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Any object of an interpreter's experience and reasoning, no matter how vaguely and casually it initially appears, up to and including the merest appearance of a sign, is already, by virtue of these very circumstances, on its way to becoming the object of a formalized sign, so long as the signs are made available to denote it.  The reason for this is rooted in each agent's capacity for reflection on its own experience and reasoning, and the critical question is only whether these transient reflections can come to constitute signs of a more permanent use.

Any object of an interpreter's experience and reasoning, no matter how vaguely and casually it initially appears, up to and including the merest appearance of a sign, is already, by virtue of these very circumstances, on its way to becoming the object of a formalized sign, so long as the signs are made available to denote it.  The reason for this is rooted in each agent's capacity for reflection on its own experience and reasoning, and the critical question is only whether these transient reflections can come to constitute signs of a more permanent use.

The immediate purpose of the arch operation is to equip the text with a syntactic mechanism for constructing higher order signs, that is, signs denoting signs.  But the step of reflection that the arch device marks corresponds to a definite change on the part of the interpreter, affecting the ''pragmatic stance'' or the ''intentional attitude'' that the interpreter takes up with respect to the affected signs.  Accordingly, because of its connection to the interpreter's capacity for critical reflection, the arch operation, whether signified by arches or quotes, opens up a topic of wide importance to the larger question of inquiry.  Unfortunately, there is much to do before this issue can be taken up in detail, and immediate concerns make it necessary to break off further discussion for now.

The immediate purpose of the "arch" operation is to equip the text with a syntactic mechanism for constructing "higher order" (HO) signs, that is, signs denoting signs.  But the step of reflection that the arch device marks corresponds to a definite change on the part of the interpreter, affecting the "pragmatic stance" or the "intentional attitude" that the interpreter takes up with respect to the affected signs.  Accordingly, because of its connection to the interpreter's capacity for critical reflection, the arch operation, whether signified by arches or quotes, opens up a topic of wide importance to the larger question of inquiry.  Unfortunately, there is much to do before this issue can be taken up in detail, and immediate concerns make it necessary to break off further discussion for now.

A general understanding of HO signs would not depend on the special devices that are used to construct them, but would define them as any signs that behave in certain ways under interpretation, that is, as any signs that are interpreted in a particular manner, yet to be specified.  A proper definition of HO signs, including a generic description of the operations that construct them, cannot be achieved at the present stage of discussion.  Doing this correctly depends on carrying out further developments in the theories of formal languages and sign relations.  Until this discussion reaches that point, much of what it says about HO signs will have to be regarded as a provisional compromise.

A general understanding of higher order signs would not depend on the special devices that are used to construct them, but would define them as any signs that behave in certain ways under interpretation, that is, as any signs that are interpreted in a particular manner, yet to be specified.  A proper definition of higher order signs, including a generic description of the operations that construct them, cannot be achieved at the present stage of discussion.  Doing this correctly depends on carrying out further developments in the theories of formal languages and sign relations.  Until this discussion reaches that point, much of what it says about higher order signs will have to be regarded as a provisional compromise.

The development of reflection on interpretation leads to the generation of "higher order" (HO) signs that denote "lower order" (LO) signs as their objects.  This process is illustrated by the following "eponymy" of progressively HO signs, all of which stem from a plain precursor and ultimately refer back to their "eponymous ancestor":

The development of reflection on interpretation leads to the generation of higher order signs that denote lower order signs as their objects.  This process is illustrated by the following sequence of progressively higher order signs, all of which stem from a plain precursor and ultimately refer back to their initial ancestor, in this case, <math>x.\!</math>

x, <x>, <<x>>, <<<x>>>,  ...

| <math>x, ~ {}^{\langle} x {}^{\rangle}, ~ {}^{\langle\langle} x {}^{\rangle\rangle}, ~ {}^{\langle\langle\langle} x {}^{\rangle\rangle\rangle}, ~ \ldots</math>

The intent of this succession, as interpreted in FL environments, is that <<x>> denotes or refers to <x>, which denotes or refers to x.  Moreover, its computational realization, as implemented in CL environments, is that <<x>> addresses or evaluates to <x>, which addresses or evaluates to x.

The intent of this succession, as interpreted in FL environments, is that <<x>> denotes or refers to <x>, which denotes or refers to x.  Moreover, its computational realization, as implemented in CL environments, is that <<x>> addresses or evaluates to <x>, which addresses or evaluates to x.