An argument against global no miracles arguments

Abstract

Howson famously argues that the no-miracles argument, stating that the success of science indicates the approximate truth of scientific theories, is a base rate fallacy: it neglects the possibility of an overall low rate of true scientific theories. Recently a number of authors has suggested that the corresponding probabilistic reconstruction is unjust, as it concerns only the success of one isolated theory. Dawid and Hartmann, in particular, suggest to use the frequency of success in some field of research $\mathcal {R}$ to infer a probability of truth for a new theory from $\mathcal {R}$. I here shed doubts on the justification of this and similar moves and suggest a way to directly bound the probability of truth. As I will demonstrate, my bound can become incompatible with the assumption specific testing and Dawid and Hartmann’s estimate for success.

Appendix

We here quickly prove eqn. (6). For simplicity, we write $\tilde{p}(x)\equiv p(x|\hat{s} \& \hat{c})$ for any x below. From (5), the adapted versions of (i*) and (ii*), and ($*$), we have

$$\begin{aligned} \tilde{p}(s) = \underbrace{\tilde{p}(s|t)}_{=g}\underbrace{\tilde{p}(t)}_{\le 1/k} + \underbrace{\tilde{p}(s|\lnot t)}_{<f}\underbrace{\tilde{p}(\lnot t)}_{=1-\tilde{p}(t)}. \end{aligned}$$

(22)

Now assume that $\tilde{p}(s)$ is greater or equal to $g/k+f$. Then we have

$$\begin{aligned} g\underbrace{\big [\tilde{p}(t)-\frac{1}{k}\big ]}_{\equiv \delta \le 0} \ge f-\tilde{p}(s|\lnot t)\tilde{p}(\lnot t) = f-\underbrace{\tilde{p}(s|\lnot t)}_{<f}\underbrace{(1-\tilde{p}(t))}_{\le 1} >0, \end{aligned}$$

(23)

which implies

$$\begin{aligned} 0\ge g\delta >0. \end{aligned}$$

(24)

Contradiction. $\square$