Changes

&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; 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 <math>L(A)\!</math> and <math>L(B),\!</math> various kinds of higher order signs are exemplified by considering a series of higher order sign relations 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.

&sect; 11. &nbsp; In this section the tools that come with the theory of higher order sign relations are applied to an illustrative exercise, roughing out the shape of a complex form of interpreter.

&sect; 11. &nbsp; In this section the tools that come with the theory of higher order sign relations are applied to an illustrative exercise, roughing out the shape of a complex form of interpreter.

The next three sections (§§ 32 34) discuss how the identified styles of usage bear on three important issues in the usage of a technical language, namely, the respective theoretical statuses of "signs", "sets", and "variables".

The next three sections (&sect;&sect;&nbsp;32&ndash;34) discuss how the identified styles of usage bear on three important issues in the usage of a technical language, namely, the respective theoretical statuses of signs, sets, and variables.

&sect; 12. &nbsp; The Status of Signs

&sect; 12. &nbsp; The Status of Signs

&sect; 13. &nbsp; The Status of Sets

&sect; 13. &nbsp; The Status of Sets

&sect; 14. &nbsp; At this point the discussion touches on an topic, concerning the being of a so called "variable", that issues in many unanswered questions.  Although this worry over the nature and use of a variable may seem like a trivial matter, it is not.  It needs to be remembered that the first adequate accounts of formal computation, Schonfinkel's combinator calculus and Church's lambda calculus, both developed out of programmes intended to clarify the concept of a variable, indeed, even to the point of eliminating it altogether as a primitive notion from the basis of mathematical logic (van Heijenoort, 355 366).

&sect; 14. &nbsp; At this point the discussion touches on an topic, concerning the being of a so called ''variable'', that issues in many unanswered questions.  Although this worry over the nature and use of a variable may seem like a trivial matter, it is not.  It needs to be remembered that the first adequate accounts of formal computation, Schönfinkel's combinator calculus and Church's lambda calculus, both developed out of programmes intended to clarify the concept of a variable, indeed, even to the point of eliminating it altogether as a primitive notion from the basis of mathematical logic (van Heijenoort, 355&ndash;366).

The pragmatic theory of sign relations has a part of its purpose in addressing these same questions about the natural utility of variables, and even though its application to computation has not enjoyed the same level of development as these other models, it promises in good time to have a broader scope.  Later, I will illustrate its potential by examining a form of the combinator calculus from a sign relational point of view.

The pragmatic theory of sign relations has a part of its purpose in addressing these same questions about the natural utility of variables, and even though its application to computation has not enjoyed the same level of development as these other models, it promises in good time to have a broader scope.  Later, I will illustrate its potential by examining a form of the combinator calculus from a sign relational point of view.

&sect; 15. &nbsp; There is an order of logical reasoning that is typically described as "propositional" or "sentential" and represented in a type of formal system that is commonly known as a "propositional calculus" or a "sentential logic" (SL).  Any one of these calculi forms an interesting example of a formal language, one that can be used to illustrate all of the preceding issues of style and technique, but one that can also serve this inquiry in a more instrumental fashion.  This section presents the elements of a calculus for propositional logic that I described in earlier work (Awbrey, 1989 & 1994).  The imminent use of this calculus is to construct and analyze logical representations of sign relations, and the treatment here focuses on the concepts and notation that are most relevant to this task.

&sect; 15. &nbsp; There is an order of logical reasoning that is typically described as "propositional" or "sentential" and represented in a type of formal system that is commonly known as a "propositional calculus" or a "sentential logic" (SL).  Any one of these calculi forms an interesting example of a formal language, one that can be used to illustrate all of the preceding issues of style and technique, but one that can also serve this inquiry in a more instrumental fashion.  This section presents the elements of a calculus for propositional logic that I described in earlier work (Awbrey, 1989 & 1994).  The imminent use of this calculus is to construct and analyze logical representations of sign relations, and the treatment here focuses on the concepts and notation that are most relevant to this task.