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==Categories of structured individuals==

==Categories of structured individuals==

We have at this point enough material to begin thinking about the forms of analogy, iconicity, metaphor, morphism, whatever you want to call it, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''entitative'' and ''existential''.

We have at this point enough material to begin thinking about the forms of analogy, iconicity, metaphor, or morphism that arise in the interpretation of logical graphs as logical propositions, in particular, the logically dual modes of interpretation that Peirce developed under the names of ''entitative graphs'' and ''existential graphs''.

By way of providing a conceptual-technical framework for organizing that discussion, let me introduce the concept of a ''category of structured individuals'' (COSI).  There may be some cause for some readers to rankle at the very idea of a ''structured individual'', for taking the notion of an individual in its strictest etymology would render it absurd that an atom could have parts, but all we mean by ''individual'' in this context is an individual by dint of some conversational convention currently in play, not an individual on account of its intrinsic indivisibility.  Incidentally, though, it will also be convenient to take in the case of a class or collection of individuals with no pertinent inner structure as a trivial case of a COSI.

By way of providing a conceptual-technical framework for organizing that discussion, let me introduce the concept of a ''category of structured individuals'' (COSI).  There may be some cause for some readers to rankle at the very idea of a ''structured individual'', for taking the notion of an individual in its strictest etymology would render it absurd that an atom could have parts, but all we mean by ''individual'' in this context is an individual by dint of some conversational convention currently in play, not an individual on account of its intrinsic indivisibility.  Incidentally, though, it will also be convenient to take in the case of a class or collection of individuals with no pertinent inner structure as a trivial case of a COSI.

For some reason — that I won't stop to examine right now — I tend to think ''category of structured individuals'' when the individuals in question have a whole lot of structure, but ''collection of structured items'' when the individuals have a minimal amount of internal structure.  For example, any set is a COSI, so any relation in extension is a COSI, but a 1-adic relation is just a set of 1-tuples, that are in some lights indiscernible from their single components, and so its structured individuals have far less structure than the k-tuples of k-adic relations, when k exceeds one.  This spectrum of differentiations among relational models will be useful to bear in mind when the time comes to say what distinguishes relational thinking proper from 1-adic and 2-adic thinking, that constitute its degenerate cases.

It seems natural to think ''category of structured individuals'' when the individuals in question have a whole lot of internal structure but ''collection of structured items'' when the individuals have a minimal amount of internal structure.  For example, any set is a COSI, so any relation in extension is a COSI, but a 1-adic relation is just a set of 1-tuples, that are in some lights indiscernible from their single components, and so its structured individuals have far less structure than the k-tuples of k-adic relations, when k exceeds one.  This spectrum of differentiations among relational models will be useful to bear in mind when the time comes to say what distinguishes relational thinking proper from 1-adic and 2-adic thinking, that constitute its degenerate cases.

Still on our way to saying what brands of iconicity are worth buying, at least when it comes to graphical systems of logic, it will useful to introduce one more distinction that affects the types of mappings that can be formed between two COSI's.

Still on our way to saying what brands of iconicity are worth buying, at least when it comes to graphical systems of logic, it will useful to introduce one more distinction that affects the types of mappings that can be formed between two COSI's.

Because I plan this time around a somewhat leisurely excursion through the primordial wilds of logics that were so intrepidly explored by C.S. Peirce and again in recent times revisited by George Spencer Brown, let me just give a few extra pointers to those who wish to run on ahead of this torturous tortoise pace:

Because I plan this time around a somewhat leisurely excursion through the primordial wilds of logics that were so intrepidly explored by C.S. Peirce and again in recent times revisited by George Spencer Brown, let me just give a few extra pointers to those who wish to run on ahead of this torturous tortoise pace:

* Jon Awbrey, "Propositional Equation Reasoning Systems (PERS)", [http://forum.wolframscience.com/printthread.php?threadid=297&perpage=35 Eprint].

:* Jon Awbrey, "Propositional Equation Reasoning Systems (PERS)", [http://forum.wolframscience.com/printthread.php?threadid=297&perpage=35 Online].

* Lou Kauffman, "Box Algebra, Boundary Mathematics, Logic, and Laws of Form", [http://www.math.uic.edu/~kauffman/Arithmetic.htm Eprint].

Two paces back I used the word "category" in a way that will turn out to be not too remote a cousin of its present day mathematical bearing, but also in way that's not unrelated to Peirce's theory of categories.

 

Two paces back I used the word ''category'' in a way that will turn out to be not too remote a cousin of its present day mathematical bearing, but also in way that's not unrelated to Peirce's theory of categories.

When I call to mind a ''category of structured individuals'' (COSI), I get a picture of a certain form, with blanks to be filled in as the thought of it develops, that can be sketched at first like so:

When I call to mind a ''category of structured individuals'' (COSI), I get a picture of a certain form, with blanks to be filled in as the thought of it develops, that can be sketched at first like so: