## Abstract

I. preliminary remarks.The writers of text-books of logic commonly take it for granted that there is a single theory, a set of rules embodying what are called the ‘laws of thought’; and that these laws are universally the same. This conception seems scarcely to accord with the present level of knowledge. For we do already possess distinct logics — if this term is used to denote precisely elaborated formalized systems: e.g., logics including or excluding a Theory of Types, systems admitting or barring the law of excluded middle, etc. Perhaps one might add that the rise of a conventionalistic mode of thinking — emanating from mathematics — today favours attempts to construct novel logics. Here two ways present themselves.

## Keywords

Free Variable Propositional Calculus Partial Negation Double Negation Strong Negation## Preview

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## Notes

- 1.For a symbolic system with a
*simple*hierarchy of types see: F. P. Ramsey, ‘The Foundations of Mathematics’,*Proc. Lond.Math. Soc*., vol. 25, 1945.Google Scholar - For a symbolic system
*without*a hierarchy of types see: A. Church, ‘A Set of Postulates for the Foundation of Logic’,*Annals of Math*., 33, 1932.Google Scholar - His system has been proved inconsistent by S. C. Koeene and J. B. Rosser,
*Annals of Math*., 36, 1935.Google Scholar - The impossibility of a system along such lines has not been proved either. Cf. K. Gödel, ‘Russell’s Mathematical Logic’, in
*The Philosophy of Bertrand Russell*, Evanston, 1944, p. 150.Google Scholar - 2.L. E. J. Brouwer, ‘Über die Bedeutung des Satzes von ausgeschlossenen Dritten …’,
*J. Math*., 154, 1925.Google Scholar - A. Heyting, ‘Die formalen Regeln der intuitionistischen Logik’,
*Ber. Akad*Berlin, 1930. See*Intuitionism*, Amsterdam, 1956.Google Scholar - 1.G. Birkhoff and J. V. Neumann, ‘The Logic of Quantum Mechanics’,
*Ann. of Math*., 37, 1936.Google Scholar - M. Strauss, ‘Zur Begrndung der statistischen “Transformationstheorie” der Quantenphysik’,
*Ber. Akad*., Berlin, 1936.Google Scholar