Changes

[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.

[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 <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:

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:

# 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 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.

# 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.

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In the formalized examples of IRs to be presented in this work, I will keep to the level of logical reasoning that is usually referred to as "propositional calculus" or "sentential logic" (SL).

<p align="center">'''Fragments'''</p>

The constrast between ERs and IRs is strongly correlated with another dimension of interest in the study of inquiry, namely, the tension between empirical and rational modes of inquiry.

In the formalized examples of IRs to be presented in this work, I will keep to the level of logical reasoning that is usually referred to as ''propositional calculus'' or ''sentential logic''.

This section begins the explicit discussion of ERs by taking a second look at the sign relations A and B.  Since the form of these examples no longer presents any novelty, this second presentation of A and B provides a first opportunity to introduce some new material.  In the process of reviewing this material, it is useful to anticipate a number of incidental issues that are on the point of becoming critical, and to begin introducing the generic types of technical devices that are needed to deal with them.

The contrast between ERs and IRs is strongly correlated with another dimension of interest in the study of inquiry, namely, the tension between empirical and rational modes of inquiry.

Therefore, the easiest way to begin an explicit treatment of ERs is by recollecting the Tables of the sign relations A and B and by finishing the corresponding Tables of their dyadic components.  Since the form of the sign relations A and B no longer presents any novelty, I can use the second presentation of these examples as a first opportunity to examine a selection of their finer points, previously overlooked.

 

Therefore, the easiest way to begin an explicit treatment of ERs is by recollecting the Tables of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> and by finishing the corresponding Tables of their dyadic components.  Since the form of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> no longer presents any novelty, I can use the second presentation of these examples as a first opportunity to examine a selection of their finer points, previously overlooked.

Starting from this standpoint, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out their missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light.

Starting from this standpoint, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out their missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light.

Because of the perfect parallelism that the literal coding contrives between individual signs and grammatical categories, this arrangement illustrates not so much a code transformation as a re interpretation of the original signs under different headings.

Because of the perfect parallelism that the literal coding contrives between individual signs and grammatical categories, this arrangement illustrates not so much a code transformation as a re-interpretation of the original signs under different headings.

===6.24. Literal Intensional Representations===

===6.24. Literal Intensional Representations===