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===6.31. Generic Orders of Relations===
===6.31. Relations in General===
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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 defined by 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 Section 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 square bracket 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 as indications of partially formed analogues or as partially informed descriptions of their corresponding species. I refer to these alternatives as the "object theoretic" and the "sign theoretic" options, respectively. The first interpretation assumes that vague and general references still have denotations, merely to vague and general objects. The second interpretation ascribes the partialities of information to the characters of the signs and expressions that are doing the denoting. In most 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 choice often fails utterly, and whenever the utmost care is required it will usually be the attention to signs that saves the day.
In order to speak of generalized orders of relations I need to outline the dimensions of variation along which I intend the characters of already familiar orders of relations to be broadened. Generally speaking, the taxonomic features of n place relations that I wish to liberalize can be read off from their "local incidence properties" (LIP's).
Definition. A "local incidence property" of an n place relation R is one that is based on the following sorts of data. Suppose R c X1x...xXn. Pick an element x in one of the domains Xi of R. Let "R&x@i" denote a subset of R called the "flag of R with x at i", or the "x@i flag of R". The "local flag" R&x@i c R is defined as follows:
R&x@i = {<x1, ... , xi, ... , xn> C R : xi = x}.
Any property P of R&x@i constitutes a "local incidence property" of R with reference to the locus "x at i".
Definition. An n place relation R c X1x...xXn is called "P regular at i" iff every flag of R with x at i is P, letting x range over the domain Xi, in symbols, iff P(R&x@i) is true for all x C Xi.
Of particular interest are the local incidence properties of relations that can be calculated from the cardinalities of their local flags, and these are naturally called "numerical incidence properties" (NIP's).
For example, R is said to be "k regular at i" or "k regular at Xi" if and only if the cardinality |R&x@i| = k for all x C Xi. In a similar fashion, one can define the NIP's "<k regular at i", ">k regular at i", and so on. For ease of reference, I record a few of these definitions here:
R is k regular at i iff |R&x@i| = k for all x C Xi.
R is <k regular at i iff |R&x@i| < k for all x C Xi.
R is >k regular at i iff |R&x@i| > k for all x C Xi.
The definition of "local flags" can be broadened to give a definition of "regional flags". Suppose R c X1x...xXn and choose a subset M c Xi. Let "R&M@i" denote a subset of R called the "flag of R with M at i", or the "M@i flag of R", defined as:
R&M@i = {<x1, ... , xi, ... , xn> C R : xi C M}.
Returning to dyadic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let R c SxT be an arbitrary dyadic relation. The following properties of R can then be defined:
R is total at S iff R is 1 regular at S.
R is total at T iff R is 1 regular at T.
R is tubular at S iff R is 1 regular at S.
R is tubular at T iff R is 1 regular at T.
If R is tubular at S, then R is called a "partial function" or "prefunction" from S to T, often indicated by writing R = p : S ~> T.
R = p : S ~> T iff R is tubular at S.
If R is a prefunction p : S ~> T that happens to be total at S, then R is called a "function" from S to T, indicated by writing R = f : S > T.
R = f : S > T iff R is 1 regular at S.
f is surjective iff f is total at T.
f is injective iff f is tubular at T.
f is bijective iff f is 1 regular at T.
A few more comments on terminology are needed in further preparation. One of the constant practical demands encountered in this project is to have available a language and a calculus for relations that can permit discussion and calculation to range over functions, dyadic relations, and n place relations with a minimum amount of trouble in making transitions from subject to subject and in drawing the appropriate generalizations.
Up to this point in the discussion, the analysis of the A and B dialogue has concerned itself almost exclusively with the relationship of triadic sign relations to the dyadic relations obtained from them by taking their projections onto various relational planes. In particular, a major focus of interest was the extent to which salient properties of sign relations can be gleaned from a study of their dyadic projections.
