c)=(x=_{t}e \wedge t\leqslant t_{5})\vee (x=_{t}f\wedge \lnot t\leqslant t_{5}),\mathbb {T}(x=_{t} d)=(x=_{t}f \wedge t\leqslant t_{5})\vee (x=_{t}e\wedge \lnot t\leqslant t_{5}), \mathbb {T}(\textrm{Emerald}(x,t))=(\textrm{Emerire}(x,t)\wedge t\leqslant t_{5})\vee (\textrm{Sapphard}(x,t)\wedge \lnot t\leqslant t_{5}),\) \(\mathbb {T}(\textrm{Sapphire}(x,t))=(\textrm{Sapphard}(x,t)\wedge t\leqslant t_{5})\vee (\textrm{Emerire}(x,t)\wedge \lnot t\leqslant t_{5})\}\).
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Acknowledgements
Thanks to Tristram McPherson, Erica Shumener, the MetaScience Group at the University of Bristol, the Metaphysics Reading Group at Rutgers University, the LLC group in Turin, the anonymous referees for this journal, as well as audiences at UT Austin, Utrecht University, Boston University, the National Autonomous University of Mexico, and, the University of Parma. This research was supported by a PASPA-DGAPA research fellowship awarded by the National Autonomous University of Mexico.
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Torza, A. Laws of Nature and Theory Choice. Synthese 200, 459 (2022). https://doi.org/10.1007/s11229-022-03950-1
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DOI: https://doi.org/10.1007/s11229-022-03950-1