Changes

Several important facts about the class of higher order sign relations in general are illustrated by these examples.  First, the notations appearing in the object columns of <math>\operatorname{HI}^1 L(A)\!</math> and <math>\operatorname{HI}^1 L(B)\!</math> are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them.  The sign relations <math>L(A)\!</math> and <math>L(B),\!</math> as extended by the transactions of <math>\operatorname{HI}^1 L(A)\!</math> and <math>\operatorname{HI}^1 L(B),\!</math> respectively, are still restricted to their original syntactic domain <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}.\!</math>  This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as higher order.  Indeed, the sign relations <math>\operatorname{HI}^1 L(A)\!</math> and <math>\operatorname{HI}^1 L(B)\!</math> are not very discriminating in their descriptions of the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.

Several important facts about the class of higher order sign relations in general are illustrated by these examples.  First, the notations appearing in the object columns of <math>\operatorname{HI}^1 L(A)\!</math> and <math>\operatorname{HI}^1 L(B)\!</math> are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them.  The sign relations <math>L(A)\!</math> and <math>L(B),\!</math> as extended by the transactions of <math>\operatorname{HI}^1 L(A)\!</math> and <math>\operatorname{HI}^1 L(B),\!</math> respectively, are still restricted to their original syntactic domain <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}.\!</math>  This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as higher order.  Indeed, the sign relations <math>\operatorname{HI}^1 L(A)\!</math> and <math>\operatorname{HI}^1 L(B)\!</math> are not very discriminating in their descriptions of the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.

In practice, it does an interpreter little good to have the higher import signs for referring to triples of objects, signs, and interpretants if it does not also have the higher ascent signs for referring to each triple's syntactic portions.  Consequently, the higher order sign relations that one is likely to observe in practice are typically a mixed bag, having both higher ascent and higher import sections.  Moreover, the ambiguity involved in having signs that refer equivocally to simple world elements and also to complex structures formed from these ingredients would most likely be resolved by drawing additional information from context and fashioning more distinctive signs.

In practice, it does an interpreter little good to have the HI signs for referring to triples of objects, signs, and interpretants if it does not also have the HA signs for referring to each triple's syntactic portions.  Consequently, the HO sign relations that one is likely to observe in practice are typically a mixed bag, having both HA and HI sections.  Moreover, the ambiguity involved in having signs that refer equivocally to simple world elements and also to complex structures formed from these ingredients would most likely be resolved by drawing additional information from context and fashioning more distinctive signs.

These reflections raise the issue of how articulate a HO sign relation is in its depiction of its object signs and object sign relations.  For now, I can do little more than note the dimension of articulation as a feature of interest, contributing to the scale of aesthetic utility that makes some sign relations better than others for a given purpose, and serving as a drive that motivates their continuing development.

These reflections raise the issue of how articulate a higher order sign relation is in its depiction of its object signs and its object sign relations.  For now, I can do little more than note the dimension of articulation as a feature of interest, contributing to the scale of aesthetic utility that makes some sign relations better than others for a given purpose, and serving as a drive that motivates their continuing development.

The technique illustrated here represents a general strategy, one that can be exploited to derive certain benefits of set theory without having to pay the overhead that is needed to maintain sets as abstract objects.  Using an identified type of a sign as a canonical form that can refer indifferently to all the members of a set is a pragmatic way of making plural reference to the members of a set without invoking the set itself as an abstract object.  Of course, it is not that one can get something for nothing by these means.  One is merely banking on one's recurring investment in the setting of a certain sign relation, a particular set of elementary transactions that is taken for granted as already funded.

The technique illustrated here represents a general strategy, one that can be exploited to derive certain benefits of set theory without having to pay the overhead that is needed to maintain sets as abstract objects.  Using an identified type of a sign as a canonical form that can refer indifferently to all the members of a set is a pragmatic way of making plural reference to the members of a set without invoking the set itself as an abstract object.  Of course, it is not that one can get something for nothing by these means.  One is merely banking on one's recurring investment in the setting of a certain sign relation, a particular set of elementary transactions that is taken for granted as already funded.

As a rule, it is desirable for the grammatical system that one uses to construct and interpret HO signs, that is, signs for referring to signs as objects, to mesh in a comfortable fashion with the overall pragmatic system that one uses to assign syntactic codes to objects in general.  For future reference, I call this requirement the problem of creating a "conformally reflective extension" (CRE) for a given sign relation.  A good way to think about this task is to imagine oneself beginning with a sign relation R c OxSxI, and to consider its denotative component DenR = ROS c OxS.  Typically one has a "naming function", call it "Nom", that maps objects into signs:

As a rule, it is desirable for the grammatical system that one uses to construct and interpret higher order signs, that is, signs for referring to signs as objects, to mesh in a comfortable fashion with the overall pragmatic system that one uses to assign syntactic codes to objects in general.  For future reference, I call this requirement the problem of creating a ''conformally reflective extension'' (CRE) for a given sign relation.  A good way to think about this task is to imagine oneself beginning with a sign relation <math>L \subseteq O \times S \times I,\!</math> and to consider its denotative component <math>\operatorname{Den}_L = L_{OS} \subseteq O \times S.\!</math> Typically one has a ''naming function'', say <math>\operatorname{Nom},\!</math> that maps objects into signs:

Nom c  DenR  c  OxS,  such that   Nom : O -> S.

| <math>\operatorname{Nom} \subseteq \operatorname{Den}_L \subseteq O \times S ~\text{such that}~ \operatorname{Nom} : O \to S.\!</math>

Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects.  This is the reason for creating HO signs, which are just signs for making objects out of signs.  One effect of progressive reflection is to extend the initial Nom through a succession of new naming functions Nom', Nom'', and so on, assigning unique names to larger allotments of the original and subsequent signs.  With respect to the difficulties of construction, the "hard" core or the "adamant" part of creating an extended naming function is in the initial portion Nom that maps objects of the "external world" into signs in the "internal world".  The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and "easy", perhaps on account of the "nominal" nature of signs themselves.

Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects.  This is the reason for creating HO signs, which are just signs for making objects out of signs.  One effect of progressive reflection is to extend the initial Nom through a succession of new naming functions Nom', Nom'', and so on, assigning unique names to larger allotments of the original and subsequent signs.  With respect to the difficulties of construction, the "hard" core or the "adamant" part of creating an extended naming function is in the initial portion Nom that maps objects of the "external world" into signs in the "internal world".  The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and "easy", perhaps on account of the "nominal" nature of signs themselves.