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On the semantic non-completeness of certain Lewis calculi

Published online by Cambridge University Press:  12 March 2014

Sören Halldén
Sören Halldén*
Affiliation:
Upsala University
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Extract

I intend to show that for S1, S2, and S3 the class of true formulas cannot coincide with the class of theorems. This is to hold irrespective of the meaning assigned to “◊”. when only “∼”, “▪”, and the variables are interpreted in the customary manner.

The idea of the proof is extremely simple and can be illustrated by the following argument concerning S3. The formula ~(◊(p▪~p)⥽▪p▪~p) ∨ (◊(q▪~q)⥽▪q▪~q) is a theorem of S3. Suppose now that all S3-theorems are true. Then either ~(◊(p▪~p)⥽▪p▪~p) is true (for every p) or (◊(q▪~q)⥽▪q▪~q) is true (for every q). But none of these formulas is an S3-theorem. Then some true S3-formula is not an S3-theorem. Hence, if all S3-theorems are true, some true S3-formula is not an S3-theorem. Then the class of S3-theorems cannot be identical with the class of true S3-formulas.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 16 , Issue 2 , June 1951 , pp. 127 - 129
Copyright
Copyright © Association for Symbolic Logic 1951

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References

1 Cf. Lewis, and Langford, , Symbolic logic, New York 1932, p. 492 ffGoogle Scholar.

2 See my paper A note concerning the paradoxes of strict implication and Lewis's system S1, this Journal, vol. 13 (1948)Google Scholar.

3 Every theorem of the two-valued propositional calculus is a theorem of S1. See Symbolic logic, p. 140.

4 See p. 493.

5 McKinsey, , A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology, this Journal, vol. 6 (1941), p. 126 (theorem 7)Google Scholar.

6 Symbolic logic, p. 501.

7 See Parry's, Modalities in the “Survey” system of strict implication, this Journal, vol. 4 (1939), p. 141 (theorem 32.1), and p. 148Google Scholar.

8 Nelson, , Intensional relations, Mind, n.s. vol. 39 (1930)Google Scholar.

9 McKinsey, and Tarski, , Some theorems about the sentential calculi of Lewis and Heyting, this Journal, vol. 13 (1948), p. 12 (theorem 4.4)Google Scholar.