Inference to the More Robust Explanation

Inference to the More Robust Explanation ABSTRACT There is a new argument form within theoretical biology. This form takes as input competing explanatory models; it yields as output the conclusion that one of these models is more plausible than the others. The driving force for this argument form is an analysis showing that one model exhibits more parametric robustness than its competitors. This article examines these inferences to the more robust explanation, analysing them as variants of inference to the best explanation. The article defines parametric robustness and distinguishes it from more familiar kinds of robustness. The article also argues that parametric robustness is an explanatory virtue not subsumed by more familiar explanatory virtues, and that the plausibility verdicts in the conclusions of inferences to the more robust explanations are best interpreted as guidance for research activity, rather than claims about likely truth. 1 Introducing Inference to the More Robust Explanation 2 Inference to the More Robust Explanation in the Study of Apoptosis 2.1 Regulating apoptosis 2.2 Competing models and evidential indecision 2.3 Measuring robustness 2.4 Robustness as a guide to plausibility 2.5 Varieties of robustness 3 Inference to the More Robust Explanation as Inference to the Best Explanation 3.1 The structure of inference to the best explanation 3.2 Parametric robustness as an explanatory virtue rather than an explanandum 3.3 Relation of parametric robustness to other explanatory virtues 4 Epistemological Significance of Inference to the More Robust Explanation 4.1 Plausibility in practice 4.2 Plausibility in principle 5 Conclusion 1 Introducing Inference to the More Robust Explanation In 2002, Morohashi and colleagues analysed the parameter sensitivities of competing models for explaining mitotic phase oscillations in early Xenopus embryos. Their analysis showed that ‘the structure of the ′98 model is capable of providing highly robust oscillatory behaviour in a manner far exceeding the capabilities of the ′91 model' ([2002], p. 28). They inferred, in consequence, that the ′98 model is more plausible than the ′91 model because its parameters are less sensitive to variations. This argument marks the first appearance in the biological literature of scientists treating parametric robustness as a measure of plausibility. I shall call it an inference to the more robust explanation, or IMRE for short. This argument form has been received favourably. For example, comparing competing models for explaining mitochondrial outer membrane permeabilization (MOMP) during apoptosis, Chen and colleagues ([2007], p. 5148) argue that one of the models is more plausible than the other because it ‘confers a better robustness' with respect to parameter variations. IMREs bear close similarities to more familiar inferences to the best explanation (IBEs). Both take as input a collection of explanatory hypotheses. Both provide as output a verdict that one of the explanatory hypotheses is better than the others, either by virtue of being more plausible (IMREs) or by virtue of being more likely to be correct (IBEs). Both also warrant this output by comparing the competing explanatory hypotheses directly to each other: IMREs compare how much robustness to parameter variation each hypothesis has; IBEs, how much explanatory power each hypothesis has. There are dissimilarities, too. For example, IMREs are restricted to comparing mathematical models; IBEs are not. Despite these differences, I propose to analyse IMREs as a kind of IBE. This is, in part, for the sake of better understanding the logical structure and epistemic force of IMREs. Specifically, with respect to logical structure, extant instances of IMRE are not clear about the role of parametric robustness: Is it supposed to be an explanandum phenomenon, so that models conferring better robustness thereby explain more? Or is it, at least descriptively, a kind of explanatory virtue in its own right, so that models conferring better robustness might explain better without necessarily explaining more? With respect to epistemic force, moreover, extant instances of IMRE are not clear about the epistemological significance of comparative plausibility verdicts: does plausibility correlate with likelihood of truth, or does it have some other significance? Beyond aiming to better understand a relatively new argument form in contemporary theoretical biology, I also propose to analyse IMREs for the sake of better understanding the kinds of robustness that interest contemporary biologists. Philosophers of biology are likely more familiar with robustness in Levins's ([1966]) sense, where a model is robust insofar as its derivational consequences are also consequences of similar models with different state variables or functional structures. There is some debate about whether this kind of robustness indicates likely truth (see Weisberg [2006]). But IMREs seem to involve a different kind of robustness. A model is robust to parameter variations insofar as changing the values of its parameters, without changing its state variables or functional structure, does not change its derivational consequences. Prima facie, models that are robust in this sense need not be robust in Levins's sense. Accordingly, I address two issues about robustness: first, whether and how parametric robustness differs from Levins’s kinds of robustness; and second, whether parametric robustness indicates likely truth. I begin in Section 2 with a case study of IMRE. The case centres on attempts to understand apoptosis. I use it to explicate the kind of robustness at work in IMREs, and to tether subsequent analysis to scientific practice. I proceed, in Section 3, to reconstruct IMREs on the model of IBE. I argue that parametric robustness is not an explanandum and that those using IMRE do not treat parametric robustness as a datum to be explained. I argue, further, that parametric robustness is an explanatory virtue in its own right and that it is not subsumed by a standard range of other explanatory virtues. In Section 4, on the epistemological significance of IMRE, I argue that the plausibility verdicts in the conclusions of IMREs are best interpreted as guidance for research activity rather than claims about likely truth. I also explore arguments from scientific realists about the connection between explanatory power and truth.1 2 IMRE in the Study of Apoptosis 2.1 Regulating apoptosis Apoptosis is programmed cell death for multicellular organisms. It helps to make room for cell proliferation, limit pathogen spread, and eliminate cells that can trigger autoimmune responses or overwhelm support systems. In contrast to necrosis, a passive process during which mitochondria swell and rupture the plasma membrane, apoptosis occurs when nuclei and cytoplasm—including mitochondria—shrink, become encased in ‘apoptotic bodies’, and send signals for being engulfed by nearby phagocytic cells. But apoptosis must be regulated: too much promotes ischemic conditions and neurodegeneration; too little promotes cancer and autoimmune diseases (Saikumar et al. [1999], pp. 489–91; Czabotar et al. [2014], p.49). The principle components regulating the intracellular mechanism for apoptosis belong to the B-cell lymphoma 2 (Bcl-2) protein family. Some proteins in this family—Bcl-2, Bcl-xL, Bcl-w, A1, Mcl-1—inhibit apoptosis. Others promote it. The BH3-only proteins—including Bid, Bim, Puma, Bad, and Noxa—sense and integrate cellular damage signals. The multi-domain proteins Bax and Bak form the mitochondrial apoptosis channel and make permeable the mitochondrial outer membrane. Bax and Bak are especially important, because MOMP makes cell death inevitable. Other Bcl-2 family proteins are important insofar as they help or hinder the activation of Bax and Bak. There remains disagreement concerning how Bcl-2-type proteins inhibit MOMP and thereby preserve cell viability, how BH3-only proteins facilitate MOMP and thereby promote cell death, how the entire Bcl-2 switching network regulates Bax–Bak activation, and how Bcl-2 family proteins are controlled by signals from proteins outside their family (Chen et al. [2007], p. 5143; Czabotar et al. [2014], pp. 50–5). In 2006–7, there was a flurry of research activity about proposed mechanisms for the Bcl-2 apoptotic switch. The issues of contention were, first, whether BH3-only proteins activate Bax–Bak directly and, second, how anti-apoptosis proteins inhibit Bax–Bak activation. 2.2 Competing models and evidential indecision Researchers offered for consideration two types of model. While advocates for each type reported new experimental evidence in their favour, the cumulative effect was a situation in which contradictory evidence failed to support either model type over its rival (see Van Delft and Huang [2006]). Some researchers favoured a ‘direct' activation model, according to which BH3-only proteins directly activate Bax–Bak while anti-apoptotic proteins inhibit this activation by sequestering the BH3-only proteins (see Galonek and Hardwick [2006]). Other researchers favoured an ‘indirect' activation model, according to which BH3-only proteins indirectly activate Bax–Bak through binding and inhibiting the anti-apoptotic proteins, and in which these anti-apoptotic proteins inhibit Bax–Bak activation by sequestering the Bax–Bak proteins (see Youle [2007]). Both models were taken to have a variety of experimental evidence in their favour. Kim et al. ([2006]) provided experiment-driven arguments for each component of the direct activation model. For example, they argued that certain BH3-only proteins directly activate Bax–Bak because (i) they effectively release cytochrome-c—a central component of the electron transport mechanism for mitochondria—from isolated mitochondria and thereby act as potent death stimulators; (ii) retroviral transduction of these proteins in mouse embryonic fibroblasts resulted in a 65% death rate after twenty-four hours; and (iii) substituting or deleting amino-acid residues from these proteins abolishes their cytochrome-c releasing activity (Kim et al. [2006], p. 1349). They argued, further, that Bax and Bak are not themselves inhibited by anti-apoptotic proteins because (iv) a mutant form of Bak, unable to interact with the anti-apoptotic proteins, does not introduce apoptosis without other death stimuli, and so these proteins do not inhibit an otherwise active-by-default Bak; and (v) over-expressing the anti-apoptotic proteins in double-knockout cells reconstituted with mutant forms of Bak and Bax—forms unable to interact with those proteins—prevents apoptosis, and so the proteins do not prohibit apoptosis by directly interacting with Bax–Bak (Kim et al. [2006], pp. 1350–1). However, Willis and colleagues ([2007]) provided experiment-driven arguments for a competing indirect activation model. For example, they argued that BH3-only proteins can initiate apoptosis without binding directly to Bax–Bak because (i) in lysates of human embryonic kidney cells, BH3-only proteins bind to anti-apoptotic proteins, but not to endogeneous Bak; and (ii) mutant forms of Bid and Bim, able to bind to anti-apoptotic proteins but not to Bax, kill mouse embryonic fibroblasts just as effectively as wild-type Bid and Bim (Willis et al. [2007], p. 857). They argued, further, that apoptosis occurs when BH3-only proteins inhibit anti-apoptotic proteins, because (iii) over-expressing Bad and Noxa—so-called ‘sensitizer' BH3-only proteins that neutralize anti-apoptotic proteins but lack the capacity to initiate apoptosis on their own—induces apoptosis and cytochrome-c release in mouse embryonic fibroblasts, and this happens when the so-called ‘activator' BH3-only Bim and Bid proteins are absent (Willis et al. [2007], p. 858). Because the experimental results reported by Kim and colleagues lent support to the direct activation model while the results reported by Willis and colleagues lent support to the indirect activation model, and because neither research group offered evidence to undermine or overturn the assumptions that the other group used to interpret their experimental data, the total evidential situation failed to support either model over its rival. Chen and colleagues ([2007], p. 5143) tactfully summarized the situation as one where ‘proving which model is more telling by means of experimental approaches seems difficult thus far'. Their diagnosis for the situation was two-fold. First, the complexity of biological signalling networks means that wet lab experiments tend to have little probative power. Second, competing research groups often examine different experimental systems (Chen et al. [2007], p. 5148). But rather than recommend an agnostic attitude toward the models until better evidence is available, and rather than develop some kind of hybrid alternative model, Chen and colleagues use an IMRE in order to infer that the direct activation model is more plausible than the indirect activation model. 2.3 Measuring robustness Chen and colleagues are interested in the direct activation model and its indirect activation competitor insofar as both purport to explain how the Bcl-2 switching network regulates Bax–Bak activation. They grant that each model (potentially) explains this phenomenon, presumably because each specifies a mechanism for how members of the Bcl-2 protein family regulate Bax–Bak activity. Moreover, while each model explains some but not all of the total then-available evidence, Chen and colleagues do not prefer one model over the other on the basis of which pieces of evidence are more important to accommodate. Nor do they prefer one over the other on the basis of which is more likely relative to the total evidence: because each model is inconsistent with some evidence (as they interpret it), each has a null posterior probability relative to the total evidence. Nor, finally, do they appeal to intuitions, dismissing them as unreliable ‘because of the complexity of the system' (Chen et al. [2007], p. 5148). Instead, Chen and colleagues use analytical considerations about robustness in order to measure each model’s plausibility. They provide, in fact, several robustness-based measures of comparative plausibility. Each measure involves (i) a mathematical criterion for whether a model exhibits some important and empirically well-attested feature, (ii) a mathematical criterion for determining the magnitude of parameter variation on some mathematical model, and (iii) a mathematical criterion for measuring how much robustness a model has for exhibiting some selected feature. Chen and colleagues argue that most of their measures favour the direct activation model over the indirect activation model. I discuss one measure in detail, for the sake of clarifying the general argumentative strategy. I then only mention the others, proceeding instead to discuss Chen and colleagues' results. First, Chen and colleagues observe that ultra-sensitivity is an important, and empirically well-attested, feature of the Bcl-2 apoptotic switch. The ultra-sensitivity of this switch means that apoptotic stimuli produce small responses initially, and increasing the stimuli produces increasingly stronger responses. Chen et al. ([2007], Supplementary Material) propose that a model exhibits ultra-sensitivity when its relative amplification coefficient, nR, is greater than one. This coefficient compares, in essence, the response ratio for the model—defined as RSX = S/X × dX/dS for stimulus level S and model-determined response level X—to the response ratio for some reference model (namely, the Michaelis–Menten equation). When nR>1, the model's response to a given stimulus is larger than the reference model's response to that same stimulus. Chen and colleagues take this to mean that the model exhibits ultra-sensitivity, because the model is more sensitive to the stimulus than the reference model. Second, Chen and colleagues compare the direct activation and indirect activation models with respect to their capacity for exhibiting ultra-sensitivity. Both models are sets of differential equations. For example, the direct activation model includes the equation   d[InBax]/dt=((kBax×[Bax])−(kInBax×[Act]×[InBax])), where the quantities in brackets represent concentrations of various mechanism components and subscripted k's represent various parameters for rate constants (see Chen et al. [2007], p. 5144 and Supplementary Material). The concentrations for each mechanism component, moreover, are given by further parameterized equations. For instance, Chen et al. [2007], p. 5145) define the concentration of BH3-only proteins as   [BH3]=([BH3]0+F)×[BH3]0, where [BH3]0 is a parameter representing the initial concentration of BH3-only proteins and F represents an m-fold increase of that concentration (0 ≥ m ≥ 20) with respect to some input stimulus. For the sake of tractability, Chen et al. [2007], pp. 5144–5) consider simplified variants of each model, preserving core information but reducing computational complexity. For example, they treat all anti-apoptotic proteins as if they are the same and they treat the ‘activator' BH3-only proteins (Bax and Bak) as if they are the same protein. The (simplified) direct activation and indirect activation models yield specific results (for purposes of explanation and prediction) only when the parameters for each model have specific values. Different parameter values, moreover, yield different results. Varying a parameter value involves changing some reference value for a parameter to some different value. Chen et al. ([2007], p. 5145) define the magnitude of parameter variation, PV, for the ith parameter of a model as follows:   PV=pi/pi,ref, where pi,ref is the value of the ith parameter as given by a fixed set of reference values and pi is the altered value of that same parameter. Chen et al. [2007], p. 5144, Table 2) fix the reference values for each parameter by either extracting values from existing literature or else estimating values with educated guessing. Given these reference values, they define the total order of magnitude of parameter variation, TPV, as the sum of (the absolute value of the common logarithm of) the parameter variations PV for each of the parameters in a model ([2007], p. 5146). (For example, for a model with two parameters, A and B, reference values pA,ref = 1 and pB,ref = 1, and altered values pA = 10 and pB = 10, TPV would equal two.) This allows them to calculate how much variation there is between a set of reference values for a model's parameters and a set of altered values for those same parameters. When TPV is greater than zero, there is some degree of variation between parameter value sets; as TPV increases, the amount of this variation also increases. Third, given reference values for the parameters in each of the direct activation and indirect activation models, given TPV as a measure for the amount of variation on these values, and given the condition that a model exhibits ultra-sensitivity whenever nR > 1, Chen and colleagues calculate how much robustness each model has for exhibiting ultra-sensitivity. For the sake of computational tractability, they randomly select 3000 well-defined parameter sets for each model. They choose each set from a range of possible values varying across four orders of magnitude (for example, from 0.01 to 1000), subject to the condition that TPV always computes as greater than zero. Then, for each set, they calculate the model's relative amplification coefficient, nR, with the specific parameter values. The result is 3000 calculated values of nR for the direct activation model, and 3000 calculated values of nR for the indirect activation model. The amount of robustness a model has for exhibiting ultra-sensitivity is then a straightforward sum of the number of parameter sets out of 3000 for which the model yields nR > 1. 