Changes

Things referred to the objective world are not things that one expects to have much control over, at least, not at first, even though a reason for developing a language is to gain more control over events in time.  Things referred to the realm of signs are things that one thinks oneself to have under control, at least, at first, even though their complexity can evolve in time beyond one's powers of oversight.

Things referred to the objective world are not things that one expects to have much control over, at least, not at first, even though a reason for developing a language is to gain more control over events in time.  Things referred to the realm of signs are things that one thinks oneself to have under control, at least, at first, even though their complexity can evolve in time beyond one's powers of oversight.

In an ordinary mathematical context, when one writes out the expression for a finite set in the form <math>\{ x_1, \ldots, x_n \},</math> one expects to see the names of objects appearing between the braces.  Furthermore, even if these additional expectations are hardly ever formalized, these objects are typically expected to be the terminal objects of denotative value in the appropriate context of discussion and to inhabit a single order of objective existence.  In other words, it is common to assume that all the objects named have the same type, with no relations of consequence, functional, semantic, or otherwise, obtaining among them.  As soon as these assumptions are made explicit, of course, it is obvious that they do not have to be so.

In an ordinary mathematical context, when one writes out the expression for a finite set in the form <math>\{ x_1, \ldots, x_n \},\!</math> one expects to see the names of objects appearing between the braces.  Furthermore, even if these additional expectations are hardly ever formalized, these objects are typically expected to be the terminal objects of denotative value in the appropriate context of discussion and to inhabit a single order of objective existence.  In other words, it is common to assume that all the objects named have the same type, with no relations of consequence, functional, semantic, or otherwise, obtaining among them.  As soon as these assumptions are made explicit, of course, it is obvious that they do not have to be so.

In formal language contexts, when a set is taken as the alphabet or the lexicon of a formal language, then the objects named are themselves signs, but it is still only their names that are subject to appearing between the braces.  Often one seeks to handle this case by saying that what really appears between the braces are signs of sort that can suffice to represent themselves, and thus that these signs literally constitute their own names, but this is not ultimately a sensible tactic to try.  As always, only the tokens of signs can appear on the page, and these come and go as the pages are turned.  Although these tokens, by representing the types that encase them, partly succeed in referring to themselves, what they denote on principle is something much more abstract, general, and invariant than their own concrete, particular, and transient selves.  Nevertheless, the expectation that all of the elements in the set reside at the same level of syntactic existence is still in effect.

In formal language contexts, when a set is taken as the alphabet or the lexicon of a formal language, then the objects named are themselves signs, but it is still only their names that are subject to appearing between the braces.  Often one seeks to handle this case by saying that what really appears between the braces are signs of sort that can suffice to represent themselves, and thus that these signs literally constitute their own names, but this is not ultimately a sensible tactic to try.  As always, only the tokens of signs can appear on the page, and these come and go as the pages are turned.  Although these tokens, by representing the types that encase them, partly succeed in referring to themselves, what they denote on principle is something much more abstract, general, and invariant than their own concrete, particular, and transient selves.  Nevertheless, the expectation that all of the elements in the set reside at the same level of syntactic existence is still in effect.

There are numerous devices that one can use to assist with the task of reflection.  Rather than trying to divert the customary connections of informal language use and the conventional conduct of its interpretation, it is easier to introduce a collection of markedly novel signs, analogous to those already in use but whose interpretation is both free enough to be changed and controlled through a series of experimental variations and flexible enough to be altered when fitting and repaired when faulty.

There are numerous devices that one can use to assist with the task of reflection.  Rather than trying to divert the customary connections of informal language use and the conventional conduct of its interpretation, it is easier to introduce a collection of markedly novel signs, analogous to those already in use but whose interpretation is both free enough to be changed and controlled through a series of experimental variations and flexible enough to be altered when fitting and repaired when faulty.

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1. The "nominal resource" ("nominal alphabet" or "nominal lexicon") for X is a set of signs that is notated and defined as follows:

1. The "nominal resource" ("nominal alphabet" or "nominal lexicon") for X is a set of signs that is notated and defined as follows: