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, 21:36, 20 November 2012→6.33. Sign Relational Complexes: move fragments to talk page
===6.33. Sign Relational Complexes===
===6.33. Sign Relational Complexes===
I would like to record here, in what is topically the appropriate place, notice of a number of open questions that will have to be addressed if anyone desires to make a consistent calculus out of this link notation. Perhaps it is only because the franker forms of liaison involved in the couple <math>a \widehat{~} b\!</math> are more subject to the vagaries of syntactic elision than the corresponding bindings of the anglish ligature <math>(a, b),\!</math> but for some reason or other the circumflex character of these diacritical notices are much more liable to suggest various forms of elaboration, including higher order generalizations and information-theoretic partializations of the very idea of <math>n\!</math>-tuples and sequences.
One way to deal with the problems of partial information …
'''Relational Complex?'''
{| align="center" cellspacing="8" width="90%"
| <math>L ~=~ L^{(1)} \cup \ldots \cup L^{(k)}\!</math>
|}
'''Sign Relational Complex?'''
{| align="center" cellspacing="8" width="90%"
| <math>L ~=~ L^{(1)} \cup L^{(2)} \cup L^{(3)}\!</math>
|}
Linkages can be chained together to form sequences of indications or <math>n\!</math>-tuples, without worrying too much about the order of collecting terms in the corresponding angle brackets.
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{matrix}
a \widehat{~} b \widehat{~} c
& = &
(a, b, c)
& = &
(a, (b, c))
& = &
((a, b), c).
\end{matrix}</math>
|}
These equivalences depend on the existence of natural isomorphisms between different ways of constructing <math>n\!</math>-place product spaces, that is, on the associativity of pairwise products, a not altogether trivial result (Mac Lane, CatWorkMath, ch. 7).
Higher Order Indications (HOIs)?
{| align="center" cellspacing="8" width="90%"
|
<math>\begin{matrix}
\widehat{~} x & = & (~, x) & ?
\\[4pt]
x \widehat{~} & = & (x, ~) & ?
\\[4pt]
\widehat{~}~\widehat{~} x & = & (~, (~, x)) & ?
\\[4pt]
x \widehat{~}~\widehat{~} & = & ((x, ~), ~) & ?
\end{matrix}</math>
|}
In talking about properties and classes of relations, one would like to refer to ''all relations'' as forming a topic of potential discussion, and then take it as a background for contemplating …
In talking and thinking, often in just that order, about properties and classes of relations, one is always invoking, explicitly or implicitly, a particular background, a limited field of experience, actual or potential, against which each object of ''discussion and thought'' figures. Expressing the matter in the idiom of logical inquiry, one brings to mind a preconceived universe of discourse <math>U\!</math> or a restricted domain of discussion <math>X,\!</math> and then contemplates …
This direction of generalization expands the scope of PIRs by means of an analogical extension, and can be charted in the following manner. If the name of a relation can be taken as a PIR to elementary relations, that is, if the formula of an <math>n\!</math>-place relation can be interpreted as a proposition about <math>n\!</math>-tuples, then a PIR to relations themselves can be formulated as a proposition about relations and thus as a HOPE about elementary relations or <math>n\!</math>-tuples.
One way to extend the generic brand of partiality among relations in a non-trivial direction can be charted as follows. If the name or formula of a relation is a PIR to elementary relations, that is, if a sign or expression of an <math>n\!</math>-place relation is interpreted as a proposition about <math>n\!</math>-tuples, then a PIR to relations …
==Scrap Area==
==Scrap Area==
