## Abstract

The well-known formal semantics of conditionals due to Stalnaker (in: Rescher (ed) Studies in logical theory, Blackwell, Oxford, 1968), Lewis (Counterfactuals, Blackwell, Oxford, 1973a), and Gärdenfors (in: Niiniluoto, Tuomela (eds) The logic and 1140 epistemology of scientific change, North-Holland, Amsterdam, 1978, Knowledge in flux, MIT Press, Cambridge, 1988) all fail to distinguish between trivially and nontrivially true indicative conditionals. This problem has been addressed by Rott (Erkenntnis 25(3):345–370, 1986) in terms of a strengthened Ramsey Test. In this paper, we refine Rott’s strengthened Ramsey Test and the corresponding analysis of explanatory relations. We show that our final analysis captures the presumed asymmetry between explanans and explanandum much better than Rott’s original analysis.

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## Notes

- 1.
The study of belief base changes has been originated by Sven Ove Hansson. Much of what we are going to say about such revisions draws on his

*Textbook of Belief Dynamics*(Hansson 1999). - 2.
This definition is inspired by Brewka (1991), but the resulting belief revision operation is not equivalent with the one defined there.

- 3.
- 4.
For the list of AGM belief revision postulates, see for example Gärdenfors (1988, pp. 54–55).

- 5.
Although the latter because sentence seems totally fine in a context, in which the agent performs a so-called

*inference to the best explanation*: repeatedly playing football may be the best explanation for occasional injuries. This reasoning*towards*(as opposed to*from*) the putative explanatory ‘causes’ seems to justify the usage of ‘because’ in the other direction. So peculiar as natural language is, we do not want to ban this usage of ‘because’ from natural language. For now, we just want to focus on the one usage showing the asymmetry without being entirely sure that this usage is strictly asymmetric. For example, ‘because*p*and*q*are true, \(p \wedge q\) is true’ does not seem to preclude ‘because \(p \wedge q\) is true,*p*and*q*are true’. - 6.
‘Antecedent’ is here a generalisation of the antecedent of a conditional sentence. It stands for ‘subordinate clause’ of the respective sentence. This mirrors Rott’s view that all of the mentioned conjunctions are derived from a framework of universal conditionals. In detail, the indicative and subjunctive ifs and ‘because’ fall into the category of universal pro-conditionals, ‘though’ into the category of universal contra-conditionals, and ‘even if’ into the category of universal un-conditionals. See Rott (1986, pp. 355–363).

- 7.
Considerations of how to systematically categorise conditionals result in the schemes of universal contra- and un-conditionals as well.

- 8.
This idea has recently been exploited in an analysis of evidential support by Chandler (2013).

- 9.
(\(\hbox {SRT}_R\)) structurally resembles Lewis (1973b)’s notion of causal dependence in terms of counterfactual conditionals. Using \(\Rightarrow \) for causal dependence, we can transcribe Lewis’s idea into the notation of belief revision: \(\alpha \Rightarrow \gamma \in K\) iff \(\gamma \in K * \alpha \) and \(\lnot \gamma \in K * \lnot \alpha \). Note that Lewis’s causal dependence requires a stronger version of difference making than (\(\hbox {SRT}_R\)), viz. the adoption of \(\lnot \gamma \) in \(K * \lnot \alpha \) in contrast to the mere retraction of \(\gamma \). Moreover, Lewis might say that ‘\(\gamma \) because \(\alpha \)’ means \(\gamma \) is causally dependent on \(\alpha \), when \(\alpha \) and \(\gamma \) are (believed to be) true. Given Lewis (1973a)’s semantics for counterfactuals, we obtain the following implication paralleling (1): \({\text{If }} \alpha , \gamma \in K {\text{, then }} [ \alpha \Rightarrow \gamma \in K\;{\text{ iff }}\;\lnot \gamma \in K * \lnot \alpha ]\).

- 10.
See Hansson (1999, p. 96) for a brief justification of why law-like statements should—in most cases—be epistemically more entrenched than factual statements.

- 11.
The example is similar to the famous tower-shadow scenario, for which there is wide agreement that the height of the tower together with the altitude of the sun explain the length of the shadow, but not vice versa. However, see Van Fraassen (1980, pp. 132–34) for an interesting challenge of this agreement involving the notion of relevance. Note that we simplified the original tower-shadow scenario such that a wider class of examples succumbs to the asymmetry problem of Rott’s (\(\hbox {Because}_R\)).

