User:Jon Awbrey/Sign Relations Archive

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Contents

Sign Relations

SR. Selection 1


| 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 itself stands to ''C''.
|
| C.S. Peirce, NEM 4, pp. 20-21 and cf. p. 54, online at:
| http://www.cspeirce.com/menu/library/bycsp/l75/l75.htm
| http://www.cspeirce.com/menu/library/bycsp/l75/ver1/toc.htm
| http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-05.htm#m12
|
| ''The New Elements of Mathematics'', Volume 4,
| Carolyn Eisele (ed.), Mouton, The Hague, 1976.

More details on how the definition of a sign relation bears on
the definition of logic are given in the contexts of this text:

| 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 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.  (C.S. Peirce, 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.  (C.S. Peirce, NEM 4, 54).
|
| ''The New Elements of Mathematics'', Volume 4,
| Carolyn Eisele (ed.), Mouton, The Hague, 1976.

SR. Selection 2


| How often do we think of the thing in algebra?
| When we use the symbol of multiplication we do not
| even think out the conception of multiplication, we think
| merely of the laws of that symbol, which coincide with the
| laws of the conception, and what is more to the purpose,
| coincide with the laws of multiplication in the object.
| Now, I ask, how is it that anything can be done with
| a symbol, without reflecting upon the conception,
| much less imagining the object that belongs to it?
| It is simply because the symbol has acquired a nature,
| which may be described thus, that when it is brought before
| the mind certain principles of its use -- whether reflected on
| or not -- by association immediately regulate the action of the
| mind;  and these may be regarded as laws of the symbol itself
| which it cannot 'as a symbol' transgress.
|
| C.S. Peirce, 'Chronological Edition', CE 1, p. 173.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

SR. Selection 3


| Logic, in its general sense, is, as I believe I have shown, only another
| name for 'semiotic' [Greek: 'semeiotike'], the quasi-necessary, or formal,
| doctrine of signs.  By describing the doctrine as "quasi-necessary", or
| formal, I mean that we observe the characters of such signs as we know,
| and from such an observation, by a process which I will not object to
| naming Abstraction, we are led to statements, eminently fallible, and
| therefore in one sense by no means necessary, as to what 'must be' the
| characters of all signs used by a "scientific" intelligence, that is to say,
| by an intelligence capable of learning by experience.  As to that process of
| abstraction, it is itself a sort of observation.  The faculty which I call
| abstractive observation is one which ordinary people perfectly recognize,
| but for which the theories of philosophers sometimes hardly leave room.
| It is a familiar experience to every human being to wish for something
| quite beyond his present means, and to follow that wish by the question,
| "Should I wish for that thing just the same, if I had ample means to gratify it?"
| To answer that question, he searches his heart, and in doing so makes what I term
| an abstractive observation.  He makes in his imagination a sort of skeleton diagram,
| or outline sketch, of himself, considers what modifications the hypothetical state
| of things would require to be made in that picture, and then examines it, that is,
| 'observes' what he has imagined, to see whether the same ardent desire is there to
| be discerned.  By such a process, which is at bottom very much like mathematical
| reasoning, we can reach conclusions as to what 'would be' true of signs in all
| cases, so long as the intelligence using them was scientific.  (CP 2.227).
|
| Charles Sanders Peirce, 'Collected Papers', CP 2.227,
| Editor Data:  From An Unidentified Fragment, c. 1897.

SR. Selection 4


| A 'Sign', or 'Representamen', is a First which stands
| in such a genuine triadic relation to a Second, called
| its 'Object', as to be capable of determining a Third,
| called its 'Interpretant', to assume the same triadic
| relation to its Object in which it stands itself to
| the same Object.
|
| The triadic relation is 'genuine', that is, its three members are
| bound together by it in a way that does not consist in any complexus
| of dyadic relations.  That is the reason the Interpretant, or Third,
| cannot stand in a mere dyadic relation to the Object, but must stand
| in such a relation to it as the Representamen itself does.
|
| Nor can the triadic relation in which the Third stands be merely similar
| to that in which the First stands, for this would make the relation of the
| Third to the First a degenerate Secondness merely.  The Third must indeed
| stand in such a relation, and thus must be capable of determining a Third
| of its own;  but besides that, it must have a second triadic relation in
| which the Representamen, or rather the relation thereof to its Object,
| shall be its own (the Third's) Object, and must be capable of determining
| a Third to this relation.  All this must equally be true of the Third's
| Third and so on endlessly;  and this, and more, is involved in the familiar
| idea of a Sign;  and as the term Representamen is here used, nothing more
| is implied.
|
| A 'Sign' is a Representamen with a mental Interpretant.
|
| Possibly there may be Representamens that are not Signs.
|
| 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.
|
| But 'thought' is the chief, if not the only, mode of representation.
|
| C.S. Peirce, "Syllabus" (c. 1902), 'Collected Papers', CP 2.274

SR. Selection 5


| But not to follow this subject too far, we have now established three species
| of representations:  'copies', 'signs', and 'symbols';  of the last of which
| only logic treats.  A second approximation to a definition of it then will
| be, the science of symbols in general and as such.  But this definition
| is still too broad;  this might, indeed, form the definitiun of a
| certain science which would be a branch of Semiotic or the general
| science of representations which might be called Symbolistic, and
| of this logic would be a species.  But logic only considers
| symbols from a particular point of view.
|
| C.S. Peirce, 'Chronological Edition', CE 1, p. 174.
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

SR. Selection 6


| I have spoken of real relations as reactions.
| It may be asked how far I mean to say that all
| real relations are reactions.  It is seldom that
| one falls upon so fascinating a subject for a train
| of thought [as] the analysis of that problem in all
| its ramifications, mathematical, physical, biological,
| sociological, psychological, logical, and so round to the
| mathematical again.  The answer cannot be satisfactorily given
| in a few words;  but it lies hidden beneath the obvious truth
| that any exact necessity is expressible by a general equation;
| and nothing can be added to one side of a general equation without
| an equal addition to the other.  Logical necessity is the necessity
| that a sign should be true to a 'real' object;  and therefore there
| is 'logical' reaction in every real dyadic relation.  If 'A' is in
| a real relation to 'B', 'B' stands in a logically contrary relation
| to 'A', that is, in a relation at once converse to and inconsistent
| with the direct relation.  For here we speak [not] of a vague sign of
| the relation but of the relation between two individuals, 'A' and 'B'.
| This very relation is one in which 'A' alone stands to any individual,
| and it to 'B' only.  There are, however, 'degenerate' dyadic relations, --
| 'degenerate' in the sense in which two coplanar lines form a 'degenerate'
| conic, -- where this is not true.  Namely, they are individual relations
| of identity, such as the relation of 'A' to 'A'.  All mere resemblances
| and relations of reason are of this sort.
|
| C.S. Peirce, ["Kaina Stoicheia"], NEM 4, 241
|
| C.S. Peirce, ["Kaina Stoicheia"], MS 517 (1904), pp. 235-263 in:
| Carolyn Eisele (ed.), 'The New Elements of Mathematics by
| Charles S. Peirce, Volume 4, Mathematical Philosophy',
| Mouton, The Hague, 1976.
|
| Cf. "New Elements", pp. 300-324 in 'The Essential Peirce, Volume 2 (1893-1913)',
| Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1998.

