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A logic for the discovery of deterministic causal regularities

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Abstract

We present a logic, \(\mathbf {ELI^r}\), for the discovery of deterministic causal regularities starting from empirical data. Our approach is inspired by Mackie’s theory of causes as INUS-conditions, and implements a more recent adjustment to Mackie’s theory according to which the left-hand side of causal regularities is required to be a minimal disjunction of minimal conjunctions. To derive such regularities from a given set of data, we make use of the adaptive logics framework. Our knowledge of deterministic causal regularities is, as Mackie noted, most often gappy or elliptical. The adaptive logics framework is well-suited to explicate both the internal and the external dynamics of the discovery of such gappy regularities. After presenting \(\mathbf {ELI^r}\), we first discuss these forms of dynamics in more detail. Next, we consider some criticisms of the INUS-account and show how our approach avoids them, and we compare \(\mathbf {ELI^r}\) with the CNA algorithm that was recently proposed by Michael Baumgartner.

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Notes

  1. Pearl’s IC-algorithm has served as the basis for \(\mathbf {ALIC}\), an adaptive logic for causal discovery (Leuridan 2009). Since \(\mathbf {ALIC}\) and the adaptive logic to be presented here start from strongly different theories of causation we will not pay any further attention to how these logics relate.

  2. For the Causal Markov Condition, see Spirtes et al. (2000, p. 54) and Pearl (2000, p. 30); for the Faithfulness Condition or Stability, see Spirtes et al. (2000, p. 56) and Pearl (2000, p. 48) respectively.

  3. In this quote they discuss the Markov Condition, but their claim applies to the Causal Markov Condition as well.

  4. Mackie’s original formulation is relativized to a causal field F. A causal field is the context in which, or the background against which, the causing takes place. What conditions are taken to be part of the field and what conditions are said to be causes of this-event-in-this-field, is partly pragmatic and a matter of choice (Mackie 1974, pp. 34–35). We have omitted this reference to F. In the application of our logic we will assume that an appropriate field is chosen.

  5. In the interest of a causal interpretation of this condition, Mackie (1974, p. 69) adds further clauses pertaining to the spatio-temporal contiguity of its occurrences and those of P. We will gloss over this point here, but return to it in Sect. 2.3.

  6. (Mackie 1974, pp. 83–87); see also (Psillos 2002, p. 90) for a succinct formulation of the problem.

  7. See Sect. 3.3.3 where we explain this second point.

  8. We are indebted to an anonymous referee, who proposed various (equivalent) simplifications of our original definitions in Sect. 3.2. Of course, all remaining unclarities are ours.

  9. This assumption that all predicates are logically and semantically independent serves the same goal as Baumgartner’s (2008, Appendix), viz. to exclude semantic regularities, yet without needing to stipulate that causal regularities concern different spatio-temporal events.

  10. More precisely, the number of formulas in RF is bounded by \(|\mathcal {V}|\times 2^{2^{|\mathcal {P}|\times 2}}\), i.e.  the number of variables \(\alpha \) multiplied by the number of subsets of \(\wp (\mathcal {L}^\alpha )\). This follows from the fact that every A in RF can be rewritten as a formula of the form \(\bigvee _{1\,\le \, i\,\le \, n} \bigwedge \varTheta _i\) where each \(\varTheta _i\) is a finite subset of \(\mathcal {L}^\alpha \) and \(\varTheta _i\ne \varTheta _j\) whenever \(i\ne j\).

  11. To be exact, where \(A = (C^1_1\wedge \ldots \wedge C^1_{m_1}) \vee \ldots \vee (C^n_1\wedge \ldots \wedge C^n_{m_n})\), we have \(|\mathcal {S}(A)| \le 2^{m_1+\ldots +\,m_n}\).

  12. The adaptive logics framework is a framework for defining logics capable of capturing defeasible reasoning forms such as inductive inference, abductive inference, or reasoning in the presence of inconsistencies. For a general introduction to adaptive logics, see Batens (2001, 2007).

  13. \(\mathbf {ELI^r}\) is a close relative of \(\mathbf {LI^r}\), the adaptive logic for inductive generalization from Batens (2006) and Batens and Haesaert (2001).

  14. This is a purely technical term, commonly used to define adaptive logics in standard format.

  15. A logic with consequence relation \(\vdash \) is supra-classical iff for all \(\varGamma ,A\), if \(\varGamma \vdash _{\mathbf {CL}} A\), then \(\varGamma \vdash A\) (but not conversely).