- 12.
“A cause is an object precedent and contiguous to another, and so united with it, that the idea of the one determines the mind to form the idea of the other, and the impression of the one to form a more lively idea of the other.” (Hume (1739/1978, p. 170) One might wonder whether the temporal order of cause and effect is a properly non-epistemic condition in the context of Hume’s work, but this is a question that need not concern us here.

- 13.
We deal with ‘causation’ in a follow-up paper.

- 14.
See Hansson (1999) for a very comprehensive study of belief base revisions and contractions, including a detailed comparison to belief set revisions and contractions.

- 15.
- 16.
The first step is reminiscent of Edmund Husserl (1913, \(\S \S \)31–33)’s Pyrrhonian epoché. This phenomenological epoché denotes the method of suspending or bracketing (German:

*Einklammerung*) the acceptance status of one’s beliefs about the world. We apply the Pyrrhonian idea with a—by far—smaller scope: we demand an agent to suspend her respective belief status of the particular antecedent and consequent under consideration. We call the bracketing or suspension of antecedent and consequent ‘agnostic move’ and credit Pyrrho by labelling our Strengthened Ramsey Test (\(\hbox {SRT}_P\)). - 17.
- 18.
The distinction between conjunctive and disjunctive explanatory scenarios is taken from an analogous distinction in the literature on actual causation [cf. Halpern and Pearl (2005, Sec. 3)]. Disjunctive scenarios amount to cases of overdetermination if more than one of the antecedent conditions is satisfied.

- 19.
For the list of AGM belief contraction postulates, see for example Gärdenfors (1988, pp. 61–62).

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## Acknowledgments

Thanks to Andrew Irvine, Hannes Leitgeb, and Hans Rott for very valuable comments on earlier versions of the paper. Special thanks go to the anonymous referees of Erkenntnis for challenging but very constructive comments. Moreover, we are grateful for the opportunity to present parts of this paper at the “2nd Munich Graduate Workshop in Mathematical Philosophy: Formal Epistemology” hosted by the Munich Center for Mathematical Philosophy (LMU Munich). This research has been supported by the Graduate School of Systemic Neurosciences.

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## Appendix: Proofs

### Appendix: Proofs

###
**Proposition 2**

*Let**K**be a non-absurd belief set and*\(\gamma \)*a non-tautology. Then* (UPC) *and*\(\alpha , \gamma \in K\)*implies* (\(\hbox {Because}_R\)).

###
*Proof*

The proof presents the simplification of (UPC) to (\(\hbox {Because}_R\)). Assume \(K\ne K_\bot \) (K is not the absurd belief set) and \(\gamma \) is not a tautology. Further, suppose \(\alpha , \gamma \in K\). Then, by Gärdenfors’s contraction postulate \((K^{-}4)\), \(\gamma \not \in K - \gamma \).^{Footnote 19} Moreover, since \(\gamma \in K\) and \(K\ne K_\bot \), \(\lnot \gamma \not \in K\). By (\(K^-2)\) this implies that \(\lnot \gamma \not \in (K - \gamma )\). By Proposition 1, \(\gamma \not \in (K - \gamma )\) implies \(\gamma \not \in (K - \gamma ) * \alpha \) or \(\gamma \not \in (K - \gamma ) * \lnot \alpha \), and \(\lnot \gamma \not \in (K - \gamma )\) implies \(\lnot \gamma \not \in (K - \gamma ) * \alpha \) or \(\lnot \gamma \not \in (K - \gamma ) * \lnot \alpha \). Hence, (i) \(\gamma \not \in (K - \gamma ) * \alpha \) or \(\gamma \not \in (K - \gamma ) * \lnot \alpha \) and (ii) \(\lnot \gamma \not \in (K - \gamma ) * \alpha \) or \(\lnot \gamma \not \in (K - \gamma ) * \lnot \alpha \). (i) implies that (iii), if \(\gamma \in (K - \gamma ) * \alpha \), then \(\gamma \not \in (K - \gamma ) * \lnot \alpha \). (ii) implies that (iv) if \(\lnot \gamma \in (K - \gamma ) *\lnot \alpha \), then \(\lnot \gamma \not \in (K - \gamma ) * \alpha \). From (iii), (iv), and (UPC), we can infer (\(\hbox {Because}_R\)). \(\square \)

