Changes

The poset <math>\operatorname{Pos}(W)\!</math> of interest here is the power set <math>\mathcal{P}(W) = \operatorname{Pow}(W).\!</math>

The poset <math>\operatorname{Pos}(W)\!</math> of interest here is the power set <math>\mathcal{P}(W) = \operatorname{Pow}(W).\!</math>

The elements of these posets are abstractly regarded as ''properties'' or ''propositions'' that apply to the elements of <math>W.\!</math> These properties and propositions are independently given entities.  In other words, they are primitive elements in their own right, and cannot in general be defined in terms of points, but they exist in relation to these points, and their extensions can be represented as sets of points.

The elements of these posets are abstractly regarded as "properties" or "propositions" that apply to the elements of W.  These properties and propositions are independently given entities.  In other words, they are primitive elements in their own right, and cannot in general be defined in terms of points, but they exist in relation to these points, and their extensions can be represented as sets of points.

For a variety of foundational reasons that I do not fully understand, perhaps most of all because theoretically given structures have their real foundations outside the realm of theory, in empirically given structures, it is best to regard points, properties, and propositions as equally primitive elements, related to each other but not defined in terms of each other, analogous to the undefined elements of a geometry.

[Variant] For a variety of foundational reasons that I do not fully understand, perhaps most of all because theoretically given structures have their real foundations outside the realm of theory, in empirically given structures, it is best to regard points, properties, and propositions as equally primitive elements, related to each other but not defined in terms of each other, analogous to the undefined elements of a geometry.

There is a foundational issue arising in this context that I do not pretend to fully understand and cannot attempt to finally dispatch.  What I do understand I will try to express in terms of an aesthetic principle:  On balance, it seems best to regard extensional elements and intensional features as independently given entities.  This involves treating points and properties as fundamental realities in their own rights, placing them on an equal basis with each other, and seeking their relation to each other, but not trying to reduce one to the other.

[Variant] There is a foundational issue arising in this context that I do not pretend to fully understand and cannot attempt to finally dispatch.  What I do understand I will try to express in terms of an aesthetic principle:  On balance, it seems best to regard extensional elements and intensional features as independently given entities.  This involves treating points and properties as fundamental realities in their own rights, placing them on an equal basis with each other, and seeking their relation to each other, but not trying to reduce one to the other.

The discussion is now specialized to consider the IRs of the sign relations A and B, their denotative projections as the digraphs Den (A) and Den (B), and their connotative projections as the digraphs Con (A) and Con (B).  In doing this I take up two different strategies of representation:

The discussion is now specialized to consider the IRs of the sign relations <math>L(\text{A}\!</math> and <math>L(\text{B},\!</math> their denotative projections as the digraphs <math>\operatorname{Den}(L_\text{A})\!</math> and <math>\operatorname{Den}(L_\text{B}),\!</math> and their connotative projections as the digraphs <math>\operatorname{Con}(L_\text{A})\!</math> and <math>\operatorname{Con}(L_\text{B}).\!</math> In doing this I take up two different strategies of representation:

1. The first strategy is called the "literal coding", because it sticks to obvious features of each syntactic element to contrive its code, or the "O(n) coding", because it uses a number on the order of n logical features to represent a domain of n elements.

# The first strategy is called the ''literal coding'', because it sticks to obvious features of each syntactic element to contrive its code, or the ''<math>\mathcal{O}(n)\!</math> coding'', because it uses a number on the order of <math>n\!</math> logical features to represent a domain of <math>n\!</math> elements.

# The second strategy is called the ''analytic coding'', because it attends to the nuances of each sign's interpretation to fashion its code, or the ''<math>\log (n)\!</math> coding'', because it uses roughly <math>\log_2 (n)\!</math> binary features to represent a domain of <math>n\!</math> elements.

Fragments

Fragments