Two important topics for later discussion will be concerned with: (1) the sense in which every n place relation can be decomposed in terms of triadic relations, and (2) the fact that not every triadic relation can be further reduced to conjunctions of dyadic relations.
It is one of the constant technical needs of this project to maintain a flexible language for talking about relations, one that permits discussion to shift from functional to relational emphases and from dyadic relations to n place relations with a maximum of ease. It is not possible to do this without violating the favored conventions of one technical linguistic community or another. I have chosen a strategy of use that respects as many different usages as possible, but in the end it cannot help but to reflect a few personal choices. To some extent my choices are guided by an interest in developing the information, computation, and decision theoretic aspects of the mathematical language used. Eventually, this requires one to render every distinction, even that of appearing or not in a particular category, as being relative to an interpretive framework.
While operating in this context, it is necessary to distinguish "domains" in the broad sense from "domains of definition" in the narrow sense. For n place relations it is convenient to use the terms "domain" and "quorum" as references to the wider and narrower sets, respectively.
For an n place relation R c X1x...xXn, I maintain the following usages:
1. The notation "Domi (R)" denotes the set Xi, called the "domain of R at i" or the "ith domain of R".
2. The notation "Quoi (R)" denotes a subset of Xi called the "quorum of R at i" or the "ith quorum of R", defined as follows:
Quoi (R) = the largest Q c Xi such that R&Q@i is >1-regular at i,
= the largest Q c Xi such that |R&x@i| > 1 for all x C Q c Xi.
In the special case of a dyadic relation R c X1xX2 = SxT, including the case of a partial function p : S ~> T or a total function f : S > T, I will stick to the following conventions:
1. The arbitrarily designated domains X1 = S and X2 = T that form the widest sets admitted to the dyadic relation are referred to as the "domain" or "source" and the "codomain" or "target", respectively, of the relation in question.
2. The terms "quota" and "range" are reserved for those uniquely defined sets whose elements actually appear as the 1st and 2nd members, respectively, of the ordered pairs in that relation. Thus, for a dyadic relation R c SxT, I let Quo (R) = Quo1 (R) c S be identified with what is usually called the "domain of definition" of R, and I let Ran (R) = Quo2 (R) c T be identified with the usual range of R.
A "partial equivalence relation" (PER) on a set X is a relation R c XxX that is an equivalence relation on its domain of definition Quo (R) c X. In this situation, [x]R is empty for each x in X that is not in Quo (R). Another way of reaching the same concept is to call a PER a dyadic relation that is symmetric and transitive, but not necessarily reflexive. Like the "self identical elements" of old that epitomized the very definition of self consistent existence in classical logic, the property of being a self related or self equivalent element in the purview of a PER on X singles out the members of Quo (R) as those for which a properly meaningful existence can be contemplated.
A "moderate equivalence relation" (MER) on the "modus" M c X is a relation on X whose restriction to M is an equivalence relation on M. In symbols, R c XxX such that R|M c MxM is an equivalence relation. Notice that the subset of restriction, or modus M, is a part of the definition, so the same relation R on X could be a MER or not depending on the choice of M. In spite of how it sounds, a moderate equivalence relation can have more ordered pairs in it than the ordinary sort of equivalence relation on the same set.
In applying the equivalence class notation to a sign relation R, the definitions and examples considered so far only cover the case where the connotative component RSI is a total equivalence relation on the whole syntactic domain S. The next job is to adapt this usage to PER's.
If R is a sign relation whose syntactic projection RSI is a PER on S, then I still write "[s]R" for the "equivalence class of s under RSI". But now, [s]R can be empty if s has no interpretant, that is, if s lies outside the "adequately meaningful" subset of the syntactic domain, where synonymy and equivalence of meaning are defined. Otherwise, if s has an i then it also has an o, by the definition of RSI. In this case, there is a triple <o, s, i> C R, and it is permissible to let [o]R = [s]R.
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===6.32. Partiality : Selective Operations===
===6.32. Partiality : Selective Operations===