###
**Proposition 3**

*Let*\(\alpha \)*and*\(\gamma \)*be literals or conjunctions of literals. Further*, \(\alpha , \gamma \in K\). *Suppose that*\(\alpha \rightarrow \gamma \)*is a non-trivial implication in**K**and Assumption*1*holds for*\(\alpha \rightarrow \gamma \). *Then*, \(\alpha {\,{}^{a} {\Rightarrow } \,} \gamma \in K\)*and*\(\gamma {\,{}^{a} {\Rightarrow } \,} \alpha \in K\).

###
*Proof*

(\(\alpha {\,{}^{a} {\Rightarrow } \,} \gamma \in K\)) Suppose \(\alpha \rightarrow \gamma \) is a non-trivial implication in *K*. Then, by Definition 3 and Assumption 1, \(\alpha \rightarrow \gamma \) is a non-trivial implication in \(K - \gamma \). By Definition 3, \((K - \gamma ) - \lnot \alpha \vdash \alpha \rightarrow \gamma \). We obtain, by the deduction theorem, \((K - \gamma ) - \lnot \alpha , \alpha \vdash \gamma \), which we can rewrite as \(((K - \gamma ) - \lnot \alpha ) + \alpha \vdash \gamma \). By the Levi identity, we obtain \(\gamma \in (K - \gamma ) * \alpha \). Using \(\alpha , \gamma \in K\), we can infer therefrom that \(\alpha {\,{}^{a} {\Rightarrow } \,} \gamma \in K\).

(\(\gamma {\,{}^{a} {\Rightarrow } \,} \alpha \in K\)) Suppose \(\alpha \rightarrow \gamma \) is a non-trivial implication in *K*. Then, by Assumption 1, \(\alpha \rightarrow \gamma \) is a non-trivial implication in \(K - \alpha \). By contraposition, \(\lnot \gamma \rightarrow \lnot \alpha \) is a non-trivial implication in \(K - \alpha \). By Definition 3, \(\lnot \gamma \rightarrow \lnot \alpha \) is a non-trivial implication in \((K - \alpha ) - \gamma \). Via the deduction theorem, we obtain \((K - \alpha ) - \gamma , \lnot \gamma \vdash \lnot \alpha \). By the Levi identity, we obtain \(\lnot \alpha \in (K - \alpha ) * \lnot \gamma \). Using \(\alpha , \gamma \in K\), we can infer therefrom that \(\gamma {\,{}^{a} {\Rightarrow } \,} \alpha \in K\). \(\square \)

###
**Proposition 4**

*Assume a* (\(\hbox {Because}_R\)) *agent accepts all facts and the generalisation of the tower-shadow scenario*, i. e. \(t, s, sh, t \wedge s \rightarrow sh \in K\), *where the order of epistemic entrenchment is*\(t, s, sh < t \wedge s \rightarrow sh\). *Then*, *t* \({}^{a} {\Rightarrow }\) \(sh \in K\)*if*\(t \le sh\)*and**sh* \({}^{a} {\Rightarrow }\) \(t \in K\)*if*\(sh \le t\).

###
*Proof*

By (\(\hbox {Because}_R\)):

\(t, s, sh \in K\) holds by assumption. We show that (a) *t* \({}^{a} {\Rightarrow }\) \(sh \in K\) if \(t \le sh\). Let us assume \(t \le sh\). By (G-), (EE2), (EE1), this implies (i) \(t \not \in K - sh\). By the recovery postulate

\(t \in (K - sh) *sh\). By the Levi identity, the deduction theorem, and (i), this implies that \(sh \rightarrow t \in (K - sh)\). Hence, \(\lnot t \rightarrow \lnot sh \in (K-sh)\). Using \(t\notin K -sh\) and the Levi identity, we can infer therefrom that \( \lnot sh \in (K - sh) *\lnot t\). This entails (a) in light of (\(\hbox {Because}_R\)).

