Generality of a Theory: An Entropic Measure

Although generality is widely regarded as a fundamental characteristic of robust scientific theories, it remains under-formalized and often confined to domain- specific metrics. This work proposes a domain-independent measure of generality rooted in Mathematical Logic, Formal Concept Analysis, and Information Theory. Since theories and categories share structural similarities, a unified representa- tional scheme was argued for. Given a formal language L and a Formal context K, a theory/category can be epistemically represented by the probabilistic Galois connected pair (p, q), where p and q are probability distributions over all L-structures and L-sentences in K, respectively. Ontologically, a theory is rep- resented by the pair (S(p), S(q)), where S(p) and S(q) are the set of L-structures and L-sentences with non-zero probabilities, respectively. The uncertainty rep- resented in (p, q) is induced by the graded satisfaction between L-structures and L-sentences encoded in K through the choice of a particular probabilistic Galois connection. The central thesis defines the generality of a theory as the entropy of its induced probability distribution over the L-structure/model space H(p) which was shown to be the unique function satisfying the essential properties of symmetry, continuity, block decomposability, and consistency with the classical account. The entropic definition of generality successfully resolves the drawbacks of the classical account associated with continuous model space and non-classical and higher-order logics. Finally, the conceptual payoffs of this definition were heuristically explored by illustrating how generality provides a unified hierarchy for understanding the relationship between abstract categories, general theories, and testable data, and sets an upper limit on data compression.