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Do Ceteris Paribus Laws Exist? A Regularity-Based Best System Analysis

Abstract

This paper argues that ceteris paribus (cp) laws exist based on a Lewisian best system analysis of lawhood (BSA). Furthermore, it shows that a BSA faces a second trivialization problem besides the one identified by Lewis. The first point concerns an argument against cp laws by Earman and Roberts. The second point aims to help making some assumptions of the BSA explicit. To address the second trivialization problem, a restriction in terms of natural logical constants is proposed that allows one to describe regularities, as specified by basic generics (e.g. ‘birds can fly’) and universals (e.g. ‘all birds can fly’). It is argued that cp laws rather than strict laws might be a part of the the best system of such a regularity-based BSA, since sets of cp laws can be both (a) simpler and (b) stronger when reconstructed as generic non-material conditionals. Yet, if sets of cp laws might be a part of the best system of a BSA and thus qualify as proper laws of nature, it seems reasonable to conclude that at least some cp laws qualify as proper laws of nature.

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Notes

  1. 1.

    Laws of fundamental physics have cp conditions according to Cartwright (2002; see also Pemberton and Cartwright 2014) and Hüttemann (2014), insofar as their application requires that there are no (disturbing) influences other than the ones specified by the application.

  2. 2.

    In case no ambiguity arises, I use the terms ‘cp law’ and ‘strict law’ in the sense of ‘law-like cp statement’ and ‘law-like strict statement’, respectively.

  3. 3.

    In Strevens’s (2014) terminology, I employ a narrowness approach rather than softening approach. For a discussion of softening approaches in terms of probabilistic accounts of cp laws see for example Reutlinger (2014) and Roberts (2014).

  4. 4.

    For the sake of simplicity, I consider here only bare plural generics with a logically simple, yet possibly negated consequent as in (3a)–(4b) (see Sect. 3).

  5. 5.

    \(\mathrm{REG }\) might include statements that are closer to existentially quantified statements than to universals. I, however, do not discuss such statements here (see Fn. 5).

  6. 6.

    Some laws of nature might have a different logical form than (a) (e.g., existentially quantified). However, it seems faithful to reconstruct strict laws as discussed in the literature on both cp laws and laws of nature as laws of the form (a).

  7. 7.

    The classical approach to cp laws described above is a completer approach in the sense of Fodor (1991). This is due to the fact that (b) guarantees that \( (Ba \& \lnot H_1a \& \ldots \& \lnot H_na) \rightarrow Fa\) holds for any entity \(a\). In fact, \( \lnot H_1a \& \dots \& \lnot H_na\) is a completer for (a) in the sense that it allows one to specify an exceptionless regularity based on (a) for any entity \(a\) (cf. Fodor 1991, p. 23; see also Reutlinger et al. 2011, §5.1).

  8. 8.

    The argument goes back to Lange (1993). However, Lange took a different starting point and argued that one has to rely on cp laws, since it not possible to state all of our scientific practices explicitly. Despite this, the structure of Lange’s version concurs with the version above.

  9. 9.

    For example, Schurz (2002, p. 365) argues that an important subclass of non-lazy cp laws can be confirmed or disconfirmed (see also Schurz 2014, Sect. 3), contrary to Earman and Roberts.

  10. 10.

    For the sake of simplicity, I do not discuss the notion of fit. Fit refers to the degree to which chances—objective single-case probabilities—concur with the history of our world.

  11. 11.

    Lewis does not presuppose that our world has a single best system. A single best system exists only if nature is kind (Lewis 1999a, p. 232f).

  12. 12.

    Lewis’s (1999a, b) approach differs from his earlier account in that point.

  13. 13.

    Sider (2012) seems to reach a similar conclusion, when he writes:

    “[I]n the Lewisian theory of laws, the logical expressions in the lawmaking language must be required to be joint-carving; otherwise cheap simplicity can be obtained with rigged logical expressions just as through rigged predicates” (p. 87).

    However, Sider does not explain how cheap simplicity can be obtained.

