SCIENCE AND SANITY - online book

An Introduction To Non-aristotelian Systems And General Semantics.

Home | About | Philosphy | Contact | Search




A PHYSICO-MATHEMATICAL RIGOUR             753
what might be called the (3+l)-dimensional el, A -system, which divides 'apace' and 'time'. , (an attitude which is carried all through the system), into a four-dimensional non-el,-system (an attitude which is also carried all through the system).                                                                                      (24)
Q) From (8)it follows that statements about statements represent results of new neurological processes, that their content varies, and that we must discriminate and not identify these different meanings. In other words, only through consciousness of abstracting which represents the most general s.r of discrimination, or the elimination of identification, can we assign single values to words which have an essentially many-Valued character. Identification confuses these many meanings into one.                                                        (25)
R) We must differentiate between descriptive and inferential words and phrases, and never use inferential terms as descriptive, without realizing that we are doing so.                                                                                              (26)
S) Certain words or phrases used to speak about languages, such as 'all statements', 'proposition about all propositions'. , lead to self-contradictions. We cannot speak about 'all' propositions without some limitations, if we proceed introducing new propositions. Even St. Paul felt the necessity for limiting the values of 'all'.* We are compelled to introduce some equivalents to the biblical 'illegitimate totalities' or the theory of types of Russell. (27)
T) Analysis finds that certain of the most important terms we use; such as, 'yes', 'no', 'true', 'false', 'all', 'fact', 'reality', 'existence', 'definition', 'relation', 'structure', 'order', 'number', 'is', 'has', 'there is', 'variable', 'infinite', 'abstraction', 'property', 'meaning', 'value', 'love', 'hate', 'knowing', 'doubt'. , . , may apply to all verbal levels and in each particular case may have a different content or meanings and so in general no single content or meaning. I call such terms multiordinal terms {m.o). The definition of such terms is always given in other m.o terms preserving their fundamental multiordinality. In other words, a m.o term represents a many-valued term. If the many values are identified, or disregarded, or confused, we treat a fundamentally many-valued term as one-valued, and we must have every kind of paradox through such an identification. All known paradoxes in mathematics and life can be manufactured by the disregard of this fundamental multiordinality. Vice versa, by formulating the general semantic problem of multiordinality we gain means to discriminate between the many meanings and so assign a single meaning in a given context. A m.o term represents a variable in general, and becomes constant or one-valued in a given context, its value being given by that context. Here we find the main importance of the semantic fact established by Skarzenski,** that the 'logical' freedom from contradiction becomes a semantic
♦Professor Cassius J. Keyser drew my attention to a passage in the first letter of St. Paul to the Corinthians, Chapter 15, line 27. 'For he hath put all things under his feet. But when he saith all things are put under him, it is manifest that he is excepted, which did put all things under him.' Italics are mine.
♦♦Quoted by Chwistek in his Neue Grtmdlagen der Logic und Mathematik.