SR. Selection 7


| Of signs there are two different degenerate forms.
| But though I give them this disparaging name, they
| are of the greatest utility, and serve purposes that
| genuine signs could not.
|
| The more degenerate of the two forms (as I look upon it)
| is the 'icon'.  This is defined as a sign of which the
| character that fits it to become a sign of the sort
| that it is, is simply inherent in it as a quality
| of it.
|
| For example, a geometrical figure drawn on paper may
| be an 'icon' of a triangle or other geometrical form.
|
| If one meets a man whose language one does not know
| and resorts to imitative sounds and gestures, these
| approach the character of an icon.  The reason they
| are not pure icons is that the purpose of them is
| emphasized.
|
| A pure icon is independent of any purpose.  It serves as a sign
| solely and simply by exhibiting the quality it serves to signify.
| The relation to its object is a degenerate relation.  It asserts
| nothing.  If it conveys information, it is only in the sense in
| which the object that it is used to represent may be said to
| convey information.  An 'icon' can only be a fragment of
| a completer sign.
|
| C.S. Peirce, ["Kaina Stoicheia"], NEM 4, 241-242
|
| C.S. Peirce, ["Kaina Stoicheia"], MS 517 (1904), pp. 235-263 in:
| Carolyn Eisele (ed.), 'The New Elements of Mathematics by
| Charles S. Peirce, Volume 4, Mathematical Philosophy',
| Mouton, The Hague, 1976.
|
| Cf. "New Elements", pp. 300-324 in 'The Essential Peirce, Volume 2 (1893-1913)',
| Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1998.

SR. Selection 8


| The other form of degenerate sign is to be termed an 'index'.
| It is defined as a sign which is fit to serve as such by
| virtue of being in a real reaction with its object.
|
| For example, a weather-cock is such a sign.  It is fit to
| be taken as an index of the wind for the reason that it is
| physically connected with the wind.  A weather-cock conveys
| information;  but this it does because in facing the very
| quarter from which the wind blows, it resembles the wind
| in this respect, and thus has an icon connected with it.
| In this respect it is not a pure index.
|
| A pure index simply forces attention to the object
| with which it reacts and puts the interpreter into
| mediate reaction with that object, but conveys no
| information.
|
| As an example, take an exclamation "Oh!"
|
| The letters attached to a geometrical figure are another case.
|
| Absolutely unexceptionable examples of degenerate forms must not be expected.
| All that is possible is to give examples which tend sufficiently in towards
| those forms to make the mean suggest what is meant.
|
| It is remarkable that while neither a pure icon nor a pure index
| can assert anything, an index which forces something to be an 'icon',
| as a weather-cock does, or which forces us to regard it as an 'icon',
| as the legend under a portrait does, does make an assertion, and forms
| a 'proposition'.  This suggests the true definition of a proposition,
| which is a question in much dispute at this moment.  A proposition
| is a sign which separately, or independently, indicates its object.
|
| No 'index', however, can be an 'argumentation'.  It may be what many
| writers call an 'argument;  that is, a basis of argumentation;  but an
| argument in the sense of a sign which separately shows what interpretant
| it is intended to determine it cannot be.
|
| C.S. Peirce, ["Kaina Stoicheia"], NEM 4, 242
|
| C.S. Peirce, ["Kaina Stoicheia"], MS 517 (1904), pp. 235-263 in:
| Carolyn Eisele (ed.), 'The New Elements of Mathematics by
| Charles S. Peirce, Volume 4, Mathematical Philosophy',
| Mouton, The Hague, 1976.
|
| Cf. "New Elements", pp. 300-324 in 'The Essential Peirce, Volume 2 (1893-1913)',
| Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1998.

SR. Selection 9


| It will be observed that the icon is very perfect in respect to signification,
| bringing its interpreter face to face with the very character signified.*  For
| this reason, it is the mathematical sign 'par excellence'.  But in denotation
| it is wanting.  It gives no assurance that any such object as it represents
| really exists.
|
| The index on the other hand does this most perfectly, actually bringing to the
| interpreter the experience of the very object denoted.  But it is quite wanting
| in signification* unless it involves an iconic part.
|
| C.S. Peirce, ["Kaina Stoicheia"], NEM 4, 242-243
|
| C.S. Peirce, ["Kaina Stoicheia"], MS 517 (1904), pp. 235-263 in:
| Carolyn Eisele (ed.), 'The New Elements of Mathematics by
| Charles S. Peirce, Volume 4, Mathematical Philosophy',
| Mouton, The Hague, 1976.
|
| Cf. "New Elements", pp. 300-324 in 'The Essential Peirce, Volume 2 (1893-1913)',
| Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1998.

* [JA].  Peirce is here using "signifies" and "signification" in the sense
  of "connotes" and "connotation", as he stipulates earlier in the article:

| In addition however to 'denoting' objects every sign sufficiently
| complete 'signifies characters, or qualities.  (NEM 4, p. 239).

SR. Selection 10


| We now come to the genuine sign for which I propose the
| technical designation 'symbol', following a use of that
| word not infrequent among logicians including Aristotle.
| A symbol is defined as a sign which is fit to serve as
| such simply because it will be so interpreted.
|
| To recapitulate:
|
|               )                                          ( it possesses
|    An icon    }                                          ( the quality
|               )                                          ( signified.
|               )                                          (
|               )                                          ( it is in real
|               )                                          ( reaction
|    An index   > is a sign fit to be used as such because < with the
|               )                                          ( object
|               )                                          ( denoted.
|               )                                          (
|               )                                          ( it determines
|    A symbol   )                                          ( the interpretant
|               )                                          ( sign.
|
| C.S. Peirce, ["Kaina Stoicheia"], NEM 4, 243
|
| C.S. Peirce, ["Kaina Stoicheia"], MS 517 (1904), pp. 235-263 in:
| Carolyn Eisele (ed.), 'The New Elements of Mathematics by
| Charles S. Peirce, Volume 4, Mathematical Philosophy',
| Mouton, The Hague, 1976.
|
| Cf. "New Elements", pp. 300-324 in 'The Essential Peirce, Volume 2 (1893-1913)',
| Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1998.

Sign Relations • Comments

SR. Comment 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

To represent a 3-tuple <x, y, z> such that xy = z in a group G,
I used to draw a picture of something like the following shape:

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

Then I would chain these together in trees and cycles to represent the
more complex constructions that I had to think about in a given setting.

