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, …, xn) is recursively definite so is (Ey)R(x1, …, xn−1, y) and (y)R(x1, …, xn−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, …, xm, y1, …, ym, S(x1, …, xm, y1, …, ym) is true if and only if there is a finite sequence of sequences (z11, …, zm1), (z12, …, Zm2), …, (z1k, …, Zmk) such that R(Z11, …, Zm1, z12, …, zm2), R(z12, …, zm2, z13, …, zm3), …, R(z1,k−1, z1k, …, Zmk) are all true, where (z11, …, zm1) is (x1, …, xm) and (z1k, …, zmk) is (y1, …, ym).

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.

The calculus generated by the addition of the postulate

to S2 will, after Miss Alban, be called “S6”, the calculus generated by the addition of the same postulate to S3, “S7”, and the calculus generated by the addition of the postulate

to S3, “S8”. No interesting interpretation of the calculi S6–8 is known, but they are of some indirect interest because of the connections between them and the five calculi S1–S5. Certain questions concerning them will be treated in the following. In §1 it will be shown that every method of decision for S7 can be turned into a method of decision for S3, and in §2 that the number of complete extensions of S3 is equal to the number of complete extensions of S7 plus one. In §3 it will be shown that McKinsey's method of decision for S2 and S4 can be modified so as to cover S6.

The letters “P”, “Q”, “R”, and “S” will be employed as syntactic variables denoting formulas. Logical expressions will sometimes be used as selfdenotative. “C” and “C” stand for arbitrary calculi. “C + P” is the calculus which is the result of adding to C as new postulates all formulas which can be derived from P by substitution. With “substitution” I mean the operation performed on propositional variables.

In an earlier article, I proposed a way of determining the relative simplicity of different sets of extralogical primitives. The calculations assumed a fully platonistic logic, committed to an indefinite hierarchy of classes, with sequences and relations defined as classes. Recently it has been shown that a nominalistic logic, countenancing no entities other than individuals, can be made to serve many of the purposes for which a platonistic logic had been thought necessary. The question naturally arises how we are to determine the simplicity of extralogical bases of systems founded upon a nominalistic logic. In such bases, the only extralogical predicates will be predicates (of one or more places) of individuals. The present paper offers a general method of measuring the simplicity of such bases.

Emil Post has shown that the word problem for Thue systems is unsolvable. A Thue system is a system of strings of letters x1, x2, …, xr, where a string or word is either void or a succession of letters a1a2 … at with each ai some xi. We are given a finite number of pairs of strings Ai, Bii = 1, …, m. Operations in the Thue system consist of the replacements or “productions.”

An (m + n)-valued propositional calculus2 may happen to be a subsystem of an m-valued propositional calculus , though the converse is never true. This fact may give us the impression that, as m grows, the content of becomes meagre. The present treatment is intended to remove this impression by constructing a complete, m-valued sub-system of any (m + n)-valued propositional calculus.

In the following we adopt the customary, autonymous mode of speech according to which symbols belonging to the object calculi or languages are used in the syntactic language as names for themselves, and juxtaposition serves to denote juxtaposition.

2.1 = df stands for definational identity in the syntactic language.

2.11 ≡ stands for definational identity in the object calculi.

2.2 ∊, ⊂, ∩, {x1, …, xn} are used in their meanings as customarily employed in the theory of sets—∊ for class membership, ⊂ for proper inclusion, ∩ for the product operation of classes, {x1, …, xn} for the class with x1, …, xn as its only elements.

2.3 x, y, z are used as unspecified natural numbers including 0. m, n, i, j are used as unspecified natural numbers other than 0.

2.401 Definition. δ = df as the function of two variables defined for any x, y such that

2.41 Definition. For m ≧ 2, ιm = df the function of two variables denned on the set {0, …, m − 1} such that ιm (x, y) = y − x for x ≦ y and ιn(x, y) = 0 for x > y.

Of the several methods for proving the completeness of sets of axioms for the prepositional calculus perhaps the simplest is due to Kalmár, although it does not appear to be widely known. In this paper we generalize Kalmár's method to indicate how to obtain a complete axiomatization of any fragment of the propositional calculus which includes material implication. We shall carry through the description and proofs for the case where, in addition to a symbol for implication, there is just one other primitive truth-function symbol. For systems in which there are no other function symbols, or more than one other such symbol, notational changes but not conceptual changes will be required.