2.4 Robustness as a guide to plausibility Chen et al. [2007], p. 5147) find that 996 of 3000 randomly selected parameter sets exhibit ultra-sensitivity (nR > 1) for the direct activation model, while only 217 of 3000 randomly selected parameter sets exhibit ultra-sensitivity for the indirect activation model. They infer that ‘the direct activation model confers a better robustness of ultrasensitivity with respect to global parameter variations' than does the indirect activation model ([2007], p. 5147). This means that there are more sets of parameter values for which the direct activation model exhibits ultra-sensitivity. Because epistemic limitations and environmental variations make it likely that set of reference values for each model's parameters includes some degree of error, a ‘better robustness of ultrasensitivity with respect to global parameter variations' means that a model is more likely to hold for real biological systems. For even if the actual values for various parameters differ from the reference values, more robustness means that the model is more likely to exhibit ultra-sensitivity even when its parameters have their actual values (as opposed to the incorrect reference values). Chen and colleagues realize that a model might be more robust for exhibiting ultra-sensitivity and yet less robust for exhibiting some different but similarly important and empirically well-attested feature. Rather than argue that one particular feature, such as ultra-sensitivity, trumps all others in importance, Chen and colleagues instead identify a handful of important and empirically well-attested features of the Bcl-2 apoptotic switch. These are features that both the direct activation and indirect activation models exhibit when the parameters for those models have their respective reference values. Chen et al. ([2007], pp. 5145-7) group names the features as follows: Ultra-sensitivity: Apoptotic stimuli produce small responses initially, and increasing the stimuli produces increasingly stronger responses. Range: There is some threshold such that, in non-apoptotic conditions, the basal activation level of Bax is below this threshold while, when apoptotic stimuli are present, the basal activation level for Bax is above this threshold. Inhibition: There is some threshold such that, when apoptotic stimuli are present, the activity of Bax exceeds this threshold, while when anti-apoptotic proteins are present, the activity of Bax is below this threshold. Insensitivity: There is relatively low Bax activity in non-apoptotic conditions, even with small perturbations of the apoptotic stimuli. Following the procedure outlined for the case of ultra-sensitivity, Chen et al. ([2007], p. 5147) calculate the amount of robustness each model has for exhibiting each of these features. I summarize the results of their computations in Table 1, where the number in each cell indicates, for a given model and feature, how many parameter sets for that model, from amongst 3000 randomly selected possibilities, exhibit that feature. Each number indicates how robust the model is for exhibiting the selected feature, and higher numbers mean more robustness. Table 1. Single-feature robustness measures   Ultra-sensitivity  Range  Inhibition  Insensitivity  Direct activation model  996  891  744  729  Indirect activation model  217  153  1344  630    Ultra-sensitivity  Range  Inhibition  Insensitivity  Direct activation model  996  891  744  729  Indirect activation model  217  153  1344  630  The number in each cell indicates how many parameter sets, from a randomly selected set of 3000 sets, exhibit the column-heading feature (based upon Chen et al. [2007], pp. 5147–8) For most features, the direct activation model exhibits better robustness than the indirect activation model. For the sake of having a more comprehensive robustness measure, and for providing a more absolute ranking of the models, Chen and colleagues also compute how many parameter sets simultaneously exhibit all four features. This yields a ranking with no ambiguity: the direct activation model exhibits all four features in 316 of 3000 sets, while the indirect activation model exhibits all four features in only 31 of 3000 sets. Chen and colleagues argue that the unified comparison is less likely to be misleading, because while all four features are important, the presence of one does not guarantee the presence of the others. Accordingly, they use the results of the unified comparison to conclude that ‘the direct activation model confers a better robustness in contrast to the indirect activation model' ([2007], p. 5148) . Chen and colleagues use IMRE to infer that the direct activation model is more plausible than its indirect activation competitor. Their underlying assumption is that ‘faithful models should not rely on fine tuned parameters to reflect biological systems' (Chen et al. [2007], p. 5148). This is another way of saying that behaviours of biochemical networks should be robust to parameter variations. Why this might be so, and what Chen and colleagues might mean in characterizing the most parametrically robust model as more plausible than its competitor, is an issue to which I return in Section 4. 2.5 Varieties of robustness When inferring that the direct activation model is more plausible than its indirect activation competitor, Chen and colleagues invoke a kind of robustness that, following Weisberg and Reisman ([2008], p. 115), I shall call ‘parametric robustness’. A model is parametrically robust for some feature of interest when the model exhibits that feature for two or more sets of parameter values. The more sets of parameter values for which the model exhibits the feature, the more robust the model is for that feature. A model is said to be parametrically robust without qualification, moreover, whenever and to the extent that it is parametrically robust for all features of interest. According to Chen and colleagues' analysis, for example, the direct activation and indirect activation models are robust for ultra-sensitivity and inhibition; the direct activation model is more robust for ultra-sensitivity and the indirect activation model is more robust for inhibition; and the direct activation model is more robust in general than is the indirect activation model (see Table 1). Determining the extent to which a model is parametrically robust requires holding fixed the model's state variables and the functional structure among those variables. This differs from Levins’s ([1966]) sense of robustness, according to which a model is robust insofar as its derivation consequences are also consequences of similar models with different state variables or functional structures. For example, the Lotka–Volterra model is robust in Levins’s sense for exhibiting the Volterra principle, according to which prey populations increase relative to the predator population upon pesticide application. For this principle is a derivational consequence of the Lotka–Volterra model, and it is also a consequence of a more complex model that includes a state variable for prey population carrying capacity (see Weisberg [2006], pp. 734–6). This kind of robustness—Weisberg and Reisman ([2008], p. 116) call it ‘structural robustness’—is distinct from parametric robustness. Moreover, both structural and parametric robustness are distinct from a third kind of robustness that Levins also seems to have in mind. Weisberg and Reisman ([2008], p. 120) call this third kind ‘representational robustness’ because it involves varying functional relationships among state variables, rather than the state variables themselves or their associated parameter values. (For similar distinctions among kinds of robustness, in a more mathematically oriented context, see Gunawardena [2010], p. 22.) Whether the direct activation and indirect activation models are robust in either of Levins’s senses depends upon whether ultra-sensitivity and various other features (or mathematical representations thereof) are derivational consequences of models with altered differential equations. But Chen and colleagues change neither the state variables nor functional structures in the differential equations for the direct activation and indirect activation models. They change only the values of parameters in those equations. They do not, accordingly, investigate whether the direct activation and indirect activation models are structurally or representationally robust. Hence, their inference from model robustness to model plausibility differs from the kinds of robustness-driven inferences more often of interest to philosophers of biology (see Orzack and Sober [1993]; Weisberg [2006]; Weisberg and Reisman [2008]; Odenbaugh [2011]). 3 Inference to the More Robust Explanation as Inference to the Best Explanation Chen and colleagues argue that because the direct activation model of the Bcl-2 apoptotic switch is more robust than the competing indirect activation model, the direct activation model is more plausible than its competitor. This argument exemplifies what I call IMRE. I take the discussion in the prior section as sufficient for illustrating how IMREs typically appear in scientific practice and how their central comparative criterion—parametric robustness—differs from more familiar kinds of structural and representational robustness. In this section, I transition from reporting about IMRE to philosophizing about it. The goal is to better understand the inferential structure of IMREs, and in particular how greater parametric robustness is supposed to support greater plausibility. My method for achieving this goal is to treat IMREs as variants of more familiar IBEs. 3.1 The structure of inference to the best explanation When Chen and colleagues infer that the direct activation model of the Bcl-2 apoptotic switch is more plausible than the competing indirect activation model, they do so in part because they take the direct activation model to be more parametrically robust than its competitor. But this is only one of their reasons. They also consider each competing model to have some minimal antecedent plausibility, and they do so because each model explains various important and empirically well-attested features of the Bcl-2 apoptotic switch (namely, ultra-sensitivity, range, inhibition, and insensitivity). Insofar as Chen and colleagues' inference is paradigmatic, we can reconstruct typical instances of IMRE as instantiating the following pattern: Some set of features, F, obtains in some real system(s) of interest. Among competing mathematical models, M1,…, Mn, each would explain F were it true. Mi is more parametrically robust (for exhibiting F) than its available competitors. Therefore, Mi is more plausible than its available competitors. This pattern bears a strong resemblance to a standard pattern for IBE: Some set of phenomena, P, obtain in some real system(s) of interest. Among competing hypotheses, H1,…, Hm, each would explain P were it true. None of its competitors can explain P as well as Hj can. Therefore, Hj is probably true. (Those who endorse this pattern for IBE include Harman ([1965], p. 89), Thagard ([1978], p. 77), Josephson ([1994], p. 5), and Psillos ([2002], p. 614). Lipton ([2004], p. 58) adds that the best explanation also must be ‘good enough for us to make any inference at all'; I suppress this caveat in what follows.) There are several salient differences between these patterns. First, the conclusion of an IMRE is a claim about comparative plausibility. The conclusion of a typical IBE, however, is a claim about (probable or approximate) truth. Harman ([1965], p. 89), for example, characterizes IBE as a form of reasoning in which ‘one infers, from the premise that a given hypothesis would provide a “better” explanation for the evidence than would any other hypothesis, to the conclusion that the given hypothesis is true'. This raises the issue of how to best interpret the significance of comparative plausibility claims. Does being more plausible entail being more likely to be true? If the best explanatory hypotheses are likely to be true and IMRE is a variant of IBE, are models with more parametric robustness also more likely to be true? These questions concern the epistemological significance of IMREs. I postpone them for the subsequent section. Their answers are better motivated after clarifying certain details about the logical structure of IMREs. Those details emerge by considering a further difference between IMREs and IBEs. A second salient difference between the patterns for IMRE and IBE concerns the third premise in each pattern. In an IMRE, this premise is a claim about the comparative parametric robustness of competing models. In an IBE, the premise is a claim about the comparative explanatory power of competing models. I can imagine two potential reasons for this difference, each of which reconstructs the logical structure of IMRE in a different manner. I shall call the first of these reconstructions ‘robustness-as-explanandum’; the second, ‘robustness-as-explanatory-virtue’. I consider each in turn, arguing that the second is more plausible. 