  14. 14.

    However, see also Fn. 12.

  15. 15.

    For a more detailed discussion of this point see Schrenk (2007, pp. 81–83). Note that Schrenk (2007) argues that Lewis’s miracles—point-wise exceptions to laws of nature—count in favor of non-strict laws in Lewis’s philosophy.

  16. 16.

    Schrenk’s account, as described here, presupposes that only a finite number of exceptions for each law exist (see also Schrenk 2014, Fn. 9). I do not further address this point here.

  17. 17.

    The syntactic form of GNCs is the same as that of Schurz’s (2002) normic laws (p. 364f). However, my interpretation of generic statements differs from Schurz’s.

  18. 18.

    This interpretation concurs with Pearl’s (1988, p. 475) all-I-know interpretation of generic conditionals (Unterhuber and Schurz 2013, p. 1416).

  19. 19.

    For the sake of simplicity, I assume here that a single set of most regular worlds always exists.

  20. 20.

    Pelletier and Asher (1997, p. 1144) argue that a semantics such as Delgrande’s must be refined in order to adequately represent natural language generics (see also Leslie 2008, p. 8; Asher and Pelletier 2012, pp. 313–316). In contrast, for the present purpose, such a semantics seems to suffice (see Sect. 6).

  21. 21.

    I owe the idea of distinguishing between exceptional and non-exceptional properties based on frequency information to Gerhard Schurz.

  22. 22.

    See Spohn (2012, Ch. 13.2) for a general account of \(n\)-fold exceptions (exceptions of degree \(n\)), i.e. exceptions of exceptions etc.

  23. 23.

    In fact, the proof theory of GNCs is strictly weaker than the classical one, since unlike in classical logic, for example, inferences from are not allowed to be valid. Otherwise one could not specify exceptions in the first place.

  24. 24.

    Earman and Roberts (1999, p. 461) call this type of cp law ‘improper’. I agree with Earman and Roberts that if one were only to use cp laws in this sense, cp laws of this sort would earn the name ‘improper’. However, in the cases discussed here it is quite likely that GNCs and the corresponding strict laws diverge.

  25. 25.

    I assume here that DMP keeps track of all GNCs with conflicting consequents. DMP only needs additional steps in case of conflicts and would then need to determine whether each instantiated conflicting GNC is blocked by a GNC (see Sect. 5).

  26. 26.

    Definition 1 is akin to Delgrande’s (1988, p. 73) notion of default support among conditionals.

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Acknowledgments

This work was carried out as a part of the DFG Research Unit FOR 1063 and supported by the DFG Grant SCHU1566/7-1. I thank Alexander Reutlinger, Markus Schrenk, Ludwig Fahrbach, Gerhard Schurz, Paul Thorn, Nancy Cartwright, Michael Strevens, Wolfgang Spohn, Kevin Kelly, Vera Hoffmann-Kolss, and Andreas Hüttemann for their valuable comments. In addition, I greatfully acknowledge two anonymous reviewers for their substantial comments, which pushed me towards a clearer and more thorough exposition of the present ideas.

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Correspondence to Matthias Unterhuber.

Appendix: The Default Modus Ponens

Appendix: The Default Modus Ponens

This appendix gives a definition of a default modus ponens (DMP; see Sect. 5). To this end, two languages \(\mathfrak {L}_{{\mathbf{GNC}}}\) and \(\mathfrak {L}_{\mathrm{m }}\) are employed, where \(\mathfrak {L}_{{\mathbf{GNC}}}\) and \(\mathfrak {L}_{\mathrm{m }}\) refer to sets of GNC formulæ and material formulæ, respectively. Both languages admit only monadic predicates \(P,\,P_1,\,P_2\), ...  Open literals and closed literals are expressions of the form (1) \(Px\) and \(\lnot Px\) as well as (2) \(Pa\) and \(\lnot Pa\), respectively, for arbitrary predicates \(P\), individual variables \(x\), and individual constants \(a\). Language \(\mathfrak {L}_{{\mathbf{GNC}}}\) allows for formulæ of the form where \(\alpha \) and \(\beta \) are conjunctions of open literals and single open literals, respectively. In contrast, \(\mathfrak {L}_{\mathrm{m }}\) contains arbitrary Boolean combinations of closed literals. Formula \(\alpha [a/x]\) results from uniformly substituting \(x\) in \(\alpha \) with \(a\).

