Phenomenalism

Positivism requires [39] that a theory does not contain postulates that are incapable of being contradicted by experience. If the truth or falsehood of a claim has no observable, or potentially observable, effects, that claim cannot form part of positivist knowledge. Examples, from physics, of assertions which would be ruled out of positivist knowledge by this principle, would be claims about [58,25,65] the absolute value of the phase of a quantum mechanical wave-function, or about [7] the absolute value of an electric potential. This principle is known as phenomenalism.

Phenomenalism can be described simply in terms of probabilities. It excludes from knowledge any theory $T$, in which the degree of belief in $T$ is unaffected by any possible evidence $E$, i.e. any $T$ for which

$$P(T\vert E)=P(T)\textrm{,}$$ (9)

for all conceivable pieces of evidence $E$. Bayes' theorem (equation 7) reveals that this is equivalent to

$$P(E\vert T)=P(E)$$ (10)

$$\iff{}P(E\vert T) = P(E\vert\bar{T})\textrm{,}$$ (11)

where $\bar{T}$ represents the negation of $T$, i.e. $\bar{T}$ is true if and only if $T$ is false.

There are two reasons for a theory $T$ to obey equation 9, and thereby to be ruled out of positivist knowledge. Firstly, it may be an intrinsic characteristic of $T$ that it fails to generate any likelihoods $P(E\vert T)$, which differ from the likelihoods $P(E\vert\bar{T})$; it may be a theory that neglects to mention any observable consequences at all. While the Bayesian view does not include an explicit prohibition on such theories, it is clear that such a theory will always have a posterior probability equal to its prior probability, and that belief or disbelief in it will therefore remain an arbitrary and subjective choice (section 3.1.) Secondly, the initial prior probability distribution may have been set up to produce $P(T\vert E)=P(T)$ for all $E$, most straightforwardly by having either $P(T)=0$ or $P(T)=1$. This initial prior probability distribution seems unreasonable, because it assigns the certainty of either truth or falsehood to a theory before any evidence has been considered. Therefore, positivism's first rule for knowledge does not appear to place any particularly strong restrictions on theories, as compared with Bayesian inference. Garrett [24] calls phenomenalism testability, and also notes that it is a reasonable principle for a Bayesian to adopt.