Dear Psycho-History Group Members,

As promised in my earlier post here on "Axioms for Dialectical Logic", this post contains a worked example of a 'Dialectical-Mathematical Model', using the language of the "purely-qualitative", "purely connotative" mathematics -- the 'Dialectical Ideography' -- of that axiomatized Dialectical Logic.

The example for this post is that of the philosopher William of Ockham's theory of science.

Ockham gave to terms of language which refer to the reality outside of the reality of language, the descriptor "categorematic". Thus, the "term" or word "rock" refers to that "external reality".

Ockham gave to terms of language which refer to categorematic terms the descriptor "syncategorematic". Thus, the "terms" or words "all", "some", "none", and "not" are such "word-referring words", or "term-referring terms".

Ockham also called terms that refer to things belonging to the reality outside of language "terms of the first intension".

He called terms which refer to terms of the first intention "terms of the second intension".

In a sense, then, syncategorematic terms are terms of the second intension.

However, Ockham distinguishes mere "terms" of language -- e.g., individual words -- from those 'meta-terms', or 'meta-words', each one made up out of a heterogeneous multiplicity of "mere" "terms, or of "mere' "words", which are named "complete sentences", or "well-formed propositions", and Ockham does not ontologically "reduce" the latter to the former.

Thus, for Ockham, categorematic terms in combination with syncategorematic terms, in the form of well-formed sentences or propositions, and thus, functioning together, refer more adequately to things belonging to the reality outside of language, than do single, isolated "mere" terms, or single, isolated words, such as those terms/words which are "names", or "nouns".

The sentence/proposition "All men are mortal." is thus first intentional.

But when we use terms such as "Genos", and "species", we are using terms of "second intension".

On the basis of these distinctions, Ockham divides the "Genos" of the sciences into two "specuies", or sub-"Gene" --



"Genos": Science, or, in F.E.D.'s terms, "Scientia Universalis".

"species" 1: "Scientia realis", encompassing the sciences of "real things". For example, the science of physics would form a sub-"species" of this "species". This "species" is "first-intensional".

"species" 2: "Scientia rationalis", encompassing the sciences of 'language-things'. For example, the science of formal logic would form a sub-"species" of this "species". This "species" is "second-intensional".

For further background on Ockham's theory, see Dictionary of Philosophy and Religion, Eastern and Western Thought, by W. L. Reese, 1980, pp. 627 ff..



To model Ockham's taxonomy of this world of human ideas, using the "Dialectical Logic" axiomatized in the earlier post here, we "interpret", or "assign" -- the mental action denoted by the ideogram '(---])' -- the first of the "meta-Natural meta-Numbers" of that "N\Q", or "unquantifiable Qualifier", "Dialectical Ideography", as follows --

q1 (---] q-subscript-r or R, such that R "reality"; the universe of "real things", outside of human language.



The model then captures the sequential presentation of Ockham's ideas via the recurrently "self-reflexive" function, or "self-iteration" expressed by --

R^(2(^s))

-- wherein s, as the higher exponent of the exponent 2, counts the "steps" or "stages" of that presentation of ideas / of philosophical categories.


0. Then, for the 0th step, s = 0 --

R^(2(^0)) = R^1 = R

-- which simply reiterates the starting point, or