Abstract
Elliott Sober is both an empiricist and an instrumentalist. His empiricism rests on a principle called actualism, whereas his instrumentalism violates this. This violation generates a tension in his work. We argue that Sober is committed to a conflicting methodological imperative because of this tension. Our argument illuminates the contemporary debate between realism and empiricism which is increasingly focused on the application of scientific inference to testing scientific theories. Sober’s position illustrates how the principle of actualism drives a wedge between two conceptions of scientific inference and at the same time brings to the surface a deep conflict between empiricism and instrumentalism.
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Notes
The PA or what is often called the “conditionality principle” is a corner-stone in Bayesian inference. For an excellent treatment of Bayesian inference, see (Howson and Urbach 2006). The paper does not try to defend or criticize the PA.
Godfrey-Smith has pointed out that when Sober’s earlier view on simplicity is domain-specific; his current use of the AIC framework that exploits the notion of simplicity is, however, “domain-independent.” (Godfrey-Smith 1999, p. 181).
Evans (2013) has recently questioned whether the likelihood principle is at all a “must” for a Bayesian. He argues that it is completely irrelevant to Bayesianism.
Hereafter we have dropped the use of “epistemological” features of the PA when we address the PA unless otherwise stated. We return to this usage when we distinguish epistemological features of the PA from its decision-based feature in Sect. 4.
How to spell out “observable” and the terms employed to define “observable” have generated a great deal of discussion in recent philosophy of science. To appreciate the complex nature of modal arguments in this context, here are some references (which are by no means exhaustive) to this literature. Ladyman thinks that there is a tension in constructive empiricism in trying to incorporate both modality and observability together (Ladyman 2000). In contrast, Hanna has defended constructive empiricism against Ladyman’s objection (see Hanna 2004). For the question of observability, see also van Fraassen (1980 and 1989). For a critique of van Fraassen’s work, see (Kitcher 1993, 2001).
See footnotes 5 and 7 for the question of observability.
One might worry that anyone including van Fraassen will ever disagree with Sober regarding the implications of the following example. In an email communication, Joseph Hanna wrote to one of the authors of the paper the following. He writes that a theory is empirically adequate, according to van Fraassen, if and only if it saves all the actual observables, past, present and future. So, empirical adequacy depends on all sorts of “observables” no one has ever, in fact, observed. But these observables are actual events that occur in the history of the world. On the other hand, the actual status of a theory—depends only on actual observations. So, Hanna concludes that there is nothing in van Fraassen’s empiricism that violates PA and respects PPE (see Hanna 2004 for his stance toward constructive empiricism). We sympathasize with Hanna along with those who might share the same concern with him that Sober errors in mistakenly attributing the PPE to van Fraassen. However, we are not interested in whether Sober is right in thinking that van Fraassen’s position violates PA. We are only interested in the kind of argument Sober employs to defend the principle.
One worry against Sober’s work here is that “the notion of evidence” being possible “is slippery here”. To do a full justice to this worry we refer the reader to footnote 9 as the ensuing discussion will help to understand both this point and our stance toward Sober’s view.
This is in continuation with the previous footnote regarding why one might think that the notion of “possible” in the notion of evidence being possible is a “slippery slope”. The worry is that if we imagine that there is not actually any test for X, then there is a sense in which evidence for the test is impossible. But such a non-actual test is still possible (in a broader sense) and so the evidence from it is possible. There is a scope of ambiguity concerning the example of the test for X being possible in one sense and impossible in others. The reason for this is that to which does the variable “X” refer? Does “X” refer to the (infallible) possible test which has not been carried out for diabetes or the test for small-pox which does not exist in Sober’s example, yet it is still possible in a broader sense? However, a charitable interpretation of the worry might be able to reveal its true spirit. It is likely that the imaginary critic who presents this worry means by “the test for X,” a possible test for small-pox which does not exist at this point, but it is still possible in some sense. We agree with the critic that although Sober’s example clearly assumes that there is no such (infallible) test possible for small-pox, such a test for small-pox is undeniably “possible” in some broader sense. The relevant question, however, is “does such a test for X being possible pose a threat to Sober’s criticism of constructive empiricism”? We have two comments here. First, if we allow such a test for X to be possible then this will force us to revise the PPE in terms of the Revised Principle of Possible Evidence (RPPE). According to the RPPE, if one’s actual evidence for the hypothesis H2 is about as good as one’s actual evidence for another hypothesis H1, and although it is possible to have evidence for both H1 and H2, but it is only possible to have better evidence for H1, but not for H2, we should form the judgment about the correctness of H1 solely based on better possible, nonfactual evidence. However, the RPPE will still be regarded by Sober as dubious as it violates the PA. In short, this revision will complicate the PPE, without disputing his rejection of the principle. Our second comment has to do with Sober’s own example (see, Sober 1993). Since we would to like to follow him in this regard, we would continue to stick to his usage by saying that there is a possible infallible test for diabetes, but there is no such “possible” test for small-pox; consequently, the PPE is the principle to be investigated and not any of its variants. The point of this along with footnote 7 is to discuss how Sober has construed the PPE and its relationship to the PA without trying to evaluate whether his argument against constructive empiricism is tenable.
