Abstract
Probabilistic dependence and independence are among the key concepts of Bayesian epistemology. This paper focuses on the study of one specific quantitative notion of probabilistic dependence. More specifically, section 1 introduces Keynes’s coefficient of dependence and shows how it is related to pivotal aspects of scientific reasoning such as confirmation, coherence, the explanatory and unificatory power of theories, and the diversity of evidence. The intimate connection between Keynes’s coefficient of dependence and scientific reasoning raises the question of how Keynes’s coefficient of dependence is related to truth, and how it can be made fruitful for epistemological considerations. This question is answered in section 2 of the paper. Section 3 outlines the consequences the results have for epistemology and the philosophy of science from a Bayesian point of view.
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Notes
Keynes (1921) attributes this measure to William Ernest Johnson’s manuscript Cumulative Formula which was unpublished at that time. The author could not verify whether the latter manuscript has been published since.
Wheeler (2009) calls this the Wayne–Shogenji correlation measure, not because Wayne or Shogenji invented it, but because of the conflicting interpretations that have recently been attached to it by these authors. Wayne (1995) tentatively suggests \({{\mathfrak {pd}}}\) as a similarity measure. Shogenji (1999) interprets it as a coherence measure. In the following no such interpretation is presupposed. Since many philosophers before Shogenji and Wheeler used this measure—such as Keynes (1921), Mackie (1969) and Horwich (1982)—I refrain from following Wheeler (2009) in calling it the Wayne–Shogenji correlation measure. In contrast to Wheeler (2009) and Brössel (2013c) I also cautiously refrain from calling it a measure of correlation (I am grateful to an anonymous referee for comments which have made me cautious on this score). Finally, it is worth mentioning that one can also find alternative measures in the literature. In information theory the most common way to measure the distance between probability distributions is the Kullback–Leibler Divergence. Applied as a measure of deviation from independence, it measures the distance between the actual probability of the conjunction and the independent distribution. In particular, as a measure of deviation from independence the Kullback–Leibler Divergence looks like this:
Definition (Probabilistic Dependence: Kullback–Leibler Divergence)
$${{\mathfrak {pd}}}_{KL}(A_1,\ldots , A_n)=\big [\Pr (A_1\cap \cdots \cap A_n)\times \log [{\mathfrak {pd}}_K(A_1,\ldots , A_n)]\big ]. $$As one can see, this measure of probabilistic dependence depends crucially on Keynes’s coefficient of dependence \({\mathfrak {pd}}_K\). Thus, studying Keynes’s coefficient of dependence is more fundamental. A third measure of deviation from independence is the following:
Definition (Deviation From Independence: Difference Measures)
$${\mathfrak {pd}}_{d}(A_1, \ldots , A_n)=\Pr (A_1\cap \cdots \cap A_n)\Pr (A_1)\times \cdots \times \Pr (A_n) $$The difference measure is usually applied when one is interested in the deviation from independence of two propositions. Its generalization for measuring the dependence of two random variables is known as the measure of covariance between the random variables. Unfortunately, discussing and comparing these measures of probabilistic dependence goes beyond the scope of the present paper. This topic will be the content of subsequent research.
Keynes (1921) actually discusses primarily the conditional variant of \({\mathfrak {pd}}_K\).
The following corollary and all other corollaries and theorems are proved in the Appendices.
The original formulation of Popper’s (1959) measure of explanatory power is this: \(EP_{P}(T,E)=\frac{\Pr (ET)\Pr (T)}{\Pr (ET)+\Pr (T)}.\)
According to Hájek and Joyce (2008), a measure ordinally equivalent to \(EP_2\) can itself be considered a confirmation measure. However, apart from this paper, I am not aware of any other philosophical text which suggests this.
This has been noted already by Shogenji (1999). Of course, Shogenji is discussing this property in the context of Bayesian coherentism and he takes \({\mathfrak {pd}}_k\) to be a measure of coherence.
Branden Fitelson argued to this effect in a private email exchange.
Unfortunately, in such a short paper it is not possible to provide a detailed exposition of the mathematically intricate convergence theorems. Accordingly, this paper will presuppose previous acquaintance with these convergence theorems. I refer the interested reader to Hawthorne’s (2014) very intelligible exposition of an approach to arrive at one of these convergence results.
