[This post is part of a series on Bayesian epistemology; see index here
The last assumption of the core of Bayesianism is that the plausibility of (logically) compound propositions depends, in a particular way, on the plausibilities of the propositions that constitute them. I will write it in the following form* (using Boolean Algebra):
Assumption 3: Compound Plausibilities: The plausibility of the logical conjunction (“A and B” or “AB”) and the logical dijunction (“A or B” or “A+B”) is a universal function of the plausibilities that constitute it, and their complements, under all relevant conditions of knowledge.
The functions are “universal” in the sense that they do not depend on the content of the propositions or the domain of discourse. The claim is that the plausibility of logical conjunction or disjunction – and thereofre of every complicated consideration of the basic propositions – depends on the plausibilities of the basic propositions, not on what they’re talking about.The assumption of universality is clearly correct in the cases of total certainty or denial. If A is true and B false, for example, we know that A+B is true – regardless of the content of the claims A and B, the topic they discuss, or so on. It is less clear why universality should be maintained for intermediate degrees of certainty. Some** suggest to consider it a hypothesis – let’s assume that there are general laws of thought, and let’s see what these are.
Another aspect of assumption 3 is that the universal functions depend only on their components. But assuming the functions are universal – what else can they depend on? They can only depend on some plausibilities. They cannot depend on the plausibility of an unrelated claim, for then it will not be possible to identify it in different domains of discourse. They must depend at least on the plausibilities of their components as they do under the extreme cases of utter certainty or rejection. It is perhaps possible to conjecture that in addition they may depend on some other compund proposition that is composed out of the basic propositions of the conjunction/disjunction, but this would surely be very strange. The decomposition into constituents therefore appears very simple and “logical” – I don’t know of any that object to it.
Let us proceed, then, under assumption 3.
It is a cumbersome assumption as each function depends on lots of variables. Fortunately, we can reduce their number. Consider the case where B is the negation of A, that is B=A. In this case
so that F depends on only two variables, (A|X) and (A|X). On the other hand, logic dictates that this plausibility must have a constant value,
Assuming that the universal function F is not constant, the only way we can maintain a constant value when we change (A|X) is to change (A|X) simultaneously. We are forced to conclude that the plausibility of a proposition is tied to that of the proposition’s negation by a universal function,
Theorem 3.1: The plausibility of A is tied to the plausibility of its negation by a universal function, (A|X)=S(A|X).
We will determine S explicitly later. For now, it is enough that it exists.
Something very important just happened – from the assumption that there are genreal rules for thought, we concluded that the plausibility of the negation of a proposition (A|X) is measured by the plausibility of the claim itself (A|X). It is therefore enough to just keep track of one plausibility, (A|X), to asses both. As we have said previously, this is an inherent part of the Bayesian analysis, and we see here that it is derived directly from the assumption of universality. The main alternative theory, the Dempster-Shafer theory, considers the measure of support that propositions have and requires a separate measure for the support of the proposition’s negation. The existence of S implies that Dempster-Shafer must reject the universality of their own theory! There cannot be a universal way to determine the support for compound propositions from the support we have for the basic propositions, and even within a particular domain if this can be done then the theory only reverts back to the Baysian one. Unsurprisingly, Shafer indeed doubts the existence of general rules of induction.
Let’s move on. The existence of S allows us to throw out the complements A and B from the parameters of the universal functions, as they are themselves a function of the propositions that they complement (A and B).
Theorem 3.2: Simple Shapes: The universal functions F and G can be written without an explicit dependence on the plausibilities of the complements.
The functions are still rather complicated, but they can be made even simpler. Consider the case where A is a tautology. In this case there is no meaning for the expression (B|A,X) – no information in the world can determine that a tautology is wrong. This expression is just not defined. But that plausibility of (AB|X)=(B|X) must still be well defined! There must therefore be a way to write G in a way that does not depend on the undefined variable (B|A,X). We should notice here that the parallel variable (B|A,X) is actually well defined in this case, so G might still depend on it. A similar situation occurs for the pair of variables (A|B,X) and (A|B,X). We can therefore conclude that we can write the universal functions in a manner that does not depend on half of each of these pairs.
Theorem 3.3: Simpler Forms: The universal functions F and G can be written without explicit dependence on the information that A or B are wrong.
These forms cannot be used when A or B are contradictions, but otherwise they should be applicable. With just four variables, they are simple enough to server as a basis from which we can prove Cox’s Theorem – the foundation of Bayesianism.
* My variant on this assumption is somewhat more general than that usually given.