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An Introduction To Non-aristotelian Systems And General Semantics.

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HIGHER ORDER ABSTRACTIONS
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go to the roots of the A semantic difficulty. They do not pay attention to the A, 'logical', 'philosophical', and 'psychological' elementalistic method and language involving the 'is' of identity, in which the Introduction of the Principia is written. Doctor Alonzo Church is the first, as far as my knowledge goes, to suggest that, following Peano, numbers should be defined in the language of abstractions. He does not carry his analysis further, however, and does not state that it involves a language of entirely different A structure.8 If we abandon the term 'class' and accept the language of 'abstractions of different orders', then we are led to the rejection of the 'is' of identity and to the present system, of which the theory of mathematical types becomes a necessary part. The problems of 'class' cease to be an 'assumption', as the different orders of abstractions are descriptions of experimental facts; and so the 'axiom of reducibility' becomes unnecessary. In my language, this axiom is also an aristotelian description of the experimental fact that we can abstract in different orders.
Section B. Multiordinal terms.
In the examples given in Section A, we used words such as 'proposition', which were applied to all higher order abstractions. We have already seen that such terms may have different uses or meanings if applied to different orders of abstractions. Thus originates what I call the multiordinality of terms. The words 'yes', 'no', 'true', 'false', 'function', 'property', 'relation', 'number', 'difference', 'name', 'definition', 'abstraction', 'proposition', 'fact', 'reality', 'structure', 'characteristic', 'problem', 'to know', 'to think', 'to speak', 'to hate', 'to love', 'to doubt', 'cause', 'effect', 'meaning', 'evaluation', and an endless array of the most important terms we have, must be considered as multiordinal terms. There is a most important semantic characteristic of these m.o terms; namely, that they are ambiguous, or oo-valued, in general, and that each has a definite meaning, or one value, only and exclusively in a given context, when the order of abstraction can be definitely indicated.
These issues appear extremely simple and general, a part and parcel of the structure of 'human knowledge' and of our language. We cannot avoid these semantic issues, and, therefore, the only way left is to face them explicitly. The test for the multiordinality of a term is simple. Let us make any statement and see if a given term applies to it ('true', 'false', 'yes', 'no', 'fact', 'reality', 'to think', 'to love',.). If it does, let us deliberately make another statement about the former statement and test if the given term may be used again. If so, it is a safe assertion that this term should be considered as m.o. Any one can test such a m.o
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