A PHYSICO-MATHEMATICAL RIGOUR 749
In a-system, the 'logical' problems of freedom from contradiction become also semantic problems of one-valued meanings made possible only under oo-valued,, non-el general semantics, and the recognition of the A multiordinality of terms ,. A-system introduces some fundamental innovations, such as completely rejecting identity, elementalism . , and becomes based on m.o structure and order, and so ultimately becomes non-el. The A, (3+1)-dimensional el, (in the main) intensional system becomes a four-dimensional, non-el, (in the main) extensional system. In such a system we cannot use the formulations of elementalislic 'logics' and 'psychologies', but must have , non-el general semantics, which when generalized become an entirely general discipline applicable to all life, as well as to generalized mathematics. For the above reasons I shall use the word 'logic', in its el sense, with quotation marks; and use the term general semantics for a non-el, discipline corresponding to the el, A. or 'logics'.
Investigations show that the primitive man (and the 'mentally' ill) use one-valued semantics which have left more or less marked traces in all of us, reflected even in science and mathematics. The elimination of these primitive traces clears the foundation for an adult civilization, a theory of sanity, and the elimination of the scientific and mathematical paradoxes.
To assume that because a many-valued 'logic' has been produced, all the problems of mathematical infinity, irrational numbers, continuity, mathematical induction, validity of mathematical proof, mathematical existence . , have been solved, would be a mistake. The aim of the present paper is to analyse some of the fundamental complexities produced by the unconscious operation of the one-valued semantic identification concealed in the formulation of the 'law of identity', which have escaped notice until now, and which would make the application of a many-valued 'logic' or oo-valued semantics and agreement impossible. Here, as in the and systems, only the most general formulations help us to discriminate between the particular cases, and so to eliminate the undesirable traces of one-valued semantics by building a ^[-system, of which the A and pre- represent only particular cases.
Let me recall the 'philosophical grammar' of our language which we solemnly call the 'laws of thought', as given by Jevons:2
1) The law of identity. Whatever is, is.
2) The law of contradiction. Nothing can both be, and not be.
3) The law of excluded third. Everything must either be, or not be. These 'laws' have different 'philosophical' interpretations which help very
little and for my purpose it is enough to emphasize that: (1) The second 'law' represents a negative statement of the first, and the third represents a corollary of the former two; namely, no third possible between two contradictories. (2) The verb 'to be', or 'is', and 'identity' play a most fundamental role in these formulations. We should not be surprised to find that the investigation of these terms may give us a long sought solution. Such an investigation is very laborious and difficult. 'The complete attempt to deal with the term is would go to the form and matter of everything in existence, at least, if not to