It is useful to start by showing how the same Bayesian model outlined in section two treats analogical inferences of a different kind from the ones discussed in the previous section: ‘predictive’ analogical arguments (Sect. 3.1). The case of disconfirmation in analogical argument will then be discussed (Sect. 3.2). Finally, some comments will be made about the notion of weight of similarities and dissimilarities and its relation to graded similarity (Sect. 3.3). These extensions are evidence of the high generalizability and non-adhocness of the proposed Bayesian analysis.
The inferences reviewed in the previous section are all instances of analogical arguments ‘from effects to causes’– ‘abductive’ analogies in Bartha’s (2009) taxonomy: from the similarities between some effects (e.g., the wavy line patterns), one reasons that the causes may well be similar (e.g., that the Moon has an irregular morphology like the Earth’s). Examples of this kind have been discussed in detail by Sober (2015, p. 118), who has put forward a more complex representation of them in probabilistic terms. The simpler recipe indicated above, which consists in identifying a ‘bridge’ hypothesis (such as G) positively correlated to both the evidence contained in the analogical argument (E) and the hypothesis that stands as its conclusion (H), can be easily adapted to the examples of abductive analogical arguments that Sober discusses. Since the application to the analogical arguments considered by Sober is a fairly uninteresting extension of the discussion of the previous section, we will simply omit it in what follows.
A more interesting result for the purposes of this paper is that the Bayesian representation offered in section two may also fit analogical arguments of a different kind than those discussed by Sober (2015) and exemplified by Galileo’s case. We have in mind analogical arguments ‘from causes to effects’—‘predictive’ analogical arguments in Bartha’s (2009) classification. An example would be cases of reasoning from a simulation: from the known similarities in ‘causes’ between the simulation and the target system being simulated, one reasons that roughly the same effect observed in the simulation may well obtain in the unfamiliar target system.Footnote 21 As a way of illustrating how this category of analogical inferences may be represented by the same Bayesian model of section two, let’s consider a somewhat simplified version of an actual episode from the history of medicine: specifically, of how evidence about the efficacy of inactivated polioviruses (the cause of poliomyelitis) in a population of rhesus monkeys confirmed Salk’s (1955) yet untested hypothesis of the efficacy and safety of the same vaccine in human populations.Footnote 22
As a preliminary point, we note that Hesse’s (1963) requirements for analogical arguments capable of inductive support (1–3) plausibly hold for the new case-study. (Hesse’s conditions are stated precisely so as to apply to both abductive and predictive analogies.) First, the similarities between rhesus monkeys and humans figuring in the analogical argument were all resemblances in scientifically respectable properties, such as sharing similar immune systems. Second, the similarities figuring as the premises of the analogical argument were plausibly relevant to the prediction that the vaccine is efficacious in human beings, since the biological features in which humans and rhesus monkeys are similar may be causally connected to the vaccine response. Finally, no critical differences between humans and rhesus monkeys were known with regards to the reaction to the poliovirus: indeed, rhesus monkeys were known to suffer from the very same symptoms as humans (paralysis) when the poliovirus was transmitted to them. These features make it plausible that confirmation by analogy occurs, meaning that the observed efficacy of the vaccine in rhesus monkeys is at least some evidence for its efficacy in human beings.
An important difference with abductive analogical arguments remains, which is crucial for the purpose of building an appropriate Bayesian model. This has to do with precisely how inferences from predictive analogical arguments work. As Galileo’s case-study illustrates, the similarities and dissimilarities which figure in an abductive analogical argument are relevant to the extent that they bear on the hypothesis (e.g., G) that roughly the same kinds of causes that obtain in the source also obtain in the target; this, in turn, makes it plausible to expect further similarities between source and target. For instance, the similarities between lunar wavy lines and terrestrial shadows make plausible the hypothesis that the same irregular morphology that we observe on Earth is realized on the Moon, from which we can in turn derive that the Moon has an irregular morphology.Footnote 23 In predictive analogies, instead, the similarities and dissimilarities in, e.g., the immune system between rhesus monkeys and humans, are relevant to the extent that they bear on the hypothesis that a given effect is relatively robust to differences (within a certain range), from which we can defeasibly infer that the effect observed in the source will have an analogue in the target: e.g., humans will respond to the vaccine in the same way as monkeys. Hence, the focus in predictive analogical arguments is mostly on the robustness of the effect.Footnote 24
With this in mind, let’s consider how an argument from the efficacy of a vaccine in rhesus monkeys to a similar outcome in human beings can be represented in the Bayesian framework. For this purpose, let’s stipulate that ‘O’ is new evidence that inactivated polioviruses are safe and efficacious on rhesus monkeys.Footnote 25 Let ‘T’ be the prediction of a similar outcome in human populations. We now introduce the bridge hypothesis ‘X’, defined as follows:
Within the range of variation that contains rhesus monkeys and humans, the vaccine’s effects are relatively robust to differences at the level of proprietary biological features.