3.2 Parametric robustness as an explanatory virtue rather than an explanandum One potential reason for the difference between the comparative claims in IMREs and IBEs is that scientists consider parametric robustness to be an explanandum. Call this the ‘robustness-as-explanandum’ approach.2 Because explananda are features of a model's representational target, this approach treats robustness not only as a formal property of mathematical models but also as an empirical property of target systems. Moreover, if robustness is an explanandum, the most parametrically robust model is more plausible than its competitors because it explains something better than its competitors, namely, the parametric robustness of its target system. Psillos ([2002], pp. 615–6) notes that one hypothesis has more explanatory power than another when, ceteris paribus, it explains more salient phenomena. Hence, according to the robustness-as-explanandum approach, IMREs are only special cases of IBE, in which the most parametrically robust models have the most explanatory power by virtue of better explaining a particularly salient kind of explanandum. There is an alternative approach to IMREs, however, according to which IMREs might qualify as a novel class of IBE rather than a special subclass of more familiar kinds. For there is a second potential reason for the difference between the comparative claims in IMREs and IBEs, namely, that scientists consider parametric robustness to be an explanatory virtue. Call this the ‘robustness-as-explanatory-virtue’ approach. Explanatory virtues are properties that enhance the explanatory power of hypotheses. These virtues often are described as comparative—that is, as virtues that an explanatory hypothesis has in relation to its competitors. Psillos ([2002], pp. 615-6) provides a representative list that includes: Completeness: The hypothesis is the only one amongst its competitors to explain all targeted explananda. Consilience: The hypothesis fits with background knowledge better than its competitors. Importance: The hypothesis explains more salient phenomena than its competitors. Parsimony: The hypothesis uses a proper subset of assumptions from its competitors. Precision: The hypothesis offers more precise explanations than its competitors. None of these virtues are phenomena to be explained by hypotheses. For example, if importance is a virtue, hypotheses that explain more salient phenomena than their competitors have (ceteris paribus) more explanatory power than their competitors, because importance confers that power; they do not have more explanatory power by virtue of explaining why certain target phenomena are salient. Similarly, according to the robustness-as-explanatory-virtue approach, models with more parametric robustness have (ceteris paribus) more explanatory power than their competitors because parametric robustness confers that power; contrary to the robustness-as-explanandum approach, they do not have more explanatory power by virtue of explaining why certain target phenomena are robust. If the robustness-as-explanatory-virtue approach is correct, whether IMREs qualify as a novel kind of IBE depends upon whether standard explanatory virtues subsume the virtue of parametric robustness. If they do—if, for example, exhibiting more parametric robustness is only a special way of exhibiting more precision—then IMREs are mere variants of more familiar IBEs. I shall argue, in Section 3.3, that a range of standard explanatory virtues does not subsume parametric robustness. Parametric robustness, I shall contend, is an explanatory virtue in its own right. Before making that argument, however, I conclude this section by arguing against the robustness-as-explanandum approach. My first argument against the robustness-as-explanandum approach is an argument from ignorance. When Chen and colleagues infer that the direct activation model of Bcl-2 apoptotic switching is more plausible than the indirect activation model, their argument turns almost entirely upon formal mathematical properties of the respective models. Neither their published text nor supplementary material refers to any kind of parametric robustness for actual systems that exhibit Bcl-2 apoptotic switching. The closest they come to such a reference is when they note that actual systems seem to exhibit certain important features, namely, ultra-sensitivity, range, inhibition, and insensitivity. They do not claim that actual systems exhibit any kind of parametric robustness for those features. Other instances of IMRE exhibit a similar absence of evidence. For instance, when Morohashi et al. ([2002]) infer that one model of mitotic phase oscillations in early Xenopus embryos is more plausible than a competing model, their argument is primarily mathematical. They appeal to empirical evidence about certain features of mitotic phase oscillations in early Xenopus embryos. They do not explicitly appeal to any kind of parametric robustness for these features or for the mitotic phase oscillations themselves. Indeed, they characterize themselves as using ‘Ockham's Razor to distinguish between any two models which may match experimental observations equally well: the model with the greatest parameter robustness […] is the more plausible!' ([2002], p. 22). If the robustness-as-explanandum approach were correct, we would expect to find some evidence of scientists appealing to parametric robustness of target systems when making IMREs. I find no such evidence in my encounters with IMRE. Arguments from ignorance are cogent only insofar as the search for evidence is competent and thorough (see Boone [1999], p. 15). Because the scientific literature is massive, I make no claim to have performed a thorough search for instances of IMRE. I claim only that after a dedicated and competent search of the biological literature (guided primarily by following citations), I find no evidence to support the robustness-as-explanandum approach. I take this as sufficient for at least shifting the burden of proof in favour of the robustness-as-explanatory virtue approach and for motivating further inquiry about the details of that approach. For those unsatisfied by burden-shifting arguments, I offer a second argument. This second argument against the robustness-as-explanandum approach turns upon a contention about the nature of certain parameters. Suppose, for the sake of argument, that parametric robustness is an empirical property of target systems, rather than only a formal property of mathematical models. I am not aware of any formal definitions for this kind of robustness, but presumably this will do: a target system exhibits parametric robustness in reality if for some property represented by a parameter value and some feature of interest about the target system, variations of that property do not affect whether the target system exhibits the feature of interest. Gunawardena ([2010], p. 17) discusses something like this under the heading of ‘robustness in reality’. There is a problem with this notion of parametric robustness in reality. Parameters for differential equations are often rate constants. For example, Chen et al. ([2007], p. 5145) use the law of mass action to derive differential equations for their models of apoptotic switching. This results in their differential equations having rate constants as parameters. As Gunawardena ([2010], p. 15–6) notes, association rates, disassociation rates, catalytic rates, and other rate constants often are […] intrinsic features of the corresponding [components] and would not be expected to change except through alterations of their amino acid sequences. This could happen on an evolutionary time scale, so that different species may have different parameter values, but this would not be expected to happen in different cells of the same organism or tissue or clonal populations of cells in cell culture. Hence, at least for Chen and colleagues' IMRE, their target system cannot exhibit parametric robustness in reality, because the properties represented by their models' rate constants do not vary in reality (on a relevant time scale). So the robustness-as-explanandum approach is not appropriate for reconstructing their inference. Nor, for the same reason, is the robustness-as-explanandum approach appropriate for reconstructing IMRE's about other models in which parameters for determining robustness represent system-invariant properties. Contrary to how the robustness-as-explanandum approach would have it, the robustness of system-invariant parameters (parameters representing system-invariant properties) is not desirable for an epistemic reason (namely, for tracking a kind of robustness in reality). Instead, for system-invariant parameters, robustness is desirable for a pragmatic reason (namely, for minimizing errors due to faulty parameter estimations). Ma and Iglesias ([2002], p. 10), who endorse IMRE-type reasoning, elaborate upon this rationale for valuing parametric robustness: In modelling biological networks, it is important that this robustness [to variations in individual parameter values] also be in evidence. The particular behaviour being characterized by the model should not rely on precise values of the model's parameters—for example, reaction rate constants or protein concentrations. In particular, a precise measurement of these constants is difficult whereas protein concentrations will vary from one cell to another or throughout the lifetime of any individual. Deviations from the nominal model parameter values should not result in a loss of the network's performance; thus, parameter sensitivity can be used to validate mathematical models of biochemical system. That is, the more insensitive the system response is to the accuracy of the parameter, the more faith we should have in the model. Notably, Ma and Iglesias extend the pragmatic reason for desiring robustness of system-invariant parameters to all parameters. They include, in particular, protein concentrations and other parameters that represent properties capable of varying in reality. This extension mitigates a weakness in the second argument against the robustness-as-explanandum approach. The second argument against the robustness-as-explanandum approach has only limited value, because parameters for differential equations sometimes correspond to properties that vary in reality. Some parameters, for example, depend upon temperature or other environmental factors. Because such environmental factors vary in reality, these parameters are system-variable (see Morohashi et al. [2002], p. 20). Hence, the robustness-as-explanandum approach is appropriate for reconstructing IMREs about models in which the parameters for determining robustness represent system-variable properties. B.-S. Chen et al. ([2005], p. 2704), when formulating a measure for parametric robustness, concur with this reasoning: Real biochemical networks must be sufficiently robust to tolerate parameter and environmental variations or else they cannot respond efficiently to small but persistent parameter and environmental perturbations. Therefore, if a model of a biochemical network has a small robustness measure, it is often a sign of structural inadequacies of the model. B.-S. Chen and colleagues do not use their measure in an IMRE. But their argument couples parameter variations with environmental variations, and thereby treats parametric robustness as a marker of empirical adequacy. Models with more parametric robustness, on their view, better represent systems that are robust in reality. Hence, in agreement with the robustness-as-explanandum approach, more parametrically robust models are more plausible than their competitors because they explain something better than their competitors, namely, the robustness in reality of their target system. There is nothing particularly objectionable about the details of this defence for the robustness-as-explanandum approach. The defence is compatible with finding no evidence to support the approach, because the IMREs with which I am familiar take as input models where at least some of the key parameters are system-invariant. In principle, I grant that the robustness-as-explanandum approach might be appropriate for reconstructing some IMREs. But, as I noted in my first argument against that approach, I find no actual IMREs that meet the conditions for appropriateness. Moreover, even if there were IMREs in which all key parameters were system-variable, it is possible to reconstruct those inferences using the robustness-as-explanatory-virtue approach and, following Ma and Iglesias, to understand parametric robustness as desirable for the pragmatic reason of minimizing errors due to faulty parameter estimation. The interpretive choice, accordingly, is between a piecemeal strategy for reconstructing IMREs—treating parametric robustness as an explanandum in some cases, but an explanatory virtue in others—or a unified strategy that always treats parametric robustness as an explanatory virtue. I prefer unified interpretive strategies over piecemeal ones. So I shall proceed as though the robustness-as-explanatory-virtue approach is correct. I invite those with a different preference to qualify what follows as restricted to only certain kinds of IMREs. For, no matter one's preference, there are some IMREs for which parametric robustness is an explanatory virtue—namely, those that take as input models in which some of the key parameters are system-invariant. 3.3 Relation of parametric robustness to other explanatory virtues Because there is no evidence favouring the robustness-as-explanandum approach, and because that approach is inappropriate for IMREs in which the robust parameters represent invariant system properties, I prefer the robustness-as-explanatory-virtue approach to reconstructing the logical structure of IMREs. According to this approach, more parametric robustness is an explanatory virtue: ceteris paribus, models with greater parametric robustness thereby have greater explanatory power. Hence, by this approach, we can refine our prior analysis for the logical structure of IMREs (from Section 3.1) by distinguishing two inferential parts: Part 1: (1) Some set of features, F, obtains in some real system(s) of interest. (2) Among competing mathematical models, M1,…, Mn, Mi is more parametrically robust for exhibiting F than its available competitors. (3) Parametric robustness is an explanatory virtue. (4) Hence, Mi has more explanatory power than its competitors. Part 2: (4) Some set of features, F, obtains in some real system(s) of interest. (5) Among competing mathematical models, M1,…, Mn, each would explain F were it true. (6) Mi has more explanatory power than its competitors (via part 1). (7) Therefore, Mi is more plausible than its available competitors. Parameter robustness analysis establishes the second premise in Part 1; the robustness-as-explanandum approach secures the third; and the conclusion of Part 1 figures as a central premise for Part 2. So reconstructed, an IMRE resembles a typical IBE, albeit with a slightly different conclusion. This resemblance, in itself, is no surprise; the analysis of logical structure results from attempting to understand the inferential structure of IMREs as variants of more familiar IBEs. In the next section, I consider how best to interpret the conclusion of IMREs. In the remainder of this section, however, I shall argue that IMRE qualifies as a novel form of IBE because parametric robustness, interpreted as an explanatory virtue, is not subsumed by explanatory virtues familiar from extant discussions of IBE. In denying that standard explanatory virtues subsume parametric robustness, I mean that parametric robustness is not a special case of those virtues. This is not a trivial contention. Some virtues subsume others. For example, Thagard ([1978], p. 91) identifies analogy as an explanatory virtue. A hypothesis exhibits analogy when it is similar to other hypotheses already established—it invokes mechanisms, entities, and concepts familiar in antecedently accepted explanations. Psillos’s virtue of consilience subsumes analogy, because being similar to antecedently accepted explanations is a specific way to fit with background knowledge. For the sake of tractability, I take the list from Section 3.2 as a representative sample of standard explanatory virtues. My strategy is to argue that, for each sampled virtue, exhibiting parametric robustness is not a special way to exhibit that virtue. When possible, I use the case study from Chen and colleagues as support. Completeness: A model exhibits completeness when it is the only one from the field of competitors to explain all relevant explananda. When Chen and colleagues argue that the direct activation model for Bcl-2 apoptotic switching is more parametrically robust than the indirect activation model, they do so because neither of those models explains all relevant explananda. Each model fails to explain at least some phenomena that proponents of the competing model claim as experimental support (see Section 2.2). Hence, parametric robustness is not a special case of completeness. Consilience: A model exhibits more consilience than its competitors if it better fits with background knowledge. Parametric robustness neither enhances nor lessens fit with background knowledge. For example, when Chen and colleagues consider the direct activation and indirect activation models, those models fit biochemical background knowledge equally well. Both invoke familiar kinds of mechanism components and interactions; both rely upon concepts familiar from other accepted explanations; and both fit different portions of existing experimental evidence. Nonetheless, the direct activation model is more parametrically robust than its competitor. So consilience does not subsume parametric robustness as an explanatory virtue. Importance: A model exhibits more importance than its competitors if it explains more salient phenomena. If parametric robustness were an explanandum, more parametrically robust models might exhibit more importance. But, granting the arguments from Section 3.2, parametric robustness is not an explanandum. Nor do models with more parametric robustness always explain more salient phenomena than their competitors. For example, the direct activation and indirect activation models for Bcl-2 apoptotic switching both explain selected qualitative features of apoptotic switching, namely, ultra-sensitivity, range, inhibition, and insensitivity (see Section 2.3). Moreover, each model explains somewhat different experimental phenomena (see Section 2.2 for details). But Chen and colleagues do not treat some of these phenomena as more salient than others. They note that ‘proving which model is more telling by means of experimental approaches seems difficult thus far' ([2007], p. 5143) . Moreover, they infer that the direct activation model is more plausible because it is more parametrically robust than its competitor, and not because it explains more salient phenomena. So importance does not subsume parametric robustness. Parsimony: A model exhibits more parsimony than its competitors if it uses a proper subset of assumptions from its competitors. The direct activation and indirect activation models for apoptotic switching do not stand in this kind of subset relationship. So parsimony does not subsume parametric robustness. Nonetheless, sometimes the models that figure in an IMRE have assumptions related in something like a subset manner. For example, when Marlovits and colleagues make an IMRE about models for mitotic phase oscillations, they compare two models: the first has a single direct feedback mechanism; the second is identical to the first except for replacing that feedback mechanism with several indirect feedback loops (see Marlovits et al. [1998], pp. 171–2). This second model contains much of the first as a proper subset, and so there is a sense in which the first exhibits more parsimony than the second. However, Morohashi et al. ([2002], p. 28) show that the second has more parametric robustness. Hence, far from subsuming parametric robustness, parsimony sometimes competes with parametric robustness as a source of explanatory power. Precision: A model exhibits more precision than its competitors if it offers more precise explanations. Precision, in this sense, refers not only to the level of detail for a model's derivational consequences, but also to the level of detail for the causal–nomological mechanisms embedded within the model (Psillos [2002], p. 616). Chen et al. ([2007], pp. 5145–7) are interested the ability of the direct activation and indirect activation models to explain qualitative explananda: the general trend of responses to stimuli; the existence of certain thresholds; the general level of various activities. Those models, moreover, consist of similarly detailed differential equations derived from similarly detailed biochemical mechanisms. Neither model, accordingly, exhibits more precision than the other. Yet the direct activation model exhibits more parametric robustness. So precision does not subsume parametric robustness. 