Changes

§ 5.   The symbolic forms employed in the construction of a RIF are found at the nexus of several different interpretive influences.  This section picks out three distinctive styles of usage that this work needs to draw on throughout its progress, usually without explicit notice, and discusses their relationships to each other in general terms.  These three styles of usage, distinguished according to whether they encourage an ''ordinary language'' (OL), a ''formal language'' (FL), or a ''computational language'' (CL) approach, have their relevant properties illustrated in the next three sections (§§ 6–8), each style being exemplified by a theoretical subject that thrives under its guidance.

§ 5.   The symbolic forms employed in the construction of a RIF are found at the nexus of several different interpretive influences.  This section picks out three distinctive styles of usage that this work needs to draw on throughout its progress, usually without explicit notice, and discusses their relationships to each other in general terms.  These three styles of usage, distinguished according to whether they encourage an ''ordinary language'' (OL), a ''formal language'' (FL), or a ''computational language'' (CL) approach, have their relevant properties illustrated in the next three sections (§§ 6–8), each style being exemplified by a theoretical subject that thrives under its guidance.

§ 6.   For ease of reference, the basic ideas of group theory used in this project are separated out and presented in this section.  Throughout this work as a whole, the subject of group theory serves in both illustrative and instrumental roles, providing, besides a rough stock of exemplary materials to work on, a ready array of precision tools to work with.

§ 6.   For ease of reference, the basic ideas of group theory used in this project are separated out and presented in this section.  Throughout this work as a whole, the subject of group theory serves in both illustrative and instrumental roles, providing, besides a rough stock of exemplary materials to work on, a ready array of precision tools to work with.

Group theory, as a methodological subject, is used to illustrate the "mathematical language" (ML) approach, which ordinarily takes it for granted that signs denote something, if not always the objects intended.  It is therefore recognizable as a special case of the OL style of usage.

Group theory, as a methodological subject, is used to illustrate the ''mathematical language'' (ML) approach, which ordinarily takes it for granted that signs denote something, if not always the objects intended.  It is therefore recognizable as a special case of the OL style of usage.

To the basic assumption of the OL approach the ML style adds only the faith that every object one desires to name has a unique proper name to do it with, and thus that all the various expressions for an object can be traded duty free and without much ado for a suitably compact name to denote it.  This means that the otherwise considerable work of practical computation, that is needed to associate arbitrarily obscure expressions with their clearest possible representatives, is not taken seriously as a feature that deserves theoretical attention, and is thus ignored as a factor of theoretical concern.  This is appropriate to the mathematical level, which abstracts away from pragmatic factors and is intended precisely to do so.

To the basic assumption of the OL approach the ML style adds only the faith that every object one desires to name has a unique proper name to do it with, and thus that all the various expressions for an object can be traded duty free and without much ado for a suitably compact name to denote it.  This means that the otherwise considerable work of practical computation, that is needed to associate arbitrarily obscure expressions with their clearest possible representatives, is not taken seriously as a feature that deserves theoretical attention, and is thus ignored as a factor of theoretical concern.  This is appropriate to the mathematical level, which abstracts away from pragmatic factors and is intended precisely to do so.

§ 8.   The notion of computation that makes sense in this setting is one of a process that replaces an arbitrary sign with a better sign of the same object.  In other words, computation is an interpretive process that improves the indications of intentions.  To deal with computational processes it is necessary to extend the pragmatic theory of signs in a couple of new but coordinated directions.  To the basic conception of a sign relation is added a notion of progress, which implies a notion of process together with a notion of quality.

§ 8.   The notion of computation that makes sense in this setting is one of a process that replaces an arbitrary sign with a better sign of the same object.  In other words, computation is an interpretive process that improves the indications of intentions.  To deal with computational processes it is necessary to extend the pragmatic theory of signs in a couple of new but coordinated directions.  To the basic conception of a sign relation is added a notion of progress, which implies a notion of process together with a notion of quality.

&sect; 9. &nbsp; This section introduces "higher order" sign relations, which are used to formalize the process of reflection on interpretation.  The discussion is approaching a point where multiple levels of signs are becoming necessary, mainly for referring to previous levels of signs as the objects of an extended sign relation, and thereby enabling a process of reflection on interpretive conduct.  To begin dealing with this issue, I take advantage of a second look at A and B to introduce the use of "raised angle brackets" (< >), also called "supercilia" or "arches", as quotation marks.  Ordinary quotation marks (" ") have the disadvantage, for formal purposes, of being used informally for many different tasks.  To get around this obstacle, I use the "arch" operator to formalize one specific function of quotation marks in a computational context, namely, to create distinctive names for syntactic expressions, or what amounts to the same thing, to signify the generation of their godel numbers.

&sect; 9. &nbsp; This section introduces ''higher order'' sign relations, which are used to formalize the process of reflection on interpretation.  The discussion is approaching a point where multiple levels of signs are becoming necessary, mainly for referring to previous levels of signs as the objects of an extended sign relation, and thereby enabling a process of reflection on interpretive conduct.  To begin dealing with this issue, I take advantage of a second look at <math>A\!</math> and <math>B\!</math> to introduce the use of ''raised angle brackets'' <math>({}^{\langle}~{}^{\rangle}),</math> also called ''supercilia'' or ''arches'', as quotation marks.  Ordinary quotation marks <math>({}^{\backprime\backprime}~{}^{\prime\prime})</math> have the disadvantage, for formal purposes, of being used informally for many different tasks.  To get around this obstacle, I use the "arch" operator to formalize one specific function of quotation marks in a computational context, namely, to create distinctive names for syntactic expressions, or what amounts to the same thing, to signify the generation of their gödel numbers.

&sect; 10. &nbsp; Returning to the sign relations A and B, various kinds of HO signs are exemplified by considering a selection of HO sign relations that are based on these two examples.

&sect; 10. &nbsp; Returning to the sign relations A and B, various kinds of HO signs are exemplified by considering a selection of HO sign relations that are based on these two examples.