Keynes’s Coefficient of Dependence Revisited

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.

Appendices

Appendix 1: Proofs of Corollaries

Proof of Corollary 1

• 1. According to Definition 3:

  $$\begin{aligned} d(T,E)&=\Pr (T|E)-\Pr (T)\\&=\left[ \frac{\Pr (T|E)}{\Pr (T)}-1\right] \times \Pr (T)\\&=\left[ {\mathfrak {pd}}_K(T,E)-1\right] \times \Pr (T) \end{aligned}$$

• 2. According to Definition 3:

  $$\begin{aligned} M(T,E)&=\Pr (E|T)-\Pr (E)\\&=\left[ \frac{\Pr (E|T)}{\Pr (E)}-1\right] \times \Pr (E)\\&=\left[ {\mathfrak {pd}}_K(T,E)-1\right] \times \Pr (E) \end{aligned}$$

• 3. According to Definition 3:

  $$\begin{aligned} S(T,E)&=\Pr (T|E)-\Pr (T|(\overline{E}))\\&=\frac{\Pr (T|E)-\Pr (T)}{\Pr (\overline{E})}\\&=\left[ \frac{\frac{\Pr (T|E)}{\Pr (T)}-1}{\Pr (\overline{E})}\right] \times \Pr (T)\\&=\left[ \frac{{\mathfrak {pd}}_K(T,E)-1}{1-\Pr (E)}\right] \times \Pr (T) \end{aligned}$$

• 4. According to Definition 3:

  $$\begin{aligned} M(T,E)&=\Pr (E|T)-\Pr (E|(\overline{T}))\\&=\frac{\Pr (E|T)-\Pr (E)}{\Pr (\overline{T})}\\&=\left[ \frac{\frac{\Pr (E|T)}{\Pr (E)}-1}{\Pr (\overline{T})}\right] \times \Pr (E)\\&=\left[ \frac{{\mathfrak {pd}}_K(T,E)-1}{1-\Pr (T)}\right] \times \Pr (E) \end{aligned}$$

• 5. According to Definition 3:

  $$\begin{aligned} r(T,E)=\log \left[ \frac{\Pr (T|E)}{\Pr (T)}\right] , \end{aligned}$$

  if $\Pr (T)>0$ and $\Pr (E)>0$. Since $\frac{\Pr (T|E)}{\Pr (T)}= \frac{\Pr (T\cap E)}{\Pr (T) \times \Pr (E)}$ 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:

  $$\begin{aligned} Z(T,E)&= {\left\{ \begin{array}{ll}\frac{d(T,E)}{1- \Pr (T)} &\quad \hbox{if}\, \Pr (T|E )\ge \Pr (T)>0\\ \frac{d(T,E)}{1- \Pr (\overline{T})} &\quad \hbox{if}\, \Pr (T|E )< \Pr (T)\\ 1 &\quad \hbox{if}\, \Pr (T)=0 \end{array}\right. }\\&= {\left\{ \begin{array}{ll}\frac{\left[ {\mathfrak {pd}}_K(T,E)-1\right] \times \Pr (T)}{1- \Pr (T)} &\quad \hbox{if}\, \Pr (T|E )\ge \Pr (T)>0\\ \frac{\left[ {\mathfrak {pd}}_K(T,E)-1\right] \times \Pr (T)}{\Pr (T)} &\quad \hbox{if}\, \Pr (T|E )< \Pr (T)\\ 1 &\quad \hbox{if}\, \Pr (T)=0 \end{array}\right. }\\&= {\left\{ \begin{array}{ll}\left[ {\mathfrak {pd}}_K(T,E)-1\right] \times \frac{\Pr (T)}{1- \Pr (T)} & {\hbox{if}}\, \Pr (T|E )\ge \Pr (T)>0\\ \left[ {\mathfrak {pd}}_K(T,E)-1\right] \times \frac{\Pr (T)}{\Pr (T)} & {\hbox{if }}\, \Pr (T|E )< \Pr (T)\\ 1 & {\hbox{if}}\, \Pr (T)=0 \end{array}\right. }\\\end{aligned}$$

• 7. According to Definition 3:

  $$\begin{aligned} l(T,E)=\log \left[ \frac{\Pr (E|T)}{\Pr (E|\overline{T})}\right] , \end{aligned}$$

  if $1>\Pr (T)>0$ and $\Pr (E)>0$. It holds however that:

  $$\begin{aligned} \log \left[ \frac{\Pr (E|T)}{\Pr (E|\overline{T})}\right] =\log \left[ \frac{\frac{\Pr (E|T)}{\Pr (E)}}{\frac{\Pr (E|\overline{T}}{\Pr (E)})}\right] . \end{aligned}$$