When I used this tactic to represent the elementary sign relations among
the sunflowers and the sun, I ended up with pictures something like this:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `s_1` `s_2` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o o` `o o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `|` \ / `|` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` s_8 o--o` `o` `o--o s_3 ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` \ `|` / ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `o o o` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o--o O o--o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` `o o o` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` / `|` \ ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` s_7 o--o` `o` `o--o s_4 ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `|` / \ `|` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o o` `o o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `s_6` `s_5` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
 
Of course, it could be any multitude of sunflowers, not just eight.

SR. Comment 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
| 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/library/bycsp/l75/l75.htm

Here is an extract of the second version:

| 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'.

It is important to note that the "correspondence" referred to here is
a "triple correspondence", what might be called a "3-place transaction"
in database terms.  Let us refer to this 3-place relation as L.  To say
that "A brings B to correspond with C in the same way that A corresponds
with C" is simply to say that A brings B into the same 3-place relation L
with something else B' and C, in that order, as A occupies in the 3-place
relation L with B and C, in that order.

To make this clearer, I will draw out the inference over several steps:

Using the acronym "SOC" for "sort of correspondence",
I form the following paraphrase of the main condition:

   A brings something into the same SOC with C
   as the SOC in which A stands to C.

In order to expand the expression correctly, one has to ask:

   What is the SOC in which A stands to C?

The SOC in which A stands to C is given by Formula 1.

   1.  A brings something into some SOC with C,
       namely, the SOC in which A stands to C.

If B enters into the same SOC with C as A stands in, that is to say,
if B takes up the same role that A had in Formula 1, then we obtain:

   2.  B brings something into some SOC with C,
       namely, the SOC in which A stands to C.

Let us suppose that B' is the something in Formula 2.
Now, B' can either be something old or something new.
If B' is something old, then it belongs to {A, B, C}.
If B' is something new, we leave it with the name B'.

In any case, we have:

   3.  B brings a thing B' into some SOC with C,
       namely, the SOC in which A stands to C.

In sum, the bringing of a new, possibly old, thing
into the relation is part of being in the relation.

SR. Comment 3


For ease of reference, I repeat here the explication
of the sign relation definition that I gave last time:

CSP: | 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'.

JA: | It is immportant to note that the "correspondence" referred to here is
    | a "triple correspondence", what might be called a "3-place transaction"
    | in database terms.  Let us refer to this 3-place relation as L.  To say
    | that "A brings B to correspond with C in the same way that A corresponds
    | with C" is simply to say that A brings B into the same 3-place relation L
    | with something else B' and C, in that order, as A occupies in the 3-place
    | relation L with B and C, in that order.
    |
    | To make this clearer, I will draw out the inference over several steps:
    |
    | Using the acronym "SOC" for "sort of correspondence",
    | I form the following paraphrase of the main condition:
    |
    |    A brings something into the same SOC with C
    |    as the SOC in which A stands to C.
    |
    | In order to expand the expression correctly, one has to ask:
    |
    |    What is the SOC in which A stands to C?
    |
    | The SOC in which A stands to C is given by Formula 1.
    |
    |    1.  A brings something into some SOC with C,
    |        namely, the SOC in which A stands to C.
    |
    | If B enters into the same SOC with C as A stands in, that is to say,
    | if B takes up the same role that A had in Formula 1, then we obtain:
    |
    |    2.  B brings something into some SOC with C,
    |        namely, the SOC in which A stands to C.
    |
    | Let us suppose that B' is the something in Formula 2.
    | Now, B' can either be something old or something new.
    | If B' is something old, then it belongs to {A, B, C}.
    | If B' is something new, we leave it with the name B'.
    |
    | In any case, we have:
    |
    |    3.  B brings a thing B' into some SOC with C,
    |        namely, the SOC in which A stands to C.
    |
    | In sum, the bringing of a new, possibly old, thing
    | into the relation is part of being in the relation.

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.

SR. Comment 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.

All in all, sign relations are not limited to purely linguistic types of systems.
They encompass the data of the senses, natural signs, and plastic representation,
just to name some randomly-chosen species of this very widely disseminated genus.

SR. Comment 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

SR. Comment 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.

SR. Comment 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).

SR. Comment 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.

SR. Comment 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

SR. Comment 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)

SR. Comment 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.

SR. Comment 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.

SR. Comment 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.

SR. Comment 14


My computer was down for the last couple of weeks, and I may have
lost some email to boot, or not to boot, as the case may be, and
so it may take me a while to catch up with the current state of
discussion.  In the interval I was forced to the extremity of
writing out my more irrepressible thoughts by means of pen
on paper, and so it seems like a good idea to emit those
patently pending quanta right away, before some dire
calamity overtakes that medium, too.

A motive of economy almost led me to recycle the disused acronym "EVINT",
here recontented as "Earned Virtuality Is Notoriously Testable", for the
sake of tying up assorted loose ends from previous discussions, but then
I thought better of it, and decided to keep the present thread under the
most generic heading that befits it.

To the purpose at hand, I am thinking of the many discussions that have been
left dangling in states of uncertainty because we found no way to pragmatize
the nearly equivalent concepts of ground, justification, reason, respect, or
virtue that were invoked in them.

For a pivotal instance, when we say that an icon is a sign
that receives its interpretant by virtue of a property that
it shares with its object, what exactly do we mean by that?
And how can we tell, in practical and operational terms, when
we have such a thing and when we do not?  In the sequel, I'll
stake out an approach to answering these and related questions.

SR. Comment 15


We take up the question of virtue in the case of an icon,
where we say that an icon is a sign that gets or begets
its interpretant sign by virtue of a property that it
bears in common with its object.

By way of starting out, experience teaches us to set aside a couple of meanings
that we probably don't mean.  For one thing, we probably don't mean to take the
sign itself as an admissible property of the object, even though this may well
be a conceivable interpretation of the word "property", say, in the sense of
accessory possessions, acquired encumbrances, or incidental real estate.
The reason that we exclude this interpretation is that we probably want
to count only absolute qualities of things, in particular, properties
that objects and signs possess "independently" of each other, in the
sense that we can tell whether something has such a property solely
by considering that thing, without consideration of anything else.
Whether anything at all has any such properties is another good
question, but for the sake of the present argument let us take
it for granted that it makes sense to speak of such qualities.
For another thing, continuing more generally along these very
same lines, the brands of properties that we are conceiving
the iconic object and the iconic sign to share are all of
them restricted to absolute, or non-relative properties.
There's a couple of catches to this, but I think that
it's best to leave them to another phase of inquiry.

SR. Comment 16


With the enabling provisions already set out,
let us consider the following organizational
chart for the primary offices of our concern:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `Quality q` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` . . ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `.` `.` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` . ` ` . ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `.` ` ` `.` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` . ` ` ` ` . ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `.` ` ` ` ` `o Sign y ` ` ` ` ` |
| ` ` ` ` ` ` ` ` . ` ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `.` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` . ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `.` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` . ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` Object x o-----------O` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o Interpretant z ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

The diagram will serve to remind us of the principal elements of
the idea in view, to wit:  An icon is a sign y that receives its
interpretant z by virtue of a property q that it shares with its
object x.  There are other words that can be substituted for the
appearances of "receives" and "by virtue of" here, but I imagine
that most readers will recall the rather lengthy variorum on the
same theme that we pursued earlier in the year.  If the vagaries
of vulgate paraphrase become too obstructive, as they very often
do, we can always fall back on the more exact formal definitions
of terms like "sign", "correspondence", "determines", "property",
and heaven help us, "prescindible".  But let's leave all that to
heaven, Plato's or otherwise, for now.