  16. Mind that this notation presupposes that \(\varTheta \) is finite. If \(\varTheta \) contains just one element A, \( Dab (\varTheta )=A\). Where \(\varTheta = \emptyset \), we let \( Dab (\varTheta ) = \bot \).

  17. A stage of a proof is a sequence of lines and a proof is a sequence of stages. Every proof starts off with stage 1. Adding a line to a proof by applying one of the rules of inference brings the proof to its next stage, which is the sequence of all lines written up to that point.

  18. The marking definition we use in this paper is that of the so-called reliability strategy—hence the superscript “\(\mathbf {r}\)” in \(\mathbf {ELI^r}\). Other strategies are available within the adaptive logics framework. Each of these comes with its own marking definition, and may lead to a slightly different consequence set. See Batens (2007) for some alternatives.

  19. This is immediate in view of (a) the supra-classicality of \(\mathbf {ELI^r}\) mentioned above, and (b) Corollary 3 in Batens (2007). This corollary states a property for all adaptive logics in standard format. In the present case, it implies that if \(\varGamma \) is \(\mathbf {CL}\)-consistent, it is also \(\mathbf {ELI^r}\)-consistent.

  20. We thank one of the referees for pointing out this example, which illustrates the subtleties of Definition 7. In Batens (2009), this definition is interpreted in game-theoretic terms.

  21. Some readers may wonder what happens in the other example \(\varGamma _{2}\). It is impossible to derive any of the disjuncts of the \( Dab \)-formula derived at line 6, so the disjunction is a minimal \( Dab \)-formula in any extension of the proof. Consequently, lines 4 and 5 are marked in any extension of the proof. More generally, one can show by means of the procedure spelled out in Sect. 3.3.3 that the formulas on these lines are not finally \(\mathbf {ELI^r}\)-derivable from \(\varGamma _{2}\).

  22. See e.g. (Batens 2007, Theorem 6).

  23. An anonymous referee claims that \(\mathbf {ELI^r}\) is decidable in the infinite case as well (i.e. even when we do not assume that \(\mathcal {L}\) has a finite signature). We leave the proof of this more general claim for further research.

  24. For some results on goal-directed proof-theories, see Batens and Provijn (2001).

  25. By “addition” we mean: to infer \(A\vee B\) from A.

  26. The terms “internal dynamics” and “external dynamics” are adopted from Batens (2001). This distinction is also found in Pollock (2008), where the terms “diachronic defeasibility”, resp. “synchronic defeasibility” are used to refer to the internal dynamics, resp. external dynamics of human reasoning.

  27. A logic \(\mathbf {L}\) is monotonic iff for all sets of \(\mathbf {L}\)-formulas \(\varGamma \) and \(\Delta \), and for all \(\mathbf {L}\)-formulas A, if \(\varGamma \vdash _{\mathbf {L}}A\) then \(\varGamma \cup \Delta \vdash _{\mathbf {L}}A\).

  28. This is a requirement \(\mathbf {ELI}^\mathbf{r}\) shares with Baumgartner’s CNA (see below, Sect. 4.3) and with Grasshoff and May’s 2001 approach.

  29. Our discussion on the heuristic merits of \(\mathbf {ELI}^\mathbf{r}\) is inspired by the discussion in Batens (2004), where a more detailed account is provided of how a logic for inductive generalization can evoke further tests and lead to new conjectures.

References

  • Allison, P. D. (2001). Missing data. Thousand Oaks: SAGE Publications.

    Google Scholar 

  • Batens, D. (2001). A general characterization of adaptive logics. Logique & Analyse, 173–175, 45–68.

    Google Scholar 

  • Batens, D. (2004). The basic inductive schema, inductive truisms, and the research-guiding capacities of the logic of inductive generalization. Logique & Analyse, 185–188, 53–84.

    Google Scholar 

  • Batens, D. (2006). On a logic of induction. Logic and Philosophy of Science, 4(1), 3–32.

    Google Scholar 

  • Batens, D. (2007). A universal logic approach to adaptive logics. Logica Universalis, 1, 221–242.

    Article  Google Scholar 

  • Batens, D. (2009). Towards a dialogic interpretation of dynamic proofs. In C. Dégremont, L. Keiff, & H. Rückert (Eds.), Dialogues, logics and other strange things (pp. 27–51)., Essays in honour of Shahid Rahman London: College Publications.