  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:

  $$\begin{aligned} EP_1(T,E)&=\frac{\Pr (E|T)}{\Pr (E)}\\&=\frac{\Pr (E\cap T)}{\Pr (E)\times \Pr (T)}\\&={\mathfrak {pd}}_K(T,E)\,{\hbox{(with\, to\, Definition\, 1)}} \end{aligned}$$

• 2. According to Definition 7:

  $$\begin{aligned} EP_2(T,E)&= \left[ \frac{\Pr (T|E)-\Pr (T|\overline{E})}{\Pr (T|E)+\Pr (T|\overline{E})}\right] \\&= \left[ \frac{\frac{\Pr (T|E)-\Pr (T|\overline{E})}{\Pr (T)}}{\frac{\Pr (T|E)+\Pr (T|\overline{E})}{\Pr (T)}}\right] \\&= \left[ \frac{\frac{\Pr (T|E)}{\Pr (T)}-\frac{\Pr (T|\overline{E})}{\Pr (T)}}{\frac{\Pr (T|E)}{\Pr (T)}+\frac{\Pr (T|\overline{E})}{\Pr (T)}}\right] \\&= \frac{{\mathfrak {pd}}_K(E,T)-{\mathfrak {pd}}_K(\overline{E},T)}{{\mathfrak {pd}}_K(E,T)+{\mathfrak {pd}}_K(\overline{E},T)} \end{aligned}$$

Proof of Corollary 3

According to Definition 8:

$$\begin{aligned} UP(e_1,e_2;T)&= \log \left[ \frac{\frac{\Pr (e_1\cap e_2|T)}{\Pr (e_1|T)\times \Pr (e_2|T)}}{\frac{\Pr (e_1\cap e_2)}{\Pr (e_1)\times \Pr (e_2)}}\right] \\&= \log \left[ \dfrac{{\mathfrak {pd}}_K(e_1,e_2|T)}{{\mathfrak {pd}}_K(e_1,e_2)}\right] \end{aligned}$$

Proof of Corollary 4

According to Definition 9:

$$\begin{aligned} SE(e_1,e_2)&=\frac{\Pr (e_1\cap e_2)}{\Pr (e_1)\times \Pr (e_2)}\\&={\mathfrak {pd}}_K(e_1,e_2) \end{aligned}$$

Proof of Corollary 5

$$\begin{aligned} {\mathfrak {pd}}_K(T,e_1\cap \cdots \cap e_m)&= \frac{\Pr (T\cap e_1\cap \cdots \cap e_m)}{\Pr (T)\times \Pr (e_1\cap \cdots \cap e_m)}\\&= \frac{\frac{\Pr (T\cap e_1\cap \cdots \cap e_m)}{\Pr (T)\times \Pr (e_1)\times \cdots \times \Pr (e_m) }}{\frac{\Pr (e_1\cap \cdots \cap e_m)}{\Pr (e_1)\times \cdots \times \Pr (e_m)}}\\&= \frac{{\mathfrak {pd}}_K(T,e_1, \ldots , e_m)}{{\mathfrak {pd}}_K(e_1, \ldots , e_m)} \end{aligned}$$

Proof of Corollary 6

$$\begin{aligned}& {\mathfrak {pd}}_K(T_1,e_1\cap \cdots \cap e_m)> {\mathfrak {pd}}_K(T_2,e_1\cap \cdots \cap e_m) \\ &\quad \Leftrightarrow \frac{{\mathfrak {pd}}_K(T_1,e_1, \ldots , e_m)}{{\mathfrak {pd}}_K(e_1, \ldots , e_m)}> \frac{{\mathfrak {pd}}_K(T_2,e_1, \ldots , e_m)}{{\mathfrak {pd}}_K(e_1, \ldots , e_m)} \\ &\quad \Leftrightarrow {\mathfrak {pd}}_K(T_1,e_1,\ldots , e_m)> {\mathfrak {pd}}_K(T_2,e_1,\ldots , e_m). \end{aligned}$$

Proof of Corollary 7

Corollary 7 follows trivially from Corollary 1 for the $l$ confirmation measure and Theorem 3.