One other preliminary that I can foresee some cause to mention:
A diagram like this has its uses, the ones that I listed above,
but it has one terribly misleading defect if we try to take it
literally as a picture of an iconic sign relation, and this is
that, taken literally in a point-by-point fashion, it does not
suggest nearly enough triples of the form <x, y, z> to make up
anything like a non-trivial sign relation, iconic or otherwise.
It will be necessary to keep that "feature" constantly in mind.

SR. Comment 17


Referring to my thumbnail sketch of the characters in this play,
what does it mean to say that the sign y has its interpretant z
by virtue of its quality q, one that y shares with its object x?

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `Quality q` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` . . ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `.` `.` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` . ` ` . ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `.` ` ` `.` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` . ` ` ` ` . ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `.` ` ` ` ` `o Sign y ` ` ` ` ` |
| ` ` ` ` ` ` ` ` . ` ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `.` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` . ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `.` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` . ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` Object x o-----------O` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o Interpretant z ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

One thing that we must keep in mind is that functional and relational
properties, like those involved in a sign relation, are not in general
determined by a single datum, for instance, the single triple <x, y, z>.

To illustrate this principle in a vastly simpler frame of reference
than sign relations, consider the ordered pair <2, 4>.  What is the
relation between 2, the first element, and 4, the second element of
the pair?  No uniquely satisfactory answer, independent of or as we
say "prescinded from" all context, can be given.  In particular the
single datum <2, 4> is a member in equally good standing of the set
{<x, 2x>}, that we may call the function "doubles" because it sends
x to 2x, and also the set {<x, x^2>}, that we may call the function
"squares" because it maps x to x^2.

Thus we find the datum <2, 4> at the nexus
of two distinct contexts at the very least:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ... ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 0:0 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 1:2 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ... ` 0:0 ` 1:1 ` 2:4 ` 3:9 ` 4:16` ... ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 3:6 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 4:8 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ... ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

That will be enough to meditate on for a little while.

SR. Comment 18


By way of making the gentlest possible transition from my simple
functional example, where we may of course view functions of the
form f : x ~> f(x) or {<x, y> : y = f(x)} as 2-adic relations of
a special kind, let's use the doubling and squaring functions to
build a couple of 3-adic relations that are naturally associated
with them.  These constructions need not yield us sign relations,
at least not the most natural examples of ones, in order to give
us some idea of the systematic structural properties that emerge
to play significant roles in our investigation of sign relations.

Consider the 3-adic relations D (for "doubles") and S (for "squares")
that are suggested by the following finite samples of their contents:

   D  =  0:0:d + 1:2:d + 2:4:d + 3:6:d + 4:8:d + ...

   S  =  0:0:s + 1:1:s + 2:4:s + 3:9:s + 4:16:s + ...

Here I am using a style of notation that Peirce actually employed,
where x:y:z is the same thing as the ordered triple <x, y, z>, and
where "+" designates an operation of "logical aggregation", such as
we employ to form a class, a collection, or a set from its elements.

Given this set-up, the following forms of statement can be made:

   1.  In reference to the 3-adic relation D,
       a "4" in the second place is assigned
       a "d" in the third place by virtue of
       the fact that 4 is the double of the
       2 in the first place.

   2.  In reference to the 3-adic relation S,
       a "4" in the second place is assigned
       an "s" in the third place by virtue of
       the fact that 4 is the square of the
       2 in the first place.

The markers that we adjoined to the ordered pairs of our initial functions
thus serve to identify the contexts or the relations that we are choosing
to contemplate the ordered pairs within from one discussion to the next.
This is very analogous to one of the roles that interpretant signs play.

SR. Comment 19


I think that we have now provisioned ourselves well enough to
attempt the construction of something approaching a simplest
possible non-trivial example of an iconic sign relation.
A non-trivial sign relation is one that contains more
than one sign relational triple, and the question is
what other conditions must it satisfy in order for
us to say, for sooth, that at least some of the
signs in the sign relation are interpreted in
the sign relation as denoting their objects
by virtue of sharing specified characters
in common with their objects.

For the sake of comparison and contrast, let us consider the
following collection of 3-adic relations, all constructed on
the same three domains, X = Y = {"[", "]"} and Z = {"0", "1"}.
I put the elements of these domains in quotations to emphasize
their purely formal character, but I'll omit the extra clutter
except in cases where it becomes critical to some end in view.

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| x y z | ` ` | x y z | ` ` | x y z | ` ` | x y z |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| [ [ 0 | ` ` | [ [ 0 | ` ` | [ [ 1 | ` ` | [ [ 0 |
| [ ] 1 | ` ` | [ ] 0 | ` ` | [ ] 0 | ` ` | [ ] 1 |
| ] [ 0 | ` ` | ] [ 0 | ` ` | ] [ 0 | ` ` | ] [ 1 |
| ] ] 0 | ` ` | ] ] 1 | ` ` | ] ] 1 | ` ` | ] ] 0 |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o

By way of mnemonics, we have the following associations:

   1.  P is a relation where "1" in the third place is
       predicated on "paired" brackets in the first two.

   2.  R is a relation where "1" in the third place is
       predicated on recognizing two "right" brackets.

   3.  S is a relation where "1" in the third place is
       predicated on "similarity" of the two brackets.

   4.  U is a relation where "1" in the third place is
       predicated on "unlikeness" of the two brackets.

SR. Comment 20


It will eventually be useful to consider the 3-adic relations P, R, S, U
at a slightly higher level of abstraction, where all three domains X, Y, Z
are equal to B = {0, 1} and where the former elements "[" and "]" have been
replaced with the boolean values 0 and 1, respectively.  Let's not bother with
devising new names for the revised editions, but simply leave it to context to
determine the case.  Here are the heterogeneous and the homogeneous versions of
the 3-adic relations P, R, S, U c X x Y x Z, for appropriate choices of X, Y, Z.

