Abstract
In their 2010 (Erkenntnis 73:393–412) paper, Dizadji-Bahmani, Frigg, and Hartmann (henceforth ‘DFH’) argue that the generalized version of the Nagel–Schaffner model that they have developed (henceforth ‘the GNS’) is the right one for intertheoretic reduction, i.e. the kind of reduction that involves theories with largely overlapping domains of application. Drawing on the GNS, DFH (Synthese 179:321–338, 2011) presented a Bayesian analysis of the confirmatory relation between the reducing theory and the reduced theory and argued that, post-reduction, evidence confirming the reducing theory also confirms the reduced theory and evidence confirming the reduced theory also confirms the reducing theory, which meets the expectations one has about theories with largely overlapping domains. In this paper, I argue that the Bayesian analysis presented by DFH (Synthese 179:321–338, 2011) faces difficulties. In particular, I argue that the conditional probabilities that DFH introduce to model the bridge law entail consequences that run against the GNS. However, I also argue that, given slight modifications of the analysis that are in agreement with the GNS, one is able to account for these difficulties and, moreover, one is able to more rigorously analyse the confirmatory relation between the reducing and the reduced theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Note that on the GNS one theory reduces to the other in virtue of empirical propositions (i.e. laws that a theory has). However, proponents of the GNS do not commit themselves to the view that “a theory just is [DFH’s italics] a set of laws, i.e. \(\mathcal {\mathbf {T_{_{A}}}}\) is not identified with \(\mathcal{T}_{_{A}}\)” (DFH 2011, p. 323).
An example of the bridge law can be found in Sect. 2.2. Another example of the bridge law is \(\mathbf {V}=\mathbf {E}\), where \(\mathbf {V}\) is the light vector from the physical optical theory of light and \(\mathbf {E}\) is the electric force vector from the theory of electromagnetic radiation. This bridge law is used to derive a number of laws of the physical optical theory of light from Maxwell’s equations (see Schaffner 2012, pp. 551–559).
Feynman et al. (1964, chapter 39) additionally provide a more comprehensive argument for why the mean kinetic energies of the two gases ought to be equal using only the concepts from the kinetic theory of gases and the definition of equilibrium. For the purposes of this paper, however, we need not go into such detail.
It is worth pointing out that in both entity identification and property identification we simplify our previously held ontology not by eliminating unnecessary entities or properties of the reduced theory (for instance, eliminating light waves and temperature), but rather by assimilating these entities and properties via identification to the corresponding entities and properties of the reducing theory. So, there are still light waves in the world, but instead of two classes of entities—light waves and electromagnetic waves—there is only one (see Sklar 1967, p. 121, 1993, pp. 361–362).
Throughout the article, random variables in the network are binary; that is, some random variable A (denoted by italicized letters) can take two values A or \(\lnot \)A (denoted by non-italicized letters).
Bayes’ Theorem: \(P(\mathrm {H}\mid \mathrm { E})=\frac{P(\mathrm {E}\mid \mathrm {H})P(\mathrm {H})}{P(\mathrm {E})}=\frac{P(\mathrm {E}\mid \mathrm {H})P(\mathrm {H})}{P(\mathrm {E}\mid \mathrm { H})P(\mathrm {H})+P(\mathrm {E}\mid \lnot \mathrm { H})P(\lnot \mathrm { H})}\).
‘
’ encodes the information that A and B are conditionally independent given C. By definition, A and B are conditionally independent given C, i.e. 
, if and only if \(P(\mathrm {A}\mid \text { B, C})=P(\mathrm {A}\mid \mathrm {C})\).d-separation is a property of Bayesian networks by which one can track down all the independences (conditional and unconditional ones) in the Bayesian network:
if and only if A and B are d-separated by \(\{C\}\). Two nodes A and B are d-separated by \(\{C\}\) if all the paths in the network between A and B are blocked by \(\{C\}\). For more details on d-separation see Neapolitan
(2003, pp. 70ff.).Although a, in principle, can take any value in the open interval (0, 1), it seems more plausible that it assumes a rather low value since we do not expect to often find that \(\mathrm {T}^*_\mathrm {P}\) holds and that \(\lnot \mathrm { T}^*_\mathrm {F}\) or \(\lnot \mathrm { B}\) hold.
Interestingly, but perhaps unsurprisingly, one can show that in DFH’s original analysis the two theorems hold in exactly the same form (see Theorem 7’ and Theorem 8’ in “Appendix”).
The situation does not differ much given somewhat different value assignments for a, b, and \(t_F\).
