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SR. Commentary Note 1

| Thus, if a sunflower, in turning towards the sun, becomes by that very act
| fully capable, without further condition, of reproducing a sunflower which
| turns in precisely corresponding ways toward the sun, and of doing so with
| the same reproductive power, the sunflower would become a Representamen of
| the sun.
|
| C.S. Peirce, "Syllabus" (c. 1902), 'Collected Papers', CP 2.274

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SR. Commentary Note 2

I record here my analysis of one of Peirce's
definitions of a sign relation that I gave
a few years back.

Here are two versions of the definition,
given in the context of defining logic:

| On the Definition of Logic [Version 1]
|
| Logic will here be defined as 'formal semiotic'.
| A definition of a sign will be given which no more
| refers to human thought than does the definition
| of a line as the place which a particle occupies,
| part by part, during a lapse of time. Namely,
| a sign is something, 'A', which brings something,
| 'B', its 'interpretant' sign determined or created
| by it, into the same sort of correspondence with
| something, 'C', its 'object', as that in which it
| itself stands to 'C'. It is from this definition,
| together with a definition of "formal", that I
| deduce mathematically the principles of logic.
| I also make a historical review of all the
| definitions and conceptions of logic, and show,
| not merely that my definition is no novelty, but
| that my non-psychological conception of logic has
| 'virtually' been quite generally held, though not
| generally recognized. (CSP, NEM 4, 20-21).
|
| On the Definition of Logic [Version 2]
|
| Logic is 'formal semiotic'. A sign is something,
| 'A', which brings something, 'B', its 'interpretant'
| sign, determined or created by it, into the same
| sort of correspondence (or a lower implied sort)
| with something, 'C', its 'object', as that in
| which itself stands to 'C'. This definition no
| more involves any reference to human thought than
| does the definition of a line as the place within
| which a particle lies during a lapse of time.
| It is from this definition that I deduce the
| principles of logic by mathematical reasoning,
| and by mathematical reasoning that, I aver, will
| support criticism of Weierstrassian severity, and
| that is perfectly evident. The word "formal" in
| the definition is also defined. (CSP, NEM 4, 54).
|
| Charles Sanders Peirce,
|'The New Elements of Mathematics', Volume 4,
| Carolyn Eisele (ed.), Mouton, The Hague, 1976.
|
| Also available at the Arisbe website:
| http://members.door.net/arisbe/menu...csp/l75/l75.htm

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SR. Commentary Note 3

At this point it may be useful to remark that most of the
complexity of the above unpacking is due to the fact that
we put the burden of maintaining the entire sign relation
on the backs of the individual signs, as if it were their
job to carry the load all by themselves. Perhaps this is
true in actual point of fact, but for the sake of logical
analysis it is much more convenient and achieves much the
same effect to think of the sign relation separately as a
structure that is preserved invariant under the allowable
processes of sign relational transformation, or semiosis.

Jon Awbrey

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SR. Commentary Note 4

It is frequently useful to approach the concept of an inquiry process
as a specialization of a sign relation, in the following three phases:

1. A "sign relation" simpliciter, L c O x S x I, could be just about
   any 3-adic relation on the arbitrary domains O, S, I, so long as
   it satisfies one of the adequate definitions of a sign relation.
   
   
2. A "sign process" is a sign relation plus a significant sense of transition.
   This means that there is a definite, non-trivial sense in which a sign can
   be said to determine in the fullness of time one or more interpretant signs
   with regard to its objects. We often find ourselves writing "<o, s, i>" as
   "<o, s, s'> in such cases, where the semiotic transition s ~> s' takes place
   in respect of the object o.
   
   
3. An "inquiry process" is a sign process that has value-directed transitions.
   This means that there is a property, a quality, or a scalar value that can
   be associated with a sign in relation to its objects, and that the transit
   from a sign to an interpretant in regard to an object occurs in such a way
   that the value is increased in the process. For example, semiotic actions
   like inquiry and computation are directed in such a way as to increase the
   qualities of alacrity, brevity, or clarity of the signs on which they work.
   