See Miller 1987 for a critique of Sober’s account of simplicity. Miller argues that one needs to incorporate causal simplicity in theory choice. In this case, as in many others, formal simplicity which Sober has advanced must be distinguished from causal simplicity. As Richard Miller (1987, 247) reminds us, “by adding a variety of novel propositions, without any corresponding simplification, evolutionary theory reduces the formal simplicity of science. [But] An enormous gain in causal simplicity results.” According to Miller, the regularities that we observe in the variation of species, for example, have causal explanations only when the evolutionary explanations are added.
Whether Sober’s view is correct here goes beyond the objective of this paper.
For his recent position about the AIC, see (Sober 2008).
This objection has been raised by a critic after reading the previous version of the paper.
Sober defends a likelihood approach to both evidence and scientific inference. Although there are fundamental differences between a likelihoodist and a Bayesian, for our present purpose, those differences are unimportant.
Van Fraassen’s constructive empiricism that combines empiricism with instrumentalism also leads to a contradiction (see Bandyopadhyay 1997).
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Acknowledgments
We would like to thank Prajit Basu, John G. Bennett, Robert Boik, Abhijit Dasgupta, Michael Evans, Roberto Festa, Dan Flory, Malcolm Forster, Jayanta Ghosh, Dan Goodman, Jason Grossman, Joseph Hanna, Bijoy Mukherjee, Megan Raby, Tasneem Sattar, Mark Taper, Susan Vineberg, and C. Andy Tsao for discussion/comments regarding the content of the paper. We are thankful to three anonymous referees of this journal and a dozen other referees from different journals for their helpful feedback. We owe special thanks to Elliott Sober for serving as an official commentator for our paper at the APA divisional meetings, and John G. Bennett for several correspondences regarding the issues raised in the paper. The research for the paper has been funded by our university’s NASA’s Astrobiology Center (Grant No. 4w1781) with which some of the authors of this paper are affiliated.
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Appendix A: Proof of equivalent ∇AICs if the equivalent condition for two experiments (PAT) holds
Appendix A: Proof of equivalent ∇AICs if the equivalent condition for two experiments (PAT) holds
Recall the PAT: two experiments, E1 and E2, provide equal evidential support for a model, that is, the parameter in question, if and only if their likelihood functions are proportional to each other as functions of the models, and therefore, any inference about the model based on these experiments should be identical. If we assume that under two different experimental designs, E1 and E2, their likelihoods have the same proportionality under two different models regarding the same parameter, then, it follows from the PAT that both models provide equal evidential support for the parameter. First, define \( L_{1} (\hat{\theta }) \) and \( L_{1} (\theta ) \) as the likelihoods under E1 for the model using the MLE of θ and the model with θ = 0.5, respectively. We could define \( L_{2} (\hat{\theta }) \) and \( L_{2} (\theta ) \) similarly. We also define the likelihood ratio for each experiment as the constants c1 and c2 in \( L_{1} (\hat{\theta }) =\text{c}_{1} L_{1} (\theta ) \) and \( L_{2} (\hat{\theta })=\text{c}_{2} L_{2} (\theta ) \). These constants are also the likelihood ratios reported in the Table (1) above, which weighs the evidence between the two models in either E1 and E2. Then if c1 = c2 = c, then we find equal evidential support using the likelihood ratio, satisfying the PAT.
Now we show how if the PAT holds, then ∇AIC will also provide equal evidence between the hypotheses. First, we assume that the AIC is reduced for the unconstrained model. Also, we use dim(θ) to indicate the number of parameters that are estimated. Then consider the definition of ∇AIC for, say, E1:
But \( {{L_{1} (\theta )} \mathord{\left/ {\vphantom {{L_{1} (\theta )} {L_{1} (\hat{\theta })}}} \right. \kern-0pt} {L_{1} (\hat{\theta })}} = 1/c = {{L_{2} (\theta )} \mathord{\left/ {\vphantom {{L_{2} (\theta )} {L_{2} (\hat{\theta })}}} \right. \kern-0pt} {L_{2} (\hat{\theta })}} \) by the PAT. Thus, ∇AIC will be the same under E1 or E2 if the PAT holds and the difference in the dimensions of \( \theta \) and \( \hat{\theta } \) are the same across the experiments. Note that this is a stronger condition than just assuming that the PAT holds since it also involves the number of estimated parameters in the constrained and unconstrained models, in addition to proportionality of the likelihoods.