In other words, X is the bridge hypothesis that the vaccine’s effects will tend, under a variety of changes in background conditions, to remain the same (in this sense, they are ‘relatively robust’).
Given these stipulations, the following inequalities plausibly hold in this case-study:
0 < P (T), P (O) < 1,
0 < P (X) < 1, where:
P (T | X) > P (T)
P (T | O & X) ≥ P (T | X)
P (T | O & ¬ X) ≥ P (T | ¬ X)
P (O | X) > P (O | ¬ X).
C1–C3 correspond respectively to the three informal requirements by Hesse (1–3 above) on analogical arguments capable of inductive support. C1 is a probabilistic rendition of the requirement of materiality, stating that the similarities figuring in the analogical argument must be in scientifically respectable features. This gets translated into the claim that both T and O possess well-defined, non-extremal probabilities given the background knowledge.
C2 encodes probabilistically Hesse’s requirement of relevance: non-negligible credence to the bridge hypothesis X is justified, which ties the evidence O together with the prediction T from the standpoint of confirmation. In this case, as with B2(a), we do not have an entailment between X and T, which would make the C2(a) trivially satisfied; however, conditioning on the claim that the vaccine’s effects tend to remain the same under a large variety of antecedent conditions, as required for satisfying the ‘robustness’ claim, makes the hypothesis that the vaccine is efficacious and safe in humans likely. Hence, C2(a) is plausible. The remaining assumptions, C2(b) and (c) are also plausible in the example. As for C2(b), new evidence that inactivated polioviruses are safe and efficacious on monkeys is arguably irrelevant to T if we already know that the vaccine’s efficacy is robust to differences between monkeys and humans; hence P (T | O & X) = P (T | X). As for C2(c), the hypothesis that the vaccine is safe and efficacious in humans is arguably not made less probable by new evidence that inactivated polioviruses are safe and efficacious on monkeys, even if the vaccine’s effects are not robust to differences between monkeys and humans; plausibly, then, P (T | O & ¬ X) ≥ P (T | ¬ X).
Finally, C3 encodes probabilistically one’s acceptance of the no-critical-difference condition. Specifically, our background knowledge about the features of monkeys and humans and the fact that no critical differences were known regarding the expected reaction of rhesus monkeys to poliovirus compared to humans made it plausible to set P (O | X) greater than P (O | ¬ X).Footnote 26
Altogether, C1-C3 entail that P (T | O) > P (T), which is exactly what we would expect from a predictive analogical argument that meets Hesse’s requirements for inductive support.
As the reader may appreciate by comparing C1-C3 to A1-A3, the conditions for confirmation by analogy are formally the same. The main difference with the Bayesian model of section two lies with the content of the ‘bridge’ hypothesis. However, we contend that this is not an ad hoc fix. It is supported by the observation that abductive and predictive analogies have different inferential profiles (cf. Bartha, 2009, p. 167): in abductive analogies, the inference from the observed to the merely predicted similarities is mediated by the claim that some effect in the target has causes of the same kind as those in the source (e.g., the same kind of irregular morphology); in predictive analogies, instead, the inference from observed to predicted similarities proceeds through the intermediate claim that some salient effect (e.g., vaccine efficacy) is robust to low-level differences in causes between source and target (e.g., differences in the specific details of the immune system between monkeys and humans). The Bayesian model relies on this plausible observation to describe the mechanism of confirmation for each kind of analogical argument.
To sum up, what we have sketched in C1-C3 is a plausible account of how a predictive analogical argument in science can be confirmatory from BC’s standpoint. Although the extension to other realistic examples will have to be left for a separate occasion, we hope that the illustration above helps clarify the general recipe that we are proposing. In each case, the idea is to identify a plausible bridge hypothesis stating that a salient effect (the one figuring in the argument’s conclusion) is robust to differences in some low-level detail between source and target and such that confirmation of the bridge hypothesis can in turn bear on the argument’s conclusion.Footnote 27 Based on its correspondence with independently plausible ideas about when and how inductive support by predictive analogical argument occurs, we have reasonable confidence that this recipe may be widely applicable. In order to complete our case that the Bayesian analysis proposed in this paper deserves serious consideration, in what follows we will highlight a few more aspects of the logic of analogy that the current proposal may be able to capture.
