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===6.33. Sign Relational Complexes===
===6.33. Sign Relational Complexes===
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In a computational framework, indeed, in any constructively analytic and practically applied setting, the problem of working with insufficient information to fully determine one's object is a constant feature that goes with the territory of "finite information constructions" (FIC's). The fineness of detail that is able to be specified by formal symbols is continually bedeviled by the frustrating truncations of every signal to a finite code and by the resistive constrictions of every intention to the restrictive confines of what can actually be conducted. Of course, one tries to get around the more finessible limitations, but the figurative extensions that one hopes to achieve by recourse to quasi circular definitions and by reversion to parable and hyperbole — all of these tactics appeal to a pre established aptness of reception on the part of interpreters that begs the very question of a determinate understanding and that risks falling short of the exact attitude needed for success. At any rate, the indirect strategy of approach relies on such large reserves of enthymeme to fuel its course that the grasp of a period to set bounds on its argument and fix a term to its conclusion is often found diverging in ways that both underreach and overreach its object, and well founded or not the search for a generic method of definition typically ends so completely dumbfounded that it often trails off into the inescapable vacuity of a quasi terminal ellipsis ...
This section treats the problems of insufficient information and indeterminate objects under the heading of "partializations", using this as a briefer term for the information theoretic generalizations of the relevant object domains that take the use of indeterminate denotations, or partial determinations of objects, explicitly into account.
In working with "partializations" or information theoretic generalizations of any subject matter, one has a choice between two options:
1. Under the "object theoretic" alternative one views the "partiality" as something attaching to the objects of discussion. Consequently, one operates as if the problems distinctive of the extended subject matter were questions of managing ordinary information about a strange new breed of partial objects.
2. Under the "sign theoretic" alternative one takes the "partiality" as something affecting only the signs used in discussion. Accordingly, one approaches the task as a matter of handling partial information about ordinary objects, namely, the same domains of objects initially given at the outset of discussion.
But a working maxim of information theory says that "Partial information is your ordinary information". Applied to the principle regulating the sign theoretic convention this means that the adjective "partial" is swallowed up by the substantive "information", so that the ostensibly more general case is always already subsumed within the ordinary case. Because partiality is part and parcel to the usual nature of information, it is a perfectly typical feature of the signs and expressions bearing it to provide normally only partial information about ordinary objects.
The only time when a finite sign or expression can give the appearance of determining a perfectly precise content or a post finite amount of information, for example, when the symbol “e” is used to denote the number also known as “the unique base of the natural logarithms” — this can only happen when interpreters are prepared, by dint of the information embodied in their prior design and preliminary training, to accept as meaningful and be terminally satisfied with what is still only a finite content, syntactically speaking. Every remaining impression that a perfectly determinate object, an "individual" in the original sense of the word, has nevertheless been successfully specified — this can only be the aftermath of some prestidigitation, that is, the effect of some pre arranged consensus, for example, of accepting a finite system of definitions and axioms that are supposed to define the space R and the element e within it, and of remembering or imagining that an effective proof system has once been able or will yet be able to convince one of its demonstrations.
Ultimately, one must be prepared to work with probability distributions that are defined on entire spaces O of the relevant objects or outcomes. But probability distributions are just a special class of functions f : O > [0, 1] c R, where R is the real line, and this means that the corresponding theory of partializations involves the dual aspect of the domain O, dealing with the "functionals" defined on it, or the functions that map it into "coefficient" spaces. And since it is unavoidable in a computational framework, one way or another every type of coefficient information, real or otherwise, must be approached bit by bit. That is, all information is defined in terms of the either or decisions that must be made to really and practically determine it. So, to make a long story short, one might as well approach this dual aspect by starting with the functions f : O > B = {0, 1}, in effect, with the logic of propositions.
I turn now to the question of "partially specified" (PS) relations, or “partially informed relations” (PIR's), 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 n place PIR and the supporting notion of a PS n 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" (PIC's) 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 PIC's 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 PIR's 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.
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" (CI's) and "localized addresses" (LA's), 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 "caret" (^) device works well in any situation where one desires to accentuate the fact that a formal subscript is being reclaimed and elevated to the status of an actual parameter. By way of the operation indicated by the caret character the index bound to an object term can be rehabilitated as a full fledged component of an elementary relation, thereby schematically embedding the indicated object in the experiential space of a typical agent.
The form of the caret notation is intended to suggest the use of "pointers" and "views" in computational frameworks, letting one interpret "j^x" in any one of many various ways, for example:
j^x = j's indication of x, j's access to x,
j's information on x, j's allusion to x,
j's copy of x, j's view of x.
Presently, the distinction between indirect pointers and direct pointers, that is, between virtual copies and actual views of an objective domain, is not yet relevant here, being a dimension of variation that the discussion is currently abstracting over.
I would like to record here, in what is topically the appropriate place, notice of a number of open questions that will have to be addressed if anyone desires to make a consistent calculus out of this caret notation. Perhaps it is only because the franker forms of liaison involved in the caret couple a^b are more subject to the vagaries of syntactic elision than the corresponding bindings of the anglish ligature <a, b>, but for some reason or other the circumflex character of these diacritical notices are much more liable to suggest various forms of elaboration, including higher order generalizations and information theoretic partializations of the very idea of n tuples and sequences.
One way to deal with the problems of partial information ...
Relational complex?
R = R(1) U ... U R(n)
Sign relational complex?
R = R(1) U R(2) U R(3)
1. Carets linkages can be chained together to form sequences of indications or n tuples, without worrying too much about the order of collecting terms in the corresponding angle brackets.
a^b^c = <a, b, c> = <a, <b, c>> = <<a, b>, c>.
These equivalences depend on the existence of natural isomorphisms between different ways of constructing n place product spaces, that is, on the associativity of pairwise products, a not altogether trivial result (MacLane, CatWorkMath, ch. 7).
2. Higher order indications (HOI's)?
^x = < , x> x^ = <x, > ?
^^x = < , < , x>> x^^ = <<x, >, > ?
Fragments
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 ...
This direction of generalization expands the scope of PIR's 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.
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 ...
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===6.34. Set Theoretic Constructions===
===6.34. Set Theoretic Constructions===