The expression ‘s-g axiom’ stands for semi-graphoid axiom. For more details on semi-graphoid axioms see Pearl (1988, pp. 84ff).
’ encodes the information that A and B are conditionally independent given C. By definition, A and B are conditionally independent given C, i.e. 
, if and only if 
if and only if A and B are d-separated by References
Aerts, D., & Rohrlich, F. (1998). Reduction. Foundations of Science, 1, 27–35.
Ager, T. A., Aronson, J. L., & Weingard, R. (1974). Are bridge laws really necessary? Noûs, 8(2), 119–134.
Batterman, R. W. (2002). The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence. Oxford: Oxford University Press.
Bovens, L., & Hartmann, S. (2003). Bayesian epistemology. Oxford: Oxford University Press.
Darden, L., & Maull, N. (1977). Interfield theories. Philosophy of Science, 44(1), 43–64.
Dizadji-Bahmani, F. (2011). Neo-Nagelian reduction: A statement, defence, and application. Ph.D. Thesis, The London School of Economics and Political Science (LSE). Retrieved from http://etheses.lse.ac.uk/355/.
Dizadji-Bahmani, F., Frigg, R., & Hartmann, S. (2010). Who’s afraid of Nagelian reduction? Erkenntnis, 73, 393–412.
Dizadji-Bahmani, F., Frigg, R., & Hartmann, S. (2011). Nagelian reduction. Synthese, 179, 321–338.
Earman, J. (1992). Bayes or bust? A critical examination of Bayesian confirmation theory. Cambridge, MA: The MIT Press.
Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman lectures on physics (Vol. 1). Reading, MA: Addison-Wesley.
Fitelson, B. (1999). The plurality of Bayesian measures of confirmation and the problem of measure sensitivity. Philosophy of Science, 66, S362–S378.
Greiner, W., Heise, L., & Stöcker, H. (1997). Thermodynamics and Statistical mechanics. New York, NY: Springer.
Háyek, A., & Hartmann, S. (2010). Bayesian epistemology. In J. Dancy, E. Sosa, & M. Steup (Eds.), A companion to epistemology (pp. 93–105). Oxford: Wiley-Blackwell.
Hartmann, S., & Sprenger, J. (2011). Bayesian epistemology. In S. Bernecker & D. Pritchard (Eds.), The Routledge companion to epistemology (pp. 609–620). New York, NY and London: Routledge.
Kuipers, T. A. F. (1982). The reduction of phenomenological to kinetic thermostatics. Philosophy of Science, 49(1), 107–119.
Nagel, E. (1961). The structure of science. London: Routledge and Keagan Paul.
Neapolitan, R. E. (2003). Learning Bayesian networks. Upper Saddle River, NJ: Prentice Hall.
Pauli, W. (1973). Pauli lectures on physics: Thermodynamics and the kinetic theory of gases (Vol. 3). Cambridge, MA and London: The MIT Press.
Pearl, J. (1988). Probabilistic reasoning in intelligent systems: Networks of plausible inference. San Francisco, CA: Morgan Kauffman.
Primas, H. (1998). Emergence in the exact sciences. Acta Polytechnica Scandinavica, 91, 83–98.
Rohrlich, F. (1989). The logic of reduction: The case of gravitation. Foundations of Physics, 19(10), 1151–1170.
Sarkar, S. (2015). Nagel on reduction. Studies in History and Philosophy of Science, 53, 43–56.
Schaffner, K. F. (1967). Approaches to reduction. Philosophy of Science, 34(2), 137–147.
Schaffner, K. F. (2006). Reduction: The Cheshire cat problem and a return to roots. Synthese, 151, 377–402.
Schaffner, K. F. (2012). Ernest Nagel and reduction. The Journal of Philosophy, 109, 534–565.
Sklar, L. (1967). Types of inter-theoretic reduction. The British Journal for the Philosophy of Science, 18(2), 109–124.
Sklar, L. (1993). Physics and chance: Philosophical issues in the foundations of statistical mechanics. Cambridge: Cambridge University Press.
van Riel, R. (2011). Nagelian reduction beyond the Nagel model. Philosophy of Science, 78(3), 353–375.
van Riel, R. (2013). Identity, asymmetry, and the relevance of meanings for models of reduction. The British Journal for the Philosophy of Science, 64, 747–761.
van Riel, R. (2014). The concept of reduction. Dordrecht: Springer.
van Riel, R., & Van Gulick, R. (2016). Scientific reduction. In E. N. Zalta (Ed.), The Stanford encyclopaedia of philosophy. Retrieved from https://plato.stanford.edu/archives/win2016/entries/scientific-reduction/.