4 Epistemological Significance of IMRE Because parametric robustness is not a special case of more familiar explanatory virtues, and because it is not an explanandum, it qualifies as an explanatory virtue in its own right—at least with respect to the project of understanding IMRE as a variant of IBE. For the sake of clarity, it is worth being explicit about the conditions under which a mathematical model exhibits this virtue: Parametric robustness: From among an n-membered collection of suitably selected parameter value sets for a specific mathematical model and n-membered collections of similarly selected parameter value sets for each of the model's competitors, the model exhibits certain and empirically well-attested features for more of its selected parameter value sets than does its competitors. Because parametric robustness is an explanatory virtue, models with more parametric robustness than their competitors have, ceteris paribus, more explanatory power than their competitors. Moreover, because it is an explanatory virtue in its own right, it is possible for a mathematical model to gain explanatory power by virtue of exhibiting parametric robustness, but lose explanatory power by virtue of failing to some other explanatory virtue. For current purposes, I set aside issues about how to weigh parametric robustness against other explanatory virtues. For the models that Chen et al. ([2007]) consider for their IMRE, parametric robustness seems to be the only one that discriminates among competing models. For the models that Morohashi et al. ([2002]) consider for their IMRE, parametric robustness seems to trump certain other virtues such as parsimony. Carlson and Doyle ([2000], p. 2532) suggest that, for certain kinds of domain-specific models, there is a general trade-off between maximizing parametric robustness and minimizing complexity. Presumably context often matters. Sorting out details requires further case studies. Rather than provide those details, I shall consider instead the epistemological significance of IMREs. The conclusions of IMREs appear to differ from typical conclusions of standard IBEs. An IMRE concludes with the claim that one model is more plausible than its competitors. A standard IBE, in contrast, concludes with the claim that one hypothesis is probably true (see Section 3.1). This raises an epistemological issue. Does a model's superior plausibility, as conferred by an IMRE, mean that the model is probably true—or at least more likely to be true than its competitors? If so, the resemblance between IMREs and standard IBEs is strengthened. But if not, what epistemic attitude does superior plausibility warrant? I shall approach this issue from two directions. First, I report some of the stances practicing scientists take on the issue with respect to their own IMREs. These stances tend to treat superior plausibility as indicating only superior suitability for continuing research efforts. However, people do not always infer what their premises warrant. So, second, I consider whether some prominent realist arguments about explanatory power support a stronger, truth-oriented stance toward the conclusions of IMREs. I argue that they do not. 4.1 Plausibility in practice Haefner ([1996], p. 152), when discussing the difference between model verification and model validation, has this to say about the adjective ‘plausible' as applied to biological models: For my part, in light of the rather small number of well-tested models in biology and the generally low rigor of the tests, I think the adjective plausible more accurately reflects the nature of tested biological models and the skeptical attitude we should adopt. To a more cynical observer, the dictionary definition of ‘specious’ might also come to mind. Haefner aims his remarks at those who, in order to avoid connotations about correctness, prefer to speak of validated models as confirmed or corroborated. For Haefner, these alternatives also have improper connotations about correctness. Linguistic preferences aside, his point seems to be that validated models—that is, plausible models—need only be well-founded, tractable, and fulfill the purposes for which they are formulated (see Carson et al. [1983], p. 217). They need not be correct, and the most plausible models need not be the most likely to be true. At least some scientists who use IMRE seem to have Haefner's meaning of plausibility in mind when characterizing models with superior parametric robustness as plausible. For example, Chen and colleagues use IMRE to infer that the direct activation model for Bcl-2 apoptotic switching is more plausible than its indirect activation competitor. They seem to think this conclusion warrants focusing future research efforts on developing the direct activation model and incorporating into it new experimental findings, for they go on to do just this (see Cui et al. [2008]). Similarly, when Tokar and Ulicny use IMRE to argue that the direct activation model is more plausible than certain ‘hybrid' models (which take direct and indirect activation mechanisms to be at work simultaneously), they note that despite their findings, ‘we cannot completely rule out the indirect model interactions' ([2012], p. 6219). Their more nuanced conclusion, accordingly, is that while the direct activation model is more plausible, their robustness analysis shows only that ‘indirect model interactions, if present, must proceed at very low rates […] and provide other important system properties' ([2012], p. 6219). Their attitude is cautious, because their robustness analysis focuses upon only some of the many important features for Bcl-2 apoptotic switching. Relative to the purpose of assessing plausibility for their limited focus, the direct activation model is most plausible. Because they do not know what their analysis might show for a different or more comprehensive focus, they do not make stronger claims about likely truth. Morohashi and colleagues more explicitly treat plausibility as indicating validity but not correctness. Comparing two models for explaining the mitotic phase oscillations of early Xenopus embryos, they argue that ‘the structure of the ′98 model is capable of providing highly robust oscillatory behaviour in a manner far exceeding the capabilities of the ′91 model […]' (Morohashi et al. [2002], p. 28). They infer from this, following the pattern for IMRE, that the ′98 model is more plausible than the ′91 model. But they note that superior plausibility does not preclude the existence of yet more plausible models, and they provide a slight variant to the ′98 model that is more parametrically robust than even the ′98 model ([2002], pp. 28–9). Hence, far from meaning that the ′98 model is true, their saying that the ′98 model is more plausible than the ′91 model seems to mean only that it would be worthwhile to focus future research efforts on developing the ′98 model. 4.2 Plausibility in principle Even if, in practice, scientists mean for the conclusions of IMREs to be about which models are most well-founded, tractable, and worth further exploration, the premises of IMREs might warrant stronger conclusions about which models are most likely to be true. The philosophical literature on IBE and scientific realism contains several arguments purporting to show that superior explanatory power indicates truth. If any of those arguments hold for IMREs, there might be good reason to interpret claims about plausibility as claims about correctness. I consider two of these arguments and argue that neither holds for extant instances of IMRE. Boyd offers a kind of transcendental argument for supposing that models—‘theories’, in the jargon of his day—with the most explanatory power are at least approximately true. Speaking of IBE in its standard form (as yielding, for a conclusion, a claim about the truth or approximate truth of the best explanation), he writes: The rejection of abduction or inference to the best explanation would place quite remarkable strictures on intellectual inquiry. In particular, it is by no means clear that students of the sciences—whether philosophers or historians—would have any methodology left if abduction were abandoned. If the fact that a theory provides the best available explanation for some important phenomenon is not a justification for believing the theory is at least approximately true, then it is hard to see how intellectual inquiry could proceed. (Boyd [1983], p. 74) Boyd presumes, as a datum, that intellectual inquiry often produces at least approximately true theories. His central premise is that this datum could not happen unless theories with the most explanatory power were approximately true. I set aside the issue of whether Boyd's premises are true or his reasoning cogent. Even if his argument succeeds for IBE generally, it does not show that theories with the most parametric robustness are thereby at least approximately true. IMREs are a relatively new kind of inference among biologists. (Morohashi et al. ([2002]) make the first explicit IMRE in the literature for theoretical biology.) These inferences, moreover, are not essential to general biological inquiry. So Boyd's central premise does not apply for IMRE: biological inquiry would continue to produce approximately true theories (presuming it produces them at all) even if the most parametrically robust theories were not approximately true. In the case of Bcl-2 apoptotic switching, for example, inquiry about further details of the direct activation model, as well as about hybrid competitors to this model, continued without any explicit commitment from Chen and colleagues, or from Tokar and Ulicny, about the approximate truth of that model (see Section 4.1). Gasper ([1990]) offers a practice-based argument for treating IBEs as yielding conclusions about empirical adequacy. This is a weaker argument than Boyd's, because empirically adequate theories need not be true about unobservable entities. He writes: […] why is explanatory power desirable? […] Recall that scientists do not typically independently identify empirically adequate theories and then compare them with respect to explanatory power. Rather, judgments of explanatory power play a central role in the choice of reliable theories to guide further research. This claim finds confirmation in the fact that explanatory power is routinely taken as a guide to theory acceptance in successful areas of science […] Realists insist that we take this practice seriously. Explanatory power […] is at least a guide to which theories are empirically adequate. In other words, explanatory power is evidential. (Gasper [1990], pp. 293–4) Gasper's argument elides the difference between deciding to use a theory to guide further research and deciding to endorse a theory as empirically adequate. But the difference matters. Chen and colleagues, for example, decide on the basis of an IMRE to use the direct activation model to guide further research about apoptotic switching. However, they endorse that model only as plausible. As I have argued, this is not an endorsement about correctness (see Section 4.1). Taking practice seriously in this case does not support Gasper's elision. 5 Conclusion Harman characterizes standard IBE as a form of reasoning in which ‘one infers, from the premise that a given hypothesis would provide a “better” explanation for the evidence than would any other hypothesis, to the conclusion that the given hypothesis is true' (Harman [1965], p. 89). So interpreting claims about plausibility in the conclusions of IMREs as claims about well-foundedness, tractability, and suitability for further exploration, rather than claims about correctness or truth, means that IMREs are not straightforward instances of standard IBEs. This is not a reason to deny that IMREs are variants of standard IBEs. There are non-standard accounts of IBE as well. Katzav ([2013]), for example, argues that some IBEs do not infer to approximate or probable truth because sometimes best explanations are idealized and therefore false. Dawes ([2013]) argues that IBEs yield as conclusions categorical claims to the effect that the best explanation is acceptable to use as a premise in knowledge-directed theoretical or practical reasoning. Either of these variants for standard approaches to IBE suffices for understanding IMREs as IBEs and interpreting plausibility claims as not making claims about truth or correctness. Peirce ([1935] Paragraph 525), moreover, characterizes abduction (a form of inference that includes what we now call IBE) as ‘the first starting of a hypothesis and the entertaining of it, whether as a simple interrogation or with any degree of confidence'. Provided that Haefner's kind of plausibility suffices for this ‘first starting', there is at least some historical precedent for a non-standard interpretation of IBE (see Hookway [2012], pp. 75–9). Understanding IMREs as variants of standard IBEs—or as instances of certain non-standard IBEs—presumes, of course, that it is appropriate to understand IMREs as some kind of IBE. The motivation for this presumption is a formal resemblance between the pattern of IMREs and the pattern of IBEs (Section 3.1). The presumption is also heuristically useful, by virtue of directing inquiry about IMRE in a way that clarifies the role of parametric robustness and the meaning of plausibility (Sections 3.2–4.2). Understanding IMREs on the model of some other inferential form might support further insights about IMRE. But that is a project for another time. I conclude, instead, with one benefit the present approach to IMRE has for our understanding of IBE. Searching through the recent literature on IBE, one finds intuitions about how IBEs ought to work and about what conclusions they ought to warrant. Lipton ([2004], p. 115), for example, appeals to intuitions about what ‘the defender of Inference to the Best Explanation' should say. Also prominent are claims about the prevalence of IBEs. For instance, Dawes ([2013], p. 62) claims that IBEs are frequent in the sciences and everyday life. But, at least recently, there are only a handful of studies examining specific instances of IBE from scientific practice. These include Weber’s ([2009]) study of IBE in the context of using experiments to explain data patterns, Tulodziecki’s ([2011]) reconstruction of John Snow’s reasoning about cholera pathology and transmission, and Katzov’s ([2013]) examination of competing climate science models. Understanding IMREs as a kind of IBE adds to philosophers' stable of cases one further study, namely, competing biochemical models of apoptotic switching (Section 2). This case, moreover, helps to highlight the way in which certain realist arguments for the truth-conduciveness of IBE depend upon contentions about scientific practice that do not hold in all contexts (Section 4.2). Acknowledgements I thank several anonymous referees for their considerable help in reorganizing and refocusing this article. A distant ancestor draft owes its origins to sabbatical work while in residence as a visiting fellow in the Center for Philosophy of Science at the University of Pittsburgh. I thank fellow residents at the Center for stimulating discussions, as well as the Center itself and University of Alabama in Huntsville for financial support. I also thank Bill Roche for helpful comments on that ancestor draft and John Norton for making salient to me the unfortunate dearth of case studies for IBE in scientific practice. 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[ 2007]: ‘Apoptosis Initiated When BH3 Ligands Engage Multiple Bcl-2 Homologs, Not Bax or Bak’, Science , 315, pp. 856– 9. Google Scholar CrossRef Search ADS PubMed  Youle R. J. [ 2007]: ‘Cellular Demolition and the Rules of Engagement’, Science , 315, pp. 776– 7. Google Scholar CrossRef Search ADS PubMed  © The Author 2016. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. For Permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The British Journal for the Philosophy of Science Oxford University Press

