Changes

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 <math>{}^{\backprime\backprime} e {}^{\prime\prime}\!</math> is used to denote the number also known as &ldquo;the unique base of the natural logarithms&rdquo; &mdash; 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 &mdash; 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 <math>\mathbb{R}\!</math> and the element <math>e\!</math> 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.

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 <math>{}^{\backprime\backprime} e {}^{\prime\prime}\!</math> is used to denote the number also known as &ldquo;the unique base of the natural logarithms&rdquo; &mdash; 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 &mdash; 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 <math>\mathbb{R}\!</math> and the element <math>e\!</math> 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 <math>O\!</math> of the relevant objects or outcomes.  But probability distributions are just a special class of functions <math>f : O \to [0, 1] \subseteq \mathbb{R},\!</math> where <math>\mathbb{R}\!</math> is the real line, and this means that the corresponding theory of partializations involves the dual aspect of the domain <math>O,\!</math> 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 determine it.  So, to make a long story short, one might as well approach this dual aspect by starting with the functions <math>f : O \to \{ 0, 1 \} = \mathbb{B},\!</math> in effect, with the logic of propositions.

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” (PIRs), 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.

I turn now to the question of ''partially specified relations'', or ''partially informed relations'' (PIRs), 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 <math>n\!</math>-place PIR and the supporting notion of a partially specified <math>n\!</math>-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" (PICs) 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 PICs 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?

One way to deal with "partially informed categories" (PICs) 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 PICs 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?