The proof of (b) \(sh {\,{}^{a} {\Rightarrow } \,} t\) if \(sh \le t\) is completely analogous to that of (a). \(\square \)

###
**Proposition 5**

*Assume a* (\(\hbox {Because}_R\)) *agent accepts all the formulas in*\(K(H, <) = K(S)\)*for*\(H=\{ t, s, sh, t \wedge s \rightarrow sh \}\), *where the order of epistemic priority is*\(t \sim s \sim sh < t \wedge s \rightarrow sh\). *Then**t* \({}^{a} {\Rightarrow }\)\(sh \not \in K_>(S)\)*and**sh* \({}^{a} {\Rightarrow }\)\(t \in K_>(S)\).

###
*Proof*

*t* \({}^{a} {\Rightarrow }\)*sh* and *sh* \({}^{a} {\Rightarrow }\)*t* remain well-defined, when replacing *K* with *K*(*S*):

*t* \({}^{a} {\Rightarrow }\)\(sh \not \in K_>(S)\), where \(S = (H, <)\): \(t, sh \in K(S)\) is satisfied by assumption. By Definition 1, the remainder set \(H \bot sh\) contains three sets, \(H' = \{ t \wedge s \rightarrow sh, s \}\), \(H'' = \{ t \wedge s \rightarrow sh, t \}\), and \(H''' = \{ s, t \}\). By Definition 2, \(H' \le H''\) and \(H'' \le H'\), but \(H' \not \le H'''\). Hence, by (Def \(\sigma \)), both \(H'\) and \(H''\) are selected for the partial meet base contraction (PMBC) which yields \(H - sh = \bigcap \sigma (H \bot sh) = \{ t \wedge s \rightarrow sh \}\).

By (PMBR), \((H - sh) * t = \bigcap \sigma ((H - sh) \bot \lnot t) + t\). Since \(\lnot t \not \in Cn(H - sh)\), by Definition 1, \(H-sh\) is the unique member of \((H - sh) \bot \lnot t\). By Definition 2, \((H-sh) \le (H-sh)\) and by (Def \(\sigma \)), the partial meet base contraction (PMBC) yields \((H-sh) - \lnot t = \bigcap \sigma ((H - sh) \bot \lnot t) = \{ t \wedge s \rightarrow sh \}\). Notice that when \(\lnot t \not \in H-sh\), then \((H-sh) - \lnot t = H-sh\). By (H + \(\alpha \)), \((H - sh) * t = \{ t \wedge s \rightarrow sh \} \cup \{ t \}\). Hence, \((S - sh) * t = ((H-sh)*t, <')\), where \(<'\) is such that generalisations have strict priority over literals. Then \(sh \not \in K((S-sh)*t)\).

By (PMBR), \((H - sh) * \lnot t = \bigcap \sigma ((H - sh) \bot t) + \lnot t\). By similar reasoning as above, \(t \not \in H - sh\) and thus \((H - sh) - t = H - sh\). By (H + \(\alpha \)), \((H - sh) * \lnot t = \{ t \wedge s \rightarrow sh \} \cup \{ \lnot t \}\). Hence, \((S - sh) * \lnot t = ((H-sh)* \lnot t, <')\), where \(<'\) is such that generalisations have strict priority over literals. Then \(\lnot sh \not \in K((S-sh)* \lnot t)\).

\(sh {\,{}^{a} {\Rightarrow } \,} t \in K_>(S)\), where \(S = (H, <)\): \(t, sh \in K(S)\) is satisfied by assumption. By Definition 1, the remainder set \(H \bot t\) contains only \(H' = \{ t \wedge s \rightarrow sh, s, sh \}\). By Definition 2, \(H' \le H'\) and by (Def \(\sigma \)), the partial meet base contraction (PMBC) yields \((H-t) = \bigcap \sigma (H \bot t) = \{ t \wedge s \rightarrow sh, s, sh \}\). By similar reasoning, \((H-t)-sh = \{ t \wedge s \rightarrow sh, s \}\). By (H + \(\alpha \)), \(((H-t)-sh) + \lnot sh = \{ t \wedge s \rightarrow sh, s, \lnot sh \}\), which is by (PMBR) \((H-t)* \lnot sh\). Hence, \((S - t) * \lnot sh = ((H-t)* \lnot sh, <')\), where \(<'\) is such that generalisations have strict priority over literals. Then \(\lnot t \in K((S-t)* \lnot sh)\). \(\square \)

###
**Proposition 6**

*Assume a* (\(\hbox {Because}_{P}\)) *agent accepts all facts and the single, more entrenched generalisation of the tower-shadow scenario*, i. e. \(t, s, sh, t \wedge s \rightarrow sh \in K\), *where the order of epistemic entrenchment is*\(t \sim s \sim sh < t \wedge s \rightarrow sh\). *Then*\(t {\,{}^{P} {\Rightarrow }\,} sh \in K_>\)*and*\(sh {\,{}^{P} {\Rightarrow }\,} t \in K_>\).