A sequence of pieces of evidence separates the set of possibilities \(W\) if and only if for every pair of worlds \(w_i\) and \(w_j \in W\) (with \(w_i\ne w_j\)) there is one piece of evidence in the sequence such that it is true in one of the possibilities and false in the other.
Note that Theorem 2 restricts these claims: \({\mathfrak {pd}}_K\) satisfies both conditions only almost surely: it only holds for every \(w\in W^\prime \) for some \(W^\prime \) such that \(W^\prime \subseteq W\) and \(\Pr ^*(W^\prime )=1\). It does not necessarily hold for all \(w\in W\).
For a more detailed discussion of convergence theorems and their philosophical implications especially for scientific realism and antirealism see Brössel (2013b).
Not all readers think that this result fits our intuitions. Accordingly, I at least want to show that it is supported by other intuitions shared by many Bayesian epistemologists. It is supported by our intuitions about unification. More specifically, many Bayesian epistemologists share the intuition that ceteris paribus a hypothesis is the more confirmed by the evidence the more it unifies the single pieces of evidence (see Sect. 1.5) and they assume the following: (1) the unificatory power of a theory consists in its ability to increase the coefficient of dependence between the pieces of evidence (McGrew 2003; Myrvold 2003), (2) the theory is the more confirmed by the evidence the higher the coefficient of dependence between them. Hence, the more the theory in question increases the coefficient of dependence between the pieces of evidence, the higher is the coefficient of dependence between the theory and the conjunction of the pieces of evidence in question.
Again, not all readers will think this result fits our intuitions. Accordingly, I at least want to show that it is supported by other intuitions shared by many Bayesian epistemologists. It is supported by our intuitions about the diversity of evidence. More specifically, many epistemologists share the intuition that ceteris paribus hypotheses are confirmed more by diverse pieces of evidence than by uniform pieces of evidence (see Sect. 1.5) and they assume the following: (1) the evidence is the more uniform or similar the more the pieces of evidence are probabilistically dependent (e.g. Wayne 1995) and (2) the hypothesis is the more confirmed by the evidence the stronger the probabilistic dependence. Hence, ceteris paribus a higher coefficient of dependence between the pieces of evidence has a negative effect on the coefficient of dependence between the theory and the evidence.
In his paper Shogenji (2007) wants to demonstrate why the coherence (in the sense of his coherence measure \(Coh_S\)) of mutually independent pieces of evidence seems to be truthconducive for some hypothesis \(H\), even if various impossibility results demonstrated that this cannot be the case (Bovens and Hartmann 2003; Olsson 2002). For this he considers in how far the “degree of coherence between the focal piece (usually the new piece) of evidence and the rest of the evidence is a significant factor in the confirmation of the hypothesis” Shogenji (2007: 367). Accordingly, Shogenji considers the interaction between the coherence/coefficient of dependence of the independent pieces of evidence \(e_1, \ldots ,e_m\) and the coherence/coefficient of dependence of the independent pieces of evidence \(e_1, \ldots ,e_{m1}\) where \(e_m\) represents the focal or new piece of evidence and proves that
$$\begin{aligned} {\mathfrak {pd}}_K(e_1\cap \cdots \cap e_{m1},e_m)=\dfrac{{\mathfrak {pd}}_K(e_1, \ldots ,e_m)}{{\mathfrak {pd}}_K(e_1, \ldots ,e_{m1}).} \end{aligned}$$if \(\Pr (e_1\cap \cdots \cap e_m)>0\).
The conclusion Shogenji finally reaches is this:
We have uncovered that when the pieces of evidence are independent with regard to the hypothesis and the rest of the evidence supports the hypothesis, the more coherent the focal piece of evidence is with the rest of the evidence, the more strongly the focal evidence supports the hypothesis. This leaves us with the impression that coherence is truth conducive. (Shogenji 2007: 371)
In the light of Corollary 6 (which does not presuppose that the pieces of evidence are mutually independent) one can see that actually something stronger is true. Ceteris paribus the coefficient of dependence between the pieces of evidence has a negative effect on the coefficient of dependence between the theory and the conjunction of the evidence. However, ceteris paribus the coefficient of dependence between the pieces of evidence and the theory has a positive effect on the coefficient of dependence between the theory and the conjunction of the evidence.
