Alonzo Church has proposed a system of “restricted recursive arithmetic” for the analysis of computer circuits which include time delays and feedback. Restricted recursive arithmetic, which will be referred to as rra, was introduced by Church in this Journal, vol. 20 (1955), pp. 286–287, in a review of an article by Edmund C. Berkeley. It is the purpose of this paper to present some results which have been found for the system rra.

Greater economy can be effected in the primitive rules for the system K of basic logic by defining the existence operator ‘E’ in terms of two-place abstraction and the disjunction operator ‘V’. This amounts to defining ‘E’ in terms of ‘ε’, ‘έ’, ‘o, ‘ό’, ‘W’ and ‘V’, since the first five of these six operators are used for defining two-place abstraction.

We assume that the class Y of atomic U-expressions has only a single member ‘σ’. Similar methods can be used if Y had some other finite number of members, or even an infinite number of members provided that they are ordered into a sequence by a recursive relation represented in K. In order to define ‘E’ we begin by defining an operator ‘D’ such that

Here ‘a’ may be thought of as an existence operator that provides existence quantification over some finite class of entities denoted by a class A of U-expressions. In other words, suppose that ‘a’ is such that ‘ab’ is in K if and only if, for some ‘e’ in A, ‘be’ is in K. Then ‘Dab’ is in K if and only if, for some ‘e and ‘f’ in A, ‘be’ or ‘b(ef)’ is in K; and ‘a’, ‘Da’, ‘D(Da)’, and so on, can be regarded as existence operators that provide for existence quantification over successively wider and wider finite classes. In particular, if ‘a’ is ‘εσ’, then A would be the class Y having ‘σ’ as its only member, and we can define the unrestricted existence operator ‘E’ in such a way that ‘Eb’ is in K if and only if some one of ‘εσb’, ‘D(εσ)b’, ‘D(D(εσ))b’, and so on, is in K.

Logical falsehoods are usually barred in confirmation theory from serving as evidence statements. I wish to study here some three major ways in which this restriction can be lifted.

1. Consider the following axioms for the two-argument confirmation function C:

A1. 0 ≤ C(h, e) ≤ 1;

A2. If ⊦h ≡ h′ and ⊦e ≡ e′, then C(h, e) = C(h′, e′);

A3. If ⊦e ⊃ h, then C(h, e) = 1;

A4. If ⊦e ⊃ ˜(h.h′), then C(h⋁h′, e) = C(h, e)+C(h′, e);

A5. C(h.h′, e) = C(h,e)×C(h′, h.e);

where h, h′, e, and e′ are any four statements from a consistent language L and where ‘⊦’ is short for ‘is logically true in L.’ A1—A5 are consistent, as shown by the following construction:

C1. Let all four of h, h′, e, and e′ be some logical truth lt of L, and let C(h, e) equal 1 for any two h and e;

under which they become true. But they yield as consequences:

and hence

No statement e of L is logically false.

Since all the languages for which confirmation theories have been proposed contain logical falsehoods, (1) is undesirable. Restrictions must accordingly be placed upon some of A1—A5.