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A principal difficulty of using sign relations for this purpose arises from the very power of productivity they bring to bear in the process, the capacity of triadic relations to generate a welter of what are bound to be mostly arbitrary structures, with only a scattered few hoping to show any promise, but the massive profusion of which exceeds from the outset any reason's ability to sort them out and test them in practice.  And yet, as the phenomena of interest become more complex, the chances grow slimmer that adequate explanations will be found in any of the thinner haystacks.  In this respect, sign relations inherit the basic proclivities of set theory, which can be so successful and succinct in presenting and clarifying the properties of already found materials and hard won formal insights, and yet so overwhelming to use as a tool of random exploration and discovery.

A principal difficulty of using sign relations for this purpose arises from the very power of productivity they bring to bear in the process, the capacity of triadic relations to generate a welter of what are bound to be mostly arbitrary structures, with only a scattered few hoping to show any promise, but the massive profusion of which exceeds from the outset any reason's ability to sort them out and test them in practice.  And yet, as the phenomena of interest become more complex, the chances grow slimmer that adequate explanations will be found in any of the thinner haystacks.  In this respect, sign relations inherit the basic proclivities of set theory, which can be so successful and succinct in presenting and clarifying the properties of already found materials and hard won formal insights, and yet so overwhelming to use as a tool of random exploration and discovery.

The sign relations of ''A'' and ''B'', though natural in themselves as far as they go, were nevertheless introduced in an artificial fashion and presented by means of arbitrary stipulations.  Sign relations that arise in more natural settings usually have a rationale, a reason for being as they are, and therefore become amenable to classification on the basis of the distinctive characters that make them what they are.

The sign relations of <math>A\!</math> and <math>B,\!</math> though natural in themselves as far as they go, were nevertheless introduced in an artificial fashion and presented by means of arbitrary stipulations.  Sign relations that arise in more natural settings usually have a rationale, a reason for being as they are, and therefore become amenable to classification on the basis of the distinctive characters that make them what they are.

Consequently, naturally occurring sign relations can be expected to fall into species or natural kinds, and to have special properties that make them keep on occurring in nature.  Moreover, cultivated varieties of sign relations, the kinds that have been converted to social purposes and found to be viable in actual practice, will have identifiable and especially effective properties by virtue of which their signs are rendered significant.

Consequently, naturally occurring sign relations can be expected to fall into species or natural kinds, and to have special properties that make them keep on occurring in nature.  Moreover, cultivated varieties of sign relations, the kinds that have been converted to social purposes and found to be viable in actual practice, will have identifiable and especially effective properties by virtue of which their signs are rendered significant.

In the pragmatic theory of sign relations, three natural kinds of signs are recognized, under the names of ''icons'', ''indices'', and ''symbols''.  Examples of indexical or accessional signs figured significantly in the discussion of ''A'' and ''B'', as illustrated by the pronouns "i" and "u" in ''S''.  Examples of iconic or analogical signs were also present, though keeping to the background, in the very form of the sign relation Tables that were used to schematize the whole activity of each interpreter.  Examples of symbolic or conventional signs, of course, abide even more deeply in the background, pervading the whole context and making up the very fabric of this discussion.

In the pragmatic theory of sign relations, three natural kinds of signs are recognized, under the names of ''icons'', ''indices'', and ''symbols''.  Examples of indexical or accessional signs figured significantly in the discussion of <math>A\!</math> and <math>B,\!</math> as illustrated by the pronouns "i" and "u" in <math>S.\!</math> Examples of iconic or analogical signs were also present, though keeping to the background, in the very form of the sign relation Tables that were used to schematize the whole activity of each interpreter.  Examples of symbolic or conventional signs, of course, abide even more deeply in the background, pervading the whole context and making up the very fabric of this discussion.

In order to deal with the array of issues presented so far in this subsection, all of which have to do with controlling the generative power of sign relations to serve the specific purposes of understanding, I apply the previously introduced concept of an ''objective genre'' (OG).  This is intended to be a determinate purpose or a deliberate pattern of analysis and synthesis that one can identify as being active at given moments in a discussion and that affects what one regards as the relevant structural properties of its objects.

In order to deal with the array of issues presented so far in this subsection, all of which have to do with controlling the generative power of sign relations to serve the specific purposes of understanding, I apply the previously introduced concept of an ''objective genre'' (OG).  This is intended to be a determinate purpose or a deliberate pattern of analysis and synthesis that one can identify as being active at given moments in a discussion and that affects what one regards as the relevant structural properties of its objects.

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A symbol like <math>x \lessdot</math> or <math>x \gtrdot</math>, with extra spaces or dots being optional, is called a ''catenation'', where <math>x\!</math> is the ''catenand'' and <math>\lessdot</math> or <math>\gtrdot</math> is the ''catenator''.  Due to the fact that <math>^{\backprime\backprime} \lessdot ^{\prime\prime}</math> and <math>^{\backprime\backprime} \gtrdot ^{\prime\prime}</math> indicate dyadic relations, the significance of these so-called ''unsaturated'' catenations can be rationalized as follows:

A symbol like <math>^{\backprime\backprime} x \lessdot ^{\prime\prime}</math> or <math>^{\backprime\backprime} x \gtrdot ^{\prime\prime}</math> is called a ''catenation'', where <math>^{\backprime\backprime} x ^{\prime\prime}</math> is the ''catenand'' and <math>^{\backprime\backprime} \lessdot ^{\prime\prime}</math> or <math>^{\backprime\backprime} \gtrdot ^{\prime\prime}</math> is the ''catenator''.  Due to the fact that <math>^{\backprime\backprime} \lessdot ^{\prime\prime}</math> and <math>^{\backprime\backprime} \gtrdot ^{\prime\prime}</math> indicate dyadic relations, the significance of these so-called ''unsaturated'' catenations can be rationalized as follows:

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