In our example, we can see specifically how the ∇AIC provides the mathematical result of equivalent evidence between the hypotheses regardless of which experiment is performed, E1 or E2. For the Binomial model, E1, the likelihood is \( f_{1} (20|\theta ) = \left( {\begin{array}{*{20}c} {30} \\ {20} \\ \end{array} } \right)\theta^{20} (1 - \theta )^{10} \). With θ = 0.5, under the constrained hypothesis, the likelihood is \( f_{1} (20|\theta ) = \left( {\begin{array}{*{20}c} {30} \\ {20} \\ \end{array} } \right)0.5^{20} (0.5)^{10} \). The AIC is calculated as −2log(L(θ)) + 2dim(θ), which, for the constrained model is \( AIC_{\theta } = - 2\log \left[ {\left( {\begin{array}{*{20}c} {30} \\ {20} \\ \end{array} } \right)0.5^{20} (0.5)^{10} } \right] + 2\dim (\theta ) = - 2\log \left( {\begin{array}{*{20}c} {30} \\ {20} \\ \end{array} } \right) -\!2\log 0.5^{20} (0.5)^{10} + 2*0 \) \( = - 2\log \left( {\begin{array}{*{20}c} {30} \\ {20} \\ \end{array} } \right) - 2\log 0.5^{20} (0.5)^{10} \). For the unconstrained model, we find \( AIC_{{\hat{\theta }}} = - 2\log \left[ {\left( {\begin{array}{*{20}c} {30} \\ {20} \\ \end{array} } \right)\hat{\theta }^{20} (1 - \hat{\theta })^{10} } \right] + 2\dim (\theta ) = - 2\log \left( {\begin{array}{*{20}c} {30} \\ {20} \\ \end{array} } \right) - 2\log 0.67^{20} (0.33)^{10} + 2*1 \) and the ∇AIC to be \( \nabla AIC = AIC_{\theta } - AIC_{{\hat{\theta }}} = - 2\log \left( {\begin{array}{*{20}c} {30} \\ {20} \\ \end{array} } \right) - 2\log 0.5^{20} (0.5)^{10} + 0 + 2\log \left( {\begin{array}{*{20}c} {30} \\ {20} \\ \end{array} } \right) + 2\log 0.67^{20} (0.33)^{10} + 2 = - 2\log 0.5^{20} (0.5)^{10} + 2\log 0.67^{20} (0.33)^{10} + 2 \), showing that the constant which is the only difference in the likelihoods between E1 and E2, cancels when we consider ∇AIC in this situation.
For the Negative Binomial model, E2, the likelihood is \( f_{2} (20|\theta ) = \left( {\begin{array}{*{20}c} {29} \\ 9 \\ \end{array} } \right)\theta^{20} (1 - \theta )^{10} \). With θ = 0.5 under the constrained hypothesis, the likelihood is \( f_{2} (20|\theta ) = \left( {\begin{array}{*{20}c} {29} \\ 9 \\ \end{array} } \right)0.5^{20} (0.5)^{10} \). So the \( AIC_{\theta } = - 2\log \left[ {\left( {\begin{array}{*{20}c} {29} \\ 9 \\ \end{array} } \right)0.5^{20} (0.5)^{10} } \right] + 2\dim (\theta ) = - 2\log \left( {\begin{array}{*{20}c} {29} \\ 9 \\ \end{array} } \right) - 2\log 0.5^{20} (0.5)^{10} + 2 \cdot 0 \) \( = - 2\log \left( {\begin{array}{*{20}c} {29} \\ 9 \\ \end{array} } \right) -\!2\log 0.5^{20} (0.5)^{10} \). For the unconstrained model, we find \( AIC_{{\hat{\theta }}} = - 2\log \left[ {\left( {\begin{array}{*{20}c} {29} \\ 9 \\ \end{array} } \right)\hat{\theta }^{20} (1 - \hat{\theta })^{10} } \right] + 2\dim (\theta ) = - 2\log \left( {\begin{array}{*{20}c} {29} \\ 9 \\ \end{array} } \right) - 2\log 0.67^{20} (0.33)^{10} + 2 \cdot 1 \) and the ∇AIC to be \( \nabla AIC \, = \, AIC_{\theta } \, - \, AIC_{{\hat{\theta }}} \, = \, - 2\log \left( {\begin{array}{*{20}c} {29} \\ 9 \\ \end{array} } \right) - 2\log 0.5^{20} (0.5)^{10} \, + \, 0 \, + \, 2\log \left( {\begin{array}{*{20}c} {29} \\ 9 \\ \end{array} } \right) \, + \, 2\log 0.67^{20} (0.33)^{10} \, + \, 2= - 2\log 0.5^{20} (0.5)^{10} \, + \, 2\log 0.67^{20} (0.33)^{10} \, + \, 2 \) as in E1, showing that since the constant cancels in either experiment, that same ∇AIC is found. The constant also cancels in the likelihood ratio when the likelihood ratios in either E1 or E2 are considered.
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Bandyopadhyay, P.S., Greenwood, M., Brittan, G. et al. Empiricism and/or Instrumentalism?. Erkenn 79 (Suppl 5), 1019–1041 (2014). https://doi.org/10.1007/s10670-013-9567-8
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DOI: https://doi.org/10.1007/s10670-013-9567-8