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A full consideration of the geometries available for the spaces in which these levels of reflective abstraction are commonly imagined to reside leads to the conclusion that familiar distinctions of “top down” versus “bottom up” are being taken for granted in an arena that has not even been established to be orientable. Thus, it needs to be recognized that the distinction between objects and signs is relative to a definite system of interpretation. The pragmatic theory of signs is designed, in part, precisely to deal with the circumstance that thoroughly objective states of systems can be signs of each other, undermining any pretended distinction between objects and signs that one might propose to draw on essential grounds.
A full consideration of the geometries available for the spaces in which these levels of reflective abstraction are commonly imagined to reside leads to the conclusion that familiar distinctions of “top down” versus “bottom up” are being taken for granted in an arena that has not even been established to be orientable. Thus, it needs to be recognized that the distinction between objects and signs is relative to a definite system of interpretation. The pragmatic theory of signs is designed, in part, precisely to deal with the circumstance that thoroughly objective states of systems can be signs of each other, undermining any pretended distinction between objects and signs that one might propose to draw on essential grounds.
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''.
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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 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.
