 He later generalizes his confirmation function to a continuum of functions \(c_{\lambda}\). Define a family of predicates to be a set of predicates such that, for each individual, exactly one member of the set applies, and consider first-order languages containing a finite number of families. Carnap (1963) focuses on the special case of a language containing only one-place predicates. He lays down a host of axioms concerning the confirmation function \(c\), including those induced by the probability calculus itself, various axioms of symmetry (for example, that \(c(h, e)\) remains unchanged under permutations of individuals, and of predicates of any family), and axioms that guarantee undogmatic inductive learning, and long-run convergence to relative frequencies. They imply that, for a family \(\{P_n\},\) \(n = 1, \ldots,k\) \((k \gt 2){:}\)