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The present paper has its origin in a short paper that appeared as “Correlation and Truth” in: V. Karakostas, D. Dieks (eds.). EPSA11 Recent Progress in Philosophy of Science: Perspectives and Foundational Problems. Dordrecht: Springer, 41–54. For their helpful remarks and suggestions on previous versions of the paper I am very thankful to Ralf Busse, Franz Huber, Gregory Wheeler, two anonymous referees of this journal, and (especially) AnnaMaria A. Eder and Ben Young.
Appendices
Appendix 1: Proofs of Corollaries
Proof of Corollary 1

1.
According to Definition 3:
$$\begin{aligned} d(T,E)&=\Pr (TE)\Pr (T)\\&=\left[ \frac{\Pr (TE)}{\Pr (T)}1\right] \times \Pr (T)\\&=\left[ {\mathfrak {pd}}_K(T,E)1\right] \times \Pr (T) \end{aligned}$$ 
2.
According to Definition 3:
$$\begin{aligned} M(T,E)&=\Pr (ET)\Pr (E)\\&=\left[ \frac{\Pr (ET)}{\Pr (E)}1\right] \times \Pr (E)\\&=\left[ {\mathfrak {pd}}_K(T,E)1\right] \times \Pr (E) \end{aligned}$$ 
3.
According to Definition 3:
$$\begin{aligned} S(T,E)&=\Pr (TE)\Pr (T(\overline{E}))\\&=\frac{\Pr (TE)\Pr (T)}{\Pr (\overline{E})}\\&=\left[ \frac{\frac{\Pr (TE)}{\Pr (T)}1}{\Pr (\overline{E})}\right] \times \Pr (T)\\&=\left[ \frac{{\mathfrak {pd}}_K(T,E)1}{1\Pr (E)}\right] \times \Pr (T) \end{aligned}$$ 
4.
According to Definition 3:
$$\begin{aligned} M(T,E)&=\Pr (ET)\Pr (E(\overline{T}))\\&=\frac{\Pr (ET)\Pr (E)}{\Pr (\overline{T})}\\&=\left[ \frac{\frac{\Pr (ET)}{\Pr (E)}1}{\Pr (\overline{T})}\right] \times \Pr (E)\\&=\left[ \frac{{\mathfrak {pd}}_K(T,E)1}{1\Pr (T)}\right] \times \Pr (E) \end{aligned}$$ 
5.
According to Definition 3:
$$\begin{aligned} r(T,E)=\log \left[ \frac{\Pr (TE)}{\Pr (T)}\right] , \end{aligned}$$if \(\Pr (T)>0\) and \(\Pr (E)>0\). Since \(\frac{\Pr (TE)}{\Pr (T)}= \frac{\Pr (T\cap E)}{\Pr (T) \times \Pr (E)}\) it follows with Definition 1 that:
$$\begin{aligned} r(T,E)=\log \left[ {\mathfrak {pd}}_K(T,E)\right] , \end{aligned}$$if \(1>\Pr (T)>0\) and \(\Pr (E)>0\).

6.
According to Definition 3:
$$\begin{aligned} Z(T,E)&= {\left\{ \begin{array}{ll}\frac{d(T,E)}{1 \Pr (T)} &\quad \hbox{if}\, \Pr (TE )\ge \Pr (T)>0\\ \frac{d(T,E)}{1 \Pr (\overline{T})} &\quad \hbox{if}\, \Pr (TE )< \Pr (T)\\ 1 &\quad \hbox{if}\, \Pr (T)=0 \end{array}\right. }\\&= {\left\{ \begin{array}{ll}\frac{\left[ {\mathfrak {pd}}_K(T,E)1\right] \times \Pr (T)}{1 \Pr (T)} &\quad \hbox{if}\, \Pr (TE )\ge \Pr (T)>0\\ \frac{\left[ {\mathfrak {pd}}_K(T,E)1\right] \times \Pr (T)}{\Pr (T)} &\quad \hbox{if}\, \Pr (TE )< \Pr (T)\\ 1 &\quad \hbox{if}\, \Pr (T)=0 \end{array}\right. }\\&= {\left\{ \begin{array}{ll}\left[ {\mathfrak {pd}}_K(T,E)1\right] \times \frac{\Pr (T)}{1 \Pr (T)} & {\hbox{if}}\, \Pr (TE )\ge \Pr (T)>0\\ \left[ {\mathfrak {pd}}_K(T,E)1\right] \times \frac{\Pr (T)}{\Pr (T)} & {\hbox{if }}\, \Pr (TE )< \Pr (T)\\ 1 & {\hbox{if}}\, \Pr (T)=0 \end{array}\right. }\\\end{aligned}$$ 
7.