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SR. Commentary Note 5

There are systems whose states and developments of states
bear witness to objective realities and real objectives
that transcend the mere immediacy of those states and
developments. How is this possible? When it happens
we say that some aspects of state codify or denote
the elements and properties of an object system,
which may well be the system itself or perhaps
yet another system. The study of such cases
is the subject matter of semiotics, as cast
in the focus of a system-theoretic light.

Surprisingly enough, the advance of an abstract science like
logic, mathematics, or semiotics depends on the accumulation
of a large stock of well-studied concrete examples. We have
lots of those in logic and math, but not so many non-trivial
examples as yet in semiotics. It will be necessary to begin
with some very elementary, but not quite trivial examples of
sign relations, that we may anticipate as affording us clues
to more complex examples down the line.

In this spirit, let me revive once more the "Story of A and B",
where we fix on those aspects of sign use between two people,
say, Ann and Bob, that concern their use of their own proper
names, "Ann" and "Bob", along with the pronouns "I" and "you".
For the sake of brevity, I abbreviate these four signs to the
set {"A", "B", "i", "u"}. A maximally abstract consideration
of how A and B use these signs to refer to themselves and to
each other leads to the contemplation of two sign relations,
L(A) and L(B), as employed by A and B, respectively.

The 3-adic sign relations, L(A) and L(B), each consist of eight triples
of the form <x, y, z>, where the object x belongs to the object domain
!O! = {A, B}, where the sign y belongs to the sign domain !S!, where
the interpretant sign z belongs to the interpretant domain !I!, and
where it happens in this case that !S! = !I! = {"A", "B", "i", "u"}.
In general, it is convenient to refer to the union !S! |_| !I! as
the "syntactic domain", but in this case !S! = !I! = !S! |_| !I!.

Tables 1 and 2 present the triples of L(A) and L(B), respectively.

Table 1. Sign Relation of Interpreter A
o---------------o---------------o---------------o
| Object` ` ` ` | Sign` ` ` ` ` | Interpretant` |
o---------------o---------------o---------------o
| A ` ` ` ` ` ` | "A" ` ` ` ` ` | "A" ` ` ` ` ` |
| A ` ` ` ` ` ` | "A" ` ` ` ` ` | "i" ` ` ` ` ` |
| A ` ` ` ` ` ` | "i" ` ` ` ` ` | "A" ` ` ` ` ` |
| A ` ` ` ` ` ` | "i" ` ` ` ` ` | "i" ` ` ` ` ` |
| B ` ` ` ` ` ` | "B" ` ` ` ` ` | "B" ` ` ` ` ` |
| B ` ` ` ` ` ` | "B" ` ` ` ` ` | "u" ` ` ` ` ` |
| B ` ` ` ` ` ` | "u" ` ` ` ` ` | "B" ` ` ` ` ` |
| B ` ` ` ` ` ` | "u" ` ` ` ` ` | "u" ` ` ` ` ` |
o---------------o---------------o---------------o

Table 2. Sign Relation of Interpreter B
o---------------o---------------o---------------o
| Object` ` ` ` | Sign` ` ` ` ` | Interpretant` |
o---------------o---------------o---------------o
| A ` ` ` ` ` ` | "A" ` ` ` ` ` | "A" ` ` ` ` ` |
| A ` ` ` ` ` ` | "A" ` ` ` ` ` | "u" ` ` ` ` ` |
| A ` ` ` ` ` ` | "u" ` ` ` ` ` | "A" ` ` ` ` ` |
| A ` ` ` ` ` ` | "u" ` ` ` ` ` | "u" ` ` ` ` ` |
| B ` ` ` ` ` ` | "B" ` ` ` ` ` | "B" ` ` ` ` ` |
| B ` ` ` ` ` ` | "B" ` ` ` ` ` | "i" ` ` ` ` ` |
| B ` ` ` ` ` ` | "i" ` ` ` ` ` | "B" ` ` ` ` ` |
| B ` ` ` ` ` ` | "i" ` ` ` ` ` | "i" ` ` ` ` ` |
o---------------o---------------o---------------o

