Changes

Briefly if roughly put, icons are signs that denote their objects by virtue of sharing properties with them.  To put it a bit more fully, icons are signs that receive their interpretant signs on account of having specific properties in common with their objects.

Briefly if roughly put, icons are signs that denote their objects by virtue of sharing properties with them.  To put it a bit more fully, icons are signs that receive their interpretant signs on account of having specific properties in common with their objects.

The family of related relationships that fall under the headings of analogy, icon, metaphor, model, simile, simulation, and so on forms an extremely important complex of ideas in mathematics, there being recognized under the generic idea of "structure-preserving mappings" and commonly formalized in the language of homomorphisms, morphisms, or "arrows", depending on the operative level of abstraction that's in play.

The family of related relationships that fall under the headings of analogy, icon, metaphor, model, simile, simulation, and so on forms an extremely important complex of ideas in mathematics, there being recognized under the generic idea of ''structure-preserving mappings'' and commonly formalized in the language of homomorphisms, morphisms, or ''arrows'', depending on the operative level of abstraction that's in play.

To consider how a system of logical graphs, taken together as a semiotic domain, might bear an iconic relationship to a system of logical objects that make up our object domain, we will next need to consider what our logical objects are.

To consider how a system of logical graphs, taken together as a semiotic domain, might bear an iconic relationship to a system of logical objects that make up our object domain, we will next need to consider what our logical objects are.

A popular answer, if by popular one means that both Peirce and Frege agreed on it, is to say that our ultimate logical objects are without loss of generality most conveniently referred to as Truth and Falsity.  If nothing else, it serves the end of beginning simply to go along with this thought for a while, and so we can start with an object domain that consists of just two objects or "values", to wit, '''O''' = '''B''' = {false, true}.

A popular answer, if by popular one means that both Peirce and Frege agreed on it, is to say that our ultimate logical objects are without loss of generality most conveniently referred to as Truth and Falsity.  If nothing else, it serves the end of beginning simply to go along with this thought for a while, and so we can start with an object domain that consists of just two ''objects'' or ''values'', to wit, <math>O = \mathbb{B} = \{ \text{false}, \text{true} \}.</math>

Given those two categories of structured individuals, namely, '''O''' = '''B''' = {false, true} and '''S''' = {logical graphs}, the next task is to consider the brands of morphisms from '''S''' to '''O''' that we might reasonably have in mind when we speak of the "arrows of interpretation".

Given those two categories of structured individuals, namely, <math>O = \mathbb{B} = \{ \text{false}, \text{true} \}</math> and <math>S = \{ \text{logical graphs} \},\!</math> the next task is to consider the brands of morphisms from <math>S\!</math> to <math>O\!</math> that we might reasonably have in mind when we speak of the ''arrows of interpretation''.

With the aim of embedding our consideration of logical graphs, as seems most fitting, within Peirce's theory of triadic sign relations, we have declared the first layers of our object, sign, and interpretant domains.  As we often do in formal studies, we've taken the sign and interpretant domains to be the same set, '''S''' = '''I''', calling it the ''semiotic domain'', or, as I see that I've done in some other notes, the ''syntactic domain''.

With the aim of embedding our consideration of logical graphs, as seems most fitting, within Peirce's theory of triadic sign relations, we have declared the first layers of our object, sign, and interpretant domains.  As we often do in formal studies, we've taken the sign and interpretant domains to be the same set, <math>S = I,\!</math> calling it the ''semiotic domain'', or, as I see that I've done in some other notes, the ''syntactic domain''.

Truth and Falsity, the objects that we've so far declared, are recognizable as abstract objects, and like so many other hypostatic abstractions that we use they have their use in moderating between a veritable profusion of more concrete objects and more concrete signs, in ''factoring complexity'' as some people say, despite the fact that some complexities are irreducible in fact.

Truth and Falsity, the objects that we've so far declared, are recognizable as abstract objects, and like so many other hypostatic abstractions that we use they have their use in moderating between a veritable profusion of more concrete objects and more concrete signs, in ''factoring complexity'' as some people say, despite the fact that some complexities are irreducible in fact.