Changes

From now on, I will reuse the ancient term ''gnomon'' in a technical sense to refer to the gödel numbers or code names of formal objects.  In other words, a gnomon is a gödel numbering or enumeration function that maps a domain of objects into a domain of signs, <math>\operatorname{Gno} : O \to S.\!</math>  When the syntactic domain <math>S\!</math> is contained within the object domain <math>O,\!</math> then the part of the gnomon that maps <math>S\!</math> into <math>S,\!</math> providing names for signs and expressions, is usually regarded as a ''quoting function''.

From now on, I will reuse the ancient term ''gnomon'' in a technical sense to refer to the gödel numbers or code names of formal objects.  In other words, a gnomon is a gödel numbering or enumeration function that maps a domain of objects into a domain of signs, <math>\operatorname{Gno} : O \to S.\!</math>  When the syntactic domain <math>S\!</math> is contained within the object domain <math>O,\!</math> then the part of the gnomon that maps <math>S\!</math> into <math>S,\!</math> providing names for signs and expressions, is usually regarded as a ''quoting function''.

In the pluralistic contexts that go with pragmatic theories of signs, it is no longer entirely appropriate to refer to ''the'' gnomon of any object.  At any moment of discussion, I can only have so-and-so's gnomon or code word for each thing under the sun.  Thus, apparent references to a uniquely determined gnomon only make sense if taken as enthymemic invocations of the ordinary context and all that is comprehended to be implied in it, promising to convert tacit common sense into definite articulations of what is understood.  Actually achieving this requires each elliptic reference to the gnomon to be explicitly grounded in the context of informal discussion, interpreted with respect to the conventional basis of understanding assumed in it, and relayed to the indexing function taken for granted by all parties to it.

In the pluralistic contexts that go with pragmatic theories of signs, it is no longer entirely appropriate to refer to "the" gnomon of any object.  At any moment of discussion, I can only have so and so's gnomon or code word for each thing under the sun.  Thus, apparent references to a uniquely determined gnomon only make sense if taken as enthymemic invocations of the ordinary context and of all that is comprehended to be implied in it, promising to convert tacit common sense into definite articulations of what is understood.  Actually achieving this requires each elliptic reference to the gnomon to be explicitly grounded in the context of informal discussion, interpreted with respect to the conventional basis of understanding assumed in it, and relayed to the indexing function taken for granted by all parties to it.

In computational terms, this brand of pluralism means that neither the gnomon nor the quoting function that forms a part of it can be viewed as well defined unless it is indexed, explicity or implicitly, by the name of a particular interpreter.  I will use the notations "Gnoi(x)" = "<x, i>" to indicate the gnomon of the object x with respect to the interpreter i.  The value Gnoi(x) = <x, i> C S is the "nominal sign in use" or the "name in use" (NIU) of the object x with respect to the interpreter i, and thus it constitutes a component of i's state.

In computational terms, this brand of pluralism means that neither the gnomon nor the quoting function that forms a part of it can be viewed as well-defined unless it is indexed, explicitly or implicitly, by the name of a particular interpreter.  I will use either one of the equivalent notations <math>{}^{\backprime\backprime} \operatorname{Gno}_i (x) {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime\langle} x, i {}^{\rangle\prime\prime}\!</math> to indicate the gnomon of the object <math>x\!</math> with respect to the interpreter <math>i.\!</math> The value <math>\operatorname{Gno}_i (x) = {}^{\langle} x, i {}^{\rangle} \in S\!</math> is the ''nominal sign in use'' or the ''name in use'' (NIU) of the object <math>x\!</math> with respect to the interpreter <math>i,\!</math> and thus it is a component of <math>i\!</math>'s state.

In the special case where x is a sign or expression in the syntactic domain, then Gnoi(x) = <x, i> is tantamount to the quotation of x by and for the use of the ith interpreter, in short, the nominal sign to i that makes x an object for i.  For signs and expressions, it is usually only the quoting function that makes them objects.  But nothing is an object in any sense for an interpreter unless it is an object of a sign relation for that interpreter.  Therefore, ...

In the special case where x is a sign or expression in the syntactic domain, then Gnoi(x) = <x, i> is tantamount to the quotation of x by and for the use of the ith interpreter, in short, the nominal sign to i that makes x an object for i.  For signs and expressions, it is usually only the quoting function that makes them objects.  But nothing is an object in any sense for an interpreter unless it is an object of a sign relation for that interpreter.  Therefore, ...