Orthodox Science: The Principles of Positivism

To this end, an attempt is made to build a Bayesian mathematical model of positivism. The positivist principle of phenomenalism is found to apply only mild constraints to the prior probability distribution, such as a Bayesian might use to prevent a situation from arising, in which beliefs are incapable of being swayed by evidence. The form of Occam's razor used in positivist philosophy, on the other hand, is modelled as a claim that a particular prior probability distribution, in which the prior probability of any particular theory $T$ is

$$P(T)=P_0\epsilon^n\textrm{,}$$ (3)

where $P_0$ is a normalizing constant, the same for all theories, $\epsilon$ is a constant equivalent to the likelihood value at which a theory is taken to be falsified, and $n$ is the number of postulates in theory $T$, is correct, and that any other assignment of prior probabilities is invalid. This places the positivist Occam's razor in a tradition known as logical probability, which is alien to the modern Bayesian standpoint. The positivist prohibition on the entry of prescriptive ideologies into knowledge is also found to be a restriction that is unnecessary for a Bayesian, who is permitted, but not obliged, to assess descriptive theories and prescriptive ideologies by the same inference method. Finally, the positivist assertion that knowledge from different academic disciplines must be capable of concatenation into a single body of knowledge is found to be nothing more than a prohibition on self-contradictory theories.

Various other limitations on the kinds of evidence that can be allowed to enter into knowledge, which are sometimes incorporated into positivism, are also considered in Bayesian terms. Firstly, deterministic falsificationism, in which only evidence $E$ that completely rules out a theory $T$, i.e. that has $P(T\vert E) = P(E\vert T) = 0$, is allowed to influence belief in that theory, is argued to be an unnecessary restriction, since Bayes' theorem can draw inferences about the truth of $T$ from a much wider range of evidence. Secondly, a prohibition on evidence obtained with an ideology in mind is found, in Bayesian inference, to be equivalent to setting the prior probability $P(T)$ to zero for any theory $T$ in which, for a piece of evidence $E$ that was obtained with an ideology in mind, $P(E\vert T) \neq{} P(E\vert\bar{T})$. Such a prohibition is, therefore, another example of logical probability, and is not part of the Bayesian view. Thirdly, reductionism, in which it is assumed that the single body of knowledge, which positivism predicts will be produced by the concatenation of theories from different academic disciplines, will take the form of the laws of physics, is found, depending on its interpretation, either to add nothing to the core principles of positivism described above, or to insist on the assignment of low prior probabilities to all theories that are incapable of expression in a particular kind of mathematical language, associating reductionism with logical probability and dissociating it from Bayesian statistics.