Winther, R. G. (2009). Schaffner’s model of theory reduction: Critique and reconstruction. Philosophy of Science, 76(2), 119–142.
Acknowledgements
I would like to thank Stephan Hartmann, Benjamin Eva, and anonymous reviewers for helpful and constructive comments that greatly improved the manuscript. I would also like to thank audiences in Lisbon, at the Third Lisbon International Conference on Philosophy of Science and in Dubrovnik, at the Formal Methods and Philosophy II Conference.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
To show: 
and 
entail \(P_1(\mathrm {T}_\mathrm {P}\mid \mathrm { E}_\mathrm {F})=P_1(\mathrm {T}_\mathrm {P})\)Footnote 17
\(\square \)
Similarly, we get that 
and 
entail \(P_1(\mathrm {T}_\mathrm {F}\mid \mathrm {E}_\mathrm {P})=P_1(\mathrm {T}_\mathrm {F})\).
To show: \(P_1(\mathrm {T}_\mathrm {P}\mid \mathrm { E}_\mathrm {F})=P_1(\mathrm {T}_\mathrm {P})\)—by d-separation
There is only one possible path between \(E_F\) and \(T_P\), namely \(E_F-T_F-E-T_P\), which is blocked at E by \(\varnothing \). Therefore, 
. By the definition of independence this translates into \(P_1(\mathrm {T}_\mathrm {P}\mid \mathrm { E}_\mathrm {F})=P_1(\mathrm {T}_\mathrm {P})\). \(\square \)
Similarly, we get that \(P_1(\mathrm {T}_\mathrm {F}\mid \mathrm { E}_\mathrm {P})=P_1(\mathrm {T}_\mathrm {F})\) holds by d-separation before the reduction.
To show: \(P_2(\mathrm {T}^*_\mathrm {F}\mid \mathrm { T}^*_\mathrm {P})=1\)
\(\square \)
To show: \(P_2(\mathrm {T}^*_\mathrm {F}\mid \lnot \mathrm { T}^*_\mathrm {P})=0\)
\(\square \)
To show: \(P_2(\mathrm {T}^*_\mathrm {F})=P_2(\mathrm {T}^*_\mathrm {P})\)
\(\square \)
I adopt the following convention: \(\overline{z}:=1-z\).
Theorem 3\(\ E_\mathrm {F}\) confirms \(T_\mathrm {P}\) iff \((p_F - q_F) \, (p^*_F - q^*_F) \, (p^*_P - q^*_P)>0\).
Proof
\(\square \)
Theorem 4\(E_\mathrm {P}\) confirms \(T_\mathrm {F}\) iff \((p_P - q_P) \, (p^*_F - q^*_F) \, (p^*_P - q^*_P)>0\).
Proof
\(\square \)
To show: \(0<P_3(\mathrm {T}^*_\mathrm {F}\mid \mathrm { T}^*_\mathrm {P})<1\)
Suppose \(P_3(\mathrm {T}^*_\mathrm {F}\mid \mathrm { T}^*_\mathrm {P})=0\), then
But \(P_3(\mathrm {T}^*_\mathrm {F})\) cannot be equal to 0, since by assumption all probabilities are within the open interval (0, 1) (except for the conditional ones that encode logical consequence).
Suppose \(P_3(\mathrm {T}^*_\mathrm {F}\mid \mathrm { T}^*_\mathrm {P})=1\), then
But \(P_3(\mathrm {T}^*_\mathrm {F})\) cannot be equal to 1, for the reason mentioned above.