According to Definition 3:
$$\begin{aligned} l(T,E)=\log \left[ \frac{\Pr (ET)}{\Pr (E\overline{T})}\right] , \end{aligned}$$if \(1>\Pr (T)>0\) and \(\Pr (E)>0\). It holds however that:
$$\begin{aligned} \log \left[ \frac{\Pr (ET)}{\Pr (E\overline{T})}\right] =\log \left[ \frac{\frac{\Pr (ET)}{\Pr (E)}}{\frac{\Pr (E\overline{T}}{\Pr (E)})}\right] . \end{aligned}$$This implies
$$\begin{aligned} l(T,E)=\log \left[ \dfrac{{\mathfrak {pd}}_K(T,E)}{{\mathfrak {pd}}_K(\overline{T},E)}\right] , \end{aligned}$$if \(1>\Pr (T)>0\) and \(\Pr (E)>0\).
Proof of Corollary 2

1.
According to Definition 6:
$$\begin{aligned} EP_1(T,E)&=\frac{\Pr (ET)}{\Pr (E)}\\&=\frac{\Pr (E\cap T)}{\Pr (E)\times \Pr (T)}\\&={\mathfrak {pd}}_K(T,E)\,{\hbox{(with\, to\, Definition\, 1)}} \end{aligned}$$ 
2.
According to Definition 7:
$$\begin{aligned} EP_2(T,E)&= \left[ \frac{\Pr (TE)\Pr (T\overline{E})}{\Pr (TE)+\Pr (T\overline{E})}\right] \\&= \left[ \frac{\frac{\Pr (TE)\Pr (T\overline{E})}{\Pr (T)}}{\frac{\Pr (TE)+\Pr (T\overline{E})}{\Pr (T)}}\right] \\&= \left[ \frac{\frac{\Pr (TE)}{\Pr (T)}\frac{\Pr (T\overline{E})}{\Pr (T)}}{\frac{\Pr (TE)}{\Pr (T)}+\frac{\Pr (T\overline{E})}{\Pr (T)}}\right] \\&= \frac{{\mathfrak {pd}}_K(E,T){\mathfrak {pd}}_K(\overline{E},T)}{{\mathfrak {pd}}_K(E,T)+{\mathfrak {pd}}_K(\overline{E},T)} \end{aligned}$$
Proof of Corollary 3
According to Definition 8:
Proof of Corollary 4
According to Definition 9:
Proof of Corollary 5
Proof of Corollary 6
Proof of Corollary 7
Corollary 7 follows trivially from Corollary 1 for the \(l\) confirmation measure and Theorem 3.
Proof of Corollary 8
Appendix 2: Proofs of Theorems
Proof of Theorem 1

1.
$$\begin{aligned} \Pr (A_1\cap \cdots \cap A_n)&= \Pr (A_1\cap \cdots \cap A_n)\\ \Pr (A_1\cap \cdots \cap A_n)&= \dfrac{\Pr (A_1\cap \cdots \cap A_n)}{\prod _{1\le i\le n}\Pr (A_i)}\times \prod _{1\le i\le n}\Pr (A_i)\\ \Pr (A_1\cap \cdots \cap A_n)&= {\mathfrak {pd}}_K(A_1, \ldots A_n)\times \prod _{1\le i\le n}\Pr (A_i) \end{aligned}$$

2.
