Philosophy of Science

An interesting idea was presented by Professor Kvanvig in Epistemology yesterday, one having to do with the evaluation of informational content. The notion of informational content was defined as follows:

(IC) (A > B) ↔ ((A → B) & ~(B → A))¹

In other words, A is greater in informational content than B just in case there is a one-way, left to right entailment between A and B. The example given was that you may infer from the fact that x is a rottweiler to x is a dog, but from the fact that x is a dog you could not infer that x is a rottweiler:

It is in this sense that A carries more informational content than B. Applying this idea to axiomatic systems, any set of axioms A is going to carry the informational content of the entire system the axioms purport to explain. Suppose a data set E:{p_1,…,p_n} by which an axiom system A:{a_1,…,a_n} provides a sufficient explanation. In this case, the conjunction of the members of A jointly carry the informational content of E. This would suggest that for any competing system A* purporting a sufficient explanation to E, when weighing the members of A and A* individually, the set with the least number of members would carry the individually most robust members in terms of informational content.

This might provide an interesting measure for competing scientific theories. If some theory T1 provides a sufficient explanation for some data set E, and a competing theory, T2, entered the picture, if T2 is less complex than T1 or expands E in some way, then T2 would carry more informational content than T1. The explanans is the hard, dense center of the explanandum, and (IC) provides a means of measurement parasitic on the relationship between the two. In pictorial form (MS Paint is fun!):

Here T2 > T1 in terms of informational content. The relationship between theories need not–indeed, is often not–be so direct. T2 does not have to nest in T1, for example:

In this case T1>E and T2>E, but there is no direct relationship between T1 and T2. T2, however, is denser than T1 in the sense that it provides an equivalent explanation in a smaller logical space (less axioms, for example). A third kind of relationship could occer where T1 and T2 are equal in size, but the explanatory range itself shifts, where T1>E1 and T2>E1, but T2>E2 and while E1:{p_1,…,p_n}, E2:{p_1,…,p_m} where m>n². In this case T2 would range farther than T1:

Applying this measure to epistemic concerns, it would seem that opaque contexts involving factive operators carry more information than their embedded propositions. For any intensional operator O which is logically stronger than truth, such that Op→p, Op would be greater in informational content than p. It would seem, then, that ascriptions of knowledge would carry more informational content than the propositions of which the knowledge is ascribed. Finally, granting a position-to-know closure schema, knowledge of some proposition p is going to carry more informational content than any of p’s implications.

I will have to think more about this one.