Inference to the More Robust Explanation

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© The Author 2016. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. For Permissions, please email: journals.permissions@oup.com
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0007-0882
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Abstract

ABSTRACT There is a new argument form within theoretical biology. This form takes as input competing explanatory models; it yields as output the conclusion that one of these models is more plausible than the others. The driving force for this argument form is an analysis showing that one model exhibits more parametric robustness than its competitors. This article examines these inferences to the more robust explanation, analysing them as variants of inference to the best explanation. The article defines parametric robustness and distinguishes it from more familiar kinds of robustness. The article also argues that parametric robustness is an explanatory virtue not subsumed by more familiar explanatory virtues, and that the plausibility verdicts in the conclusions of inferences to the more robust explanations are best interpreted as guidance for research activity, rather than claims about likely truth. 1 Introducing Inference to the More Robust Explanation 2 Inference to the More Robust Explanation in the Study of Apoptosis 2.1 Regulating apoptosis 2.2 Competing models and evidential indecision 2.3 Measuring robustness 2.4 Robustness as a guide to plausibility 2.5 Varieties of robustness 3 Inference to the More Robust Explanation as Inference to the Best Explanation 3.1 The structure of inference to the best explanation 3.2 Parametric robustness as an explanatory virtue rather than an explanandum 3.3 Relation of parametric robustness to other explanatory virtues 4 Epistemological Significance of Inference to the More Robust Explanation 4.1 Plausibility in practice 4.2 Plausibility in principle 5 Conclusion 1 Introducing Inference to the More Robust Explanation In 2002, Morohashi and colleagues analysed the parameter sensitivities of competing models for explaining mitotic phase oscillations in early Xenopus embryos. Their analysis showed that ‘the structure of the ′98 model is capable of providing highly robust oscillatory behaviour in a manner far exceeding the capabilities of the ′91 model' ([2002], p. 28). They inferred, in consequence, that the ′98 model is more plausible than the ′91 model because its parameters are less sensitive to variations. This argument marks the first appearance in the biological literature of scientists treating parametric robustness as a measure of plausibility. I shall call it an inference to the more robust explanation, or IMRE for short. This argument form has been received favourably. For example, comparing competing models for explaining mitochondrial outer membrane permeabilization (MOMP) during apoptosis, Chen and colleagues ([2007], p. 5148) argue that one of the models is more plausible than the other because it ‘confers a better robustness' with respect to parameter variations. IMREs bear close similarities to more familiar inferences to the best explanation (IBEs). Both take as input a collection of explanatory hypotheses. Both provide as output a verdict that one of the explanatory hypotheses is better than the others, either by virtue of being more plausible (IMREs) or by virtue of being more likely to be correct (IBEs). Both also warrant this output by comparing the competing explanatory hypotheses directly to each other: IMREs compare how much robustness to parameter variation each hypothesis has; IBEs, how much explanatory power each hypothesis has. There are dissimilarities, too. For example, IMREs are restricted to comparing mathematical models; IBEs are not. Despite these differences, I propose to analyse IMREs as a kind of IBE. This is, in part, for the sake of better understanding the logical structure and epistemic force of IMREs. Specifically, with respect to logical structure, extant instances of IMRE are not clear about the role of parametric robustness: Is it supposed to be an explanandum phenomenon, so that models conferring better robustness thereby explain more? Or is it, at least descriptively, a kind of explanatory virtue in its own right, so that models conferring better robustness might explain better without necessarily explaining more? With respect to epistemic force, moreover, extant instances of IMRE are not clear about the epistemological significance of comparative plausibility verdicts: does plausibility correlate with likelihood of truth, or does it have some other significance? Beyond aiming to better understand a relatively new argument form in contemporary theoretical biology, I also propose to analyse IMREs for the sake of better understanding the kinds of robustness that interest contemporary biologists. Philosophers of biology are likely more familiar with robustness in Levins's ([1966]) sense, where a model is robust insofar as its derivational consequences are also consequences of similar models with different state variables or functional structures. There is some debate about whether this kind of robustness indicates likely truth (see Weisberg [2006]). But IMREs seem to involve a different kind of robustness. A model is robust to parameter variations insofar as changing the values of its parameters, without changing its state variables or functional structure, does not change its derivational consequences. Prima facie, models that are robust in this sense need not be robust in Levins's sense. Accordingly, I address two issues about robustness: first, whether and how parametric robustness differs from Levins’s kinds of robustness; and second, whether parametric robustness indicates likely truth. I begin in Section 2 with a case study of IMRE. The case centres on attempts to understand apoptosis. I use it to explicate the kind of robustness at work in IMREs, and to tether subsequent analysis to scientific practice. I proceed, in Section 3, to reconstruct IMREs on the model of IBE. I argue that parametric robustness is not an explanandum and that those using IMRE do not treat parametric robustness as a datum to be explained. I argue, further, that parametric robustness is an explanatory virtue in its own right and that it is not subsumed by a standard range of other explanatory virtues. In Section 4, on the epistemological significance of IMRE, I argue that the plausibility verdicts in the conclusions of IMREs are best interpreted as guidance for research activity rather than claims about likely truth. I also explore arguments from scientific realists about the connection between explanatory power and truth.1 2 IMRE in the Study of Apoptosis 2.1 Regulating apoptosis Apoptosis is programmed cell death for multicellular organisms. It helps to make room for cell proliferation, limit pathogen spread, and eliminate cells that can trigger autoimmune responses or overwhelm support systems. In contrast to necrosis, a passive process during which mitochondria swell and rupture the plasma membrane, apoptosis occurs when nuclei and cytoplasm—including mitochondria—shrink, become encased in ‘apoptotic bodies’, and send signals for being engulfed by nearby phagocytic cells. But apoptosis must be regulated: too much promotes ischemic conditions and neurodegeneration; too little promotes cancer and autoimmune diseases (Saikumar et al. [1999], pp. 489–91; Czabotar et al. [2014], p.49). The principle components regulating the intracellular mechanism for apoptosis belong to the B-cell lymphoma 2 (Bcl-2) protein family. Some proteins in this family—Bcl-2, Bcl-xL, Bcl-w, A1, Mcl-1—inhibit apoptosis. Others promote it. The BH3-only proteins—including Bid, Bim, Puma, Bad, and Noxa—sense and integrate cellular damage signals. The multi-domain proteins Bax and Bak form the mitochondrial apoptosis channel and make permeable the mitochondrial outer membrane. Bax and Bak are especially important, because MOMP makes cell death inevitable. Other Bcl-2 family proteins are important insofar as they help or hinder the activation of Bax and Bak. There remains disagreement concerning how Bcl-2-type proteins inhibit MOMP and thereby preserve cell viability, how BH3-only proteins facilitate MOMP and thereby promote cell death, how the entire Bcl-2 switching network regulates Bax–Bak activation, and how Bcl-2 family proteins are controlled by signals from proteins outside their family (Chen et al. [2007], p. 5143; Czabotar et al. [2014], pp. 50–5). In 2006–7, there was a flurry of research activity about proposed mechanisms for the Bcl-2 apoptotic switch. The issues of contention were, first, whether BH3-only proteins activate Bax–Bak directly and, second, how anti-apoptosis proteins inhibit Bax–Bak activation. 2.2 Competing models and evidential indecision Researchers offered for consideration two types of model. While advocates for each type reported new experimental evidence in their favour, the cumulative effect was a situation in which contradictory evidence failed to support either model type over its rival (see Van Delft and Huang [2006]). Some researchers favoured a ‘direct' activation model, according to which BH3-only proteins directly activate Bax–Bak while anti-apoptotic proteins inhibit this activation by sequestering the BH3-only proteins (see Galonek and Hardwick [2006]). Other researchers favoured an ‘indirect' activation model, according to which BH3-only proteins indirectly activate Bax–Bak through binding and inhibiting the anti-apoptotic proteins, and in which these anti-apoptotic proteins inhibit Bax–Bak activation by sequestering the Bax–Bak proteins (see Youle [2007]). Both models were taken to have a variety of experimental evidence in their favour. Kim et al. ([2006]) provided experiment-driven arguments for each component of the direct activation model. For example, they argued that certain BH3-only proteins directly activate Bax–Bak because (i) they effectively release cytochrome-c—a central component of the electron transport mechanism for mitochondria—from isolated mitochondria and thereby act as potent death stimulators; (ii) retroviral transduction of these proteins in mouse embryonic fibroblasts resulted in a 65% death rate after twenty-four hours; and (iii) substituting or deleting amino-acid residues from these proteins abolishes their cytochrome-c releasing activity (Kim et al. [2006], p. 1349). They argued, further, that Bax and Bak are not themselves inhibited by anti-apoptotic proteins because (iv) a mutant form of Bak, unable to interact with the anti-apoptotic proteins, does not introduce apoptosis without other death stimuli, and so these proteins do not inhibit an otherwise active-by-default Bak; and (v) over-expressing the anti-apoptotic proteins in double-knockout cells reconstituted with mutant forms of Bak and Bax—forms unable to interact with those proteins—prevents apoptosis, and so the proteins do not prohibit apoptosis by directly interacting with Bax–Bak (Kim et al. [2006], pp. 1350–1). However, Willis and colleagues ([2007]) provided experiment-driven arguments for a competing indirect activation model. For example, they argued that BH3-only proteins can initiate apoptosis without binding directly to Bax–Bak because (i) in lysates of human embryonic kidney cells, BH3-only proteins bind to anti-apoptotic proteins, but not to endogeneous Bak; and (ii) mutant forms of Bid and Bim, able to bind to anti-apoptotic proteins but not to Bax, kill mouse embryonic fibroblasts just as effectively as wild-type Bid and Bim (Willis et al. [2007], p. 857). They argued, further, that apoptosis occurs when BH3-only proteins inhibit anti-apoptotic proteins, because (iii) over-expressing Bad and Noxa—so-called ‘sensitizer' BH3-only proteins that neutralize anti-apoptotic proteins but lack the capacity to initiate apoptosis on their own—induces apoptosis and cytochrome-c release in mouse embryonic fibroblasts, and this happens when the so-called ‘activator' BH3-only Bim and Bid proteins are absent (Willis et al. [2007], p. 858). Because the experimental results reported by Kim and colleagues lent support to the direct activation model while the results reported by Willis and colleagues lent support to the indirect activation model, and because neither research group offered evidence to undermine or overturn the assumptions that the other group used to interpret their experimental data, the total evidential situation failed to support either model over its rival. Chen and colleagues ([2007], p. 5143) tactfully summarized the situation as one where ‘proving which model is more telling by means of experimental approaches seems difficult thus far'. Their diagnosis for the situation was two-fold. First, the complexity of biological signalling networks means that wet lab experiments tend to have little probative power. Second, competing research groups often examine different experimental systems (Chen et al. [2007], p. 5148). But rather than recommend an agnostic attitude toward the models until better evidence is available, and rather than develop some kind of hybrid alternative model, Chen and colleagues use an IMRE in order to infer that the direct activation model is more plausible than the indirect activation model. 2.3 Measuring robustness Chen and colleagues are interested in the direct activation model and its indirect activation competitor insofar as both purport to explain how the Bcl-2 switching network regulates Bax–Bak activation. They grant that each model (potentially) explains this phenomenon, presumably because each specifies a mechanism for how members of the Bcl-2 protein family regulate Bax–Bak activity. Moreover, while each model explains some but not all of the total then-available evidence, Chen and colleagues do not prefer one model over the other on the basis of which pieces of evidence are more important to accommodate. Nor do they prefer one over the other on the basis of which is more likely relative to the total evidence: because each model is inconsistent with some evidence (as they interpret it), each has a null posterior probability relative to the total evidence. Nor, finally, do they appeal to intuitions, dismissing them as unreliable ‘because of the complexity of the system' (Chen et al. [2007], p. 5148). Instead, Chen and colleagues use analytical considerations about robustness in order to measure each model’s plausibility. They provide, in fact, several robustness-based measures of comparative plausibility. Each measure involves (i) a mathematical criterion for whether a model exhibits some important and empirically well-attested feature, (ii) a mathematical criterion for determining the magnitude of parameter variation on some mathematical model, and (iii) a mathematical criterion for measuring how much robustness a model has for exhibiting some selected feature. Chen and colleagues argue that most of their measures favour the direct activation model over the indirect activation model. I discuss one measure in detail, for the sake of clarifying the general argumentative strategy. I then only mention the others, proceeding instead to discuss Chen and colleagues' results. First, Chen and colleagues observe that ultra-sensitivity is an important, and empirically well-attested, feature of the Bcl-2 apoptotic switch. The ultra-sensitivity of this switch means that apoptotic stimuli produce small responses initially, and increasing the stimuli produces increasingly stronger responses. Chen et al. ([2007], Supplementary Material) propose that a model exhibits ultra-sensitivity when its relative amplification coefficient, nR, is greater than one. This coefficient compares, in essence, the response ratio for the model—defined as RSX = S/X × dX/dS for stimulus level S and model-determined response level X—to the response ratio for some reference model (namely, the Michaelis–Menten equation). When nR>1, the model's response to a given stimulus is larger than the reference model's response to that same stimulus. Chen and colleagues take this to mean that the model exhibits ultra-sensitivity, because the model is more sensitive to the stimulus than the reference model. Second, Chen and colleagues compare the direct activation and indirect activation models with respect to their capacity for exhibiting ultra-sensitivity. Both models are sets of differential equations. For example, the direct activation model includes the equation   d[InBax]/dt=((kBax×[Bax])−(kInBax×[Act]×[InBax])), where the quantities in brackets represent concentrations of various mechanism components and subscripted k's represent various parameters for rate constants (see Chen et al. [2007], p. 5144 and Supplementary Material). The concentrations for each mechanism component, moreover, are given by further parameterized equations. For instance, Chen et al. [2007], p. 5145) define the concentration of BH3-only proteins as   [BH3]=([BH3]0+F)×[BH3]0, where [BH3]0 is a parameter representing the initial concentration of BH3-only proteins and F represents an m-fold increase of that concentration (0 ≥ m ≥ 20) with respect to some input stimulus. For the sake of tractability, Chen et al. [2007], pp. 5144–5) consider simplified variants of each model, preserving core information but reducing computational complexity. For example, they treat all anti-apoptotic proteins as if they are the same and they treat the ‘activator' BH3-only