Modeling creative abduction Bayesian style
Abstract
Schurz (Synthese 164:201–234, 2008) proposed a justification of creative abduction on the basis of the Reichenbachian principle of the common cause. In this paper we take up the idea of combining creative abduction with causal principles and model instances of successful creative abduction within a Bayes net framework. We identify necessary conditions for such inferences and investigate their unificatory power. We also sketch several interesting applications of modeling creative abduction Bayesian style. In particular, we discuss usenovel predictions, confirmation, and the problem of underdetermination in the context of abductive inferences.
Keywords
Creative abduction Theoretical concepts Bayes nets Unification Novel predictions Underdetermination1 Introduction
One can basically distinguish two kinds of abductive inferences: those generating new hypotheses and those aiming at determining the best hypothesis from a set of available candidates. Let us call abductive inferences of the former kind creative, and those of the latter kind selective.^{1} While most of the philosophical literature on abduction focuses on selective abduction (see, e.g., Lipton 2004; Niiniluoto 1999; Williamson 2016), there is also an increasing interest in creative abduction (cf. Douven 2017).
In contrast to selective abduction and other kinds of inferences (such as deduction and induction), creative abduction is intended as an inference method for generating hypotheses featuring new theoretical concepts on the basis of empirical phenomena. Most philosophers of science are quite sceptical about whether a general approach toward such a logic of scientific inquiry can be fruitful. However, since theoretical concepts are intimately connected to empirical phenomena via dispositions (see, e.g., Carnap 1936, 1937), a restriction of the domain of application of such an approach to empirically correlated dispositions might be promising. Schurz (2008) differentiates between different patterns of abduction and argues for the view that at least one kind of creative abduction can be theoretically justified. In a nutshell, his approach is based on the idea that inferences to theoretical concepts unifying empirical correlations among dispositions can be justified by Reichenbach’s (1956) principle of the common cause.
In this paper we take up Schurz’ (2008) proposal to combine creative abduction and principles of causation. We model cases of successful creative abduction within a Bayes net framework which can, if causally interpreted, be seen as a generalization of Reichenbach’s (1956) ideas (cf. Glymour et al. 1991). Such a move allows us to specify general conditions which have to be satisfied in order to generate hypotheses involving new theoretical concepts and to describe their unificatory power in a more finegrained way. In addition, it can be used to shed new light on several other issues discussed within philosophy of science. In this paper we will sketch how it allows for handling cases in which we can only measure nonstrict (i.e., probabilistic) empirical dependencies among dispositions, and how it paves the way for new applications to other topics within philosophy of science. We consider our analysis of successful instances of creative abduction by means of Bayes net models as another step toward a unified Bayesian philosophy of science in the sense of Sprenger and Hartmann (forthcoming).
The paper is structured as follows: In Section 2 we introduce Schurz’ (2008) approach to creative abduction. We also explain how it allows for unifying strict empirical correlations among dispositions and how it can be justified by Reichenbach’s (1956) principle of the common cause. In Section 3 we then briefly introduce the Bayes net formalism, present our proposal how to model successful cases of creative abduction within this particular framework, and identify necessary conditions for such cases. Next we investigate the unificatory power gained by creative abduction in the Bayesian setting and draw a comparison with the unificatory power creative abduction provides in the strict setting. In Section 4 we sketch possible applications of our analysis to other topics within philosophy of science. In particular, we discuss the generation of usenovel predictions, new possible ways of applying Bayesian confirmation theory, a possible (partial) solution to the problem of underdetermination, and the connection of modeling successful instances of creative abduction Bayesian style to epistemic challenges tackled in the causal inference literature. We conclude in Section 5.
2 Creative abduction, unification, and the principle of the common cause
Note that creative abduction as discussed above can be interpreted either in a realist or an instrumentalist way. Under the latter interpretation \(\mathcal {D}\) is taken to be nothing over and above a more or less useful theoretical means to unify empirical descriptions of certain phenomena of interest that can—in principle—be replaced by any other concept with equal empirical adequacy and unificatory power. Under the realist interpretation, on the other hand, \(\mathcal {D}\) is assumed to represent a real structure; statements involving \(\mathcal {D}\) are considered to be either true or false. Schurz (2008) made a strong case in favour of a realist interpretation by endorsing Reichenbach’s (1956) common cause principle:
 (CCP)

If two properties A and B are correlated and neither A causes B nor B causes A, then A and B are effects of a common cause C.
