Learning the Semantics of Logical Content

A learner that induces logic programs from a template $L$ of $N$ candidate Horn clauses appears to face $2^N$ hypotheses, but most rule-sets deduce the same facts and are indistinguishable to any learner scored on its predictions. The true hypothesis class of a recursive template is therefore not the syntax $2^L$ but the set of realizable deductive closures $\text{Om}$, the distinct semantic meanings the template can express. Its log-count $\text{CL}={\log }_2|\text{Om}|$, the semantic information (capacity) of the template, is its learning complexity: when facts arrive online and the learner predicts whether the theory deduces each, Halving makes at most $\text{CL}$ mistakes in the realizable case, the minimax log-loss (Shtarkov) regret is exactly $\text{CL}$ once the queries separate the contents, and the agnostic minimax $0/1$ regret is $\text{O}(\sqrt{\text{CL}T})$. We compute $\text{CL}$ in closed form: $n{\log }_2{\rho }_d$ bits for the bandwidth-$d$ reachability template on $r_0,\dots ,r_n$, with ${\rho }_d$ the $d$-bonacci constant solving ${\rho }^d={\rho }^{d-1}+\cdots +\rho +1$, and $\frac{1}{4}{m}^2(1+o(1))$ bits for nonlinear transitive closure on $m$ constants. In a probabilistic reading a single content's size undergoes a sharp phase transition at one base fact per constant. One computable number thus governs counting, coding, and learning, realizing the semantic-information program of carnap1952semantic and lastras2026 as a learning rate. Empirically, on Padgett's Florentine families network the $2^{40}$ clause-sets collapse onto $|\text{Om}|=4,104$ contents and the online code length lands exactly on $\text{CL}=12$ bits; the same identity holds on WordNet.