Jon Awbrey

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SR. Commentary Note 6

Let us now discuss the way in which the sign relations
L(A) and L(B) satisfy the definition of sign relations
that was given earlier. This involves us in detailing
the forms of correspondence and determination that are
exemplified by L(A) and L(B). At the present level of
abstraction, we are dealing with sign relations simple,
with no parameters of time and causality yet specified,
and thus we have no specification of the corresponding
sign processes, though it can be said that these forms
produce all of the sign processes that conform to them.

As it happens, all of the various aspects of correspondence
and determination can be subsumed under the general concept
of "constraint", as it is understood in information theory.
For example, the fact that the sign relations L(A) and L(B)
are proper subsets of the cartesian product !O! x !S! x !I!
implies that a non-trivial constraint is present, and such
a constraint can generally be exploited somehow or another
for the sake of conveying information. More specifically,
we can say that the object A constrains or determines the
sign and the interpretant within the sign relation L(B) in
that the set of pairs from the SI plane !S! x !I! that are
compatible with having A in the object role, in other words,
that form triples of L(B) when combined with A as an object,
is a proper subset of the SI plane, specifically, the set of
four pairs {<"A", "A">, <"A", "u">, <"u", "A">, <"u", "u">}.
This is "determination" in the formal or mathematical sense,
as when we treat of algebras, geometries, or combinatorial
designs of various sorts. Thus we can speak of how signs
determine objects and interpretants, or how interpretants
determine objects and signs, all with equal facility.

Jon Awbrey

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SR. Commentary Note 7

It is well to observe that the form of determination that we find
in an abstract sign relation is not of necessity an "absolute" or
a "functional" determination. For example, given the object B in
the sign relation L(A), there two possible signs for it, namely,
"B" and "u". This amounts to a non-trivial constraint, since
the set of signs {"B", "u"} is a proper subset of !S!, but
it does not amount to an absolute determination, since
the corresponding 2-adic relation between !O! and !S!
is not a function from !O! to !S!, there being more
than one relational image of some objects in !O!.
In fact, this is true for every object in !O!.
In contrast, given any sign in the domain !S!
of L(A), there is exactly one object in the
domain !O! that corresponds to it, and so
the corresponding 2-adic relation from
!S! to !O! is a function. Thus we may
write it in the form f : !S! -> !O!.
This relation is the "denotation"
relation, and it happens in this
case to be a function for both
sign relations L(A) and L(B).

Jon Awbrey

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SR. Commentary Note 8

Let us now discuss another issue that often comes up in discussing
sign relations, the question of their possible cardinalities. The
recursive form of the usual definition of a sign relation may lead
one to think that sign relations are necessarily infinite, but the
definition is in fact perfectly consistent with the existence of
finite sign relations, that is, those that have a finite number
of triples. For example, L(A) and L(B) each have eight triples.

The definition of a sign relation does require that each sign of
a given object requires the existence of a further sign of the same
object to fill the role of its interpretant, which in turn requires
the existence of a further sign of the same object to fill the role
of its interpretant, and so on, 'ad infinitum', one might say, but
this does not require that all of these signs be different or even
that they succeed one another in time. At this highest level of
abstraction the sign relation demands nothing more than a purely
formal or logical determination, not a causal or temporal one.
Accordingly, it is best to view this form of determination as
a form of closure, one that can be filled by a finite number
of signs, interpretant signs, and sign relational triples.