(CCP) demands that every correlation among any pair of properties not standing in direct causal dependence to each other has to be explained by the existence of an independent common cause. In this sense (CCP) provides a way of causally unifying observed regularities. In the case of pairwise empirically correlated dispositions such as D_{1},...,D_{n} above, (CCP) supports a realist interpretation of the unifying higherlevel disposition \(\mathcal {D}\): The correlation among dispositions D_{1},...,D_{n} is explained by postulating a common cause \(\mathcal {D}\).
In the next section we take up the idea of combining creative abduction and principles of causation by modeling cases of successful creative abduction in a Bayes net framework. Though Bayes nets can be causally interpreted, one does not have to subscribe to a realist interpretation when employing this particular framework to model creative abduction. While the realist gets a justification for creative abductive inferences on the basis of a causal interpretation, the instrumentalist can still use the Bayes net framework without a causal interpretation as a tool for justifying abductive inferences in terms of unificatory power. In this paper we prefer to stay neutral on the realist vs. instrumentalist question. As we will show, modeling creative abduction Bayesian style comes with a couple of advantages regardless of the answer to that question.
3 Modeling creative abduction Bayesian style
 1.
\(\mathcal {D}\) is not extreme, i.e., \(0<P(\mathcal {D})<1\).
 2.
Each D_{i} depends positively on \(\mathcal {D}\), i.e., \(P(D_{i}\mathcal {D})>P(D_{i})\).
 3.
Each E_{i} depends positively on its corresponding D_{i}, i.e., P(E_{i}D_{i}) > P(E_{i}).
Conditions 1., 2., and 3. are necessary conditions for successful creative abduction: They guarantee pairwise correlations among lowerlevel dispositions that have to be inductively inferred on the basis of observed evidence and build the basis for introducing the higherlevel disposition \(\mathcal {D}\) which is then, in turn, used to explain these correlations.^{4}
It follows from the Markov factorization (Eq. 6) that these \(\binom {n}{2}\) empirical correlation statements can be unified by the 2n + 1 probabilistic statements in conditions 1., 2. and 3.: n statements of the form P(E_{i}D_{i}) > P(E_{i}) (with 1 ≤ i ≤ n), n statements of the form \(P(D_{i}\mathcal {D})>P(D_{i})\) (with 1 ≤ i ≤ n), and 1 statement \(0<P(\mathcal {D})<1\). To compare Schurz’ (2008) approach and the Bayesian approach w.r.t. their unificatory power, we introduce a simple measure u intended to capture the intuitions about unification outlined above. Given n correlated lowerlevel dispositions, u(n) measures the ratio between x(n) empirical statements to be unified and y(n) unifying theoretical statements. In order to shift the neutral case to 0, we subtract 1 from this ratio: \(u(n)=\frac {x(n)}{y(n)}1\). Its output is in the interval [− 1,∞), where a negative value means that the theoretical description is more costly than simply listing the empirical statements, 0 means that there is no gain but also no cost in providing a theoretical description, and a positive value means that the theoretical description provides unification.^{5}
As the comparison in Fig. 2 shows, the original approach proposed by Schurz (2008) and our Bayesian approach perform differently well in different settings. In the case without conditional correlations, the strict approach fares better. It provides more unificatory power and leads already to unification with only two empirically correlated dispositions, while our Bayes net approach requires at least four empirically correlated dispositions to produce positive unificatory power. In the nonstrict setting with conditional correlations, on the other hand, Schurz’ approach is not applicable. This is the setting where the Bayesian approach excels. Although the version with 2n + 1 unifying statements also requires at least four empirically correlated dispositions to produce positive unificatory power, the amount of unificatory power provided explodes. The version with n + 1 unifying statements fares even better. Note that it already provides positive unificatory power with three empiricaly correlated dispositions. These results suggest that the two approaches might rather be seen as complementing each other than as concurring accounts.
4 Possible applications and connections to other issues
In this section we outline possible applications of modeling creative abduction Bayesian style and connections to other topics from the philosophy of science literature. In particular, we discuss how abduced theoretical concepts allow for usenovel predictions, how the approach fits with a recent proposal to solve the problem of underdetermination, and how it provides new possibilities for confirmation. Finally, we briefly discuss how results from the causal discovery literature could be used to approach creative abduction from an epistemic perspective.
