Changes

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, denotative and connotative, that can affect a sign relation.

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.

<li>A sign relation <math>L\!</math> has a non-deterministic denotation if its dyadic component <math>L_{SO}\!</math> (the converse of <math>L_{OS}\!</math>) is not a function <math>L_{SO} : S \to O,\!</math> in other words, if there are signs in <math>S\!</math> with missing or multiple objects in <math>O.\!</math></li>

2. A sign relation R has a non deterministic connotation if its dyadic component RSI is not a function RSI: S >I, in other words, if there are signs in S with missing or multiple interpretants in I.  As a rule, sign relations are rife with this variety of non determinism, but it is usually felt to be under control so long as RSI remains close to being an equivalence relation.

<li>A sign relation <math>L\!</math> has a non-deterministic connotation if its dyadic component <math>L_{SI}\!</math> is not a function <math>L_{SI} : S \to I,\!</math> in other words, if there are signs in <math>S\!</math> with missing or multiple interpretants in <math>I.\!</math> As a rule, sign relations are rife with this variety of non-determinism, but it is usually felt to be under control so long as <math>L_{SI}\!</math> remains close to being an equivalence relation.</li>

Thus, it is really the denotative type of indeterminacy that is felt to be a problem in this context.

Thus, it is really the denotative type of indeterminacy that is felt to be a problem in this context.

The next two pairs of reflective extensions demonstrate that there are ways of achieving reflective closure that do not generate infinite sign relations.

The next two pairs of reflective extensions demonstrate that there are ways of achieving reflective closure that do not generate infinite sign relations.

As a flexible and fairly general strategy for describing reflective extensions it is convenient to take the following tack.  Given a syntactic domain S, there is an independent formal language F = F(S) = S<<>>, to be called "the free quotational extension of S", that can be generated from S by embedding each of its signs to any depth of quotation marks.  In F, the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations.  In other words, for every s C S, the sequence s, <s>, <<s>>, ... contains nothing but pairwise distinct elements in F no matter how far it is produced.  The set F(s) = s<<>> c F that collects the elements of this sequence is called "the subset of F generated from s by quotation".

As a flexible and fairly general strategy for describing reflective extensions it is convenient to take the following tack.  Given a syntactic domain S, there is an independent formal language F = F(S) = S<<>>, to be called "the free quotational extension of S", that can be generated from S by embedding each of its signs to any depth of quotation marks.  In F, the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations.  In other words, for every s C S, the sequence s, <s>, <<s>>, ... contains nothing but pairwise distinct elements in F no matter how far it is produced.  The set F(s) = s<<>> c F that collects the elements of this sequence is called "the subset of F generated from s by quotation".