Hence,
\(\square \)
To show: \(P_3(\mathrm {T}^*_\mathrm {P})>P_3(\mathrm {T}^*_\mathrm {F})\)
\(\square \)
Of \(P_5(\mathrm {T}^*_\mathrm {P}\mid \lnot \mathrm { T}^*_\mathrm {F},\mathrm {B})\), \(P_5(\mathrm {T}^*_\mathrm {P}\mid \mathrm {T}^*_\mathrm {F},\lnot \mathrm { B})\), and \(P_5(\mathrm {T}^*_\mathrm {P}\mid \lnot \mathrm {T}^*_\mathrm {F},\lnot \mathrm { B})\), as the most plausible candidate for assigning the value 0 is \(P_5(\mathrm {T}^*_\mathrm {P}\mid \lnot \mathrm { T}^*_\mathrm {F},\mathrm {B})\), since, one could say, given true bridge laws and false \(T^*_F\) (i.e. \(\lnot \mathrm { T}^*_\mathrm {F}\)), theory \(T^*_P\) should not come out as true. As for \(P_5(\mathrm {T}^*_\mathrm {P}\mid \mathrm { T}^*_\mathrm {F},\lnot \mathrm { B})\) and \(P_5(\mathrm {T}^*_\mathrm {P}\mid \lnot \mathrm { T}^*_\mathrm {F}, \lnot \mathrm { B})\), regard them as randomizers and assign them \(a\in (0,1)\) (see Bovens and Hartmann 2003, pp. 57ff.). However, with these probability assignments, a drawback of the original analysis recurs: \(\mathrm {T}^*_\mathrm {P}\) and \(\mathrm {B}\) entail \(\mathrm {T}^*_\mathrm {F}\). In order to show that this entailment holds, observe that in the network in Fig. 5
holds (the only two paths between \(T^*_F\) and B, i.e. \(T^*_F-T^*_P-B\) and \( T^*_F-T_F-E-T_P-T^*_P-B\), are blocked by \(\varnothing \) at \(T^*_P\) and E respectively; so, \(T^*_F\) and B are d-separated by \(\varnothing \)).
To show: \(P_5(\mathrm {T}^*_\mathrm {F}\mid \mathrm { T}^*_\mathrm {P},\mathrm {B})=1\)
\(\square \)
To show: \(0<P_4(\mathrm {T}^*_\mathrm {F}\mid \mathrm { T}^*_\mathrm {P},\mathrm {B})<1\)
\(\square \)
To show: \(P_4(\mathrm {T}^*_\mathrm {P})>P_4(\mathrm {T}^*_\mathrm {F})\) or \(P_4(\mathrm {T}^*_\mathrm {P})<P_4(\mathrm {T}^*_\mathrm {F})\) or \(P_4(\mathrm {T}^*_\mathrm {P})=P_4(\mathrm {T}^*_\mathrm {F})\)
So, when \(P_4(\mathrm {T}^*_\mathrm {F})=\frac{a}{\overline{a} \, \overline{b}+a}\), then \(P_4(\mathrm {T}^*_\mathrm {P})=P_4(\mathrm {T}^*_\mathrm {F})\) (note that for \(0<a<1\) and \(0<b<1\), \(0<\frac{a}{\overline{a} \, \overline{b}+a}<1\)). When \(P_4(\mathrm {T}^*_\mathrm {F})<\frac{a}{\overline{a} \, \overline{b}+a}\), then \(P_4(\mathrm {T}^*_\mathrm {P})>P_4(\mathrm {T}^*_\mathrm {F})\). When \(P_4(\mathrm {T}^*_\mathrm {F})>\frac{a}{\overline{a} \, \overline{b}+a}\), then \(P_4(\mathrm {T}^*_\mathrm {P})<P_4(\mathrm {T}^*_\mathrm {F})\).
\(\square \)
Theorem 5\(E_\mathrm {F}\) confirms \(T_\mathrm {P}\) iff \((p_F - q_F) \, (p^*_F - q^*_F) \, (p^*_P - q^*_P)>0\).
Proof
\(\square \)
Theorem 6\(E_\mathrm {P}\) confirms \(T_\mathrm {F}\) iff \((p_P - q_P) \, (p^*_F - q^*_F) \, (p^*_P - q^*_P)>0\).
Proof
\(\square \)
Theorem 7\(E_\mathrm {F}\) adds to \(E_\mathrm {P}\)’s confirmation of \(T_\mathrm {P}\) iff \((p_F - q_F) \, (p^*_F - q^*_F) \, (p^*_P - q^*_P)>0\).
Proof
\(\square \)
Theorem 8\(E_\mathrm {P}\) adds to \(E_\mathrm {F}\)’s confirmation of \(T_\mathrm {F}\) iff \((p_P - q_P) \, (p^*_F - q^*_F) \, (p^*_P - q^*_P)>0\).
Proof
\(\square \)
Theorem 7’\(E_\mathrm {F}\) adds to \(E_\mathrm {P}\)’s confirmation of \(T_\mathrm {P}\) iff \((p_F - q_F) \, (p^*_F - q^*_F) \, (p^*_P - q^*_P)>0\).
Proof
\(\square \)
Theorem 8’\(E_\mathrm {P}\) adds to \(E_\mathrm {F}\)’s confirmation of \(T_\mathrm {F}\) iff \((p_P - q_P) \, (p^*_F - q^*_F) \, (p^*_P - q^*_P){>}0\).