Heterogeneous Relations P, R, S, U c X x Y x Z

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| x y z | ` ` | x y z | ` ` | x y z | ` ` | x y z |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| [ [ 0 | ` ` | [ [ 0 | ` ` | [ [ 1 | ` ` | [ [ 0 |
| [ ] 1 | ` ` | [ ] 0 | ` ` | [ ] 0 | ` ` | [ ] 1 |
| ] [ 0 | ` ` | ] [ 0 | ` ` | ] [ 0 | ` ` | ] [ 1 |
| ] ] 0 | ` ` | ] ] 1 | ` ` | ] ] 1 | ` ` | ] ] 0 |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o

Homogeneous Relations P, R, S, U c X x Y x Z = B^3

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| x y z | ` ` | x y z | ` ` | x y z | ` ` | x y z |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| 0 0 0 | ` ` | 0 0 0 | ` ` | 0 0 1 | ` ` | 0 0 0 |
| 0 1 1 | ` ` | 0 1 0 | ` ` | 0 1 0 | ` ` | 0 1 1 |
| 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 1 |
| 1 1 0 | ` ` | 1 1 1 | ` ` | 1 1 1 | ` ` | 1 1 0 |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o

SR. Comment 21


Strange to say, after all these years, but there remain
as yet very few examples of non-trivial sign relations
that have been discussed to any depth in the semiotics
literature.  This is largely due to a widespread lack
of familiarity with the generic properties of 3-adic
relations, acquaintance with which is 'sine qua non'
to a firm grasp of non-trivial sign relations, not
to mention all of their many-splendored species.

Consequently, it will not hurt us one bit to acquaint ourselves
with the distinguishing marks, known associates, and aliases of
our present line-up of 3-adic relations, even if turns out that
some of them are innocent of involvement in the immediate issue.

Rogues Gallery of Triadic Relations

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| x y z | ` ` | x y z | ` ` | x y z | ` ` | x y z |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| [ [ 0 | ` ` | [ [ 0 | ` ` | [ [ 1 | ` ` | [ [ 0 |
| [ ] 1 | ` ` | [ ] 0 | ` ` | [ ] 0 | ` ` | [ ] 1 |
| ] [ 0 | ` ` | ] [ 0 | ` ` | ] [ 0 | ` ` | ] [ 1 |
| ] ] 0 | ` ` | ] ] 1 | ` ` | ] ] 1 | ` ` | ] ] 0 |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o

The question whether any of these 3-adic relations
are sign relations is simply a matter of checking
the reigning definition of a sign realation, say,
the one in Letter 75, to see if the conditions
on correspondence and determination are met.
By my lights, anyway, all of them qualify,
if a few just by the skin of their teeth
on the "lower implied sort" clause.

Cf: SR 1.  http://stderr.org/pipermail/inquiry/2004-December/002134.html
In: SR.    http://stderr.org/pipermail/inquiry/2004-December/thread.html#2134

But their mere admittance as sign relations is not in itself so important
as 'how' they qualify as sign relations, or how they behave once admitted.
So let's attend to the specific forms of correspondence and determination
that we may observe in the present set of examples.  Here it is useful to
introduce some additional language and to notice some logical connections
among the concepts thus introduced.

In formal sciences like logic and mathematics the concepts of determination,
functionality, information, and predication are "of imagination all compact".
In its mathematical aspect, determination means that something is a function
of something else.  For instance, "two points determine a line" implies that
there exists a function from pairs of distinct points to lines, nothing more.
This is a formal or you might say "informational" determination.  It clearly
presides over a higher level of abstraction than its causal and material kin.

Each of the above 3-adic relations is a function from its first two domains
to its third domain, allowing us to write P, R, S, U : X x Y -> Z if we so
choose to emphasize the point.  Considering the fact that Z = B = {0, 1},
each of these functions can be interpreted as a "predicate" on the set
of ordered pairs X x Y.

In sum, for the moment, or in this context,
the following locutions are all equivalent:

   1.  z is determined by x and y.

   2.  z is a function of x and y.

   3.  z is predicated on x and y.

SR. Comment 22


Sticking close to the heels of our immediate suspects,
let us continue to profile their traits and disguises.

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| x y z | ` ` | x y z | ` ` | x y z | ` ` | x y z |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| [ [ 0 | ` ` | [ [ 0 | ` ` | [ [ 1 | ` ` | [ [ 0 |
| [ ] 1 | ` ` | [ ] 0 | ` ` | [ ] 0 | ` ` | [ ] 1 |
| ] [ 0 | ` ` | ] [ 0 | ` ` | ] [ 0 | ` ` | ] [ 1 |
| ] ] 0 | ` ` | ] ] 1 | ` ` | ] ] 1 | ` ` | ] ] 0 |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o

We have noted that all of these relations are functions
of the form f : X x Y -> Z, indeed, they are predicates
of the form f : X x Y -> B.  It is convenient to refer
to sign relations whose interpretants are functions of
their objects and signs as "functional sign relations".
In order to emphasize the functional aspect even more,
and to call attention to the times when we draw on it,
it is frequently worthwhile to use the corresponding
lower case letters to mark the functional reading,
in this case writing p, r, s, u : X x Y -> Z.

Let's now try to settle which of the above suspects
might be implicated in the harboring of iconicities.

Our criterion for iconicity is based on the
equivalence of the following two statements:

  1.  y is an icon of x.

  2.  y receives its interpretant z
      on account of x and y sharing
      a property q.

But since we have restricted the interpretant domain Z
to the boolean domain B, saying that z is predicated on
something is the same as saying that z is a predicate of
something, therefore our criterion boils down to requiring
that z = q(x) q(y), that is, z = the conjunction q(x) & q(y).
Moreover, the predicate or property q must be what is variously
called an absolute, inherent, internal, or intrinsic property.
In practical terms, this means that we should be able to tell
whether a thing t has the property q simply by looking at t,
without looking at any other things.  The only one of the
above relations that has that property is the relation R,
so that is our candidate for an iconic sign relation.

SR. Comment 23


That was a bit fast, so let's slow down and look a little more
closely at the complex of features that lead us to suspect the
3-adic relation R of iconic tendencies.

o-------o
| ` R ` |
o-------o
| x y z |
o-------o
| [ [ 0 |
| [ ] 0 |
| ] [ 0 |
| ] ] 1 |
o-------o

Recall that we had the following set-up.  There is the finite set
or "alphabet" of brackets A = {"[", "]"} that appears in the role
of both the object domain X and the sign domain Y, and then there
is the boolean domain B = {0, 1}, whose values we take to signify
some sort of binary choice or decision that we want to make about
some issue or question before us.  Moreover, we found a predicate
q : A -> B defined this way:  q(x) = 0 when x is waxing, or equal
to "[", and q(x) = 1 when x is waning, or equal to "]".  In short,
the predicate q "recognizes" the quality of being a right bracket.

The relation R c X x Y x Z = A x A x B was observed to be a function,
permitting us to write it in the alternate fashion as r : A x A -> B,
and we observed that r(x, y) = q(x) q(y), where multiplication means
the same thing as logical conjunction, q(x) q(y) = q(x) & q(y), over
the boolean domain B.  In other words, R is "atypical", "degenerate",
or let us say "special" in two distinct ways:  first it's a function
r : X x Y -> Z, and second it's a function that factors as a product
r(x, y) = q(x) q(y).  Those are manifestly special properties within
the set of all conceivable 3-adic relations, or subsets of X x Y x Z.

