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===6.9. Higher Order Sign Relations : Introduction===

===6.9. Higher Order Sign Relations : Introduction===

When interpreters reflect on their own use of signs they require an appropriate technical language in which to pursue these reflections.  For this they need signs that refer to sign relations, signs that refer to the elements and components of sign relations, and signs that refer to the properties and classes of sign relations.  All of these additional signs can be placed under the description of '''higher order signs''', and the extended sign relations that involve them can be referred to as '''higher order sign relations'''.

When interpreters reflect on their own use of signs they require an appropriate technical language in which to pursue these reflections.  For this they need signs that refer to sign relations, signs that refer to the elements and components of sign relations, and signs that refer to the properties and classes of sign relations.  All of these additional signs can be placed under the description of '''higher order signs''', and the extended sign relations that involve them can be referred to as '''higher order sign relations'''.

Whether any forms of observation and reflection can be conducted outside the medium of language is not a question I can address here.  It is apparent as a practical matter, however, that stable and sharable forms of knowledge depend on the availability of an adequate language.  Accordingly, there is a relationship of practical necessity that binds the conditions for reflective interpretation to the possibility of extending sign relations through higher orders.  At minimum, in addition to the signs of objects originally given, there must be signs of signs and signs of their interpretants, and each of these higher order signs requires a further occurrence of higher order interpretants to continue and complete its meaning within a higher order sign relation.  In general, higher order signs can arise in a number of independent fashions, but one of the most common derivations is through the specialized devices of quotation.  This establishes a contingent relation between reflection and quotation.

Whether any forms of observation and reflection can be conducted outside the medium of language is not a question I can address here.  It is apparent as a practical matter, however, that stable and sharable forms of knowledge depend on the availability of an adequate language.  Accordingly, there is a relationship of practical necessity that binds the conditions for reflective interpretation to the possibility of extending sign relations through higher orders.  At minimum, in addition to the signs of objects originally given, there must be signs of signs and signs of their interpretants, and each of these higher order signs requires a further occurrence of higher order interpretants to continue and complete its meaning within a higher order sign relation.  In general, higher order signs can arise in a number of independent fashions, but one of the most common derivations is through the specialized devices of quotation.  This establishes a contingent relation between reflection and quotation.

Here are the species of higher order signs that can be used to discuss the structural constituents and intensional genera of sign relations:

Here are the species of higher order signs that can be used to discuss the structural constituents and intensional genera of sign relations:

# Signs that denote signs, that is, signs whose objects are signs in the same sign relation, are called '''higher ascent''' (HA) signs.

# Signs that denote signs, that is, signs whose objects are signs in the same sign relation, are called '''higher ascent''' (HA) signs.

# Signs that denote dyadic components of elementary sign relations, that is, signs whose objects are elemental pairs or dyadic actions having any one of the forms <math>(o, s),\!</math> <math>(o, i),\!</math> or <math>(s, i),\!</math> are called '''higher employ''' (HE) signs.

# Signs that denote dyadic components of elementary sign relations, that is, signs whose objects are elemental pairs or dyadic actions having any one of the forms <math>(o, s),\!</math> <math>(o, i),\!</math> or <math>(s, i),\!</math> are called '''higher&nbsp;employ''' (HE) signs.

# Signs that denote elementary sign relations, that is, signs whose objects are elemental triples or triadic transactions having the form <math>(o, s, i),\!</math> are called '''higher import''' (HI) signs.

# Signs that denote elementary sign relations, that is, signs whose objects are elemental triples or triadic transactions having the form <math>(o, s, i),\!</math> are called '''higher&nbsp;import''' (HI) signs.

# Signs that denote sign relations, that is, signs whose objects are themselves sign relations, are called '''higher upshot''' (HU) signs.

# Signs that denote sign relations, that is, signs whose objects are themselves sign relations, are called '''higher&nbsp;upshot''' (HU) signs.

# Signs that denote intensional genera of sign relations, that is, signs whose objects are properties or classes of sign relations, are called '''higher yclept''' (HY) signs.