Proof of Corollary 8

$$\begin{aligned} {\mathfrak {pd}}_K(h_1\cap \cdots \cap h_n,E)&= \frac{\Pr (h_1\cap \cdots \cap h_n\cap E)}{\Pr (h_1\cap \cdots \cap h_n)\times \Pr (E)}\\&= \frac{\frac{\Pr (h_1\cap \cdots \cap h_n\cap E)}{\Pr (h_1)\times \cdots \times \Pr (h_n) \times \Pr (E)}}{\frac{\Pr (h_1\cap \cdots \cap h_n)}{\Pr (h_1)\times \cdots \times \Pr (h_n)}}\\&= \frac{{\mathfrak {pd}}_K(h_1, \ldots , h_n,E)}{{\mathfrak {pd}}_K(h_1, \ldots , h_n)} \end{aligned}$$

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

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 $\mathcal {A}$:

$$\begin{aligned} lim_{n\rightarrow \infty }\Pr (T|E^w_n)=\mathcal {I}(T,w) \end{aligned}$$

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 $\mathcal {A}$:

$$\begin{aligned} lim_{n\rightarrow \infty }\Pr (T|E^w_n)=\mathcal {I}(T,w) \end{aligned}$$

where $\mathcal {I}(T,w)=1$, if $w\,\vDash\, T$ and 0 otherwise.

Now it holds that

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathfrak {pd}}_K(T,E^w_n)&= \lim _{n\rightarrow \infty }\dfrac{\Pr (T|E^w_n)}{\Pr (T)}\\&= \lim _{n\rightarrow \infty }\Pr (T|E^w_n)\times \lim _{n\rightarrow \infty }\dfrac{1}{\Pr (T)}\\&= \lim _{n\rightarrow \infty }\Pr (T|E^w_n)\times \dfrac{1}{\Pr (T)}\\&= {\left\{ \begin{array}{ll} \dfrac{1}{\Pr (T)}, & {\hbox{if}}\,\lim _{n\rightarrow \infty }\Pr (T|E^w_n)=1\\ 0, & {\hbox{if}}\, \lim _{n\rightarrow \infty }\Pr (T|E^w_n)=0\end{array}\right. }\\&= {\left\{ \begin{array}{ll} \dfrac{1}{\Pr (T)}, & {\hbox{if}}\,w\,\vDash\, T_1\\ 0, & {\hbox{otherwise.}} \end{array}\right. } \end{aligned}$$

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 $\mathcal {A}$:

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathfrak {pd}}_K(T,E^w_n)=\dfrac{1}{\Pr (T)}\quad{\hbox{if }}\,w\,\vDash\, T\,{\hbox{and\, 0\, otherwise.}} \end{aligned}$$

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 $m\ge n_1:$

$$\begin{aligned} {\mathfrak {pd}}_K(T_1,E^{w^\prime }_m)> {\mathfrak {pd}}_K(T_2 ,E^{w^\prime }_m) \end{aligned}$$

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 $m\ge n_1:$

$$\begin{aligned} {\mathfrak {pd}}_K(T_1,E^w_m)> {\mathfrak {pd}}_K(T_2 ,E^w_m) \end{aligned}$$

Proof of Theorem 3

$$\begin{aligned} {\mathfrak {pd}}_K(T,e_1\cap \cdots \cap e_m)&= \frac{\Pr (e_1\cap \cdots \cap e_m\cap T)}{\Pr (e_1\cap \cdots \cap e_m)\times \Pr (T)}\\&= \frac{\Pr (T|e_1)}{\Pr (T)}\times \cdots \times \frac{\Pr (T|e_m)}{\Pr (T)}\times \frac{\frac{\Pr (e_1\cap \cdots \cap e_m|T)}{\Pr (e_1\cap \cdots \cap e_m)}}{ \frac{\Pr (T|e_1)}{\Pr (T)}\times \cdots \times \frac{\Pr (T|e_m)}{\Pr (T)}}\\&= \frac{\Pr (T|e_1)}{\Pr (T)}\times \cdots \times \frac{\Pr (T|e_m)}{\Pr (T)}\times \frac{\frac{\Pr (e_1\cap \cdots \cap e_m|T)}{\Pr (e_1\cap \cdots \cap e_m)}}{\frac{\Pr (e_1|T)\times \cdots \times \Pr (e_m|T)}{\Pr (e_1)\times \cdots \times \Pr (e_m)}}\\&= \frac{\Pr (T|e_1)}{\Pr (T)}\times \cdots \times \frac{\Pr (T|e_m)}{\Pr (T)}\times \frac{\frac{\Pr (e_1\cap \cdots \cap e_m|T)}{\Pr (e_1|T)\times \cdots \times \Pr (e_m|T) }}{\frac{\Pr (e_1\cap \cdots \cap e_m)}{\Pr (e_1)\times \cdots \times \Pr (e_m)}}\\&={\mathfrak {pd}}_K(T,e_1)\times \cdots \times {\mathfrak {pd}}_K(T,e_m)\times \dfrac{{\mathfrak {pd}}_K(e_1,\ldots , e_m|T)}{{\mathfrak {pd}}_K(e_1,\ldots ,e_m)} \end{aligned}$$

