Note on a property of matrices for Lewis and Langford's calculi of propositions

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Abstract
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. ◊N = N (N the null class)◊A = {1, 2, …, n} (A any non-null class).5

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