Unity of Science

Positivism requires [39] that belief in a theory does not disregard relevant evidence, merely because it was gathered in a different discipline from that in which the study was produced. For a positivist, if the products of a method of inquiry are applicable in one discipline, they are applicable in all disciplines. A positivist will, therefore, expect that the theories accepted in different academic disciplines will be capable of concatenation into a single theory.

To develop a mathematical model of this fourth characteristic of positivist knowledge, it will first be useful to state its meaning in the following natural-language terms: for theories devised in different disciplines to be capable of concatenation into a single ``theory of everything,'' all that is required is that they are not mutually contradictory. It is a simple matter to give mathematical expression to this rule. Consider two theories $T_1$ and $T_2$, which are to be concatenated into a single theory $\{T_1,T_2\}$. If there is some piece of evidence $E$, for which, in response to some test, the performance of which is rendered possible by the concatenated theory, $P(E\vert T_1)=0$ and $P(\bar{E}\vert T_2)=0$, then a positivist will assign a prior probability $P(\{T_1,T_2\})=0$ to the concatenated theory.

While Shulman [67] has made it clear that the principle that self-contradictory theories may be assumed to be false is itself a mathematical postulate, which may be opened to question, that question is beyond the scope of this paper, and possibly beyond the scope of Bayesian statistics. For now, it will be assumed that the requirement of the fourth characteristic of positivist knowledge is a weak and sensible constraint.