Jon Awbrey

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SR. Commentary Note 9

Figures 3 and 4 present graphic representations
of the sign relational triples in L(A) and L(B).
Here, the fact that interpreter J uses a triple
of the form <x, y, z> is represented as follows:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `y` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` x o------J` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `z` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Note, however, that the full set of sixteen
triples is partitioned differently than was
shown in Tables 1 and 2, this time arranged
around their respective objects rather than
according to their respective interpreters.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `"i"` `"i"` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o o` `o o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `|` \ / `|` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` "A" o--A` `A` `A--o "A" ` ` ` ` ` ` |
| ` ` ` ` ` ` "A" o ` \ `|` / ` o "A" ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `o o o` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` A--o A o--B ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` `o o o` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` "A" o ` / `|` \ ` o "A" ` ` ` ` ` ` |
| ` ` ` ` ` ` "A" o--B` `B` `B--o "A" ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `|` / \ `|` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o o` `o o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `"u"` `"u"` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 3. Triples of L(A) and L(B) with Object A

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `"i"` `"i"` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o o` `o o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `|` \ / `|` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` "B" o--B` `B` `B--o "B" ` ` ` ` ` ` |
| ` ` ` ` ` ` "B" o ` \ `|` / ` o "B" ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `o o o` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` B--o B o--A ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` `o o o` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` "B" o ` / `|` \ ` o "B" ` ` ` ` ` ` |
| ` ` ` ` ` ` "B" o--A` `A` `A--o "B" ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `|` / \ `|` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o o` `o o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `"u"` `"u"` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 4. Triples of L(A) and L(B) with Object B

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SR. Commentary Note 10

We touched briefly on the 2-adic "denotation relation", that
we derive or project from a 3-adic sign relation by ignoring
the interpretant column of the relational table, so to speak.
In the case of L(A) and L(B), the denotation relations yield
functions of the form f : !S! -> !O! that can be pictured as
shown in Figures 5 and 6, respectively.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` "A" ` ` "i" ` ` "B" ` ` "u" ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` `o` ` ` `o` ` ` `o` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `A` ` ` ` ` ` ` `B` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 5. Denotation Relation Derived from L(A)

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` "A" ` ` "u" ` ` "B" ` ` "i" ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` `o` ` ` `o` ` ` `o` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `A` ` ` ` ` ` ` `B` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 6. Denotation Relation Derived from L(B)

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SR. Commentary Note 11

If we ignore the object column in a sign relational table, and
focus only on the 2-adic relation between signs and interpretants,
we end up with what, for the momentary lack of better name, can be
called the "connotation relation" that is derived or projected from
the sign relation in question. The interpretant sign is often said
to be an equivalent or implied sign in relation to the sign that it
interprets. Thus we may expect sign relations, if they are fully
filled out, to yield connotation relations that are equivalence
relations, that is, reflexive, symmetric, transitive relations.
Indeed, this is just what happens in the case of L(A) and L(B).

Tables 7 and 8 show the results of deleting the object columns from
Tables 1 and 2, respectively, ignoring repeated pairs in what remains.
This gives us what is called the "projection" of L(A) and L(B) on the
SI plane, which may be notated as Proj_SI (L(A)) and Proj_SI (L(B))
or more simply as L(A)_SI and L(B)_SI, respectively.

Table 7. L(A)_SI c !S! x !I!
o---------------o---------------o
| Sign` ` ` ` ` | Interpretant` |
o---------------o---------------o
| "A" ` ` ` ` ` | "A" ` ` ` ` ` |
| "A" ` ` ` ` ` | "i" ` ` ` ` ` |
| "i" ` ` ` ` ` | "A" ` ` ` ` ` |
| "i" ` ` ` ` ` | "i" ` ` ` ` ` |
o---------------o---------------o
| "B" ` ` ` ` ` | "B" ` ` ` ` ` |
| "B" ` ` ` ` ` | "u" ` ` ` ` ` |
| "u" ` ` ` ` ` | "B" ` ` ` ` ` |
| "u" ` ` ` ` ` | "u" ` ` ` ` ` |
o---------------o---------------o