$$\begin{aligned} \Pr (A_1\cap \cdots \cap A_nB)&= \Pr (A_1\cap \cdots \cap A_nB)\\ \Pr (A_1\cap \cdots \cap A_nB)&= \dfrac{\Pr (A_1\cap \cdots \cap A_nB)}{\prod _{1\le i\le n}\Pr (A_iB)}\times \prod _{1\le i\le n}\Pr (A_iB)\\ \Pr (A_1\cap \cdots \cap A_nB)&= {\mathfrak {pd}}_K(A_1, \ldots A_nB)\times \prod _{1\le i\le n}\Pr (A_iB) \end{aligned}$$
Proof of Theorem 2
The proof proceeds as follows: First, the Gaifman–Snir Theorem is presented (for a proof see Gaifman and Snir 1982). Second, Lemma 1 is proven. Third, Theorem 2 is completed.
1. The Gaifman–Snir Theorem: Let \(W\) be a set of possibilities and let \(\mathcal {A}\) be some algebra over \(W\). The elements of \(\mathcal {A}\) are interpreted as propositions expressible in some language \(\mathcal {L}\) suitable for arithmetic. In particular, let \(\mathcal {L}\) be some first order language containing the numerals ‘1’, ‘2’, ‘3’,… as names, respectively, individual constants, and symbols for addition, multiplication, identity etc. In addition, let \(\mathcal {L}\) contain finitely many relations and functional symbols. Gaifman and Snir (1982) call them the ‘empirical symbols’. Accordingly one can think of the possibilities in \(W\) as models for that language \(\mathcal {L}\) (which agree on the interpretation of the mathematical symbols but can disagree on the interpretation of the empirical symbols).
Now let \(e_1\),…, \(e_n\),… be a sequence of propositions of \(\mathcal {A}\) that separates \(W\), and for all \(w\in W\) let \(e^w_i =e_i\), if \(w\,\vDash\, e_i\) and \(\overline{e_i}\) otherwise. Let \(\Pr \) be a regular (or strict) probability function on \(\mathcal {A}\). Let \(\Pr ^*\) be the unique probability function on the smallest \(\sigma \)field \(\mathcal {A}^*\) containing the field \(\mathcal {A}\) satisfying \(\Pr ^*(A)=\Pr (A)\) for all \(A\in \mathcal {A}\).
Then there is a \(W^\prime \subseteq W\) with \(\Pr ^*(W^\prime )=1\) so that the following holds for every \(w\in W^\prime \) and all theories \(T\) of \(\mathcal {A}\):
where \(\mathcal {I}(T,w)=1\), if \(w\,\vDash\, T\) and 0 otherwise.
2. Lemma 1: Let \(W\) be a set of possibilities and let \(\mathcal {A}\) be some algebra over \(W\). The elements of \(\mathcal {A}\) are interpreted as propositions expressible in some suitable language \(\mathcal {L}\) as specified in more detail above. The possibilities in \(W\) can be interpreted as models for \(\mathcal {L}\). Let \(e_0,\ldots , e_n,\ldots \) be a sequence of propositions of \(\mathcal {A}\) that separates \(W\), and let \(e^w_i =e_i\) if \(w\,\vDash\, e_i\) and \(\overline{e_i}\) otherwise. Let \(\Pr \) be a regular (or strict) probability function on \(\mathcal {A}\). Let \(\Pr ^*\) be the unique probability function on the smallest \(\sigma \)field \(\mathcal {A}^*\) containing the field \(\mathcal {A}\) satisfying \(\Pr ^*(A)=\Pr (A)\) for all \(A\in \mathcal {A}\).
Then according to the Gaifman–Snir Theorem there is a \(W^\prime \subseteq W\) with \(\Pr ^*(W^\prime )=1\) so that the following holds for every \(w\in W^\prime \) and all theories \(T\) of \(\mathcal {A}\):
where \(\mathcal {I}(T,w)=1\), if \(w\,\vDash\, T\) and 0 otherwise.