For sets of formulæ \(\Gamma \) and formulæ \(\alpha \) of language \(\mathfrak {L}_{{\mathbf{GNC}}}\) [\(\mathfrak {L}_{\mathrm{m }}\)], \(\Gamma \vdash _{{\mathbf{GNC}}} \alpha \) [\(\Gamma \vdash _{{\mathbf{FOL}}} \alpha \)] indicates derivability in a logic \({\mathbf{GNC}}\) for GNCs [first-order logic]. System \({{\mathbf{GNC}}}\) corresponds semantically to Lewis’s (1973) sphere semantics without centering. System \({{\mathbf{GNC}}}\) is also required to make and \({\mathbf{GNC}}\)-inconsistent given \(\alpha [a/x]\) is \({\mathbf{FOL}}\)-consistent. For open formulæ \(\alpha \) and sets of open formulæ \(\Gamma \) the expression \(\Gamma \vdash _{{\mathbf{FOL}}} \alpha \) abbreviates \(\{\gamma [a/x]\): \(\gamma \in \Gamma \} \vdash _{{\mathbf{FOL}}} \alpha [a/x]\), where \(a\) is a fixed individual constant. Let ∆ be a set of of formulæ of language \(\mathfrak {L}_{\mathrm{m }}\) with the following property: ∆ contains a finite number of formulæ in which any given individual constant a occurs. The DMP can now be defined as follows:Footnote 26

Definition 1

(Default Modus Ponens) Suppose that α[a/x] is the conjunction of all formulæ in ∆ in which a occurs. Infer \(\gamma [a/x]\) from \(\alpha [a/x]\) based on \(\Gamma \) and ∆ if there is a formula \(\beta \) such that

  1. (1)
  2. (2)

    \(\{ \alpha \} \vdash _{{\mathbf{FOL}}} \beta \), and

  3. (3)

    for all \(\gamma '\) such that (i) (ii) \(\{\alpha \} \vdash _{{\mathbf{FOL}}} \beta '\), and (iii) it holds that \(\{\beta \} \vdash _{{\mathbf{FOL}}} \beta '\).

Observe that conditions (2) and (3ii) ensure that DMP inferences pertain only to GNCs with instantiated antecedents as specified by α[a/x].

Let me finally give an example, based on System CP (see Sect. 6). Suppose that \(\Gamma =\{\)(A1’), (A2’), (A3’), (A4’)\(\}\) and that one aims to infer \(P_4a\) for a given individual \(a\) from \(\Gamma \) and \( P_1a \& \lnot P_2a \& \lnot P_3a\), based on (A4’) For this inference, conditions (1) and (2) are trivially satisfied. Furthermore, \( P_1a \& \lnot P_2a \& \lnot P_3a\) implies (i) \(P_1a\), (ii) \( P_1a \& \lnot P_2a\), and (iii) \( P_1a \& \lnot P_2a\), which correspond to the antecedent formulæ of (A1’)–(A3’), respectively. Only the consequent formulæ of (A2’) and (A3’) contradict the consequent formulæ of (A4’). Thus, merely (ii) and (iii) are required to be implied by \( P_1a \& \lnot P_2a \& \lnot P_3a\), which is easily verified.

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Unterhuber, M. Do Ceteris Paribus Laws Exist? A Regularity-Based Best System Analysis. Erkenn 79, 1833–1847 (2014). https://doi.org/10.1007/s10670-014-9645-6

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Keywords

  • Logical Constant
  • Indicative Conditional
  • Trivialization Problem
  • Natural Language Generic
  • Bare Plural