... The three-valued system of propositional logic . . . is the simplest example of a consistent logical system which is as different from the ordinary two-valued system as any non-Euclidean geometry is from the Euclidean.
I think it may be said that the system mentioned is the first intuitively grounded system differing from the ordinary propositional calculus. (..) It is true that Post has investigated many-valued systems of propositional logic from a purely formal point of view, yet he has not been able to interpret them logically.1 The well-known attempts of Brouwer, who rejects the universal validity of the law of the excluded middle and also repudiates several theses of the ordinary propositional calculus, have so far not led to an intuitively based system. They are merely fragments of a system whose construction and significance are still entirely obscure.2
It would perhaps not be right to call the many-valued systems of propositional logic established by me 'non-Aristotelian' logic, as Aristotle was the first to have thought that the law of bivalence could not be true for certain propositions. Our new-found logic might be rather termed 'non-Chrysippean', since Chrysippus appears to have been the first logician to consciously set up and stubbornly defend the theorem that every proposition is either true or false. This Chrysippean theorem has to the present day formed the most basic foundation of our entire logic.
It is not easy to foresee what influence the discovery of non-Chrysippean systems of logic will exercise on philosophical speculation. However, it seems to me that the philosophical significance of the systems of logic treated here might be at least as great as the significance of non-Euclidean systems of geometry.
1 E. L. Post, 'Introduction to a general theory of elementary propositions', Am. Journ. of Math. 43 (1921), p. 1832 : '. . . . the highest dimensioned intuitional proposition space is two.'
2 Cf. e.g., L .E. J. Brouwer, 'Intuitionistische Zerlegung mathematischer Grundbegriffe', Jahresber. d. Deutsch. Math.-Vereinigung 33 (1925), pp. 251 ff. ; 'Zur Begründung der intuitionistischen Mathematik. I', Math. Ann. 93 (1925), pp. 244 ff.
Polish Logic 1920-1939, Storrs McCall,
Oxford 1967