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For these reasons I introduce the use of raised angle brackets <math>( {}^{\langle} ~ {}^{\rangle} ),</math> also called &ldquo;arches&rdquo; or &ldquo;supercilia&rdquo;, 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 <math>( {}^{\langle} ~ {}^{\rangle} ),</math> also called &ldquo;arches&rdquo; or &ldquo;supercilia&rdquo;, 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.

In IL contexts the arch quotes are construed together with their syntactic contents as forming a certain kind of term, one that achieves a naming function on syntactic elements by taking the enclosed text as a functional argument and giving a directly embedded indication of it.  In this type of setting the name of a string of length <math>k\!</math> is a string of length <math>k + 2.\!</math>

In IL contexts the arch quotes are construed together with their syntactic contents as forming a certain kind of term, one that achieves a naming function on syntactic elements by taking the enclosed text as a functional argument and giving a directly embedded indication of it.  In this type of setting the name of a string of length k is a string of length k+2.

In FL contexts the arch marker denotes a function that takes the literal syntactic element bounded by it as its argument and returns the name, code, annotation, godel number, or "unique numerical identifier" (UNI) of that syntactic element.  In this setting there need be no straightforward relationship between the size or complexity of the syntactic element and the magnitude of its numerical code or the form of its symbolic code.

In FL contexts the arch marker denotes a function that takes the literal syntactic element bounded by it as its argument and returns the name, code, annotation, gödel number, or ''unique numerical identifier'' of that syntactic element.  In this setting there need be no straightforward relationship between the size or complexity of the syntactic element and the magnitude of its numerical code or the form of its symbolic code.

In CL implementations the arch operation is intended to do exactly what the principal uses of ordinary quotes are supposed to do, except that it obeys restrictions that are necessary to make it work as a notation for a computable function on the identified syntactic domain.

In CL implementations the arch operation is intended to do exactly what the principal uses of ordinary quotes are supposed to do, except that it obeys restrictions that are necessary to make it work as a notation for a computable function on the identified syntactic domain.

One further remark on the uses of quotation marks is pertinent here.  When using HA signs with high orders of complexity and depth, it is often convenient to revert to the use of ordinary quotes at the outer boundary of a quotational expression, in this way marking a return to the ordinary context of interpretation.  For example, one observes the colloquial equivalence:  <<<x>>>  =  "<<x>>".

One further remark on the uses of quotation marks is pertinent here.  When using HA signs with high orders of complexity and depth, it is often convenient to revert to the use of ordinary quotes at the outer boundary of a quotational expression, in this way marking a return to the ordinary context of interpretation.  For example, one observes the colloquial equivalence:  <<<x>>>  =  "<<x>>".