Changes

I turn now to the question of ''partially specified relations'', or ''partially informed relations'' (PIRs), in other words, to the explicit treatment of relations in terms of the information that is logically possessed or actually expressed about them.  There seem to be several ways to approach the concept of an <math>n\!</math>-place PIR and the supporting notion of a partially specified <math>n\!</math>-tuple.  Since the term ''partial relation'' is already implicitly in use for the general class of relations that are not necessarily total on any of their domains, I will coin the term ''pro-relation'', on analogy with ''pronoun'' and ''proposition'', to denote an expression of information about a relation, a contingent indication that, if and when completed, conceivably points to a particular relation.

I turn now to the question of ''partially specified relations'', or ''partially informed relations'' (PIRs), in other words, to the explicit treatment of relations in terms of the information that is logically possessed or actually expressed about them.  There seem to be several ways to approach the concept of an <math>n\!</math>-place PIR and the supporting notion of a partially specified <math>n\!</math>-tuple.  Since the term ''partial relation'' is already implicitly in use for the general class of relations that are not necessarily total on any of their domains, I will coin the term ''pro-relation'', on analogy with ''pronoun'' and ''proposition'', to denote an expression of information about a relation, a contingent indication that, if and when completed, conceivably points to a particular relation.

One way to deal with ''partially informed categories'' of <math>n\!</math>-place relations is to contemplate incomplete relational forms or schemata.  Regarded over the years chiefly in logical and intensional terms, constructs of roughly this type have been variously referred to as ''rhemes'' or ''rhemata'' (Peirce), ''unsaturated relations'' (Frege), or ''frames'' (in current AI parlance).  Expressed in extensional terms, talking about partially informed categories of <math>n\!</math>-place relations is tantamount to admitting elementary relations with missing elements.  The question is not just syntactic &mdash; How to represent an <math>n\!</math>-tuple with empty places? &mdash; but also semantic &mdash; How to make sense of an <math>n\!</math>-tuple with less than <math>n\!</math> elements?

One way to deal with "partially informed categories" (PICs) of n place relations is to contemplate incomplete relational forms or schemata.  Regarded over the years chiefly in logical and intensional terms, constructs of roughly this type have been variously referred to as "rhemes" or "rhemata" (Peirce), "unsaturated relations" (Frege), or "frames" (in current AI literature).  Expressed in extensional terms, talking about PICs of n place relations is tantamount to admitting elementary relations with missing elements.  The question is not just syntactic — How to represent an n tuple with empty places? — but also semantic — How to make sense of an n tuple with less than n elements?

In order to deal with PIRs in a thoroughly consistent fashion, it appears necessary to contemplate elementary relations that present themselves as being "unsaturated" (in Frege's usage of that term), in other words, to consider elements of a presumptive product space that in some sense "wanna be" n tuples or "would be" sequences of a certain length, but are currently missing components in some of their places.

In order to deal with PIRs in a thoroughly consistent fashion, it appears necessary to contemplate elementary relations that present themselves as being ''unsaturated'' (in Frege's sense of that term), in other words, to consider elements of a presumptive product space that in some sense ''wanna&nbsp;be'' <math>n\!</math>-tuples or ''would&nbsp;be'' sequences of a certain length, but are currently missing components in some of their places.

To the extent that the issues of partialization become obvious at the level of symbols and can be dealt with by elementary syntactic means, they initially make their appearance in terms of the various ways that data can go missing.

To the extent that the issues of partialization become obvious at the level of symbols and can be dealt with by elementary syntactic means, they initially make their appearance in terms of the various ways that data can go missing.

The alternate notation "a^b" is provided for the ordered pair <a, b>.  This choice of representation for ordered pairs is especially apt in the case of "concrete indices" (CIs) and "localized addresses" (LAs), where one wants the lead item to serve as a pointed reminder of the itemized content, as in i^Xi = <i, Xi>, and it helps to stress the individuality of each member in the indexed family, as in G = {Gj} = {j^Gj} = {<j, Gj>}.

The alternate notation "a^b" is provided for the ordered pair <a, b>.  This choice of representation for ordered pairs is especially apt in the case of "concrete indices" (CIs) and "localized addresses" (LAs), where one wants the lead item to serve as a pointed reminder of the itemized content, as in i^Xi = <i, Xi>, and it helps to stress the individuality of each member in the indexed family, as in G = {Gj} = {j^Gj} = {<j, Gj>}.