ONT Relations In General
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RIG. Relations In General
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RIG. Note 1
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In a realistic computational framework, where incomplete and inconsistent
information is the rule, it is necessary to work with genera of relations
that are increasingly relaxed in their constraining characters but still
preserve a measure of analogy with the fundamental species of relations
that are found to be prevalent in perfect information contexts.
In the present application, the kinds of relations of primary interest are
functions, equivalence relations, and other species of relations that are
defined by their axiomatic properties. Thus, the information-theoretic
generalizations of these structures lead to partially defined functions
and partially constrained versions of these specially defined classes
of relations.
The purpose of this Subsection is to outline the kinds of generalized functions
and other families of relations that are needed to extend the discussion of the
present example. In this connection, to frame the problem in concrete terms,
I need to adapt the "equivalence class" notation for two generalizations of
equivalence relations, to be defined below. But first, a number of broader
issues need to be treated.
Generally speaking, one is free to interpret references to
generalized objects either (1) as indications of partially
formed analogues of ordinary objects, or (2) as partially
informed descriptions of the corresponding types of objects.
I refer to these alternatives as (O) the "object-theoretic"
and (S) the "sign-theoretic" options, respectively.
The object-theoretic way of reading partial signs assumes that
vague and general references nevertheless have their objective
denotations, but merely to vague and general objects.
The sign-theoretic way of reading partial signs ascribes the
partialities of information to the characters of the signs,
the expressions, and the texts that are doing the denoting.
In most of the cases that arise in casual discussion the choice
between these conventions is purely stylistic. However, in many
of the more intricate situations that arise in formal discussion,
the object-theoretic choice frequently fails utterly, and whenever
the utmost care is required it will usually be a due attention to
the partialities of signs that saves the day.
Jon Awbrey
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