Usenovel predictions
Let us illustrate how creative abduction in a Bayes net model allows for generating usenovel predictions^{8} by means of the magnet example introduced in Section 2. Our line of reasoning here is in accordance with Schurz (2008). Although regarding usenovel facts our framework does not add anything to his argumentation, we think that it is good to see that the Bayesian approach can provide usenovel predictions as well. Assume that an empirical correlation between the two dispositions of attracting iron (D_{1}) and producing electricity when being moved along a wire (D_{2}) had been established by experimenting with lodestone. It is inferred by abductive inference that this correlation is brought about by the higherorder disposition of generating an electromagnetic field (\(\mathcal {D}\)). In our approach, this means that one subscribes to a dispositional pattern captured by a Bayes net model with the structure \(D_{1}\longleftarrow \mathcal {D}\longrightarrow D_{2}\). Now assume that one finds an object that is not a lodestone, but attracts iron anyway (D_{1}). It follows from our model together with conditions 1. and 2. that this increases the probability that this object’s having disposition \(\mathcal {D}\) brought about its having disposition D_{1}. Hence, the probability for \(\mathcal {D}\) is increased as well. But since \(\mathcal {D}\) also increases the probability of this object’s having the disposition to produce electricity by being moved along a wire, also the probability of D_{2} is increased. Thus, observing that the object has disposition D_{1} predicts that P(D_{2}D_{1}) > P(D_{2}) applies to it as well. Note that this prediction is usenovel since only lodestone was used in building the theoretical model.
Confirmation
Given two dispositions D_{1} and D_{2} are empirically correlated, it seems to be commonly accepted that one can use evidence for one of these dispositions to confirm the presence of the other disposition. If, for example, one finds that an object attracts iron (E_{1}), then one tends to accept this as evidence that it has the disposition of producing electricity when being moved along a wire (D_{2}) as well. So E_{1} can be understood as a test for whether an object has disposition D_{2}. This can be justified by help of our model as follows: Once the model’s structure \(E_{1}\longleftarrow D_{1}\longleftarrow \mathcal {D}\longrightarrow D_{2}\) has been established via creative abduction, it follows with condition 3. that observing E_{1} increases the probability for the presence of D_{1} which, in turn, by conditions 1. and 2. increases the probability of the presence of \(\mathcal {D}\). Since \(\mathcal {D}\) is a positive factor for bringing about D_{2} as well, also the probability for D_{2}’s presence will be increased. Thus, P(D_{2}E_{1}) > P(D_{2}) applies to our object and, according to Bayesian confirmation theory, E_{1} confirms D_{2}.^{9} Below we will see that a qualitative model of such confirmation, which might be considered to be a straightforward application of the theory of creative abduction based on the common cause principle (CCP), has several problems. In this sense, expanding the account by switching to the Bayes net framework seems to allow for increased applicability.
The problem of underdetermination
 i
\(\mathcal {H}\) entails H_{2} and H_{3} (but not H_{1}). Furthermore, E^{′} confirms H_{3}.
 ii
Hence: E^{′} confirms also \(\mathcal {H}\). (with i)
 iii
Hence: E^{′} confirms also H_{2}. (with i and ii)
 (CCC)

If A entails B and C confirms B, then C also confirms A.
And the underlying principle which grants the inference of iii is the socalled special consequence condition (SCC):
 (SCC)

If A entails B and C confirms A, then C also confirms B.
Hempel (1965) also demonstrated that these two principles taken together trivialize the notion of qualitative confirmation because they imply that every statement confirms every other statement. The reason for this is simple:“Special Consequence Condition: If an observation report confirms a hypothesis H, then it also confirms every consequence of H. [… The other condition is] the condition that whatever confirms a given hypothesis also confirms every stronger one. [… This principle might be called] ‘converse consequence condition’.” (Hempel 1965, pp. 31f)
 1)
Trivially, A entails A.
 2)
Hence, by (SCC): A confirms A.
 3)
Trivially also A ∧ B entails A.
 4)
Hence, by (CCC): A confirms A ∧ B.
 5)
But then, again by (SCC): A confirms B.
The epistemic challenge: search
In this paper we aimed at modeling creative abduction in the Bayes net framework. To this end we assumed that creative abduction had already been successfully applied. We did not provide an answer to the epistemic question of how and under which conditions creative abduction can be successfully applied in practice. So the epistemic challenge consists in developing reliable methods to abduce unifying dispositions on the basis of empirical data. As Glymour (2018) points out, this problem is tackled in the literature on search of latent variables (see, e.g., Silva et al. 2006; Kummerfeld and Ramsey 2016). Such procedures would, however, require continuous data rather than binary variables as we used them in this paper. So variables should rather represent the strengths of dispositions than simply the presence of such dispositions to get these approaches to work. How exactly such approaches to latent variable search fit with the classical literature on abduction within philosophy of science has to be investigated in future research.
