Changes

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 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 higher order 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 number of additional definitions are relevant to sign relations whose connotative components constitute equivalence relations, if only in part.

A number of additional definitions are relevant to sign relations whose connotative components constitute equivalence relations, if only in part.

A ''dyadic relation on a single set'' (DROSS) is a non-empty set of points plus a set of ordered pairs on these points.  Until further notice, any reference to a ''dyadic relation'' is intended to be taken in this sense, in other words, as a reference to a DROSS.  In a typical notation, the dyadic relation <math>G = (X, G) = (G^{(1)}, G^{(2)})\!</math> is specified by giving the set of points <math>X = G^{(1)}\!</math> and the set of ordered pairs <math>G = G^{(2)} \subseteq X \times X</math> that go together to define the relation.  In contexts where the set of points is understood, it is customary to call the whole relation <math>G\!</math> by the name of the set <math>G.\!</math>

A ''dyadic relation on a single set'' (DROSS) is a non-empty set of points plus a set of ordered pairs on these points.  Until further notice, any reference to a ''dyadic relation'' is intended to be taken in this sense, in other words, as a reference to a DROSS.  In a typical notation, the dyadic relation <math>\mathfrak{G} = (X, G) = (G^{(1)}, G^{(2)})</math> is specified by giving the set of points <math>X = G^{(1)}\!</math> and the set of ordered pairs <math>G = G^{(2)} \subseteq X \times X</math> that go together to define the relation.  In contexts where the set of points is understood, it is customary to call the whole relation <math>\mathfrak{G}</math> by the name of the set <math>G.\!</math>

A ''subrelation'' of a dyadic relation <math>G = (X, G) = (G^{(1)}, G^{(2)})\!</math> is a dyadic relation <math>H = (Y, H) = (H^{(1)}, H^{(2)})\!</math> that has all of its points and pairs in <math>G,\!</math> more precisely, that has all of its points <math>Y \subseteq X</math> and all of its pairs <math>H \subseteq G.</math>

A ''subrelation'' of a dyadic relation <math>\mathfrak{G} = (X, G) = (G^{(1)}, G^{(2)})</math> is a dyadic relation <math>\mathfrak{H} = (Y, H) = (H^{(1)}, H^{(2)})</math> that has all of its points and pairs in <math>\mathfrak{G}</math> more precisely, that has all of its points <math>Y \subseteq X</math> and all of its pairs <math>H \subseteq G.</math>

The ''induced subrelation on a subset'' (ISOS), taken with respect to the dyadic relation <math>G \subseteq X \times X</math> and the subset <math>Y \subseteq X,</math> is the maximal subrelation of <math>G\!</math> whose points belong to <math>Y.\!</math>  In other words, it is the dyadic relation on <math>Y\!</math> whose extension contains all of the pairs of <math>Y \times Y</math> that appear in <math>G.\!</math>  Since the construction of an ISOS is uniquely determined by the data of <math>G\!</math> and <math>Y,\!</math> it can be represented as a function of these arguments, as in the notation <math>\operatorname{ISOS} (G, Y),</math> which can be denoted more briefly as <math>G_Y.\!</math>.  Using the symbol <math>\cap</math> to indicate the intersection of a pair of sets, the construction of <math>G_Y = \operatorname{ISOS} (G, Y)</math> can be defined as follows:

The ''induced subrelation on a subset'' (ISOS), taken with respect to the dyadic relation <math>G \subseteq X \times X</math> and the subset <math>Y \subseteq X,</math> is the maximal subrelation of <math>G\!</math> whose points belong to <math>Y.\!</math>  In other words, it is the dyadic relation on <math>Y\!</math> whose extension contains all of the pairs of <math>Y \times Y</math> that appear in <math>G.\!</math>  Since the construction of an ISOS is uniquely determined by the data of <math>G\!</math> and <math>Y,\!</math> it can be represented as a function of these arguments, as in the notation <math>\operatorname{ISOS} (G, Y),</math> which can be denoted more briefly as <math>\mathfrak{G}_Y.\!</math>.  Using the symbol <math>\cap</math> to indicate the intersection of a pair of sets, the construction of <math>\mathfrak{G}_Y = \operatorname{ISOS} (G, Y)</math> can be defined as follows:

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