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This book is very clear that logical levels are created by inclusion of logical categories within larger, more encompassing categories. However, I found the discussion of logical types confusing. It defines two sets as being of the same logical type if there is isomorphic mapping between them. (pp. 295-301) But since no use whatsoever is given for the term logical type, I have no idea how this proposed distinction is of any use.
Bertrand Russell used the term logical types interchangeably with logical levels, in declaring that a class (at one logical level) cannot also be a member of itself (at a smaller level).
G. Spencer Brown in the preface to Laws of Form (1974) has shown that Russells theory of logical types is not only unnecessary, but if accepted, would deprive us of the branch of mathematics dealing with imaginary numbers, which is very useful in electronics and in calculations involving sine waves and other trigonometric functions! The theory of logical types would make impossible the many useful self-referential (and sometimes paradoxical) messages which people do, in fact, make and respond to. It would also outlaw important and interesting phenomena such as the self-concept, which describes itself recursively, including itself in its description.
The theory of logical types (and any conclusions derived from it by Bateson, Dilts, Hall and others) was declared brain-dead by Bertrand Russell himself in 1967, as reported by G. Spencer Brown (also in the preface to Laws of Form): The theory was, he said, the most arbitrary thing he and Whitehead had ever had to do, not really a theory but a stopgap, and he was glad to have lived long enough to see the matter resolved. Please let us hear no more about the Theory of Logical Types.
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