Rational Expectation in Multinomial Contexts

Carnap constructed the continuum of inductive methods (i.e., the set of C-systems, 0 < λ < ∞) for application to the sentences of a first-order language, containing a finite family of (Q —) predicates and a countable number of individual constants. Let K be such a family of Q-predicates Q1, Q2,… Q k . We consider first the case that there are only finitely many individual constants: U N = a1, a2, …, aN. Let U n indicate the set containing the first n individuals a1, a2, …, a n . A state-description of U n is a sentence of the form Q1 (a1 & Q i2 (a2) & … & Q in (an) and will be indicated here by e(U n ). The number of occurrences of Q i in e(U n ) will be indicated by n i (e(U n )), or simply n i .