Now it holds that
3. Proof of Theorem 2: Let \(W\) be a set of possibilities and let \(\mathcal {A}\) be some algebra over \(W\). The elements of \(\mathcal {A}\) are interpreted as propositions expressible in some suitable language \(\mathcal {L}\) as specified in more detail above. The possibilities in \(W\) can be interpreted as models for \(\mathcal {L}\). Let \(e_0,\ldots , e_n,\ldots \) be a sequence of propositions of \(\mathcal {A}\) that separates \(W\), and let \(e^w_i =e_i\) if \(w\,\vDash\, e_i\) and \(\overline{e_i}\) otherwise. Let \(\Pr \) be a regular (or strict) probability function on \(\mathcal {A}\). Let \(\Pr ^*\) be the unique probability function on the smallest \(\sigma \)field \(\mathcal {A}^*\) containing the field \(\mathcal {A}\) satisfying \(\Pr ^*(A)=\Pr (A)\) for all \(A\in \mathcal {A}\).
Then according to Lemma 1 there is a \(W^\prime \subseteq W\) with \(\Pr ^*(W^\prime )=1\) so that the following holds for every \(w\in W^\prime \) and all theories \(T\) of \(\mathcal {A}\):
1. Let \(T_1\) and \(T_2\) be two theories of \(\mathcal {A}\) and suppose additionally that \(w \,\vDash\, T_1\) and \(w \,\vDash\, \overline{T_2}\).
We know that \(\lim _{n\rightarrow \infty }{\mathfrak {pd}}_K(T_1,E^w_n)=\frac{1}{\Pr (T_1)}\) since \(w \,\vDash\, T_1\). We also know that \(\lim _{n\rightarrow \infty }{\mathfrak {pd}}_K(T_2,E^w_n)=0\) since \(w \,\vDash\, \overline{T_2}\).
Let \(\epsilon =\frac{\frac{1}{\Pr (T_1)}}{2}\). By the definition of \(\lim \) it holds that: \(\exists n\in \mathbb {N}\forall m\ge n : \frac{1}{\Pr (T_1)}{\mathfrak {pd}}_K(T_1,E^{w^\prime }_m)<\epsilon \) and \(\exists n^\prime \in \mathbb {N} \forall m\ge n^\prime 0{\mathfrak {pd}}_K(T_2,E^{w^\prime }_m)<\epsilon \).
Now let \(n_1=\)max.\(\{n, n^\prime \}\). Then it holds for all \(m\ge n_1:\)
2. Now assume that \(w \,\vDash\, T_1\) and \(w \,\vDash\, T_2\) and \(T_1\,\vDash\, T_2\) but \(T_2\,\nvDash \,T_1\).
Because of Lemma 1 we know that \(\lim _{n\rightarrow \infty }{\mathfrak {pd}}_K(T_1,E^w_n)=\frac{1}{\Pr (T_1)}\) since \(w \,\vDash\, T_1\). We also know that \(\lim _{n\rightarrow \infty }{\mathfrak {pd}}_K(T_2,E^w_n)=\frac{1}{\Pr (T_2)}\) since \(w \,\vDash\, T_2\).
The assumption is that \(\Pr \) is a strict probability function and \(T_1\,\vDash\, T_2\) but \(T_2\,\nvDash\, T_1\). It follows that \(\Pr (T_1)<\Pr (T_2)\) and \(\frac{1}{p(T_1)}> \frac{1}{p(T_2)}\).
Now let \(\epsilon =\frac{\frac{1}{p(T_1)}\frac{1}{p(T_2)}}{2}\). Then it holds that: \(\exists n\in \mathbb {N}\forall m\ge n : \frac{1}{p(T_1)}{\mathfrak {pd}}_K(T_1,E^w_m)<\epsilon \) and \(\exists n^\prime \in \mathbb {N} \forall m\ge n^\prime \frac{1}{p(T_2)}{\mathfrak {pd}}_K(T_2,E^w_m)<\epsilon \).