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

$$\begin{aligned} &{\mathfrak {pd}}_K\left(T_1,\bigcap _{0\le i\le m}e^w_i\right)> {\mathfrak {pd}}_K\left(T_2,\bigcap _{0\le i\le m}e^w_i\right) \\ &\quad \Leftrightarrow {\mathfrak {pd}}_K\left(T_1,e^w_1,\ldots , e^w_m\right)> {\mathfrak {pd}}_K\left(T_2,e^w_1,\ldots , e^w_m\right). \end{aligned}$$

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

$$\begin{aligned} {\mathfrak {pd}}_K(h_1\cap \cdots \cap h_n,E)&= \frac{\Pr (h_1\cap \cdots \cap h_n\cap E)}{\Pr (h_1\cap \cdots \cap h_n)\times \Pr (E)}\\&= \frac{\Pr (h_1|E)}{\Pr (h_1)}\times \cdots \times \frac{\Pr (h_n|E)}{\Pr (h_n)}\times \frac{\frac{\Pr (h_1\cap \cdots \cap h_n|E)}{\Pr (h_1\cap \cdots \cap h_n)}}{ \frac{\Pr (h_1|E)}{\Pr (h_1)}\times \cdots \times \frac{\Pr (h_n|E)}{\Pr (h_n)}}\\&= \frac{\Pr (h_1|E)}{\Pr (h_1)}\times \cdots \times \frac{\Pr (h_n|E)}{\Pr (h_n)}\times \frac{\frac{\Pr (h_1\cap \cdots \cap h_n|E)}{\Pr (h_1\cap \cdots \cap h_n)}}{\frac{\Pr (h_1|E)\times \cdots \times \Pr (h_n|E)}{\Pr (h_1)\times \cdots \times \Pr (h_m)}}\\&={\mathfrak {pd}}_K(h_1,E)\times \cdots \times {\mathfrak {pd}}_K(h_n,E)\times \dfrac{{\mathfrak {pd}}_K(h_1,\ldots , h_n|E)}{{\mathfrak {pd}}_K(h_1,\ldots ,h_n)} \end{aligned}$$

Proof of Theorem 6

$$\begin{aligned} {\mathfrak {pd}}_K(h_1\cap \cdots \cap h_n,e_1\cap \cdots \cap e_m)&= \frac{\Pr (h_1\cap \cdots \cap h_n \cap e_1\cap \cdots \cap e_m)}{\Pr (h_1\cap \cdots \cap h_n)\times \Pr (e_1\cap \cdots \cap e_m)}\\&= \frac{\frac{\Pr (h_1\cap \cdots \cap h_n \cap e_1\cap \cdots \cap e_m)}{\Pr (h_1)\times \cdots \times \Pr (h_n)\times \Pr (e_1)\times \cdots \times \Pr (e_m)}}{\frac{\Pr (h_1\cap \cdots \cap h_n)}{\Pr (h_1)\times \cdots \times \Pr (h_n)}\times \frac{\Pr (e_1\cap \cdots \cap e_m)}{\Pr (e_1)\times \cdots \times \Pr (e_m)}}\\&=\dfrac{{\mathfrak {pd}}_K(h_1, \ldots ,h_n,e_1,\ldots , e_m)}{{\mathfrak {pd}}_K(h_1,\ldots , h_n)\times {\mathfrak {pd}}_K(e_1, \ldots , e_m)} \end{aligned}$$

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:

$$\begin{aligned}&{\mathfrak {pd}}_K(h_1\cap \cdots \cap h_n,E^w_l)> {\mathfrak {pd}}_K(h^\prime _1 \cap \cdots \cap h^\prime _m ,E^w_l)\\ &\quad\Leftrightarrow\frac{{\mathfrak {pd}}_K(h_1, \ldots , h_n,E)}{{\mathfrak {pd}}_K(h_1, \ldots , h_n)}>\frac{{\mathfrak {pd}}_K(h^\prime _1, \ldots , h^\prime _m,E)}{{\mathfrak {pd}}_K(h^\prime _1, \ldots , h^\prime _m)} \end{aligned}$$

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 $E^w_l=\bigcap _{0\le i\le l}e^w_i$.