SR. Comment 24


Returning to my initial sketch of the question,
a question about the possible linkages between
the quality q and the interpretant sign z that
might play a role in making the initial sign y
an icon of the object x, we can say that we've
at least considered a number of plausibilities.

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `Quality q` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` . . ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `.` `.` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` . ` ` . ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `.` ` ` `.` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` . ` ` ` ` . ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `.` ` ` ` ` `o Sign y ` ` ` ` ` |
| ` ` ` ` ` ` ` ` . ` ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `.` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` . ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `.` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` . ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` Object x o-----------O` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o Interpretant z ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

Of course, we do not imagine the relationships
that are indicated by the dotted lines to form
as fast a link as the algebraic alchemy or the
elective affinity of the covalent bonds in the
sign relation itself, but it's conventional to
recognize at least some order of force in what
has been writ above the proverbial dotted line.

As these notes were written out just recently in a time of flux,
and the ink is still literally not as dry as it will be in time,
I will keep what I've thought to discover about icons remaining
in a markedly provisional state, and proceed to explore some of
the topics that come to mind in the vicinity of these musements.

SR. Comment 25


For the next phase of exploration let's take up the slightly more
abstract derivatives of the 3-adic relations that we started with.
These are formed by replacing the more concrete members of the set
A = {"[", "]"} with their B-values under the predicate q : A -> B
that values a left bracket "[" as 0 and a right bracket "]" as 1.

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| x y z | ` ` | x y z | ` ` | x y z | ` ` | x y z |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| 0 0 0 | ` ` | 0 0 0 | ` ` | 0 0 1 | ` ` | 0 0 0 |
| 0 1 1 | ` ` | 0 1 0 | ` ` | 0 1 0 | ` ` | 0 1 1 |
| 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 1 |
| 1 1 0 | ` ` | 1 1 1 | ` ` | 1 1 1 | ` ` | 1 1 0 |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o

What we lose in concreteness here we gain in the ability to recognize
and specify more general patterns of relational structure by means of
logical formulas, and this becomes more and more critical a facility
as the complexity of sign relations and 3-adic relations in general
increases to more realistic proportions than these first examples.
This will be important if we want to keep from being overwhelmed
by the complexity of detail that very quickly develops in the
vast and richly interconnected realms of 3-adic relations.

For example, 3-adic relations on the order of the above examples
can all be expressed by logical formulas in many different ways.
As subsets of the cartesian power B^3 = B x B x B -- and here
the "x" that we find ourselves forced to use for a cartesian
product symbol should not be confused with the "x" that we
normally use as a variable name -- the 3-adic relations
P, R, S, U are just particular sets of triples of the
form <x, y, z> in B^3.

A sufficiently general way to specify a subset U of any set V
is to formulate a predicate f : V -> B such that f values the
members of U as 1 and the remainder of the elements in V as 0.
This predicate is called the "characteristic function" or the
"indicator function" of U in V, being written as f_U : V -> B.

Let's keep this fact in reserve for now, as the more specific
fact that all of our current examples are functional on X x Y
allows us to write slightly more compact logical formulas for
this particular collection of 3-adic relations, as shown here:

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| x y z | ` ` | x y z | ` ` | x y z | ` ` | x y z |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| 0 0 0 | ` ` | 0 0 0 | ` ` | 0 0 1 | ` ` | 0 0 0 |
| 0 1 1 | ` ` | 0 1 0 | ` ` | 0 1 0 | ` ` | 0 1 1 |
| 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 1 |
| 1 1 0 | ` ` | 1 1 1 | ` ` | 1 1 1 | ` ` | 1 1 0 |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
z = (x) y ` ` z `=` x y ` ` z=((x,y)) ` ` z = (x,y)

I'm using here a syntax for propositional logic that amounts
to a slight extension of Peirce's alpha graphs, specifically
in their existential interpretation, and as best they can be
rendered in character string form.  In this syntax, the form
"x y" means "x and y", the form "(x)" means "not x", and the
form "(x, y)" means "x not equal to y".  This has the result
that the form "((x, y))" means "x equal to y" as values in B.

In sum, we have just observed that each of the above examples
can be described by a logical equation of the form z = f<x, y>,
where f : B^2 -> B is one the 16 possible "2-adic connectives".

SR. Comment 26


One of the fonder illusions of nominal thinking is that there is
an absolute distinction between generals and individuals, beyond
any that are relative to the conventions of a particular context.
On the contrary, the relativity of this distinction to a context
of interpretation, in other words, to a particular sign relation,
means that 3-adic relations are the minimal context in which one
can talk about qualities and reactions.

SR. Comment 27


Re: SR-COM 25.  http://stderr.org/pipermail/inquiry/2005-September/003061.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-September/thread.html#3028

I've been meaning to get back to the gang of four 3-adic relations that
we were looking at in connection with the question of iconic minimalism,
that is, our inquiry into the simplest possible sign relations that can
be devised that might conceivably qualify as containing iconic signs.

One of the things that makes this type of investigation interesting,
over and above the focal question of icons, is that it illustrates
some of the themes of recursive inquiry, in particular, the ways
that an inquiry into one class of sign relations, that we might
call the "object" class, relies on our ability to use as ways,
means, and tools another class of sign relations, the "base"
or "resource" class.  For example, in the present case, we
are using the sign relations of propositional expressions
as a "base camp" for the ascent on iconic sign relations.

We can see this theme more clearly now if we look at the relationship
between the following portraits of 3-adic relations and their legends,
the propositional expressions that are inscribed beneath their frames:

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| x y z | ` ` | x y z | ` ` | x y z | ` ` | x y z |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| 0 0 0 | ` ` | 0 0 0 | ` ` | 0 0 1 | ` ` | 0 0 0 |
| 0 1 1 | ` ` | 0 1 0 | ` ` | 0 1 0 | ` ` | 0 1 1 |
| 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 1 |
| 1 1 0 | ` ` | 1 1 1 | ` ` | 1 1 1 | ` ` | 1 1 0 |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
z = (x) y ` ` z `=` x y ` ` z=((x,y)) ` ` z = (x,y)

Indeed, acting on Peirce's discovery that graphical calculi are extremely
useful for the logical grasp of many subjects, largely by virtue of their
x-ray facility for revealing the functional articulations of propositions,
it might be useful to examine the graphical forms that these propositions
take, at least, as they allow of rendering into the present medium by way
of the "parse graphs" that are topologically dual to a moderate extension
of Peirce's own "alpha graphs".

To make a long story short, here is my first try
at graphing the above collection of propositions:

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` ` ` | ` ` | ` ` ` | ` ` | x ` y | ` ` | ` ` ` |
| ` ` ` | ` ` | ` ` ` | ` ` | o---o | ` ` | ` ` ` |
| ` ` ` | ` ` | ` ` ` | ` ` | `\`/` | ` ` | x ` y |
| ` o x | ` ` | ` ` ` | ` ` | ` o ` | ` ` | o---o |
| ` | ` | ` ` | `x`y` | ` ` | ` | ` | ` ` | `\`/` |
| z=O y | ` ` | z=O ` | ` ` | z=O ` | ` ` | z=O ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
z = (x) y ` ` z `=` x y ` ` z=((x,y)) ` ` z = (x,y)

These graphs are "mixed mode", inasmuch as they use
the non-graphical symbol of equality "=", and thus
they will need to be converted into a non-hybrid
syntax before they become fully satisfactory as
logical graphs.

