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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:

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The various glyphs of this picturesque hierarchy serve to remind us that a COSI in general consists of many individuals, which in spite of their calling as such may have specific structures involving the ordering of their component parts.  Of course, this generic picture may have degenerate realizations, as when we have a 1-adic relation, that may be viewed in most settings as nothing different than a set:

The various glyphs of this picturesque hierarchy serve to remind us that a COSI in general consists of many individuals, which in spite of their calling as such may have specific structures involving the ordering of their component parts.  Of course, this generic picture may have degenerate realizations, as when we have a 1-adic relation, that may be viewed in most settings as nothing different than a set:

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The practical use of Peirce's categories is simply to organize our thoughts about what sorts of formal models are demanded by a material situation, for instance, a domain of phenomena from atoms to biology to culture.  To say that "k-ness" is involved in a phenomenon is simply to say that we need k-adic relations to model it adequately, and that the phenomenon itself appears to demand nothing less.  Aside from this, Peirce's realization that k-ness for k = 1, 2, 3 affords us with a sufficient basis for all that we need to model is a formal fact that depends on a particular theorem in the logic of relatives.  If it weren't for that, there would hardly be any reason to single out three.

The practical use of Peirce's categories is simply to organize our thoughts about what sorts of formal models are demanded by a material situation, for instance, a domain of phenomena from atoms to biology to culture.  To say that "k-ness" is involved in a phenomenon is simply to say that we need k-adic relations to model it adequately, and that the phenomenon itself appears to demand nothing less.  Aside from this, Peirce's realization that k-ness for k = 1, 2, 3 affords us with a sufficient basis for all that we need to model is a formal fact that depends on a particular theorem in the logic of relatives.  If it weren't for that, there would hardly be any reason to single out three.

Cast into the form of a 3-adic sign relation, the situation before us can now be given the following shape:

Cast into the form of a 3-adic sign relation, the situation before us can now be given the following shape:

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The interpretation maps ''En'', ''Ex'' : ''Y'' &rarr; ''X'' are factored into a shared syntactic part:

The interpretation maps ''En'', ''Ex'' : ''Y'' &rarr; ''X'' are factored into a shared syntactic part:

The more Peirce-systent among you, on contemplating that last picture, will 1st or 2nd or 3rd-naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"

The more Peirce-systent among you, on contemplating that last picture, will 1st or 2nd or 3rd-naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"

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The answer is that the last bastion of 3-adic irreducibility presidios precisely in the duality of the dual interpretations ''En''_sem and ''Ex''_sem.  To see this, consider the consequences of there being, contrary to all that we've assumed up to this point, some ultimately compelling reason to assert that the clean slate, the empty medium, the vacuum potential, whatever one wants to call it, is inherently more meaningful of either Falsity or Truth.  This would issue in a conviction forthwith that the 3-adic sign relation involved in this case decomposes as a composition of a couple of functions, that is to say, reduces to a 2-adic relation.

The answer is that the last bastion of 3-adic irreducibility presidios precisely in the duality of the dual interpretations ''En''_sem and ''Ex''_sem.  To see this, consider the consequences of there being, contrary to all that we've assumed up to this point, some ultimately compelling reason to assert that the clean slate, the empty medium, the vacuum potential, whatever one wants to call it, is inherently more meaningful of either Falsity or Truth.  This would issue in a conviction forthwith that the 3-adic sign relation involved in this case decomposes as a composition of a couple of functions, that is to say, reduces to a 2-adic relation.

At the level of the ''primary arithmetic'' (PAR), we have a set-up like this:

At the level of the ''primary arithmetic'' (PAR), we have a set-up like this:

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The object domain '''O''' is the boolean domain '''B''' = {Falsity, Truth}, the semiotic domain '''S''' is any of the spaces isomorphic to the set of rooted trees, matched-up parentheses, or unlabeled alpha graphs, and we treat a couple of ''denotation maps'' ''D''_en, ''D''_ex : '''S''' &rarr; '''O'''.

The object domain '''O''' is the boolean domain '''B''' = {Falsity, Truth}, the semiotic domain '''S''' is any of the spaces isomorphic to the set of rooted trees, matched-up parentheses, or unlabeled alpha graphs, and we treat a couple of ''denotation maps'' ''D''_en, ''D''_ex : '''S''' &rarr; '''O'''.

The general scheme of things is suggested by the following Figure, where the mapping ''f'' from COSI ''U'' to COSI ''V'' is analyzed in terms of a mapping ''g'' that takes individuals to individuals, ignoring their inner structures, and a set of mappings ''h''_''j'', where ''j'' ranges over the individuals of COSI ''U'', and where ''h''_''j'' specifies just how the parts of ''j'' map to the parts of ''g''(''j''), its counterpart under ''g''.

The general scheme of things is suggested by the following Figure, where the mapping ''f'' from COSI ''U'' to COSI ''V'' is analyzed in terms of a mapping ''g'' that takes individuals to individuals, ignoring their inner structures, and a set of mappings ''h''_''j'', where ''j'' ranges over the individuals of COSI ''U'', and where ''h''_''j'' specifies just how the parts of ''j'' map to the parts of ''g''(''j''), its counterpart under ''g''.

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Next time we'll apply this general scheme to the ''En'' and ''Ex'' interpretations of logical graphs, and see how it helps us to sort out the varieties of iconic mapping that are involved in that setting.

Next time we'll apply this general scheme to the ''En'' and ''Ex'' interpretations of logical graphs, and see how it helps us to sort out the varieties of iconic mapping that are involved in that setting.

Here is the Figure for the Entitative interpretation:

Here is the Figure for the Entitative interpretation:

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Here is the Figure for the Existential interpretation:

Here is the Figure for the Existential interpretation:

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Note that the structure of a tree begins at its root, marked by an "O".  The objects in '''O''' have no further structure to speak of, so there is nothing much happening in the object domain '''O''' between the level of individuals and the level of structures.  In the sign domain '''S''', the individuals are the parts of the partition into referential equivalence classes, each part of which contains a countable infinity of syntactic structures, rooted trees, or whatever form one views their structures taking.  The sense of the Figures is that the interpretation under consideration maps the individual on the left (right) side of '''S''' to the individual on the left (right) side of '''O'''.

Note that the structure of a tree begins at its root, marked by an "O".  The objects in '''O''' have no further structure to speak of, so there is nothing much happening in the object domain '''O''' between the level of individuals and the level of structures.  In the sign domain '''S''', the individuals are the parts of the partition into referential equivalence classes, each part of which contains a countable infinity of syntactic structures, rooted trees, or whatever form one views their structures taking.  The sense of the Figures is that the interpretation under consideration maps the individual on the left (right) side of '''S''' to the individual on the left (right) side of '''O'''.

* ''Ex'' maps every tree on the left (right) of '''S''' to the right (left) of '''O'''.

* ''Ex'' maps every tree on the left (right) of '''S''' to the right (left) of '''O'''.

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Now, those who wish to say that these logical signs are iconic of their logical objects must not only find some reason that logic itself singles out one interpretation over the other, but, even if they succeed in that, they must then make us believe that every sign for Truth is iconic of Truth, while every sign for Falsity is iconic of Falsity.  Well, I confess that it strains my imagination, if not the over-abundant resources of theirs.

Now, those who wish to say that these logical signs are iconic of their logical objects must not only find some reason that logic itself singles out one interpretation over the other, but, even if they succeed in that, they must then make us believe that every sign for Truth is iconic of Truth, while every sign for Falsity is iconic of Falsity.  Well, I confess that it strains my imagination, if not the over-abundant resources of theirs.