Bayesianism: Consistent Plausibilities

[This post is part of a series on Bayesian epistemology; see index here]

In the previous post we set the foundational assumption of Bayesianism, which is that plausibility can be measured by a Real number. We now put forward the second assumption: consistency. We will assume that the rational plausibility of logically-equivalent propositions must be equal.

Assumption 2: Consistency: If A and B are (logically) identical propositions, then their plausibilities must be equal (A|X)=(B|X). They must also have the same affect as information, (C|A,X)=(C|B,X).
This is a trivial assumption – treating identical things differently is the height of irrationality. I don’t think anyone can object to it.

All tautologies are always true and therefore logically identical to each other and to the truth (T). Since nothing can be more certain than the truth, they must receive the maximal plausibility value (regardless of any information). The situation is similar in regards to contradictions, which are always false (F).

Theorem 2.1: If A is a tautology, then (A|X)=(T|X)=vT, where vT is the highest possible plausibility value, representing absolute certainty. If A is a contradiction, then (A|X)=(F|X)=vF, where vF is the lowest possible plausibility value, representing total rejection of the proposition.

In light of this, we can rewrite the definition we have provided for “further infomration” in a simpler manner:
Theorem 2.2: The plausibility (A|A,X) of proposition A under information A and the further information that A is correct, is that of the truth (A|A,X)=T. The plausibility (A|A,X) of proposition A under information X and the further information that A is false, is that of falsehood (A|A,X)=F.

These are pretty obvious results. The importancce of demanding consistency will be made more obvious soon, as we discuss complex propositions.
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