In the system presented by Quine in his book Mathematical logic, one can derive the Burali-Forti Paradox. It is the purpose of this paper to present the details of this derivation. For the derivation, only familiarity with Quine's book is assumed.

The present derivation is based on the derivations given by Hobson and by Whitehead and Russell.

In both these sources, the primary interest is the theory of ordinals, and the Burali-Forti Paradox is of interest only as something to be avoided. In the present paper the primary interest is the Burali-Forti Paradox, so that much of the theory of ordinals is absent from this paper and only those details remain which are relevant to the derivation of the paradox.

Sets of postulates for at least three different combinatory logics have been given ([2], [3], [4], [5], [6], [7]). The logics and their sets of postulates are quite similar. In all cases it is the proof of the completeness of the set of postulates which causes the difficulty and makes the main papers so long ([3], [7]). It is the purpose of this paper to present a method of writing down sets of postulates for combinatory logics for which the proof of completeness will be fairly simple. We assume acquaintance with certain descriptive portions of [1], namely sections 1–6, 12, 13, 15. Otherwise the present paper is self-contained.

We first solve our problem for the particular logic studied by Rosser ([7]) and then indicate how the method would apply to other logics.

We start with a set, s, of primitive terms. We define s-combination as in [1], p. 43. The precise contents of s need not concern us. It suffices that there be s-combinations I, J, and Q, with no variables in them, and having properties as described below. I and J are to have the same meaning as in [1], p. 43, namely IA is the same as A and JABCD is the same as AB(ADC). QAB is to have the meaning that A and B, considered as functions, are the same.

This note is concerned with the logical formalism with types recently introduced by Church [1] (and called (C) in this note) It was shewn in his paper (Theorem 26α) that if Yα stands for

(a form of the “axiom of infinity” for the type α), Yα can be proved formally, from Yι and the axioms 1 to 7, for all types α of the forms ι′, ι″, …. For other types the question was left open, but for the purposes of an intrinsic characterisation of the Church type-stratification given by one of us, it is desirable to have the remaining cases cleared up. A formal proof of Yα is now given for all types α containing ι, but the proof uses, in addition to Axioms 1 to 7 and Yι, also Axiom 9 (in connection with Def. 4), and Axiom 10 (in Theorem 9).