###
*Proof*

\(t {\,{}^{P} {\Rightarrow }\,} sh \in K_>\): The agnostic belief set is \(K' = K - (t \vee sh)\). By (G-), \(t, s, sh \not \in K'\), but by *recovery* (i) \((t \vee sh) \rightarrow sh \in K'\). By assumption, \(\lnot t \not \in K\) and so by \((K^-2)\), (ii) \(\lnot t \not \in K^\prime \). Using \(t \vdash t \vee sh\) and the Levi identity, we can infer from (i) and (ii) that \(sh \in K' * t\).

\(sh {\,{}^{P} {\Rightarrow }\,} t \in K_>\): The agnostic belief set is, again, \(K' = K - (t \vee sh)\) such that \(t, s, sh \not \in K'\), but by *recovery*\((t \vee sh) \rightarrow t \in K'\). Using \(sh \vdash t \vee sh\), we infer that \(t \in K' * sh\). \(\square \)

###
**Proposition 7**

*Assume a* (\(\hbox {Because}_{P}\)) *agent accepts all the formulas in*\(K(H, <) = K(S)\) for \(H=\{ t, s, sh, t \wedge s \rightarrow sh \}\), *where*\(t, s, sh < t \wedge s \rightarrow sh\)*and the order of epistemic priority is*\(t \sim s \sim sh < t \wedge s \rightarrow sh\). *Then*\(t {\,{}^{P} {\Rightarrow }\,} sh \in K_>(S)\), *but*\(sh {\,{}^{P} {\Rightarrow }\,} t \not \in K_>(S)\).

###
*Proof*

\(t {\,{}^{P} {\Rightarrow }\,} sh \in K_>(S)\), where \(S=(H, <)\): \(t, sh \in K(S)\) is satisfied by assumption. By Definition 1, the remainder set \(H \bot (t \vee sh)\) contains only \(H'' = \{ t \wedge s \rightarrow sh, s \}\). By Definition 2, \(H'' \le H''\) and by (Def \(\sigma \)), the partial meet base contraction (PMBC) yields the agnostic belief base \(H' = H - (t \vee sh) = \bigcap \sigma (H \bot (t \vee sh)) = \{ t \wedge s \rightarrow sh, s \}\).

By (PMBR), \(H' * t = \bigcap \sigma (H' \bot \lnot t) + t\). Since \(\lnot t \not \in Cn(H')\), by Definition 1, \(H'\) is the only member of \(H' \bot \lnot t\). By Definition 2 and (Def \(\sigma \)), \(\bigcap \sigma (H' \bot \lnot t) = H'\). By (H + \(\alpha \)), \(H' + t = \{ t \wedge s \rightarrow sh, s \} \cup \{ t \}\). Hence, \(S' * t = (H' + t, <')\), where \(<'\) is such that generalisations have strict priority over literals. Then \(sh \in K(S' * t)\) so that \(t \gg sh \in K_> (S)\).

\(sh {\,{}^{P} {\Rightarrow }\,} t \not \in K_>(S)\), where \(S=(H, <)\): \(t, sh \in K(S)\) is satisfied by assumption. The agnostic belief base is again \(H' = H - (t \vee sh) = \{ t \wedge s \rightarrow sh, s \}\). By (PMBR), \(H' * sh = \bigcap \sigma (H' \bot \lnot sh) + sh\). Since \(\lnot sh \not \in Cn(H')\), by Definition 1, \(H'\) is the only member of \(H' \bot \lnot sh\). By Definition 2 and (Def \(\sigma \)), \(\bigcap \sigma (H' \bot \lnot t) = H'\). By (H + \(\alpha \)), \(H' + sh = \{ t \wedge s \rightarrow sh, s \} \cup \{ sh \}\). Hence, \(S' * sh = (H' + sh, <')\), where \(<'\) is such that generalisations have strict priority over literals. Then \(t \not \in K(S' * sh)\) so that \(sh \gg t \not \in K_> (S)\). \(\square \)