An adequate treatment of dissimilarity is one of the desiderata of an account of confirmation by analogy. In what follows, we will offer hints towards accommodating the role of dissimilarities within the same Bayesian framework proposed in the previous sections. The most common case, though by no means the only one, is when a dissimilarity weakens the conclusion of an analogical argument. Consider, for instance, Darwin’s argument for natural selection in the Origin of Species: from the fact that domesticated animals possess a wide variety of traits due to human breeding, and the similarly wide trait variety found in wild animals and plants, Darwin argued that a selection process analogous to the one imposed by human breeders on domesticated animals may be responsible for the wide variety in heritable traits found in nature (cf. Kitcher, 2003:61). One fact of dissimilarity that Darwin considered in spelling out his argument was the fact that artificial selection had not been observed to produce events of speciation (Darwin, 1859:389). This dissimilarity plausibly disconfirms the hypothesis of natural selection to some degree (but see 3.3 on the weight of the dissimilarity). Let’s consider how this disconfirmation by dissimilarity can be represented within the framework outlined in the previous sections.
We will make use of the following stipulations. Let ‘W’ be Darwin’s hypothesis of natural selection. Let ‘D’ be the observation (which we assume to be new) that artificial selection has not given rise to new species of animals or plants. As Darwin’s argument has the form of an abductive analogical argument, the relevant bridge hypothesis will be defined as follows:
Trait variation in nature and in domesticated animals belong to the same class of processes (in undergoing selection).
The formal conditions for disconfirmation by dissimilarity can be derived immediately by symmetry with the conditions of confirmation by similarity (1–3) and are as follows:
0 < P (W), P (D) < 1.
D2. 0 < P (J) < 1, where:
P (¬ W | ¬ J) > P (¬ W)
P (¬ W | D & ¬ J) ≥ P (¬ W | ¬ J)
P (¬ W | D & J) ≥ P (¬ W | J)
P (D | ¬ J) > P (D | J)
Conditions D1-D3 state three plausible conditions for disconfirmation by dissimilarity that have natural explications in informal terms: namely, the dissimilarities in the argument are in scientifically respectable properties (D1), they are negatively relevant to the conclusion of the analogical argument (D2), and no critical difference is known between source and target that would undermine the conclusion of the analogical argument even before the new dissimilarity is introduced (D3). It is a trivial corollary of Roche and Shogenji’s transitivity theorem that D1-D3 together entail that P (¬ W | D) > P (¬ W), which means that D disconfirms Darwin’s hypothesis of natural selection. This is exactly what we would expect in the case in which we learn some new fact of dissimilarity that disconfirms the conclusion of an analogical argument.
Even though this covers the most typical case, the treatment of dissimilarity just outlined is not complete, since not all dissimilarities weaken an analogical argument (even though they weaken the analogy): some of them may even strengthen the argument.Footnote 28 Consider, for instance, Darwin’s point that natural selection is a much more pervasive force than artificial selection, acting at various stages of the life of an organism (e.g. at the embryonal stage as well as at adult stage), with respect to so many parts of an organism, whereas artificial selection depends on the time, commitment, and physical limitations of the breeder. This dissimilarity figures prominently in Darwin’s argument for the natural selection hypothesis in the Origin. As Darwin writes:
Why, if man can by patience select variations most useful to himself, should nature fail in selecting variations useful… to her living products? What limit can be put to this power, acting during long ages and rigidly scrutinizing [the traits of] each creature? (1859, p. 379)
In this case, it is plausible that the dissimilarity confirms, rather than disconfirms, Darwin’s hypothesis that trait variation in nature is due to an underlying process of natural selection.
The Bayesian representation of this paper gives us precise conditions for confirmation by dissimilarity. Letting ‘Z’ be the fact that natural selection is a more pervasive force than artificial selection, Z’s capacity to confirm W can be cashed out in the (by now) familiar way:
0 < P (W), P (Z) < 1.
2. 0 < P (J) < 1, where:
P (W | J) > P (W)
P (W | Z & J) ≥ P (W | J)
P (W | Z & ¬ J) ≥ P (W | ¬ J)
P (Z | J) > P (Z | ¬ J)
The relevant conditions here are E2(b), E2(c), and E3. Stated informally, the formal model tells us that Z can confirm W only if Z bears confirmation-wise on J (as E3 requires) and Z does not disconfirm W conditional on J or on ¬ J (as E2 requires). These conditions are independently credible requirements for a given dissimilarity to provide confirmation; moreover, they are all plausibly satisfied in the specific case of Darwin’s argument for the theory of natural selection.
