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Independence of the primitive symbols of Lewis's calculi of propositions

Published online by Cambridge University Press:  12 March 2014

M. J. Alban
M. J. Alban*
Affiliation:
Skidmore College
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Extract

In this paper it will be shown that the three primitive symbols used by Lewis are independent in each one of his systems S1–S6.

Lemma I. “·” is not definable in terms of “∼” and “◊” in any one of the systems S1–S6.

Proof. This is obvious, since no binary operation can be defined by unary operations alone.

Lemma II. “∼” is not definable in terms of “˙” and “◊” in any one of the systems S1–S5.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 8 , Issue 1 , 19 March 1943 , pp. 25 - 26
Copyright
Copyright © Association for Symbolic Logic 1943

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References

1 For the definitions of these systems, see Lewis, and Langford, , Symbolic logic, pp. 500–501Google Scholar. For convenience in terminology, the system arising from the use of B1–B8 together with ◊◊p will herein be termed S6.

2 See Lewis and Langford, loc. cit., p. 492. This matrix is, of course, but a special case of the matrix due to Henle. The ordinary 2-clement matrix for “˙” and “∼” with condition 4 above can just as well be used to prove the lemma in question, but any one of Henle's, matrices with 2n elements (n > 1) is more convenient for the purposes of this paper, since a slight modification of any one of them permits a proof of the lemma for S6.

3 If A and B are any two classes, “A = B” reduces to the class {1, 2} if A and B are the same class, and to the value N otherwise.

4 This matrix corresponds to one due to Parry. Vide Lewis and Langford, loc. cit., pp. 492–493.

5 A finite characteristic matrix for a logic L is a matrix, consisting of a finite number of elements, which satisfies those, and only those, provable formulas of L.

6 Dugundji, J., Note on a property of matrices for Lewis and Langford's calculi of propositions, this Journal, vol. 5 (1940) pp. 150–151.Google Scholar