Now let \(n_1=\)max.\(\{n, n^\prime \}\). Then it holds for all \(m\ge n_1:\)
Proof of Theorem 3
Proof of Theorem 4
Let \(W\) be a set of possibilities and let \(\mathcal {A}\) be some algebra over \(W\). The elements of \(\mathcal {A}\) are interpreted as propositions expressible in some suitable language \(\mathcal {L}\) as specified in more detail in the Appendix 2 in connection with Theorem 2. The possibilities in \(W\) can be interpreted as models for \(\mathcal {L}\). Let \(e_0,\ldots , e_n,\ldots \) be a sequence of propositions of \(\mathcal {A}\) that separates \(W\), and let \(e^w_i =e_i\) if \(w\,\vDash\, e_i\) and \(\overline{e_i}\) otherwise. Let \(\Pr \) be a regular (or strict) probability function on \(\mathcal {A}\). Let \(\Pr ^*\) be the unique probability function on the smallest \(\sigma \)field \(\mathcal {A}^*\) containing the field \(\mathcal {A}\) satisfying \(\Pr ^*(A)=\Pr (A)\) for all \(A\in \mathcal {A}\).
Then according to Theorem 2 there is a \(W^\prime \subseteq W\) with \(\Pr ^*(W^\prime )=1\) so that the following holds for every \(w\in W^\prime \) and all theories \(T_1\) and \(T_2\) of \(\mathcal {A}\).

1.
If \(w\,\vDash\, T_1\) and \(w\,\vDash\, \overline{T_2}\), then:
$$\begin{aligned} \exists n \forall m\ge n{:}\, \left[{\mathfrak {pd}}_K\left(T_1, \bigcap _{0\le i\le m}e^w_i \right)>{\mathfrak {pd}}_K\left(T_2 , \bigcap _{0\le i\le m}e^w_i\right)\right] \end{aligned}$$ 
2.
If \(w\,\vDash\, T_1\cap T_2\) and \(T_1\,\vDash\, T_2\) but \(T_2\,\nvDash\, T_1\), then:
$$\begin{aligned} \exists n \forall m\ge n{:}\, \left[{\mathfrak {pd}}_K\left(T_1,\bigcap _{0\le i\le m}e^w_i\right)> {\mathfrak {pd}}_K\left(T_2 ,\bigcap _{0\le i\le m}e^w_i\right)\right]. \end{aligned}$$
According to Corollary 6 it holds that
Hence, there is a \(W^\prime \subseteq W\) with \(\Pr ^*(W^\prime )=1\) so that the following holds for every \(w\in W^\prime \) and all theories \(T_1\) and \(T_2\) of \(\mathcal {A}\).

1.
If \(w\,\vDash\, T_1\) and \(w\,\vDash\, \overline{T_2}\), then:
$$\begin{aligned} \exists n \forall m\ge n{:}\, [{\mathfrak {pd}}_K(T_1,e^w_1 ,\ldots , e^w_m )>{\mathfrak {pd}}_K(T_2 ,e^w_1 ,\ldots , e^w_m )] \end{aligned}$$ 
2.
If \(w\,\vDash\, T_1\cap T_2\) and \(T_1\,\vDash\, T_2\) but \(T_2\,\nvDash\, T_1\), then:
$$\begin{aligned} \exists n \forall m\ge n{:}\, [{\mathfrak {pd}}_K(T_1,e^w_1 ,\ldots , e^w_m )> {\mathfrak {pd}}_K(T_2 ,e^w_1 ,\ldots , e^w_m )]. \end{aligned}$$
Proof of Theorem 5
Proof of Theorem 6
Proof of Theorem 7
Let \(W\) be a set of possibilities and let \(\mathcal {A}\) be some algebra over \(W\). The elements of \(\mathcal {A}\) are interpreted as propositions expressible in some suitable language \(\mathcal {L}\) as specified in more detail in the Appendix 2 in connection with Theorem 2. The possibilities in \(W\) can be interpreted as models for \(\mathcal {L}\). Let \(e_0,\ldots , e_n,\ldots \) be a sequence of propositions of \(\mathcal {A}\) that separates \(W\), and let \(e^w_i =e_i\) if \(w\,\vDash\, e_i\) and \(\overline{e_i}\) otherwise. Let \(\Pr \) be a regular (or strict) probability function on \(\mathcal {A}\). Let \(\Pr ^*\) be the unique probability function on the smallest \(\sigma \)field \(\mathcal {A}^*\) containing the field \(\mathcal {A}\) satisfying \(\Pr ^*(A)=\Pr (A)\) for all \(A\in \mathcal {A}\).