Table 8. L(B)_SI c !S! x !I!
o---------------o---------------o
| Sign` ` ` ` ` | Interpretant` |
o---------------o---------------o
| "A" ` ` ` ` ` | "A" ` ` ` ` ` |
| "A" ` ` ` ` ` | "u" ` ` ` ` ` |
| "u" ` ` ` ` ` | "A" ` ` ` ` ` |
| "u" ` ` ` ` ` | "u" ` ` ` ` ` |
o---------------o---------------o
| "B" ` ` ` ` ` | "B" ` ` ` ` ` |
| "B" ` ` ` ` ` | "i" ` ` ` ` ` |
| "i" ` ` ` ` ` | "B" ` ` ` ` ` |
| "i" ` ` ` ` ` | "i" ` ` ` ` ` |
o---------------o---------------o

Each connotation relation takes on the structure of an "equivalence relation".
In other words, L(A)_SI and L(B)_SI are reflexive, symmetric, and transitive.
First, !S! = !I!, so we can say that both relations are subsets of !S! x !S!.
Each relation L is reflexive, since the pair <x, x> is in L for all x in !S!.
Each relation L is symmetric, because <x, y> in L implies that <y, x> is too.
Each relation L is transitive, because <x, y> and <y, z> in L => <x, z> in L.

When you have an equivalence relation in !S! x !S!, the elements of the
underlying set !S! can always be partitioned into "equivalence classes"
of elements that are all related to each other by the relation, while
elements in different classes are not so related. In this case, the
equivalence classes are {"A", "i"} and {"B", "u"} for interpreter A,
while they are {"A", "u"} and {"B", "i"} for interpreter B.

Jon Awbrey

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SR. Commentary Note 12

In cases of sign relations like the ones we are considering,
the denotative component and the connotative component exist
in a coherent relationship to one another. If we examine the
situation with the sign relations L(A) and L(B) we can see that
the denotation relations L(A)_SO and L(B)_SO map the equivalence
classes of the connotation relations L(A)_SI and L(B)_SI onto the
objects of !O! in such a way that all of the signs in a distinct
equivalence class are mapped onto the same distinct object of !O!.
For the interpreter A, the class of signs {"A", "i"} maps to the
object A and the class of signs {"B", "u"} maps to the object B.
For the interpreter B, the class of signs {"A", "u"} maps to the
object A and the class of signs {"B", "i"} maps to the object B.

Now this is very pretty, and some people get so enamored of it that
they would even say you can now do away with the objects themselves,
having "explained them away" or "reconstructed" them as equivalence
classes of syntactic entities. Some folks read Frege this way, for
instance. But there are several good reasons for stopping short of
that extreme. One reason is the non-uniqueness of the construction,
in other words, the partition into equivalence classes is different
for each interpreter. This is a very general phenomenon, betraying
a certain "point of view relativity" in the way that the structures
of an objective world are represented in the structures of language.

Jon Awbrey

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SR. Commentary Note 13

We have just seen one of the most primitive of linguistic forms in which
the widely distributed phenomenon of POV-relativity arises, namely, with
the understanding of so-called demonstratives, indexicals, pronouns, and
words like that. What's the "ontology" of words like "I", "you", "they",
"this", "that"? Silly way to ask the question. There is just no way to
understand such words by way of a simple fixed map from signs to objects.

Even when the structure of the object domain !O! is reconstructed on the
semiotic plane !S! x !I!, by partitioning signs into equivalence classes
of some sort, it's a sure bet that the details of the representation are
going to be different for different interpreters.

The kicker of the story is this: When you really begin to look at how
people actually use language, you will find that their use of words is
far more personalized, that is, far more indexed to their own peculiar
identities, and thus far more like demonstratives, indexicals, pronouns,
and so on, than most people would like to think, and so it is only with
a whole lot of luck and work that we ever begin to extract common senses
out of the massa confusa of all this initial idiosyncracy.

Jon Awbrey

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