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how sign relations can be used to clarify the very languages that
how sign relations can be used to clarify the very languages that
are used to talk about them.
are used to talk about them.
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'''1.3.10. Recurring Themes (CFR Version)'''
<pre>
The overall purpose of the next sixteen Subsections is threefold:
1. To continue to illustrate the salient properties of sign relations
in the medium of selected examples.
2. To demonstrate the use of sign relations to analyze and to clarify
a particular order of difficult symbols and complex texts, namely,
those that involve recursive, reflective, or reflexive features.
3. To begin to suggest the implausibility of understanding this order
of phenomena without using sign relations or something like them,
namely, concepts with the power of 3-adic relations.
The prospective lines of an inquiry into inquiry cannot help but meet at
various points, where a certain entanglement of the subjects of interest
repeatedly has to be faced. The present discussion of sign relations is
currently approaching one of these points. As the work progresses, the
formal tools of logic and set theory become more and more indispensable
to say anything significant or to produce any meaningful results in the
study of sign relations. And yet it appears, at least from the vantage
of the pragmatic perspective, that the best way to formalize, to justify,
and to sharpen the use of these tools is by means of the sign relations
that they involve. And so the investigation shuffles forward on two or
more feet, shifting from a stance that fixes on a certain level of logic
and set theory, using it to advance the understanding of sign relations,
and then exploits the leverage of this new pivot to consider variations,
and hopefully improvements, in the very language of concepts and terms
that one uses to express questions about logic and sets, in all of its
aspects, from syntax, to semantics, to the pragmatics of both human and
computational interpreters.
The main goals of this Section are as follows:
1. To introduce a basic logical notation and a naive theory of sets,
just enough to treat sign relations as the set-theoretic extensions
of logically expressible concepts.
2. To use this modicum of formalism to define a number of conceptual
constructs, useful in the analysis of more general sign relations.
3. To develop a proof format that is amenable to deriving facts about
these constructs in careful and potentially computational fashions.
4. More incidentally, but increasingly effectively, to get a sense
of how sign relations can be used to clarify the very languages
that are used to talk about them.
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appearing in evidence, can always be interpreted as a piece of
appearing in evidence, can always be interpreted as a piece of
evidence that some sort of sampling relation is being applied.
evidence that some sort of sampling relation is being applied.
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appearing in evidence, can always be interpreted as a piece of evidence
appearing in evidence, can always be interpreted as a piece of evidence
that some sort of sampling relation is being applied.
that some sort of sampling relation is being applied.
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=====1.3.10.2. Intermediary Notions=====
<pre>
A number of additional definitions are relevant to sign relations whose
connotative components constitute equivalence relations, if only in part.
A "dyadic relation on a single set" (DROSS) is a non-empty set of points
plus a set of ordered pairs on these points. Until further notice, any
reference to a "dyadic relation" is intended to be taken in this sense,
in other words, as a reference to a DROSS.
When the maximum precision of notation is needed, a dyadic relation !G!
will be given in the form !G! = <G(1), G(2)>, where G(1) is a non-empty
set of points and G(2) c G(1) x G(1) is a set of ordered pairs from G(1).
At other times, a dyadic relation may be specified in the form <X, G>, where
X is the set of points and where G c X x X is the set of ordered pairs that
go together to define the relation. This option is often used in contexts
where the set of points is understood, and thus it becomes convenient to
call the whole relation <X, G> by the name of its second set G c X x X.
A "subrelation" of a dyadic relation !G! = <X, G> = <G(1), G(2)>
is a dyadic relation !H! = <Y, H> = <H(1), H(2)> that has all of
its points and pairs in !G!, more precisely, that has all of its
point-set Y c X and all of its pair-set H c G.
The "induced subrelation on a subset" (ISOS), taken with respect to
the dyadic relation G c X x X and the subset Y c X, is the maximal
subrelation of G whose points belong to Y. In other words, it is
the dyadic relation on Y whose extension contains all of the pairs
of Y x Y that appear in G. Since the construction of an ISOS is
uniquely determined by the data of G and Y, it can be represented
as a function of those arguments, as in the notation ISOS(G, Y),
which can be denoted more briefly as !G!_Y. Using the symbol
"|^|" to indicate the intersection of sets, the construction
of !G!_Y = ISOS(G, Y) can be defined as follows:
!G!_Y = <Y, G_Y> = <G_Y (1), G_Y (2)>
= <Y, {<x, y> in Y x Y : <x, y> in G(2)}>
= <Y, Y x Y |^| G(2)>
These definitions for dyadic relations can now be applied in a context where
each fragment of a sign relation that is being considered satisfies a special
set of conditions. Namely, if F is the fragment under consideration, we have:
1. Syntactic Domain X = Sign Domain S = Interpretant Domain I.
2. Connotative Component = F_XX = F_SI = Equivalence Relation E.
With regard to fragments of sign relations that satisfy these conditions,
it is useful to consider further selections of a specialized sort, namely,
those that keep equivalent signs synonymous. Here is a first description:
An "arbit" of a sign relation is a more judicious fragment of it, preserving
a semblance of whatever SEP happens to rule over its signs, and respecting the
semiotic parts of the sampled sign relation, when it has such parts. That is,
an arbit suggests an act of selection that represents the parts of the original
SEP by means of the parts of the resulting SEP, that extracts an ISOS of each
clique in the SER that it bothers to select any points at all from, and that
manages to portray in at least this partial fashion all or none of every SEC
that appears in the original sign relation.
The use of these ideas will become clear when we
meet with concrete examples of their application.
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