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In the theory of sign relations, as in formal language theory, one tends to spend a lot of the time talking about signs as objects.  Doing this requires one to have signs for denoting signs and ways of telling when a sign is being used as a sign or is just being mentioned as an object.  Generally speaking, reflection on the usage of an established order of signs recruits another order of signs to denote them, and then another, and another, until a limit on one's powers of reflection is ultimately reached, and finally one is forced to conduct one's meaning in forms of interpretive practice that fail to be fully reflective in one critical respect or another.  In the last resort one resigns oneself to letting the recourse of signs be guided by casually intuited inklings of their potential senses.

In the theory of sign relations, as in formal language theory, one tends to spend a lot of the time talking about signs as objects.  Doing this requires one to have signs for denoting signs and ways of telling when a sign is being used as a sign or is just being mentioned as an object.  Generally speaking, reflection on the usage of an established order of signs recruits another order of signs to denote them, and then another, and another, until a limit on one's powers of reflection is ultimately reached, and finally one is forced to conduct one's meaning in forms of interpretive practice that fail to be fully reflective in one critical respect or another.  In the last resort one resigns oneself to letting the recourse of signs be guided by casually intuited inklings of their potential senses.

In this text a number of linguistic devices are used to assist the faculty of reflection, hopefully forestalling the relegation of its powers to its own natural resources for a long enough spell to observe its action.  A discussion of these techniques and strategies follows.

In this text a number of linguistic devices are used to assist the faculty of reflection, hopefully forestalling the relegation of its powers to its own natural resources for a long enough spell to observe its action.  Two of the most frequently used strategies toward this purpose can be described as follows:

1. In the declaration of HO signs and the specification of their uses, one can employ the same terminology and technical distinctions that are found to be effective in describing sign relations.  This turns the established terms for significant properties of world elements and the provisional terms for their relationships to each other to the ends of prescribing the relative orders of HO signs and their objects.  In short, the received theory of signs, however transient it may be at any given moment of inquiry, allows one to declare the absolute types and the relative roles that all of these entities are meant to take up.

In the declaration of higher order signs and the specification of their uses, one can employ the same terminology and technical distinctions that are found to be effective in describing sign relations.  This turns the established terms for significant properties of world elements and the provisional terms for their relationships to each other to the ends of prescribing the relative orders of higher order signs and their objects.  In short, the received theory of signs, however transient it may be at any given moment of inquiry, allows one to declare the absolute types and the relative roles that all of these entities are meant to take up.

For example, if I say that x connotes y and that y denotes z, then it means I imagine myself to have an interpretation or a sign relation in mind where x and y are both signs belonging to a single order of signs and y is a sign belonging to the next higher order of signs up from z, everything being relative to that particular moment of interpretation.  Of course, as far as wholly arbitrary sign relations go, there is nothing to guarantee that the interpretation I think myself to have in mind at one moment can be integrated with the interpretation I think myself to have in mind at another moment, or that a just order can be founded in the end by any manner of interpretation that "just follows orders" in this way.

For example, if I say that <math>x ~\text{connotes}~ y\!</math> and that <math>y ~\text{denotes}~ z,\!</math> then it means I imagine myself to have an interpretation or a sign relation in mind where <math>x\!</math> and <math>y\!</math> are both signs belonging to a single order of signs and <math>y\!</math> is a sign belonging to the next higher order of signs up from <math>z,\!</math> everything being relative to that particular moment of interpretation.  Of course, as far as wholly arbitrary sign relations go, there is nothing to guarantee that the interpretation I think myself to have in mind at one moment can be integrated with the interpretation I think myself to have in mind at another moment, or that a just order can be founded in the end by any manner of interpretation that &ldquo;just follows orders&rdquo; in this way.

2. Ordinary quotation marks ("...") function as an operator on pieces of text to create names for the signs or expressions enclosed in them.  In doing this the quotation marks delay, defer, or interrupt the normal use of their subtended contents, interfering with the referential use of a sign or the evaluation of an expression in order to create a new sign.  The use of this constructed sign is to mention the immediate contents of the quotation marks in a way that can serve thereafter to indicate these contents directly or allude to them indirectly.

Ordinary quotation marks <math>( {}^{\backprime\backprime} ~ {}^{\prime\prime} )</math> function as an operator on pieces of text to create names for the signs or expressions enclosed in them.  In doing this the quotation marks delay, defer, or interrupt the normal use of their subtended contents, interfering with the referential use of a sign or the evaluation of an expression in order to create a new sign.  The use of this constructed sign is to mention the immediate contents of the quotation marks in a way that can serve thereafter to indicate these contents directly or allude to them indirectly.

In the informal context, however, quotation marks are used equivocally for several other purposes.  In particular, they are frequently used to call attention to the immediate use of a sign, to stress it or redress it for a definitive, emphatic, or skeptical service, but without necessarily intending to interrupt or seriously alter its ongoing use.  Furthermore, ordinary quotation marks are commonly taken so literally that they can inadvertently pose an obstacle to functional abstraction.  For instance, if I try to refer to the effect of quotation as a mapping that takes signs to HO signs, thereby attempting to define its action by means of a lambda abstraction:  x > "x", then there are modes of IL interpretation that would read this literally as a constant map, one that sends every element of the functional domain into the single code for the letter "x".

In the informal context, however, quotation marks are used equivocally for several other purposes.  In particular, they are frequently used to call attention to the immediate use of a sign, to stress it or redress it for a definitive, emphatic, or skeptical service, but without necessarily intending to interrupt or seriously alter its ongoing use.  Furthermore, ordinary quotation marks are commonly taken so literally that they can inadvertently pose an obstacle to functional abstraction.  For instance, if I try to refer to the effect of quotation as a mapping that takes signs to higher order signs, thereby attempting to define its action by means of a lambda abstraction:  <math>x \mapsto {}^{\backprime\backprime} x {}^{\prime\prime},</math> then there are modes of IL interpretation that would read this literally as a constant map, one that sends every element of the functional domain into the single code for the letter <math>{}^{\backprime\backprime} x {}^{\prime\prime}.</math>

For these reasons I introduce the use of raised angle brackets (<...>), also called "arches" or "supercilia", to configure a form of quotation marker, but one that is subject to a more definite set of understandings about its interpretation.  Namely, the arch marker denotes a function on signs that takes (the name of) a syntactic element located within it as (the name of) a functional argument and returns as its functional value the name of that syntactic element.  The parenthetical operators in this statement reflect the optional readings that prevail in some cases, where the simple act of noticing a syntactic element as a functional argument is already tantamount to having a name for it.  As a result, a quoting function that is designed to operate on the signs denoting and not on the objects denoted seems to do nothing at all, but merely uses up a moment of time to do it.

For these reasons I introduce the use of raised angle brackets (<...>), also called "arches" or "supercilia", to configure a form of quotation marker, but one that is subject to a more definite set of understandings about its interpretation.  Namely, the arch marker denotes a function on signs that takes (the name of) a syntactic element located within it as (the name of) a functional argument and returns as its functional value the name of that syntactic element.  The parenthetical operators in this statement reflect the optional readings that prevail in some cases, where the simple act of noticing a syntactic element as a functional argument is already tantamount to having a name for it.  As a result, a quoting function that is designed to operate on the signs denoting and not on the objects denoted seems to do nothing at all, but merely uses up a moment of time to do it.