proteins (Bax and Bak) as if they are the same protein. The (simplified) direct activation and indirect activation models yield specific results (for purposes of explanation and prediction) only when the parameters for each model have specific values. Different parameter values, moreover, yield different results. Varying a parameter value involves changing some reference value for a parameter to some different value. Chen et al. ([2007], p. 5145) define the magnitude of parameter variation, PV, for the ith parameter of a model as follows:   PV=pi/pi,ref, where pi,ref is the value of the ith parameter as given by a fixed set of reference values and pi is the altered value of that same parameter. Chen et al. [2007], p. 5144, Table 2) fix the reference values for each parameter by either extracting values from existing literature or else estimating values with educated guessing. Given these reference values, they define the total order of magnitude of parameter variation, TPV, as the sum of (the absolute value of the common logarithm of) the parameter variations PV for each of the parameters in a model ([2007], p. 5146). (For example, for a model with two parameters, A and B, reference values pA,ref = 1 and pB,ref = 1, and altered values pA = 10 and pB = 10, TPV would equal two.) This allows them to calculate how much variation there is between a set of reference values for a model's parameters and a set of altered values for those same parameters. When TPV is greater than zero, there is some degree of variation between parameter value sets; as TPV increases, the amount of this variation also increases. Third, given reference values for the parameters in each of the direct activation and indirect activation models, given TPV as a measure for the amount of variation on these values, and given the condition that a model exhibits ultra-sensitivity whenever nR > 1, Chen and colleagues calculate how much robustness each model has for exhibiting ultra-sensitivity. For the sake of computational tractability, they randomly select 3000 well-defined parameter sets for each model. They choose each set from a range of possible values varying across four orders of magnitude (for example, from 0.01 to 1000), subject to the condition that TPV always computes as greater than zero. Then, for each set, they calculate the model's relative amplification coefficient, nR, with the specific parameter values. The result is 3000 calculated values of nR for the direct activation model, and 3000 calculated values of nR for the indirect activation model. The amount of robustness a model has for exhibiting ultra-sensitivity is then a straightforward sum of the number of parameter sets out of 3000 for which the model yields nR > 1. 2.4 Robustness as a guide to plausibility Chen et al. [2007], p. 5147) find that 996 of 3000 randomly selected parameter sets exhibit ultra-sensitivity (nR > 1) for the direct activation model, while only 217 of 3000 randomly selected parameter sets exhibit ultra-sensitivity for the indirect activation model. They infer that ‘the direct activation model confers a better robustness of ultrasensitivity with respect to global parameter variations' than does the indirect activation model ([2007], p. 5147). This means that there are more sets of parameter values for which the direct activation model exhibits ultra-sensitivity. Because epistemic limitations and environmental variations make it likely that set of reference values for each model's parameters includes some degree of error, a ‘better robustness of ultrasensitivity with respect to global parameter variations' means that a model is more likely to hold for real biological systems. For even if the actual values for various parameters differ from the reference values, more robustness means that the model is more likely to exhibit ultra-sensitivity even when its parameters have their actual values (as opposed to the incorrect reference values). Chen and colleagues realize that a model might be more robust for exhibiting ultra-sensitivity and yet less robust for exhibiting some different but similarly important and empirically well-attested feature. Rather than argue that one particular feature, such as ultra-sensitivity, trumps all others in importance, Chen and colleagues instead identify a handful of important and empirically well-attested features of the Bcl-2 apoptotic switch. These are features that both the direct activation and indirect activation models exhibit when the parameters for those models have their respective reference values. Chen et al. ([2007], pp. 5145-7) group names the features as follows: Ultra-sensitivity: Apoptotic stimuli produce small responses initially, and increasing the stimuli produces increasingly stronger responses. Range: There is some threshold such that, in non-apoptotic conditions, the basal activation level of Bax is below this threshold while, when apoptotic stimuli are present, the basal activation level for Bax is above this threshold. Inhibition: There is some threshold such that, when apoptotic stimuli are present, the activity of Bax exceeds this threshold, while when anti-apoptotic proteins are present, the activity of Bax is below this threshold. Insensitivity: There is relatively low Bax activity in non-apoptotic conditions, even with small perturbations of the apoptotic stimuli. Following the procedure outlined for the case of ultra-sensitivity, Chen et al. ([2007], p. 5147) calculate the amount of robustness each model has for exhibiting each of these features. I summarize the results of their computations in Table 1, where the number in each cell indicates, for a given model and feature, how many parameter sets for that model, from amongst 3000 randomly selected possibilities, exhibit that feature. Each number indicates how robust the model is for exhibiting the selected feature, and higher numbers mean more robustness. Table 1. Single-feature robustness measures   Ultra-sensitivity  Range  Inhibition  Insensitivity  Direct activation model  996  891  744  729  Indirect activation model  217  153  1344  630    Ultra-sensitivity  Range  Inhibition  Insensitivity  Direct activation model  996  891  744  729  Indirect activation model  217  153  1344  630  The number in each cell indicates how many parameter sets, from a randomly selected set of 3000 sets, exhibit the column-heading feature (based upon Chen et al. [2007], pp. 5147–8) For most features, the direct activation model exhibits better robustness than the indirect activation model. For the sake of having a more comprehensive robustness measure, and for providing a more absolute ranking of the models, Chen and colleagues also compute how many parameter sets simultaneously exhibit all four features. This yields a ranking with no ambiguity: the direct activation model exhibits all four features in 316 of 3000 sets, while the indirect activation model exhibits all four features in only 31 of 3000 sets. Chen and colleagues argue that the unified comparison is less likely to be misleading, because while all four features are important, the presence of one does not guarantee the presence of the others. Accordingly, they use the results of the unified comparison to conclude that ‘the direct activation model confers a better robustness in contrast to the indirect activation model' ([2007], p. 5148) . Chen and colleagues use IMRE to infer that the direct activation model is more plausible than its indirect activation competitor. Their underlying assumption is that ‘faithful models should not rely on fine tuned parameters to reflect biological systems' (Chen et al. [2007], p. 5148). This is another way of saying that behaviours of biochemical networks should be robust to parameter variations. Why this might be so, and what Chen and colleagues might mean in characterizing the most parametrically robust model as more plausible than its competitor, is an issue to which I return in Section 4. 2.5 Varieties of robustness When inferring that the direct activation model is more plausible than its indirect activation competitor, Chen and colleagues invoke a kind of robustness that, following Weisberg and Reisman ([2008], p. 115), I shall call ‘parametric robustness’. A model is parametrically robust for some feature of interest when the model exhibits that feature for two or more sets of parameter values. The more sets of parameter values for which the model exhibits the feature, the more robust the model is for that feature. A model is said to be parametrically robust without qualification, moreover, whenever and to the extent that it is parametrically robust for all features of interest. According to Chen and colleagues' analysis, for example, the direct activation and indirect activation models are robust for ultra-sensitivity and inhibition; the direct activation model is more robust for ultra-sensitivity and the indirect activation model is more robust for inhibition; and the direct activation model is more robust in general than is the indirect activation model (see Table 1). Determining the extent to which a model is parametrically robust requires holding fixed the model's state variables and the functional structure among those variables. This differs from Levins’s ([1966]) sense of robustness, according to which a model is robust insofar as its derivation consequences are also consequences of similar models with different state variables or functional structures. For example, the Lotka–Volterra model is robust in Levins’s sense for exhibiting the Volterra principle, according to which prey populations increase relative to the predator population upon pesticide application. For this principle is a derivational consequence of the Lotka–Volterra model, and it is also a consequence of a more complex model that includes a state variable for prey population carrying capacity (see Weisberg [2006], pp. 734–6). This kind of robustness—Weisberg and Reisman ([2008], p. 116) call it ‘structural robustness’—is distinct from parametric robustness. Moreover, both structural and parametric robustness are distinct from a third kind of robustness that Levins also seems to have in mind. Weisberg and Reisman ([2008], p. 120) call this third kind ‘representational robustness’ because it involves varying functional relationships among state variables, rather than the state variables themselves or their associated parameter values. (For similar distinctions among kinds of robustness, in a more mathematically oriented context, see Gunawardena [2010], p. 22.) Whether the direct activation and indirect activation models are robust in either of Levins’s senses depends upon whether ultra-sensitivity and various other features (or mathematical representations thereof) are derivational consequences of models with altered differential equations. But Chen and colleagues change neither the state variables nor functional structures in the differential equations for the direct activation and indirect activation models. They change only the values of parameters in those equations. They do not, accordingly, investigate whether the direct activation and indirect activation models are structurally or representationally robust. Hence, their inference from model robustness to model plausibility differs from the kinds of robustness-driven inferences more often of interest to philosophers of biology (see Orzack and Sober [1993]; Weisberg [2006]; Weisberg and Reisman [2008]; Odenbaugh [2011]). 3 Inference to the More Robust Explanation as Inference to the Best Explanation Chen and colleagues argue that because the direct activation model of the Bcl-2 apoptotic switch is more robust than the competing indirect activation model, the direct activation model is more plausible than its competitor. This argument exemplifies what I call IMRE. I take the discussion in the prior section as sufficient for illustrating how IMREs typically appear in scientific practice and how their central comparative criterion—parametric robustness—differs from more familiar kinds of structural and representational robustness. In this section, I transition from reporting about IMRE to philosophizing about it. The goal is to better understand the inferential structure of IMREs, and in particular how greater parametric robustness is supposed to support greater plausibility. My method for achieving this goal is to treat IMREs as variants of more familiar IBEs. 3.1 The structure of inference to the best explanation When Chen and colleagues infer that the direct activation model of the Bcl-2 apoptotic switch is more plausible than the competing indirect activation model, they do so in part because they take the direct activation model to be more parametrically robust than its competitor. But this is only one of their reasons. They also consider each competing model to have some minimal antecedent plausibility, and they do so because each model explains various important and empirically well-attested features of the Bcl-2 apoptotic switch (namely, ultra-sensitivity, range, inhibition, and insensitivity). Insofar as Chen and colleagues' inference is paradigmatic, we can reconstruct typical instances of IMRE as instantiating the following pattern: Some set of features, F, obtains in some real system(s) of interest. Among competing mathematical models, M1,…, Mn, each would explain F were it true. Mi is more parametrically robust (for exhibiting F) than its available competitors. Therefore, Mi is more plausible than its available competitors. This pattern bears a strong resemblance to a standard pattern for IBE: Some set of phenomena, P, obtain in some real system(s) of interest. Among competing hypotheses, H1,…, Hm, each would explain P were it true. None of its competitors can explain P as well as Hj can. Therefore, Hj is probably true. (Those who endorse this pattern for IBE include Harman ([1965], p. 89), Thagard ([1978], p. 77), Josephson ([1994], p. 5), and Psillos ([2002], p. 614). Lipton ([2004], p. 58) adds that the best explanation also must be ‘good enough for us to make any inference at all'; I suppress this caveat in what follows.) There are several salient differences between these patterns. First, the conclusion of an IMRE is a claim about comparative plausibility. The conclusion of a typical IBE, however, is a claim about (probable or approximate) truth. Harman ([1965], p. 89), for example, characterizes IBE as a form of reasoning in which ‘one infers, from the premise that a given hypothesis would provide a “better” explanation for the evidence than would any other hypothesis, to the conclusion that the given hypothesis is true'. This raises the issue of how to best interpret the significance of comparative plausibility claims. Does being more plausible entail being more likely to be true? If the best explanatory hypotheses are likely to be true and IMRE is a variant of IBE, are models with more parametric robustness also more likely to be true? These questions concern the epistemological significance of IMREs. I postpone them for the subsequent section. Their answers are better motivated after clarifying certain details about the logical structure of IMREs. Those details emerge by considering a further difference between IMREs and IBEs. A second salient difference between the patterns for IMRE and IBE concerns the third premise in each pattern. In an IMRE, this premise is a claim about the comparative parametric robustness of competing models. In an IBE, the premise is a claim about the comparative explanatory power of competing models. I can imagine two potential reasons for this difference, each of which reconstructs the logical structure of IMRE in a different manner. I shall call the first of these reconstructions ‘robustness-as-explanandum’; the second, ‘robustness-as-explanatory-virtue’. I consider each in turn, arguing that the second is more plausible. 