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When the subject matter of discussion is bounded by a universal set <math>X,\!</math> out of which all objects referred to must come, then every PIR to an object can be identified with the name or formula (sign or expression) of a subset <math>A \subseteq X\!</math> or else with that of its selector function <math>f_A : X \to \mathbb{B}.\!</math>  Conceptually, one imagines generating all the objects in <math>X\!</math> and then selecting the ones that satisfy a definitive test for membership in <math>A.\!</math>

When the subject matter of discussion is bounded by a universal set <math>X,\!</math> out of which all objects referred to must come, then every PIR to an object can be identified with the name or formula (sign or expression) of a subset <math>A \subseteq X\!</math> or else with that of its selector function <math>f_A : X \to \mathbb{B}.\!</math>  Conceptually, one imagines generating all the objects in <math>X\!</math> and then selecting the ones that satisfy a definitive test for membership in <math>A.\!</math>

In a realistic computational framework, however, when the domain of interest is given generatively in a genuine sense of the word, that is, defined solely in terms of the primitive elements and operations that are needed to generate it, and when the resource limitations in actual effect make it impractical to enumerate all the possibilities in advance of selecting the adumbrated subset, then the implementation of PIRs becomes a genuine computational problem.

In a realistic computational framework, however, when the domain of interest is given generatively in a genuine sense of the word, that is, defined solely in terms of the primitive elements and operations that are needed to generate it, and when the resource limitations in actual effect make it impractical to enumerate all the possibilities in advance of selecting the adumbrated subset, then the implementation of PIRs becomes a genuine computational problem.

Considered in its application to n place relations, the generic brand of partial specification constitutes a rather limited type of partiality, in that every element conceived as falling under the specified relation, no matter how indistinctly indicated, is still envisioned to maintain its full arity and to remain every bit a complete, though unknown, n tuple.  Still, there is a simple way to extend the concept of generic partiality in a significant fashion, achieving a form of PIRs to relations by making use of "higher order propositions" (HOPs).

Considered in its application to <math>n\!</math>-place relations, the generic brand of partial specification constitutes a rather limited type of partiality, in that every element conceived as falling under the specified relation, no matter how indistinctly indicated, is still envisioned to maintain its full arity and to remain every bit a complete, though unknown, <math>n\!</math>-tuple.  Still, there is a simple way to extend the concept of generic partiality in a significant fashion, achieving a form of PIRs to relations by making use of ''higher order propositions''.

Extending the concept of generic partiality, by iterating the principle on which it is based, leads to higher order propositions about elementary relations, or propositions about relations, as one way to achieve partial specifications of relations, or PIRs to relations.

Extending the concept of generic partiality, by iterating the principle on which it is based, leads to higher order propositions about elementary relations, or propositions about relations, as one way to achieve partial specifications of relations, or PIRs to relations.

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 sign or expression (name or formula) of an n place relation can be interpreted as a proposition about n tuples and thus as a PIR to an elementary relation, then a higher order proposition about n tuples is a proposition about n place relations that can be used to formulate a PIR to an n place relation.

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 sign or expression (name or formula) of an <math>n\!</math>-place relation can be interpreted as a proposition about <math>n\!</math>-tuples and thus as a PIR to an elementary relation, then a higher order proposition about <math>n\!</math>-tuples is a proposition about <math>n\!</math>-place relations that can be used to formulate a PIR to an <math>n\!</math>-place relation.

In order to formalize these ideas, it is helpful to have notational devices for switching back and forth among different ways of exemplifying what is abstractly the same contents of information, in particular, for translating among sets, their logical expressions, and their functional indications.

In order to formalize these ideas, it is helpful to have notational devices for switching back and forth among different ways of exemplifying what is abstractly the same contents of information, in particular, for translating among sets, their logical expressions, and their functional indications.

If S c X is a set contained in a universal set or domain X, then "S#", read as "S sharp" or "S selective", denotes the "selector function" of S, defined as:

If S c X is a set contained in a universal set or domain X, then "S#", read as "S sharp" or "S selective", denotes the "selector function" of S, defined as: