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=====1.3.10.1.  Preliminary Notions=====

=====1.3.10.1.  Preliminary Notions=====

The discussion in this subsection proceeds by recalling a series of basic definitions, refining them to deal with more specialized situations, and refitting them as necessary to cover larger families of sign relations.

The present phase of discussion proceeds by recalling a series of basic definitions, refining them to deal with more specialized situations, and refitting them as necessary to cover larger families of sign relations.

In this discussion the word "semantic" is being used as a generic adjective to describe anything concerned with or related to meaning, whether denotative, connotative, or pragmatic, and without regard to how these different aspects of meaning are correlated with each other.  The word "semiotic" is being used, more specifically, to indicate the connotative relationships that exist between signs, in particular, to stress the aspects of process and of potential for progress that are involved in the transitions between signs and their interpretants.  Whenever the focus fails to be clear from the context of discussion, the modifiers "denotative" and "referential" are available to pinpoint the relationships that exist between signs and their objects.  Finally, there is a common usage of the term "pragmatic" to highlight aspects of meaning that have to do with the context of use and the language user, but I reserve the use of this term to refer to the interpreter as an agent with a purpose, and thus to imply that all three aspects of sign relations are involved in the subject under discussion.

In this discussion the word ''semantic'' is being used as a generic adjective to describe anything concerned with or related to meaning, whether denotative, connotative, or pragmatic, and without regard to how these different aspects of meaning are correlated with each other.  The word ''semiotic'' is being used, more specifically, to indicate the connotative relationships that exist between signs, in particular, to stress the aspects of process and of potential for progress that are involved in the transitions between signs and their interpretants.  Whenever the focus fails to be clear from the context of discussion, the modifiers ''denotative'' and ''referential'' are available to pinpoint the relationships that exist between signs and their objects.  Finally, there is a common usage of the term ''pragmatic'' to highlight aspects of meaning that have to do with the context of use and the language user, but I reserve the use of this term to refer to the interpreter as an agent with a purpose, and thus to imply that all three aspects of sign relations are involved in the subject under discussion.

Recall the definitions of "semiotic equivalence classes" (SEC's), "semiotic partitions" (SEP's), "semiotic equations" (SEQ's), and "semiotic equivalence relations" (SER's), as in Subsection 1.3.4.3.

Recall the definitions of ''semiotic equivalence classes'' (SECs), ''semiotic partitions'' (SEPs), ''semiotic equations'' (SEQs), and ''semiotic equivalence relations'' (SERs), as in Segment 1.3.4.3.

The discussion up to this point is partial to examples of sign relations that enjoy especially nice properties, in particular, whose connotative components form equivalence relations and whose denotative components conform to these equivalences, in the sense that all of the signs in a single equivalence class always denote one and the same object.  By way of liberalizing this discussion to more general cases of sign relations, this subsection develops a number of additional concepts for describing the internal relations of sign relations and makes a set of definitions that do not take the aforementioned features for granted.

The discussion up to this point is partial to examples of sign relations that enjoy especially nice properties, in particular, whose connotative components form equivalence relations and whose denotative components conform to these equivalences, in the sense that all of the signs in a single equivalence class always denote one and the same object.  By way of liberalizing this discussion to more general cases of sign relations, this subsection develops a number of additional concepts for describing the internal relations of sign relations and makes a set of definitions that do not take the aforementioned features for granted.

The complete sign relation involved in a situation encompasses all the things that one thinks about and all the thoughts that one thinks about them while engaged in that situation, in other words, all the signs and ideas that flit through one's mind in relation to a domain of objects.  Only a rarefied sample of this complete sign relation is bound to avail itself to reflective awareness, still less of it is likely to inspire a common interest in the community of inquiry at large, and only bits and pieces of it can be expected to suit themselves to a formal analysis.  In view of these considerations, it is useful to have a general idea of the "sampling relation" that an investigator, oneself in particular, is likely to form between two sign relations:  (1) the whole sign relation that one intends to study, and (2) the selective portion of it that one is able to pin down for a formal examination.

The complete sign relation involved in a situation encompasses all the things that one thinks about and all the thoughts that one thinks about them while engaged in that situation, in other words, all the signs and ideas that flit through one's mind in relation to a domain of objects.  Only a rarefied sample of this complete sign relation is bound to avail itself to reflective awareness, still less of it is likely to inspire a common interest in the community of inquiry at large, and only bits and pieces of it can be expected to suit themselves to a formal analysis.  In view of these considerations, it is useful to have a general idea of the ''sampling relation'' that an investigator, oneself in particular, is likely to form between two sign relations:  (1) the whole sign relation that one intends to study, and (2) the selective portion of it that one is able to pin down for a formal examination.

It is important to realize that a "sampling relation", to express it roughly, is a special case of a sign relation.  Aside from acting on sign relations and creating an association between sign relations, a sampling relation is also involved in a larger sign relation, at least, it can be subsumed within a general order of sign relations that allows sign relations themselves to be taken as the objects, the signs, and the interpretants of what can be called a "higher order" (HO) sign relation.  Considered with respect to its full potential, its use, and its purpose, a sampling relation does not fall outside the closure of sign relations.  To be precise, a sampling relation falls within the denotative component of a HO sign relation, since the sign relation sampled is the object of study and the sample is taken as a sign of it.

It is important to realize that a ''sampling relation'', to express it roughly, is a special case of a sign relation.  Aside from acting on sign relations and creating an association between sign relations, a sampling relation is also involved in a larger sign relation, at least, it can be subsumed within a general order of sign relations that allows sign relations themselves to be taken as the objects, the signs, and the interpretants of what can be called a ''higher order sign relation''.  Considered with respect to its full potential, its use, and its purpose, a sampling relation does not fall outside the closure of sign relations.  To be precise, a sampling relation falls within the denotative component of a HO sign relation, since the sign relation sampled is the object of study and the sample is taken as a sign of it.

With respect to the general variety of sampling relations there are a number of specific conceptions that are likely to be useful in this study, a few of which can now be discussed.

With respect to the general variety of sampling relations there are a number of specific conceptions that are likely to be useful in this study, a few of which can now be discussed.

A "bit" of a sign relation is defined to be any subset of its extension, that is, an arbitrary selection from its set of ordered triples.

A ''bit'' of a sign relation is defined to be any subset of its extension, that is, an arbitrary selection from its set of ordered triples.

Described in relation to sampling relations, a bit of a sign relation is just the most arbitrary possible sample of it, and thus its occurring to mind implies the most general form of sampling relation to be in effect.  In essence, it is just as if a bit of a sign relation, by virtue of its appearing in evidence, can always be interpreted as a bit of evidence that some sort of sampling relation is being applied.

Described in relation to sampling relations, a bit of a sign relation is just the most arbitrary possible sample of it, and thus its occurring to mind implies the most general form of sampling relation to be in effect.  In essence, it is just as if a bit of a sign relation, by virtue of its appearing in evidence, can always be interpreted as a bit of evidence that some sort of sampling relation is being applied.