Proof
\(\square \)
Theorem 9 Given a, \(p_F\), \(q_F\), \(p^*_F\), \(q^*_F\), \(p^*_P\), \(q^*_P\), and \(t_F\) are constant and \(p_F>q_F\), \(p^*_F>q^*_F\), and \(p^*_P>q^*_P\), if b increases (decreases), then \(d(T_\mathrm {P},E_\mathrm {F})\) increases (decreases).
Proof
From the proof of the Theorem 5 above, we have that:
Observe that, given \(t_F\), a, \(p_F\), \(q_F\), \(p^*_F\), \(q^*_F\), \(p^*_P\), and \(q^*_P\) are constant and \(p_F>q_F\), \(p^*_F>q^*_F\), and \(p^*_P>q^*_P\), if b increases, then \(\overline{a} \, b \, t_F \, \overline{t_F} \, (p_F - q_F) \, (p^*_F - q^*_F) \, (p^*_P-q^*_P)\) increases. As the denominator, i.e. \(p_F \, t_F + q_F \, \overline{t_F}\), does not dependent on b, then if b increases, \(\frac{\overline{a} \, b \, t_F \, \overline{t_F} \, (p_F - q_F) \, (p^*_F - q^*_F) \, (p^*_P - q^*_P)}{p_F \, t_F + q_F \, \overline{t_F}}\) increases, i.e. \(d(\mathrm {T}_\mathrm {P},\mathrm {E}_\mathrm {F})\) increases.
\(\square \)
Theorem 10 Given a, \(p_P\), \(q_P\), \(p^*_F\), \(q^*_F\), \(p^*_P\), \(q^*_P\), and \(t_F\) are constant and \(p_P>q_P\), \(p^*_F>q^*_F\), and \(p^*_P>q^*_P\), if b increases (decreases), then \(d(T_\mathrm {F},E_\mathrm {P})\) increases (decreases).
Proof
From the proof of the Theorem 6 above, we have that:
Notice that, in contrast to the previous proof, the denominator, i.e. \(P_4(\mathrm {E}_\mathrm {P})\), is dependent on b; so, changing the value of b would also change the value of \(P_4(\mathrm {E}_\mathrm {P})\). Alternative way of writing \(P_4(\mathrm {E}_\mathrm {P})\) so that it better serves the purpose of this proof is:
Further, let us introduce the following abbreviations:
Observe that both \(\mathrm {C}\) and \(P_4(\mathrm {E}_\mathrm {P})\) increase as b increases (other values remaining constant). To see which of the two, \(\mathrm {C}\) or \(P_4(\mathrm {E}_\mathrm {P})\), increases faster with the increase of b, we calculate \(\mathrm {C}-\mathrm {D}\).
So, given \(p_P>q_P\) and \(p^*_P>q^*_P\), \(\mathrm {C}<\mathrm {D}\) (as a consequence \(\mathrm {C}<P_4(\mathrm {E}_\mathrm {P})\); so \(\frac{\mathrm {C}}{P_4(\mathrm {E}_\mathrm {P})}<1\)). Hence, with the increase of b, \(P_4(\mathrm {E}_\mathrm {P})\) increases faster than \(\mathrm {C}\), that is, the slope of \(P_4(\mathrm {E}_\mathrm {P})\) is greater than the slope of \(\mathrm {C}\). Nevertheless, even with a very large slope of \(P_4(\mathrm {E}_\mathrm {P})\) and a very small slope of \(\mathrm {C}\), \(\frac{\mathrm {C}}{P_4(\mathrm {E}_\mathrm {P})}\) still increases, as shown in Fig. 7. So, if b increases, \(d(\mathrm {T}_\mathrm {F},\mathrm {E}_\mathrm {P})\) increases. \(\square \)
Dependence of \(P_4(\mathrm {E}_\mathrm {P})\) (left figure) and \(\mathrm {C}\) and \(d(\mathrm {T}_\mathrm {F},\mathrm {E}_\mathrm {P})\) (right figure) on b
Theorem 11\(\varDelta _0=0\) iff (\(p^*_F=q^*_F\)) or (\(p^*_P=q^*_P\)). And \(\varDelta _0>0\) if (\(p^*_F>q^*_F\)) and if (\(p^*_P>q^*_P\)).
Proof
\(\square \)
Rights and permissions
About this article
Cite this article
Tešić, M. Confirmation and the generalized Nagel–Schaffner model of reduction: a Bayesian analysis. Synthese 196, 1097–1129 (2019). https://doi.org/10.1007/s11229-017-1501-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-017-1501-1