###
**Proposition 8**

*Let*\(\alpha \)*and*\(\gamma \)*be literals. Epistemic states are represented by prioritised belief bases with two levels: an upper level**G**of generalisations and a lower level**L**of literals, as explained in Sect.*2.5. *A* (\(\hbox {Because}_{P'}\)) *agent accepts* ‘\(\gamma \)*because of*\(\alpha \)’ *with respect to*\((H,<)\)*iff*\(\alpha \)*inferentially explains*\(\gamma \) – *in the sense of Definition*6 – *in the eyes of the agent accepting all members of**H*.

###
*Proof*

Suppose (i) \(\gamma \) because of \(\alpha \) is verified by an epistemic state \((H,<)\) (in the sense of (\(\hbox {Because}_{P'}\))). Let *G* be the set of generalisations of *H*, while *L* is the set of literals of *H*. Hence, (ii) there are \((H'', <'')\) and \(L^-\) such that \({(H',<')}= {( H, <)} - \bigvee L^-\) and \(\alpha \gg \gamma \in K_>(H', <')\). Therefore, (iii) there is \((H'',<'')= {( H', <')} - \alpha \vee \gamma \) such that \(H''=G'' \cup L''\), \(G''= G \cap H''\), and \(L''= L \cap H''\). Hence, the pair \((G'', L'')\) satisfies conditions (1), (2), and (4) for an agent who accepts all members of *H*. Moreover, (ii) and (iii) imply that \(G'' \cup L'', \alpha \vdash \gamma \). Hence, Condition (5) is satisfied as well for \((G'', L'')\). Finally, Condition (6) holds for \((G'', L'')\) because of (ii) and (iii). (i) implies that Condition (3) of Definition 6 is satisfied for an agent who accepts all members of *H*. Hence, all conditions of this definition are satisfied for such an agent. Thus, \(\alpha \) inferentially explains \(\gamma \)—in the sense of Definition 6—in the eyes of an agent who accepts all members of *H*.

For the other direction, suppose (i) \(\alpha \) inferentially explains \(\gamma \) in the eyes of an agent *a*, in the sense of Definition 6. Hence, there is a set *G* of generalisations and a set *L* of literals such that conditions (1)–(6) of Definition 6 are satisfied for *a*. (ii) \(H:=G \cup L \cup \{\alpha \}\). < is such that generalisations are prioritised over literals. Obviously, (iii) \(\alpha \in K(H)\). By Condition (5) of Definition 6, (iv) \(\gamma \in K(H, <)\). We show that \(\alpha \vee \gamma \notin Cn(G \cup L)\). Suppose, for contradiction, \(\alpha \vee \gamma \in Cn(G \cup L)\). This implies that (v) \(\lnot \alpha \rightarrow \gamma \in Cn(G \cup L).\) By Condition (5) of Definition 6, we know that (vi) \(\alpha \rightarrow \gamma \in Cn(G \cup L)\). Since \(\alpha \vee \lnot \alpha \in Cn(G \cup L)\), (v) and (vi) imply that \(\gamma \in Cn(G \cup L)\). This contradicts Condition (6) of Definition 6. Hence, \(\alpha \vee \gamma \notin Cn(G \cup L)\). Therefore, \(\sigma ((H, <)\bot \alpha \vee \gamma ):=\{G \cup L\}\), where \(\sigma \) is defined by (Def \(\sigma \)) and Definition 2. Using Condition (5) of Definition 6, we can infer therefrom that \(\alpha \gg \gamma \in K_>(H,<)\). Using (iii) and (iv), we can infer therefrom that ‘\(\gamma \) because of \(\alpha \)’ is verified by the epistemic state \((H,<)\) (in the sense of (\(\hbox {Because}_{P'}\))). \(\square \)

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Andreas, H., Günther, M. On the Ramsey Test Analysis of ‘Because’.
*Erkenn* **84, **1229–1262 (2019). https://doi.org/10.1007/s10670-018-0006-8

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