5 Conclusion
This paper was about modeling successful cases of creative abduction on the basis of empirically correlated dispositions within a Bayes net framework. After introducing Schurz’ (2008) strict approach in Section 2, we developed a Bayes net representation of instances of successful creative abduction in the sense of Schurz in Section 3. This move allows for a more finegrained investigation of the unificatory power gained by creative abduction. It also allows for identifying the relevant necessary conditions for successful cases of creative abduction. Note that our approach to creative abduction can, in a very limited way, be used for purposes of selective abduction as well. It suggests to penalize all dispositions of a given set of candidates that do not meet the necessary conditions for successful creative abduction, i.e., all those \(\mathcal {D}\)s that (i) are not positively correlated with one of the lowerlevel dispositions D_{1},...,D_{n} (or one of the pieces of evidence E_{1},...,E_{n}) to be explained or (ii) do not screen off all nonintersecting sets of lowerlevel dispositions (or pieces of evidence) from each other. If (i) were the case, then \(\mathcal {D}\) would not explain every lowerlevel disposition (or piece of evidence), and if (ii) were not the case, the Markov condition would be violated and \(\mathcal {D}\) would not fully explain some correlations among lowerlevel dispositions (or pieces of evidence). In both cases, there might be a better dispositional explanation available. The approach does, however, not come with a criterion for how to select the best disposition(s) \(\mathcal {D}\) of a set of rivals all satisfying these necessary conditions. For this purpose, one could use one of the approaches to selective abduction already on the market (see, e.g., Lipton 2004; Niiniluoto 1999; Williamson 2016).
In Section 4 we then discussed several possible applications of modeling creative abduction Bayesian style. In particular, we spelled out how creative abductive inferences can generate usenovel predictions in our setting. We also presented a new possibility to apply Bayesian confirmation theory: Once a higherlevel connection between lowerlevel dispositions has been established via creative abduction, one can confirm the presence of one of these lowerlevel dispositions by finding evidence for one of the other lowerlevel dispositions. Another result was that a quantitative (probabilistic) reading of Laudan and Leplin’s (1991) proposed solution to the problem of underdetermination can be supported once one is able to unify one of the competing hypotheses with an additional hypothesis via creative abduction.
This paper was about modeling successful instances of creative abduction and about which interesting conclusions one can draw from a Bayes net representation. An issue that has not been tackled is the epistemic question of how exactly theoretical concepts should be abduced on the basis of empirical data. If dispositions can be adequately represented by continuous variables, then this seems to open the door for a fruitful application of much more sophisticated search procedures from the literature on causal discovery.
Footnotes
 1.
Selective abduction is often subsumed under the term inference to the best explanation.
 2.
 3.
 4.
Note that our Bayes net account differs from Schurz’ (2015) approach to unify statistical dependencies and independencies by causal structure. While Schurz reduces a number of statistical dependencies and independencies to a (smaller) number of causal relations, our account reduces a number of correlations among different pieces of evidence to a number of statements postulating abduced dispositions.
 5.
Measuring unificatory power by counting statements, argument patterns, etc. is common in the unification literature (cf. Woodward 2017, sec. 5.4). There are, however, also other ways of measuring unificatory power. To avoid problems Bayesian measures have with common cause structures (cf. Schupbach 2005), Myrvold (2017) suggests to avoid an explicit representation of common causes. For purposes of unification, one should use hypotheses postulating such common causes instead. But since we focus on creative abduction in this paper, avoiding common causes in order to maintain a Bayesian measure for unification seems to be inappropriate for our endeavor. For this reason and in order to compare the Bayes net analysis with Schurz’ (2008) approach, we decided in favor of a simple counting measure.
 6.
The conditional probabilities \(P(E_{i}\mathcal {D})\) can be computed as \(P(E_{i}D_{i},\mathcal {D})\cdot P(D_{i}\mathcal {D})+P(E_{i}\overline {D}_{i},\mathcal {D})\cdot P(\overline {D}_{i}\mathcal {D})\).
 7.
We are indebted to an anonymous referee for pointing this out to us.
 8.
A prediction is usenovel if it predicts an empirical phenomenon that was unknown at the time of the prediction or that has not been used as evidence in constructing the theory on whose basis this phenomenon is predicted (see, e.g., Worrall 1985, 2006). The ability to produce usenovel predictions is often regarded as a requirement for empirically successful theories since it renders theories independently testable.
 9.
For a similar line of argumentation in the case of confirmation across analogical systems, see (Dardashti et al. 2017). For possible problems and an extension of this approach, see (FeldbacherEscamilla and Gebharter ms).
 10.
We are indebted to an anonymous referee for stressing this parallel between the mentioned intuitions on a qualitative notion of confirmation and the properties of a quantitative notion of confirmation in the Bayesian framework applied here.
Notes
Acknowledgements
This work was supported by Deutsche Forschungsgemeinschaft (DFG), research unit Inductive Metaphysics (FOR 2495). We would like to thank Gerhard Schurz for important discussions and two anonymous referees for valuable comments.
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