Coherence and (Likeness to) Truth

Abstract

According to a coherentist position in philosophy of science, good theories cohere with the available data and one theory is better than another if it coheres better with the available data. This paper examines that relationship with a special focus on probabilistic measures of coherence. In a first step, it is shown that existing coherence measures satisfy a number of reasonable adequacy constraints for the comparison of two rival scientific theories. In a second step, the virtue of a coherentist position in philosophy of science is considered. More specifically, it is assessed whether coherence implies verisimilitude in the sense that a higher degree of coherence between theory and evidence entails a higher degree of (estimated) truthlikeness. To this end, it is demonstrated that there is an intimate relationship in this sense if we explicate coherence by means of the so-called overlap-measure.

Appendix: A Proof of Theorem 5.1

Let $e_{\Pr }$ be a distribution of probabilities $\Pr (c\vert e)$ for each constituent $c \in \mathcal{C}$. The degree of coherence between c and an arbitrary c-theory $t = p_{i_{1}} \wedge \ldots \wedge p_{i_{k}}$ is given by the following formula:

$$\displaystyle{ \mathcal{O}(t,c,\overline{c}) =\log _{2}\big[\mathcal{O}(\bigwedge t_{c}^{+},c)\big/\mathcal{O}(\bigwedge t_{c}^{-},\overline{c})\big] }$$

(1)

It can easily be shown that the following identity holds:

$$\displaystyle\begin{array}{rcl} \mathcal{O}(t,c,\overline{c})& =& \log _{2}\left [ \frac{1} {\Pr (\bigwedge t_{c}^{+}\vert c)} + \frac{1} {\Pr (c\vert \bigwedge t_{c}^{+})} - 1\right ]^{-1}{}\end{array}$$

(2)

$$\displaystyle\begin{array}{rcl} & & -\log _{2}\left [ \frac{1} {\Pr (\bigwedge t_{c}^{-}\vert \overline{c})} + \frac{1} {\Pr (\overline{c}\vert \bigwedge t_{c}^{-})} - 1\right ]^{-1}{}\end{array}$$

(3)

Now for conjunctive theories, both $\Pr (\bigwedge t_{c}^{+}\vert c)$ and $\Pr (\bigwedge t_{c}^{-}\vert \overline{c})$ are equal to 1. Hence, Eq. 2 reduces to the following formula:

$$\displaystyle{ \mathcal{O}(t,c,\overline{c}) =\log _{2}\Pr (c\vert \bigwedge t_{c}^{+}) -\log _{ 2}\Pr (\overline{c}\vert \bigwedge t_{c}^{-}) }$$

(4)

Exploiting Bayes’ theorem, we get (remember that $\Pr$ is the logical probability $\Pr ^{{\ast}}$):

$$\displaystyle{ \mathcal{O}(t,c,\overline{c}) =\log _{2} \frac{\Pr (c)} {\Pr (\bigwedge t_{c}^{+})} -\log _{2} \frac{\Pr (\overline{c})} {\Pr (\bigwedge t_{c}^{-})} = \vert t_{c}^{+}\vert -\vert t_{ c}^{-}\vert }$$

(5)

Hence, $\mathcal{O}(t,c,\overline{c})$ is strictly increasing in | t _c ⁺ | and strictly decreasing in | t _c ⁻ | for each constituent c. The same obviously holds for Vs(t, c). Therefore, for each pair of c-theories $t,t^{{\prime}}$ and each constituent the following equivalence holds:

$$\displaystyle{ \mathcal{O}(t,c,\overline{c}) \geq \mathcal{O}(t^{{\prime}},c,\overline{c})\quad \Leftrightarrow \quad \mathrm{Vs}(t,c) \geq \mathrm{ Vs}(t^{{\prime}},c) }$$

(6)

Hence, (5), (6) and the definition of Vs together entail that

$$\displaystyle{ \sum _{c\in \mathbb{C}}\Pr (c\vert e) \times \mathcal{O}(t,c,\overline{c}) \geq \sum _{c\in \mathbb{C}}\Pr (c\vert e) \times \mathcal{O}(t^{{\prime}},c,\overline{c}) }$$

$$\displaystyle{ \Leftrightarrow }$$

$$\displaystyle{ \sum _{c\in \mathbb{C}}\Pr (c\vert e) \times \mathrm{ Vs}(t,c) \geq \sum _{c\in \mathbb{C}}\Pr (c\vert e) \times \mathrm{ Vs}(t^{{\prime}},c) }$$

Thus, we get the desired result that if $\mathbb{E}(\mathcal{O}(t,e)) > \mathbb{E}(\mathcal{O}(t,e))$, then $\mathrm{EVs}(t,e) >\mathrm{ EVs}(t^{{\prime}},e)$ (and also vice versa). □