Taking for the relation of confirmation the following obvious axioms, we obtain several more or less well-known theorems and are able to solve in a definite and strict manner several problems concerning confirmation.

Let a, b, and c be variable names of sentences belonging to a certain class, the operations a·b, a + b, and ā the (syntactical) product, the sum, and the negation of them. Let us further assume the existence of a real non-negative function c(a, b) of a and b, when b is not self-contradictory. Let us read ‘c(a, b)’ ‘degree of confirmation of a with respect to b' and take the following axioms:

Axiom I. If a is a consequence of b, c(a, b) = 1.

Axiom II. If is a consequence of c,

Axiom III. c(a·b, c) = c(a, c)·c(b, a·c).

Axiom IV. If b is equivalent to c, c(a, b) = c(a, c).

As may be easily seen, the interval of variation for c is (0, 1); this is quite conventional.

The purpose of this note is to call attention to a minor error in Lewis and Langford's Symbolic logic. On page 221, in discussing the Tarski-Łukasiewicz three-valued logic, the authors make the following assertion: “Let T(p) be any proposition, involving only one element, whose analogue holds in the two-valued system; if T(p) does not hold in the Three-valued Calculus, then pC.T(p) and Np.C.T(p) both hold.”

I shall show, by means of a counter-example, that this assertion is not true. Let T(p) be the sentence:

It is then easily verified that T(0) = T(1) = 1, and that T(½) = 0. Thus T(p) holds in the two-valued calculus, but not in the three-valued calculus. On the other hand, pC.T(p) does not hold, since ½.CT(½) = ½C0 = ½; similarly, Np.C.T(p) does not hold, since N½.C.T(½) = ½C0 = ½.

The object of this note is to show that there is no finite characteristic matrix for any one of Lewis and Langford's systems.

Theorem I. There is no finite characteristic matrix for any one of the systems S1–S5.

Proof. Let M be a matrix with less than n elements for which all the provable formulas in S1, or S2, or S3, or S4, or S5, are satisfied. Let Fn represent the formula

where ∑ stands for a ∨-chain, and the pi, are variables in any one of the calculi. Using M, there is always at least one summand in Fn where pi and pk have the same value. Therefore, Fn can always be written in the form (a = a)∨ B, and thus will give, for any B, a “designated” value, since the formula (p = p) ∨ q is provable in any one of the systems S1–S5.

Give to any one of the systems S1–S5, the following matrix, due to Henle:

1. Elements: all possible classes formed from the integers 1, 2, 3, …, n.

2. “Designated” element: the class {1, 2, 3, …,n}.

3. Boole-Schröder algebra on the elements.

4.