# Signs that denote intensional genera of sign relations, that is, signs whose objects are properties or classes of sign relations, are called '''higher&nbsp;yclept''' (HY) signs.

Analogous species of higher order signs can be used to discuss the structural constituents and intensional genera of arbitrary relations.  In order to describe them, it is necessary to introduce a few extra notions from the theory of relations.  This, in turn, occasions a recurring difficulty with the exposition that needs to be noted at this point.

Analogous species of HO signs can be used to discuss the structural constituents and intensional genera of arbitrary relations.  In order to describe them, it is necessary to introduce a few extra notions from the theory of relations.  This, in turn, occasions a recurring difficulty with the exposition that needs to be noted at this point.

The subject matters of relations, types, and functions enjoy a form of recursive involvement with one another that makes it difficult to know where to get on and where to get off the circle of explanation.  As I currently understand their relationship, it can be approached in the following order:

The subject matters of relations, types, and functions enjoy a form of recursive involvement with one another that makes it difficult to know where to get on and where to get off the circle of explanation.  As I currently understand their relationship, it can be approached in the following order:

: Relations have types.

* '''''Relations have types.'''''

: Types are functions.

* '''''Types are functions.'''''

: Functions are relations.

* '''''Functions are relations.'''''

In this setting, a "type" is a function from the "places" of a relation, that is, from the index set of its components, to a collection of sets that are called the "domains" of the relation.

In this setting, a ''type'' is a function from the ''places'' of a relation, that is, from the index set of its components, to a collection of sets that are called the ''domains'' of the relation.

When a relation is given an extensional representation as a collection of elements, these elements are called its "elementary relations" or its "individual transactions".  The "type" of an elementary relation is a function from an index set whose elements are called the "places" of the relation to a set of sets whose elements are called the "domains" of the relation.  The "arity" or "adicity" of an elementary relation is the cardinality of this index set.  In general, these cardinalities can be ranked as finite, denumerably infinite, or non denumerable.

When a relation is given an extensional representation as a collection of elements, these elements are called its ''elementary relations'' or its ''individual transactions''.  The ''type'' of an elementary relation is a function from an index set whose elements are called the ''places'' of the relation to a set of sets whose elements are called the ''domains'' of the relation.  The ''arity'' or ''adicity'' of an elementary relation is the cardinality of this index set.  In general, these cardinalities can be ranked as finite, denumerably infinite, or non-denumerable.

Elementary relations are also called the "effects" of a relation, more specifically, as its "maximal" or "total" effects, which are the kinds of effects that one usually intends in the absence of further qualification.  More generally, a "component relation" or a "partial transaction" of a relation is a projection of one of its elementary relations on a subset of its places.   

Elementary relations are also called the ''effects'' of a relation, more specifically, as its ''maximal'' or ''total'' effects, which are the kinds of effects that one usually intends in the absence of further qualification.  More generally, a ''component relation'' or a ''partial transaction'' of a relation is a projection of one of its elementary relations on a subset of its places.   

A "homogeneous relation" is a relation, all of whose elementary relations have the same type.  In this case, the type and the arity are properties that are defined for the relation itself.  The rest of this discussion is specialized to homogeneous relations.

A ''homogeneous relation'' is a relation, all of whose elementary relations have the same type.  In this case, the type and the arity are properties that are defined for the relation itself.  The rest of this discussion is specialized to homogeneous relations.

When the arity of a relation is a finite number n, then the relation is called an "n place relation".  In this case, the elementary relations are just the n tuples belonging to the relation.  In the finite case, for example, a "non trivial properly partial transaction" is a k tuple extracted from an n tuple of the relation, where 1 < k < n.  The first element of an elementary relation is called its "object" or "prelate", while the remaining elements are called its "correlates".

When the arity of a relation is a finite number n, then the relation is called an "n place relation".  In this case, the elementary relations are just the n tuples belonging to the relation.  In the finite case, for example, a "non trivial properly partial transaction" is a k tuple extracted from an n tuple of the relation, where 1 < k < n.  The first element of an elementary relation is called its "object" or "prelate", while the remaining elements are called its "correlates".