Note on an idea of Fitch

Extract

The concept of a recursively definite predicate of natural numbers was introduced by F. B. Fitch in his An extension of basic logic as follows:

Every recursive predicate is recursively definite. If R(x₁, …, x_n) is recursively definite so is (Ey)R(x₁, …, x_n−1, y) and (y)R(x₁, …, x_n−1, y). If R is recursively definite and S is the proper ancestral of R, then S is recursively definite, where the proper ancestral of a relation is defined as follows: if R is of even degree, say 2m, then the proper ancestral of R is the relation S such that for all x₁, …, x_m, y₁, …, y_m, S(x₁, …, x_m, y₁, …, y_m) is true if and only if there is a finite sequence of sequences (z₁₁, …, z_m1), (z₁₂, …, Z_m2), …, (z_1k, …, Z_mk) such that R(Z₁₁, …, Z_m1, z₁₂, …, z_m2), R(z₁₂, …, z_m2, z₁₃, …, z_m3), …, R(z₁,_k−1, z_1k, …, Z_mk) are all true, where (z₁₁, …, z_m1) is (x₁, …, x_m) and (z_1k, …, z_mk) is (y₁, …, y_m).

An arithmetic predicate is one which is definable in terms of the operations ‘+’ and ‘·’ of elementary arithmetic, the connectives of the classical prepositional calculus, and quantifiers.