    Google Scholar 

  • Batens, D. (2011). Logics for qualitative inductive generalization. Studia Logica, 97, 61–80.

    Article  Google Scholar 

  • Batens, D., & Haesaert, L. (2001). On classical adaptive logics of induction. Logique & Analyse, 173–175, 255–290.

    Google Scholar 

  • Batens, D., & Provijn, D. (2001). Pushing the search paths in the proofs. A study in proof heuristics. Logique & Analyse, 173–175, 113–134.

    Google Scholar 

  • Baumgartner, M. (2008). Regularity theories reassessed. Philosophia, 36(3), 327–354.

    Article  Google Scholar 

  • Baumgartner, M. (2009). Uncovering deterministic causal structures: A Boolean approach. Synthese, 170(1), 71–96.

    Article  Google Scholar 

  • Graßhoff, G., & May, M. (2001). Causal regularities. In W. Spohn, M. Ledwig, & M. Esfeld (Eds.), Current issues in causation (pp. 85–114). Paderborn: Mentis.

    Google Scholar 

  • Illari, P., & Russo, F. (2014). Causality, philosophical theory meets scientific practice. Oxford: Oxford University Press.

    Google Scholar 

  • Kim, J. (1971). Causes and events: Mackie on causation. Journal of Philosophy, 68(14), 426–441.

    Article  Google Scholar 

  • Leuridan, B. (2009). Causal discovery and the problem of ignorance. An adaptive logic approach. Journal of Applied Logic, 7(2), 188–205.

    Article  Google Scholar 

  • Mackie, J. L. (1974). The cement of the universe: A study of causation. Oxford: Clarendon Press.

    Google Scholar 

  • Mill, J. (1843). A system of logic. London: John W. Parker.

    Google Scholar 

  • Pearl, J. (2000). Causality. Models, reasoning, and inference. Cambridge: Cambridge University Press.

    Google Scholar 

  • Pearl, J. (2009). Causality. Models, Reasoning, and Inference (2nd ed.). Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Pollock, J. (2008). Defeasible reasoning. In J. Adler & L. Rips (Eds.), Reasoning. Studies of human inference and its foundations (pp. 451–470). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • Psillos, S. (2002). Causation and explanation. Chesham: Acumen.

    Google Scholar 

  • Reiss, J. (2015). Causation, evidence, and inference. New York: Routledge.

    Google Scholar 

  • Russo, F. (2006). The rationale of variation in methodological and evidential pluralism. Philosophica, 77, 97–123.

    Google Scholar 

  • Spirtes, P., Glymour, C., & Scheines, R. (2000). Causation, prediction, and search. Cambridge, MA: MIT Press.

    Google Scholar 

Download references

Acknowledgments

The research of Mathieu Beirlaen was supported by the project “Logics of discovery, heuristics and creativity in the sciences” (PAPIIT, IN400514-3) granted by the National Autonomous University of Mexico (UNAM), by the Programa de Becas Posdoctorales de la Coordinación de Humanidades (UNAM), and by a Sofja Kovalevskaja award of the Alexander von Humboldt Foundation, funded by the German Ministry for Education and Research. Frederik Van De Putte is a post-doctoral fellow of the Research Foundation of Flanders (FWO-Vlaanderen). We are greatly indebted to the anonymous referees for their invaluable comments, criticisms, and suggestions.

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Authors and Affiliations

  1. Research Group for Non-Monotonic Logic and Formal Argumentation, Ruhr-Universität Bochum, Universitätsstraße 150, Gebäude GA, Raum 3/39, 44801, Bochum, Germany

    Mathieu Beirlaen

  2. Centre for Philosophical Psychology, University of Antwerp, Antwerp, Belgium

    Bert Leuridan

  3. Centre for Logic and Philosophy of Science, Ghent University, Ghent, Belgium

    Bert Leuridan & Frederik Van De Putte

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  1. Mathieu Beirlaen
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  2. Bert Leuridan
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  3. Frederik Van De Putte
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Correspondence to Mathieu Beirlaen.

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Beirlaen, M., Leuridan, B. & Van De Putte, F. A logic for the discovery of deterministic causal regularities. Synthese 195, 367–399 (2018). https://doi.org/10.1007/s11229-016-1222-x

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