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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 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" (DAT) figures.  Expressing the matter in the idiom of logical inquiry, one brings to mind a preconceived universe of discourse U or a restricted domain of discussion X, and then contemplates ...

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 &hellip;

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 n place relation can be interpreted as a proposition about n tuples, then a PIR to relations themselves can be formulated as a proposition about relations and thus as a HOPE about elementary relations or n tuples.

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 an n place relation is interpreted as a proposition about n tuples, then a PIR to relations ...

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 &hellip;

===6.34. Set-Theoretic Constructions===

===6.34. Set-Theoretic Constructions===