According to Theorem 2 there is a \(W^\prime \subseteq W\) with \(\Pr ^*(W^\prime )=1\) so that the following holds for every \(w\in W^\prime \) and all hypotheses \(h_1\),… \(h_n\) and \(h^\prime _1\),… \(h^\prime _m\) of \(\mathcal {A}\).

1.
If \(w\,\vDash\, h_1\cap \cdots \cap h_n\) and \(w\,\vDash\, (\overline{h^\prime _1 \cap \cdots \cap h^\prime _m})\), then:
$$\begin{aligned} \exists k \forall l\ge k{:}\, [{\mathfrak {pd}}_K(h_1\cap \cdots \cap h_n,E^w_l)> {\mathfrak {pd}}_K(h^\prime _1 \cap \cdots \cap h^\prime _m ,E^w_l)] \end{aligned}$$ 
2.
If \(w\,\vDash\, h_1\cap \cdots \cap h_n\cap h^\prime _1 \cap \cdots \cap h^\prime _m\) and \(h_1\cap \cdots \cap h_n\,\vDash\, h^\prime _1 \cap \cdots \cap h^\prime _m\) but \(h^\prime _1 \cap \cdots \cap h^\prime _m\,\nvDash\, h_1\cap \cdots \cap h_n\), then:
$$\begin{aligned} \exists k \forall l\ge k{:}\, [{\mathfrak {pd}}_K(h_1\cap \cdots \cap h_n,E^w_l)> {\mathfrak {pd}}_K(h^\prime _1 \cap \cdots \cap h^\prime _m ,E^w_l)] \end{aligned}$$where \(E^w_l=\bigcap _{0\le i\le l}e^w_i\).
Corollary 8 implies that:
Hence, there is a \(W^\prime \subseteq W\) with \(\Pr ^*(W^\prime )=1\) so that the following holds for every \(w\in W^\prime \) and all hypotheses \(h_1\),… \(h_n\) and \(h^\prime _1\),… \(h^\prime _m\) of \(\mathcal {A}\).

1.
If \(w\,\vDash\, h_1\cap \cdots \cap h_n\) and \(w\,\vDash\, (\overline{h^\prime _1 \cap \cdots \cap h^\prime _m})\), then:
$$\begin{aligned} \exists k \forall l\ge k{:}\, \left[ \dfrac{{\mathfrak {pd}}_K(h_1, \ldots , h_n, E^w_l )}{{\mathfrak {pd}}_K(h_1, \ldots , h_n)}>\dfrac{{\mathfrak {pd}}_K(h^\prime _1, \ldots , h^\prime _m, E^w_l )}{{\mathfrak {pd}}_K(h_1^\prime , \ldots , h^\prime _n)}\right] . \end{aligned}$$ 
2.
If \(w\,\vDash\, h_1\cap \cdots \cap h_n\cap h^\prime _1 \cap \cdots \cap h^\prime _m\) and \(h_1\cap \cdots \cap h_n\,\vDash\, h^\prime _1 \cap \cdots \cap h^\prime _m\) but \(h^\prime _1 \cap \cdots \cap h^\prime _m\,\nvDash\, h_1\cap \cdots \cap h_n\), then:
$$\begin{aligned} \exists k \forall l\ge k{:}\, \left[ \dfrac{{\mathfrak {pd}}_K(h_1, \ldots , h_n, E^w_l )}{{\mathfrak {pd}}_K(h_1, \ldots , h_n)}>\dfrac{{\mathfrak {pd}}_K(h^\prime _1, \ldots , h^\prime _m, E^w_l )}{{\mathfrak {pd}}_K(h_1^\prime , \ldots , h^\prime _n)}\right] \end{aligned}$$where \(E^w_l=\bigcap _{0\le i\le l}e^w_i\).
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Brössel, P. Keynes’s Coefficient of Dependence Revisited. Erkenn 80, 521–553 (2015). https://doi.org/10.1007/s1067001496723
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DOI: https://doi.org/10.1007/s1067001496723