Of weight and degree
The proposal on offer makes yet no room for a notion that is often associated with the idea of confirmation from analogy: degree of similarity (Carnap, 1980). In itself, this need not be a defect of the account. Hesse (1963:115) already questioned the relevance of the notion of graded similarity for understanding confirmation by analogy, on grounds that dissimilarities may sometimes weigh just as much as similarities in favor of the conclusion of an analogical argument. In this sub-section, our aim is to use the formal framework developed in the previous sections to explain why there is at least a defeasible association of confirmation by analogy with greater degree of similarity, even though it is not always the case that more similarity means more confirmation. In other words, we will offer the sketch of a reductive account of graded similarity by means of a theory of weight of similarity. This account is further evidence of the capacity of the proposal on offer to correctly represent the logic of analogical inference.
To do so, we propose to introduce the notion of weight of a given similarity or dissimilarity. Paralleling Good’s (1984) standard account of the weight of evidence, we note that, for any (set of) similarities or dissimilarities S, bridge hypothesis B bearing on the conclusion of an analogical argument L, S’s weight w is plausibly given by the following ratio:
The more this ratio diverges from unity, the more we say that S weighs either favorably or unfavorably in confirmation. This proposal is in line with the informal account by Hesse (1963) that we take as our starting point: a given similarity or dissimilarity has more weight the more it bears on the claim that observations in the target have the same kinds of causes as those in the source (in abductive analogical arguments) or the claim that the effects observed in the source are robust to the known differences with the target system (in predictive analogical arguments).
With the notion of weight of similarity, we can recover a role for graded similarity indirectly (as it were) in confirmation. To adopt an intuitive illustration, consider three potential telescopic observations of shadow patterns on the Moon’s surface: let ‘M’ be a potential observation of extremely wavy lines on the surface of the Moon (much more irregularly shaped than terrestrial shadows); ‘E’ a potential observation of moderately wavy lines on the Moon, similar to the shadows cast by terrestrial mountains; finally, let ‘C’ be a potential observation of mostly elliptical wavy lines. The following inequalities plausibly hold in this scenario:
m: P (M | G) / P (M | ¬ G) ≈ 1 (read: it is almost equal to 1)
e: P (E | G) / P (E | ¬ G) > 1
c: P (C | G) / P (C | ¬ G) < 1
What this comparison is meant to show is that, under a plausible assignment of conditional probabilities, it follows that the more similar the lunar wavy lines are to terrestrial shadows, the more the evidence will favorably weigh on the bridge G.Footnote 29 Of course, this is by no means true in all cases: as discussed in Sect. 3.2, sometimes dissimilarities confirm more than similarities. The point of this illustration is merely to explain why, by considering some realistic examples, there exists at least a defeasible association between more similarity and more confirmation. This is because, as illustrated by Galileo’s example, in many cases a larger match in observable features (as in e) bears comparatively more in confirmation. We contend that this ‘reductive’ account is all what we could ever want with regards to graded similarity and its relation to confirmation.
Before leaving the discussion of the formal details, one final feature of the Bayesian representation may be noted. In some cases, an argument by analogy may be reinforced, not by pointing out some further similarities between the domains being compared, but by arguing that some of the known dissimilarities are unimportant. For instance, consider again the objection that artificial selection has not been observed to produce events of speciation. A defender of natural selection might point out that: (a) the practice of breeding is relatively recent, in geological times, when compared to the timeframe of natural selection; (b) breeders have not attempted to produce events of speciation (or may have even acted against it). We may represent this formally by noting that, although plausibly P (J | D) < P (J), it is also true that:
This gives us a way of representing how, under a reasonable choice of prior probabilities, dissimilarity D would not weigh significantly against Darwin’s bridge hypothesis J, and therefore be also relatively unimportant in disconfirming the hypothesis of natural selection.
To sum up the discussion of the previous sub-sections, the proposal of section two turns out to naturally extend to a surprising number of phenomena related to the logic of analogical inference. In particular, we have offered sketches of how the Bayesian analysis may extend to predictive analogical arguments, of how it could offer a perspicuous and compelling account of the role of dissimilarities in analogical reasoning, as well as provide a neat probabilistic rendering of the notions of weight and grade of similarity and dissimilarity (which avoids various logical pitfalls). In the next section, we will respond to two important objections to our proposal.