3.2 Parametric robustness as an explanatory virtue rather than an explanandum One potential reason for the difference between the comparative claims in IMREs and IBEs is that scientists consider parametric robustness to be an explanandum. Call this the ‘robustness-as-explanandum’ approach.2 Because explananda are features of a model's representational target, this approach treats robustness not only as a formal property of mathematical models but also as an empirical property of target systems. Moreover, if robustness is an explanandum, the most parametrically robust model is more plausible than its competitors because it explains something better than its competitors, namely, the parametric robustness of its target system. Psillos ([2002], pp. 615–6) notes that one hypothesis has more explanatory power than another when, ceteris paribus, it explains more salient phenomena. Hence, according to the robustness-as-explanandum approach, IMREs are only special cases of IBE, in which the most parametrically robust models have the most explanatory power by virtue of better explaining a particularly salient kind of explanandum. There is an alternative approach to IMREs, however, according to which IMREs might qualify as a novel class of IBE rather than a special subclass of more familiar kinds. For there is a second potential reason for the difference between the comparative claims in IMREs and IBEs, namely, that scientists consider parametric robustness to be an explanatory virtue. Call this the ‘robustness-as-explanatory-virtue’ approach. Explanatory virtues are properties that enhance the explanatory power of hypotheses. These virtues often are described as comparative—that is, as virtues that an explanatory hypothesis has in relation to its competitors. Psillos ([2002], pp. 615-6) provides a representative list that includes: Completeness: The hypothesis is the only one amongst its competitors to explain all targeted explananda. Consilience: The hypothesis fits with background knowledge better than its competitors. Importance: The hypothesis explains more salient phenomena than its competitors. Parsimony: The hypothesis uses a proper subset of assumptions from its competitors. Precision: The hypothesis offers more precise explanations than its competitors. None of these virtues are phenomena to be explained by hypotheses. For example, if importance is a virtue, hypotheses that explain more salient phenomena than their competitors have (ceteris paribus) more explanatory power than their competitors, because importance confers that power; they do not have more explanatory power by virtue of explaining why certain target phenomena are salient. Similarly, according to the robustness-as-explanatory-virtue approach, models with more parametric robustness have (ceteris paribus) more explanatory power than their competitors because parametric robustness confers that power; contrary to the robustness-as-explanandum approach, they do not have more explanatory power by virtue of explaining why certain target phenomena are robust. If the robustness-as-explanatory-virtue approach is correct, whether IMREs qualify as a novel kind of IBE depends upon whether standard explanatory virtues subsume the virtue of parametric robustness. If they do—if, for example, exhibiting more parametric robustness is only a special way of exhibiting more precision—then IMREs are mere variants of more familiar IBEs. I shall argue, in Section 3.3, that a range of standard explanatory virtues does not subsume parametric robustness. Parametric robustness, I shall contend, is an explanatory virtue in its own right. Before making that argument, however, I conclude this section by arguing against the robustness-as-explanandum approach. My first argument against the robustness-as-explanandum approach is an argument from ignorance. When Chen and colleagues infer that the direct activation model of Bcl-2 apoptotic switching is more plausible than the indirect activation model, their argument turns almost entirely upon formal mathematical properties of the respective models. Neither their published text nor supplementary material refers to any kind of parametric robustness for actual systems that exhibit Bcl-2 apoptotic switching. The closest they come to such a reference is when they note that actual systems seem to exhibit certain important features, namely, ultra-sensitivity, range, inhibition, and insensitivity. They do not claim that actual systems exhibit any kind of parametric robustness for those features. Other instances of IMRE exhibit a similar absence of evidence. For instance, when Morohashi et al. ([2002]) infer that one model of mitotic phase oscillations in early Xenopus embryos is more plausible than a competing model, their argument is primarily mathematical. They appeal to empirical evidence about certain features of mitotic phase oscillations in early Xenopus embryos. They do not explicitly appeal to any kind of parametric robustness for these features or for the mitotic phase oscillations themselves. Indeed, they characterize themselves as using ‘Ockham's Razor to distinguish between any two models which may match experimental observations equally well: the model with the greatest parameter robustness […] is the more plausible!' ([2002], p. 22). If the robustness-as-explanandum approach were correct, we would expect to find some evidence of scientists appealing to parametric robustness of target systems when making IMREs. I find no such evidence in my encounters with IMRE. Arguments from ignorance are cogent only insofar as the search for evidence is competent and thorough (see Boone [1999], p. 15). Because the scientific literature is massive, I make no claim to have performed a thorough search for instances of IMRE. I claim only that after a dedicated and competent search of the biological literature (guided primarily by following citations), I find no evidence to support the robustness-as-explanandum approach. I take this as sufficient for at least shifting the burden of proof in favour of the robustness-as-explanatory virtue approach and for motivating further inquiry about the details of that approach. For those unsatisfied by burden-shifting arguments, I offer a second argument. This second argument against the robustness-as-explanandum approach turns upon a contention about the nature of certain parameters. Suppose, for the sake of argument, that parametric robustness is an empirical property of target systems, rather than only a formal property of mathematical models. I am not aware of any formal definitions for this kind of robustness, but presumably this will do: a target system exhibits parametric robustness in reality if for some property represented by a parameter value and some feature of interest about the target system, variations of that property do not affect whether the target system exhibits the feature of interest. Gunawardena ([2010], p. 17) discusses something like this under the heading of ‘robustness in reality’. There is a problem with this notion of parametric robustness in reality. Parameters for differential equations are often rate constants. For example, Chen et al. ([2007], p. 5145) use the law of mass action to derive differential equations for their models of apoptotic switching. This results in their differential equations having rate constants as parameters. As Gunawardena ([2010], p. 15–6) notes, association rates, disassociation rates, catalytic rates, and other rate constants often are […] intrinsic features of the corresponding [components] and would not be expected to change except through alterations of their amino acid sequences. This could happen on an evolutionary time scale, so that different species may have different parameter values, but this would not be expected to happen in different cells of the same organism or tissue or clonal populations of cells in cell culture. Hence, at least for Chen and colleagues' IMRE, their target system cannot exhibit parametric robustness in reality, because the properties represented by their models' rate constants do not vary in reality (on a relevant time scale). So the robustness-as-explanandum approach is not appropriate for reconstructing their inference. Nor, for the same reason, is the robustness-as-explanandum approach appropriate for reconstructing IMRE's about other models in which parameters for determining robustness represent system-invariant properties. Contrary to how the robustness-as-explanandum approach would have it, the robustness of system-invariant parameters (parameters representing system-invariant properties) is not desirable for an epistemic reason (namely, for tracking a kind of robustness in reality). Instead, for system-invariant parameters, robustness is desirable for a pragmatic reason (namely, for minimizing errors due to faulty parameter estimations). Ma and Iglesias ([2002], p. 10), who endorse IMRE-type reasoning, elaborate upon this rationale for valuing parametric robustness: In modelling biological networks, it is important that this robustness [to variations in individual parameter values] also be in evidence. The particular behaviour being characterized by the model should not rely on precise values of the model's parameters—for example, reaction rate constants or protein concentrations. In particular, a precise measurement of these constants is difficult whereas protein concentrations will vary from one cell to another or throughout the lifetime of any individual. Deviations from the nominal model parameter values should not result in a loss of the network's performance; thus, parameter sensitivity can be used to validate mathematical models of biochemical system. That is, the more insensitive the system response is to the accuracy of the parameter, the more faith we should have in the model. Notably, Ma and Iglesias extend the pragmatic reason for desiring robustness of system-invariant parameters to all parameters. They include, in particular, protein concentrations and other parameters that represent properties capable of varying in reality. This extension mitigates a weakness in the second argument against the robustness-as-explanandum approach. The second argument against the robustness-as-explanandum approach has only limited value, because parameters for differential equations sometimes correspond to properties that vary in reality. Some parameters, for example, depend upon temperature or other environmental factors. Because such environmental factors vary in reality, these parameters are system-variable (see Morohashi et al. [2002], p. 20). Hence, the robustness-as-explanandum approach is appropriate for reconstructing IMREs about models in which the parameters for determining robustness represent system-variable properties. B.-S. Chen et al. ([2005], p. 2704), when formulating a measure for parametric robustness, concur with this reasoning: Real biochemical networks must be sufficiently robust to tolerate parameter and environmental variations or else they cannot respond efficiently to small but persistent parameter and environmental perturbations. Therefore, if a model of a biochemical network has a small robustness measure, it is often a sign of structural inadequacies of the model. B.-S. Chen and colleagues do not use their measure in an IMRE. But their argument couples parameter variations with environmental variations, and thereby treats parametric robustness as a marker of empirical adequacy. Models with more parametric robustness, on their view, better represent systems that are robust in reality. Hence, in agreement with the robustness-as-explanandum approach, more parametrically robust models are more plausible than their competitors because they explain something better than their competitors, namely, the robustness in reality of their target system. There is nothing particularly objectionable about the details of this defence for the robustness-as-explanandum approach. The defence is compatible with finding no evidence to support the approach, because the IMREs with which I am familiar take as input models where at least some of the key parameters are system-invariant. In principle, I grant that the robustness-as-explanandum approach might be appropriate for reconstructing some IMREs. But, as I noted in my first argument against that approach, I find no actual IMREs that meet the conditions for appropriateness. Moreover, even if there were IMREs in which all key parameters were system-variable, it is possible to reconstruct those inferences using the robustness-as-explanatory-virtue approach and, following Ma and Iglesias, to understand parametric robustness as desirable for the pragmatic reason of minimizing errors due to faulty parameter estimation. The interpretive choice, accordingly, is between a piecemeal strategy for reconstructing IMREs—treating parametric robustness as an explanandum in some cases, but an explanatory virtue in others—or a unified strategy that always treats parametric robustness as an explanatory virtue. I prefer unified interpretive strategies over piecemeal ones. So I shall proceed as though the robustness-as-explanatory-virtue approach is correct. I invite those with a different preference to qualify what follows as restricted to only certain kinds of IMREs. For, no matter one's preference, there are some IMREs for which parametric robustness is an explanatory virtue—namely, those that take as input models in which some of the key parameters are system-invariant. 3.3 Relation of parametric robustness to other explanatory virtues Because there is no evidence favouring the robustness-as-explanandum approach, and because that approach is inappropriate for IMREs in which the robust parameters represent invariant system properties, I prefer the robustness-as-explanatory-virtue approach to reconstructing the logical structure of IMREs. According to this approach, more parametric robustness is an explanatory virtue: ceteris paribus, models with greater parametric robustness thereby have greater explanatory power. Hence, by this approach, we can refine our prior analysis for the logical structure of IMREs (from Section 3.1) by distinguishing two inferential parts: Part 1: (1) Some set of features, F, obtains in some real system(s) of interest. (2) Among competing mathematical models, M1,…, Mn, Mi is more parametrically robust for exhibiting F than its available competitors. (3) Parametric robustness is an explanatory virtue. (4) Hence, Mi has more explanatory power than its competitors. Part 2: (4) Some set of features, F, obtains in some real system(s) of interest. (5) Among competing mathematical models, M1,…, Mn, each would explain F were it true. (6) Mi has more explanatory power than its competitors (via part 1). (7) Therefore, Mi is more plausible than its available competitors. Parameter robustness analysis establishes the second premise in Part 1; the robustness-as-explanandum approach secures the third; and the conclusion of Part 1 figures as a central premise for Part 2. So reconstructed, an IMRE resembles a typical IBE, albeit with a slightly different conclusion. This resemblance, in itself, is no surprise; the analysis of logical structure results from attempting to understand the inferential structure of IMREs as variants of more familiar IBEs. In the next section, I consider how best to interpret the conclusion of IMREs. In the remainder of this section, however, I shall argue that IMRE qualifies as a novel form of IBE because parametric robustness, interpreted as an explanatory virtue, is not subsumed by explanatory virtues familiar from extant discussions of IBE. In denying that standard explanatory virtues subsume parametric robustness, I mean that parametric robustness is not a special case of those virtues. This is not a trivial contention. Some virtues subsume others. For example, Thagard ([1978], p. 91) identifies analogy as an explanatory virtue. A hypothesis exhibits analogy when it is similar to other hypotheses already established—it invokes mechanisms, entities, and concepts familiar in antecedently accepted explanations. Psillos’s virtue of consilience subsumes analogy, because being similar to antecedently accepted explanations is a specific way to fit with background knowledge. For the sake of tractability, I take the list from Section 3.2 as a representative sample of standard explanatory virtues. My strategy is to argue that, for each sampled virtue, exhibiting parametric robustness is not a special way to exhibit that virtue. When possible, I use the case study from Chen and colleagues as support. Completeness: A model exhibits completeness when it is the only one from the field of competitors to explain all relevant explananda. When Chen and colleagues argue that the direct activation model for Bcl-2 apoptotic switching is more parametrically robust than the indirect activation model, they do so because neither of those models explains all relevant explananda. Each model fails to explain at least some phenomena that proponents of the competing model claim as experimental support (see Section 2.2). Hence, parametric robustness is not a special case of completeness. Consilience: A model exhibits more consilience than its competitors if it better fits with background knowledge. Parametric robustness neither enhances nor lessens fit with background knowledge. For example, when Chen and colleagues consider the direct activation and indirect activation models, those models fit biochemical background knowledge equally well. Both invoke familiar kinds of mechanism components and interactions; both rely upon concepts familiar from other accepted explanations; and both fit different portions of existing experimental evidence. Nonetheless, the direct activation model is more parametrically robust than its competitor. So consilience does not subsume parametric robustness as an explanatory virtue. Importance: A model exhibits more importance than its competitors if it explains more salient phenomena. If parametric robustness were an explanandum, more parametrically robust models might exhibit more importance. But, granting the arguments from Section 3.2, parametric robustness is not an explanandum. Nor do models with more parametric robustness always explain more salient phenomena than their competitors. For example, the direct activation and indirect activation models for Bcl-2 apoptotic switching both explain selected qualitative features of apoptotic switching, namely, ultra-sensitivity, range, inhibition, and insensitivity (see Section 2.3). Moreover, each model explains somewhat different experimental phenomena (see Section 2.2 for details). But Chen and colleagues do not treat some of these phenomena as more salient than others. They note that ‘proving which model is more telling by means of experimental approaches seems difficult thus far' ([2007], p. 5143) . Moreover, they infer that the direct activation model is more plausible because it is more parametrically robust than its competitor, and not because it explains more salient phenomena. So importance does not subsume parametric robustness. Parsimony: A model exhibits more parsimony than its competitors if it uses a proper subset of assumptions from its competitors. The direct activation and indirect activation models for apoptotic switching do not stand in this kind of subset relationship. So parsimony does not subsume parametric robustness. Nonetheless, sometimes the models that figure in an IMRE have assumptions related in something like a subset manner. For example, when Marlovits and colleagues make an IMRE about models for mitotic phase oscillations, they compare two models: the first has a single direct feedback mechanism; the second is identical to the first except for replacing that feedback mechanism with several indirect feedback loops (see Marlovits et al. [1998], pp. 171–2). This second model contains much of the first as a proper subset, and so there is a sense in which the first exhibits more parsimony than the second. However, Morohashi et al. ([2002], p. 28) show that the second has more parametric robustness. Hence, far from subsuming parametric robustness, parsimony sometimes competes with parametric robustness as a source of explanatory power. Precision: A model exhibits more precision than its competitors if it offers more precise explanations. Precision, in this sense, refers not only to the level of detail for a model's derivational consequences, but also to the level of detail for the causal–nomological mechanisms embedded within the model (Psillos [2002], p. 616). Chen et al. ([2007], pp. 5145–7) are interested the ability of the direct activation and indirect activation models to explain qualitative explananda: the general trend of responses to stimuli; the existence of certain thresholds; the general level of various activities. Those models, moreover, consist of similarly detailed differential equations derived from similarly detailed biochemical mechanisms. Neither model, accordingly, exhibits more precision than the other. Yet the direct activation model exhibits more parametric robustness. So precision does not subsume parametric robustness. 