SR. Comment 28


Formal reasoning of the sort that they do in mathematics, for all
its abstraction, would make very little headway without the benefit
of lots and lots of concrete examples.  And so the student who could
not provide examples of algebras, geometries, graphs, groups, spaces,
topologies, or whatever that exemplify this or that set of properties
would hardly get past the first quiz in whatever course he or she took.
The spectrum of shades from abstract to concrete is relative, of course.

Thus it presents something of a problem in trying to understand Peirce that
concrete examples, fleshed out to the level of detail that it takes to fully
exemplify his more abstract concepts, are few and far between in his writings.
They are slightly more prevalent in his earliest lectures and especially in his
mathematical papers, which is one of the reasons that I find myself eternally
returning to them for the clarification of his ideas.  So we have to do as
they commonly do in math courses, which is to make the most of the sparse
examples that are doled out to us, to work through the proofs ourselves,
often to fill in their gaps and correct their gaffes, and to recognize
the times when we have encountered, as teachers and textbook writers
frequently say, an "exercise for the reader".  In the present cases
the exercises require us to construct objects to specifications.

With all that in mind, let us return to the quest for something
nigh unto the "smallest possible iconic sign relation" (SPICON).
I think that many of the difficulties that we find in reasoning
about the relations of signs in general to the assorted species
of signs, like icons and indices, would be greatly eased by the
examination of such concretely detailed examples, if we can but
find a few.

SR. Comment 29


Before we set out in quest of the SPICON, there are
a few other ground rules that come to mind that are
probably necessary to articulate as best that I can.

First, we need to understand that we're looking for a sign relation,
a collection of 3-tuples of the form <x, y, z> in a cartesian space
X x Y x Z, for suitable choices of the sets X, Y, Z, to be known as
the "object domain", the "sign domain", the "interpretant domain"
respectively.  The choice of these three domains, along with the
choice of a subset L c X x Y x Z, are choices that are up to us
to make in our tries at constructing a sign relation L that'll
fit all of the conditions that we find ourselves led by the
accumulated definitions to lay on it.

Second, though it's possible to make a big fuss about the
distinction between axioms and definitions, I will not be
doing that in this setting.  I am more used to the way of
talking that says, for example, "a mathematical object of
type T is 'defined' by the following set of 'axioms' ...",
and that is the manner of speaking that I'll have in mind.

Third, depending on how one states ones axioms or definitions,
there may be many sorts of trivial examples of the concept in
question.  For instance, some people will allow the empty set
to be counted as a graph or a group -- I have seen some silly
fights break out at conferences, and there's actually a paper
somewhere entitled "Is the Empty Graph a Pointless Concept?" --
but sensible people will usually just stipulate "nonempty set"
as part of the definition of whatever it is they have in mind.

Fourth, I'm taking it for granted that we're limiting our
interest to "non-trivial" sign relations.  This rules out,
for instance, the empty sign relation, L = {} c X x Y x Z,
and singleton sign relations, L = {<x, y, z>} c X x Y x Z.

Acronymic accuracy would probably demand that I nomenclate it as
a "smallest possible un-trivial iconic sign relation" (SPUTICON),
but that sounds too medicinal.

SR. Comment 30


A generic comment on the point of view behind this exercise:
I will try not weary my readers with grounds that we've all
picked over a gadshillion times before, in hope at least to
worry them with relatively fresh material.  I'm used to the
notion that natural variations in human affective-cognitive
styles might just necessitate residual incommensurabilities
that will outlast my tenure on this planet, or even persist
well into the foreseeable future, but I also know that some
of my readers at least will already know what a relation is,
what a 3-adic relation is, and have a Peirce-catalyzed clue
as to what a 3-adic sign relation is, relying on compatible
ideas of "3-adic correspondence" and "3-adic determination",
all of which I've dredged up the persinent texts for on not
just a few occasions, and so I will work on that assumption.
The purpose of exercises like these is precisely to clarify
the abstract and abstruse concepts in question.  The way it
works is that we already know how to generate generic cases
under a given genus, say, 3-adic relation, and if our sense
of a species, say, sign relation, doesn't tell us forthwith
exactly which creatures of the genus fall under the species
in question, then that is prima facie evidence that we need
to do more work on sharpening up the differentials at issue.

SR. Comment 31


Perhaps one other preliminary comment on the character of determination,
as it is characterized in Peirce's and in formally akin writings, would
be in order.  Our biologically evolved natural languages are apparently
not yet evolved enough to handle relations in general with any facility.
They avail us with forms of syntax that can be patched together more or
mostly less well to handle many particular situations, to be sure, else
we would not have evolved far enough to reflect on the issue, but there
is the tendency of these languages to focus our attention on the single
case rather than the systems ruled, and, what amounts to their greatest
aberrancy of continged coloring and distorted shaping, the liability to
which they expose us of confounding the forms of objects with the forms
of syntax.  This is one of the reasons that Peirce found himself forced
by the task of understanding how science works to create a new language
for talking about sign relations, and found himself forced by this task
in turn to create a new language for talking about relations in general,
that he was well on the way to doing by the "Logic of Relatives" (1870).

As we have discussed on many occasions, determination is not a property
of single cases, single pairs of elements, or single tuples in relation.
It is a concept that makes sense only with regard to whole systems that
are affected by or participate in systematic relations of the type that
we call determinate.

Determination is not just a property of 2-adic relations, as "x determines y",
but is a concept that can be applied to arbitrary relations, even those which
do not have a definable arity or adicity.  Without going to those extremes of
generality just yet, however, we can say for the present that it also applies
to k-adic relations, as "x_1 determines x_2, ..., x_k".  We're naturally very
focused on the present application to sign relations, in the case where k = 3.

With these generalities sufficiently covered for now,
it will perhaps be helpful to concoct a few examples.

After another hit of coffee ...

