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There are many ways to extend sign relations in an effort to increase their reflective capacities.  The implicit goal of a reflective project is to achieve ''reflective closure'', <math>S \subseteq O,\!</math> where every sign is an object.

There are many ways to extend sign relations in an effort to increase their reflective capacities.  The implicit goal of a reflective project is to achieve ''reflective closure'', <math>S \subseteq O,\!</math> where every sign is an object.

Considered as reflective extensions, there is nothing unique about the constructions of <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B})\!</math> but their common pattern of development illustrates a typical approach toward reflective closure.  In a sense it epitomizes the project of ''free'', ''naive'', or ''uncritical'' reflection, since continuing this mode of production to its closure would generate an infinite sign relation, passing through infinitely many higher orders of signs, but without examining critically to what purpose the effort is directed or evaluating alternative constraints that might be imposed on the initial generators toward this end.

Considered as reflective extensions, there is nothing unique about the constructions of Ref1 (A) and Ref1 (B), but their common pattern of development illustrates a typical approach toward reflective closure.  In a sense it epitomizes the project of "free", "naive", or "uncritical" reflection, since continuing this mode of production to its closure would generate an infinite sign relation, passing through infinitely many higher orders of signs, but without examining critically to what purpose the effort is directed or evaluating alternative constraints that might be imposed on the initial generators toward this end.

At first sight it seems as though the imposition of reflective closure has multiplied a finite sign relation into an infinite profusion of highly distracting and largely redundant signs, all by itself and all in one step.  But this explosion of orders happens only with the complicity of another requirement, that of deterministic interpretation.

At first sight it seems as though the imposition of reflective closure has multiplied a finite sign relation into an infinite profusion of highly distracting and largely redundant signs, all by itself and all in one step.  But this explosion of orders happens only with the complicity of another requirement, that of deterministic interpretation.

There are two types of non determinism that can affect a sign relation, denotative and connotative.

There are two types of non-determinism that can affect a sign relation, denotative and connotative.

1. A sign relation R has a non deterministic denotation if its dyadic component RSO (the converse of ROS) is not a function RSO: S >O, that is, if there are signs in S with missing or multiple objects in O.

1. A sign relation R has a non deterministic denotation if its dyadic component RSO (the converse of ROS) is not a function RSO: S >O, that is, if there are signs in S with missing or multiple objects in O.