4 Epistemological Significance of IMRE Because parametric robustness is not a special case of more familiar explanatory virtues, and because it is not an explanandum, it qualifies as an explanatory virtue in its own right—at least with respect to the project of understanding IMRE as a variant of IBE. For the sake of clarity, it is worth being explicit about the conditions under which a mathematical model exhibits this virtue: Parametric robustness: From among an n-membered collection of suitably selected parameter value sets for a specific mathematical model and n-membered collections of similarly selected parameter value sets for each of the model's competitors, the model exhibits certain and empirically well-attested features for more of its selected parameter value sets than does its competitors. Because parametric robustness is an explanatory virtue, models with more parametric robustness than their competitors have, ceteris paribus, more explanatory power than their competitors. Moreover, because it is an explanatory virtue in its own right, it is possible for a mathematical model to gain explanatory power by virtue of exhibiting parametric robustness, but lose explanatory power by virtue of failing to some other explanatory virtue. For current purposes, I set aside issues about how to weigh parametric robustness against other explanatory virtues. For the models that Chen et al. ([2007]) consider for their IMRE, parametric robustness seems to be the only one that discriminates among competing models. For the models that Morohashi et al. ([2002]) consider for their IMRE, parametric robustness seems to trump certain other virtues such as parsimony. Carlson and Doyle ([2000], p. 2532) suggest that, for certain kinds of domain-specific models, there is a general trade-off between maximizing parametric robustness and minimizing complexity. Presumably context often matters. Sorting out details requires further case studies. Rather than provide those details, I shall consider instead the epistemological significance of IMREs. The conclusions of IMREs appear to differ from typical conclusions of standard IBEs. An IMRE concludes with the claim that one model is more plausible than its competitors. A standard IBE, in contrast, concludes with the claim that one hypothesis is probably true (see Section 3.1). This raises an epistemological issue. Does a model's superior plausibility, as conferred by an IMRE, mean that the model is probably true—or at least more likely to be true than its competitors? If so, the resemblance between IMREs and standard IBEs is strengthened. But if not, what epistemic attitude does superior plausibility warrant? I shall approach this issue from two directions. First, I report some of the stances practicing scientists take on the issue with respect to their own IMREs. These stances tend to treat superior plausibility as indicating only superior suitability for continuing research efforts. However, people do not always infer what their premises warrant. So, second, I consider whether some prominent realist arguments about explanatory power support a stronger, truth-oriented stance toward the conclusions of IMREs. I argue that they do not. 4.1 Plausibility in practice Haefner ([1996], p. 152), when discussing the difference between model verification and model validation, has this to say about the adjective ‘plausible' as applied to biological models: For my part, in light of the rather small number of well-tested models in biology and the generally low rigor of the tests, I think the adjective plausible more accurately reflects the nature of tested biological models and the skeptical attitude we should adopt. To a more cynical observer, the dictionary definition of ‘specious’ might also come to mind. Haefner aims his remarks at those who, in order to avoid connotations about correctness, prefer to speak of validated models as confirmed or corroborated. For Haefner, these alternatives also have improper connotations about correctness. Linguistic preferences aside, his point seems to be that validated models—that is, plausible models—need only be well-founded, tractable, and fulfill the purposes for which they are formulated (see Carson et al. [1983], p. 217). They need not be correct, and the most plausible models need not be the most likely to be true. At least some scientists who use IMRE seem to have Haefner's meaning of plausibility in mind when characterizing models with superior parametric robustness as plausible. For example, Chen and colleagues use IMRE to infer that the direct activation model for Bcl-2 apoptotic switching is more plausible than its indirect activation competitor. They seem to think this conclusion warrants focusing future research efforts on developing the direct activation model and incorporating into it new experimental findings, for they go on to do just this (see Cui et al. [2008]). Similarly, when Tokar and Ulicny use IMRE to argue that the direct activation model is more plausible than certain ‘hybrid' models (which take direct and indirect activation mechanisms to be at work simultaneously), they note that despite their findings, ‘we cannot completely rule out the indirect model interactions' ([2012], p. 6219). Their more nuanced conclusion, accordingly, is that while the direct activation model is more plausible, their robustness analysis shows only that ‘indirect model interactions, if present, must proceed at very low rates […] and provide other important system properties' ([2012], p. 6219). Their attitude is cautious, because their robustness analysis focuses upon only some of the many important features for Bcl-2 apoptotic switching. Relative to the purpose of assessing plausibility for their limited focus, the direct activation model is most plausible. Because they do not know what their analysis might show for a different or more comprehensive focus, they do not make stronger claims about likely truth. Morohashi and colleagues more explicitly treat plausibility as indicating validity but not correctness. Comparing two models for explaining the mitotic phase oscillations of early Xenopus embryos, they argue that ‘the structure of the ′98 model is capable of providing highly robust oscillatory behaviour in a manner far exceeding the capabilities of the ′91 model […]' (Morohashi et al. [2002], p. 28). They infer from this, following the pattern for IMRE, that the ′98 model is more plausible than the ′91 model. But they note that superior plausibility does not preclude the existence of yet more plausible models, and they provide a slight variant to the ′98 model that is more parametrically robust than even the ′98 model ([2002], pp. 28–9). Hence, far from meaning that the ′98 model is true, their saying that the ′98 model is more plausible than the ′91 model seems to mean only that it would be worthwhile to focus future research efforts on developing the ′98 model. 4.2 Plausibility in principle Even if, in practice, scientists mean for the conclusions of IMREs to be about which models are most well-founded, tractable, and worth further exploration, the premises of IMREs might warrant stronger conclusions about which models are most likely to be true. The philosophical literature on IBE and scientific realism contains several arguments purporting to show that superior explanatory power indicates truth. If any of those arguments hold for IMREs, there might be good reason to interpret claims about plausibility as claims about correctness. I consider two of these arguments and argue that neither holds for extant instances of IMRE. Boyd offers a kind of transcendental argument for supposing that models—‘theories’, in the jargon of his day—with the most explanatory power are at least approximately true. Speaking of IBE in its standard form (as yielding, for a conclusion, a claim about the truth or approximate truth of the best explanation), he writes: The rejection of abduction or inference to the best explanation would place quite remarkable strictures on intellectual inquiry. In particular, it is by no means clear that students of the sciences—whether philosophers or historians—would have any methodology left if abduction were abandoned. If the fact that a theory provides the best available explanation for some important phenomenon is not a justification for believing the theory is at least approximately true, then it is hard to see how intellectual inquiry could proceed. (Boyd [1983], p. 74) Boyd presumes, as a datum, that intellectual inquiry often produces at least approximately true theories. His central premise is that this datum could not happen unless theories with the most explanatory power were approximately true. I set aside the issue of whether Boyd's premises are true or his reasoning cogent. Even if his argument succeeds for IBE generally, it does not show that theories with the most parametric robustness are thereby at least approximately true. IMREs are a relatively new kind of inference among biologists. (Morohashi et al. ([2002]) make the first explicit IMRE in the literature for theoretical biology.) These inferences, moreover, are not essential to general biological inquiry. So Boyd's central premise does not apply for IMRE: biological inquiry would continue to produce approximately true theories (presuming it produces them at all) even if the most parametrically robust theories were not approximately true. In the case of Bcl-2 apoptotic switching, for example, inquiry about further details of the direct activation model, as well as about hybrid competitors to this model, continued without any explicit commitment from Chen and colleagues, or from Tokar and Ulicny, about the approximate truth of that model (see Section 4.1). Gasper ([1990]) offers a practice-based argument for treating IBEs as yielding conclusions about empirical adequacy. This is a weaker argument than Boyd's, because empirically adequate theories need not be true about unobservable entities. He writes: […] why is explanatory power desirable? […] Recall that scientists do not typically independently identify empirically adequate theories and then compare them with respect to explanatory power. Rather, judgments of explanatory power play a central role in the choice of reliable theories to guide further research. This claim finds confirmation in the fact that explanatory power is routinely taken as a guide to theory acceptance in successful areas of science […] Realists insist that we take this practice seriously. Explanatory power […] is at least a guide to which theories are empirically adequate. In other words, explanatory power is evidential. (Gasper [1990], pp. 293–4) Gasper's argument elides the difference between deciding to use a theory to guide further research and deciding to endorse a theory as empirically adequate. But the difference matters. Chen and colleagues, for example, decide on the basis of an IMRE to use the direct activation model to guide further research about apoptotic switching. However, they endorse that model only as plausible. As I have argued, this is not an endorsement about correctness (see Section 4.1). Taking practice seriously in this case does not support Gasper's elision. 5 Conclusion Harman characterizes standard IBE as a form of reasoning in which ‘one infers, from the premise that a given hypothesis would provide a “better” explanation for the evidence than would any other hypothesis, to the conclusion that the given hypothesis is true' (Harman [1965], p. 89). So interpreting claims about plausibility in the conclusions of IMREs as claims about well-foundedness, tractability, and suitability for further exploration, rather than claims about correctness or truth, means that IMREs are not straightforward instances of standard IBEs. This is not a reason to deny that IMREs are variants of standard IBEs. There are non-standard accounts of IBE as well. Katzav ([2013]), for example, argues that some IBEs do not infer to approximate or probable truth because sometimes best explanations are idealized and therefore false. Dawes ([2013]) argues that IBEs yield as conclusions categorical claims to the effect that the best explanation is acceptable to use as a premise in knowledge-directed theoretical or practical reasoning. Either of these variants for standard approaches to IBE suffices for understanding IMREs as IBEs and interpreting plausibility claims as not making claims about truth or correctness. Peirce ([1935] Paragraph 525), moreover, characterizes abduction (a form of inference that includes what we now call IBE) as ‘the first starting of a hypothesis and the entertaining of it, whether as a simple interrogation or with any degree of confidence'. Provided that Haefner's kind of plausibility suffices for this ‘first starting', there is at least some historical precedent for a non-standard interpretation of IBE (see Hookway [2012], pp. 75–9). Understanding IMREs as variants of standard IBEs—or as instances of certain non-standard IBEs—presumes, of course, that it is appropriate to understand IMREs as some kind of IBE. The motivation for this presumption is a formal resemblance between the pattern of IMREs and the pattern of IBEs (Section 3.1). The presumption is also heuristically useful, by virtue of directing inquiry about IMRE in a way that clarifies the role of parametric robustness and the meaning of plausibility (Sections 3.2–4.2). Understanding IMREs on the model of some other inferential form might support further insights about IMRE. But that is a project for another time. I conclude, instead, with one benefit the present approach to IMRE has for our understanding of IBE. Searching through the recent literature on IBE, one finds intuitions about how IBEs ought to work and about what conclusions they ought to warrant. Lipton ([2004], p. 115), for example, appeals to intuitions about what ‘the defender of Inference to the Best Explanation' should say. Also prominent are claims about the prevalence of IBEs. For instance, Dawes ([2013], p. 62) claims that IBEs are frequent in the sciences and everyday life. But, at least recently, there are only a handful of studies examining specific instances of IBE from scientific practice. These include Weber’s ([2009]) study of IBE in the context of using experiments to explain data patterns, Tulodziecki’s ([2011]) reconstruction of John Snow’s reasoning about cholera pathology and transmission, and Katzov’s ([2013]) examination of competing climate science models. Understanding IMREs as a kind of IBE adds to philosophers' stable of cases one further study, namely, competing biochemical models of apoptotic switching (Section 2). This case, moreover, helps to highlight the way in which certain realist arguments for the truth-conduciveness of IBE depend upon contentions about scientific practice that do not hold in all contexts (Section 4.2). Acknowledgements I thank several anonymous referees for their considerable help in reorganizing and refocusing this article. A distant ancestor draft owes its origins to sabbatical work while in residence as a visiting fellow in the Center for Philosophy of Science at the University of Pittsburgh. I thank fellow residents at the Center for stimulating discussions, as well as the Center itself and University of Alabama in Huntsville for financial support. I also thank Bill Roche for helpful comments on that ancestor draft and John Norton for making salient to me the unfortunate dearth of case studies for IBE in scientific practice. 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Published: Mar 1, 2018

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