SR. Comment 32


Let's look at one of Peirce's most endearing impressions of
determination, the example of a raindrop falling on a stone.
If reluctantly, it's our duty now to analyze its impression,
and to articulate its inherent nuances in unrecondite terms.

| To determine means to make a circumstance different from what
| it might have been otherwise.  For example, a drop of rain
| falling on a stone determines it to be wet, provided the
| stone may have been dry before.  But if the fact of
| a whole shower half an hour previous is given,
| then one drop does not determine the stone to
| be wet;  for it would be wet, at any rate.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 245-246
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
| Cf: DET 13.  http://stderr.org/pipermail/inquiry/2004-December/002209.html
| In: DET.     http://stderr.org/pipermail/inquiry/2004-December/thread.html#2197

In saying that a Rain Drop determines a Dry Stone to become a Wet Stone,
we are not talking a single sequence of occurrences merely, but about a
system, the Stone, that has distinct states, Dry and Wet, and that will
undergo definite changes of state under the action or influence of some
some kind of agency, in this case a Rain Drop, more accurately observed,
a mere sample from a more general system of activity, a Rain Storm, say,
and which in fact determines a "transformation" of the "phase space" of
the Stone into itself, that is, a function from the domain of states to
itself, having the form Rain : {Dry, Wet} -> {Dry, Wet}, as shown below.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Dry Stone ` O ` ` ` O ` Wet Stone ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Rain`Drop ` ` ` v ` v ` Rain Drop ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\| ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Dry Stone ` ` ` ` ` O ` Wet Stone ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Thus Peirce's Parable Of The Rain Drop quite well illustrates
the minimal features implued in talking about "determination".

SR. Comment 33


Correcting a few typos in the last installment, and continuing.

Let's look at one of Peirce's most endearing impressions of
determination, the example of a raindrop falling on a stone.
If reluctantly, it's our duty now to analyze its impression,
and to articulate its inherent nuances in unrecondite terms.

| To determine means to make a circumstance different from what
| it might have been otherwise.  For example, a drop of rain
| falling on a stone determines it to be wet, provided the
| stone may have been dry before.  But if the fact of
| a whole shower half an hour previous is given,
| then one drop does not determine the stone to
| be wet;  for it would be wet, at any rate.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 245-246
|
| Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
| Cf: DET 13.  http://stderr.org/pipermail/inquiry/2004-December/002209.html
| In: DET.     http://stderr.org/pipermail/inquiry/2004-December/thread.html#2197

In saying that a Rain Drop determines a Dry Stone to become a Wet Stone,
we're not speaking of a single sequence of occurrences merely, but of a
system, the Stone, that has distinct states, Dry and Wet, and that will
undergo definite changes of state under the action or influence of some
some kind of agency, in this case a Rain Drop, more accurately observed,
a mere sample from a more general system of activity, a Rain Storm, say,
and which in fact determines a "transformation" of the "phase space" of
the Stone into itself, that is, a function from the domain of states to
itself, having the form Rain : {Dry, Wet} -> {Dry, Wet}, as shown below.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Dry Stone ` O ` ` ` O ` Wet Stone ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Rain`Drop ` ` ` v ` v ` Rain Drop ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\| ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Dry Stone ` ` ` ` ` O ` Wet Stone ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Thus Peirce's Parable Of The Rain Drop quite well illustrates
the minimal features implied in talking about "determination".

This much illustration will do as a first rough sketch of the
Weather-Stone system, but closer examination reveals a number
of additional details that require articulation before we can
definitively say what's essential to determination in general.

But before we do that let's consider a rather different brand
of determination, as comparison of maximally diverse examples
can frequently help us to "triangulate" the sought for target.

Consider that nugget from the "Old Elements":  Two points determine a line.
What this means, taken within the implied context of a particular geometry,
is that every choice of two distinct points in the implied geometric space
determines a line in the same space.  Obviously, or as would be evident to
our geometric imagination, many different pairs of distinct points will be
found to determine the same line.  So we get a many-to-one mapping like so:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` `D_1` `D_2` `D_3` `D_4` `D_5` `D_6` `D_7` `D_8` `D_9` ` `
` ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` `
` ` ` `\` ` | ` `/` ` ` `\` ` | ` `/` ` ` `\` ` | ` `/` ` ` `
` ` ` ` \ ` | ` / ` ` ` ` \ ` | ` / ` ` ` ` \ ` | ` / ` ` ` `
` ` ` ` `\` | `/` ` ` ` ` `\` | `/` ` ` ` ` `\` | `/` ` ` ` `
` ` ` ` ` \ | / ` ` ` ` ` ` \ | / ` ` ` ` ` ` \ | / ` ` ` ` `
` ` ` ` ` `\|/` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` `\|/` ` ` ` ` `
` ` ` ` ` ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` `
` ` ` ` ` `L_1` ` ` ` ` ` ` `L_2` ` ` ` ` ` ` `L_3` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Adopting the term "dyad" for a pair of distinct points, the Figure
shows the "incidence relation" between dyads D_i and lines L_j for
a hypothetical case where the dyad D_1 determines the line L_1 and
so on down the line, er, series, where the dyad D_9 determines the
line L_3.  As would be the usual thing in any geometry, then, each
dyad determines just one line, but each line is determined by many
different dyads.  This can be expressed in a very succinct fashion
by saying that there is a many-to-one function J : !D! -> !L! with
!D! being the domain of dyads and !L! being the co-domain of lines.
It should be obvious that this is a formal, informational, logical,
or mathematical brand of determination, not a causal determination.

SR. Comment 34


Re: SR-COM 33.  http://stderr.org/pipermail/inquiry/2005-December/003338.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-December/thread.html#3325

After reflecting on that last picture of geometric determination for a few
moments, some readers will no doubt be asking:  What happened to the part
about determination meaning "to make a circumstance different from what
it might have been otherwise"?  What circumstance?  What difference?

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` `D_1` `D_2` `D_3` `D_4` `D_5` `D_6` `D_7` `D_8` `D_9` ` `
` ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` o ` ` `
` ` ` `\` ` | ` `/` ` ` `\` ` | ` `/` ` ` `\` ` | ` `/` ` ` `
` ` ` ` \ ` | ` / ` ` ` ` \ ` | ` / ` ` ` ` \ ` | ` / ` ` ` `
` ` ` ` `\` | `/` ` ` ` ` `\` | `/` ` ` ` ` `\` | `/` ` ` ` `
` ` ` ` ` \ | / ` ` ` ` ` ` \ | / ` ` ` ` ` ` \ | / ` ` ` ` `
` ` ` ` ` `\|/` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` `\|/` ` ` ` ` `
` ` ` ` ` ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` `
` ` ` ` ` `L_1` ` ` ` ` ` ` `L_2` ` ` ` ` ` ` `L_3` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Because Peirce illustrated the abstract concept of determination with
the richly concrete example of the Weather-Stone system, and because
the meaning of the word "concrete" is that an example has a richer
collection of properties than the minimal wealth needed to serve
as an example of the concept in question, part of what it means
to comprehend the import of the example is to recognize which
properties are being illustrated and which are merely excess
ornamentation.  So one of the questions we need to ask is:
What sort of formal or informational process is analogous
to the causal or material process that is represented in
the illustration of determination?  Answering this will
help us to grasp the kind of determination that we are
meant to abstract from the example of determination.

Well, let's all mull this question over overnight and
I will let you know what I come up with with the dawn.

SR. Comment 35


Re: SR-COM 34.  http://stderr.org/pipermail/inquiry/2005-December/003339.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-December/thread.html#3325

Arisbe List, Inquiry List,

Well, it appears that some dawns take a longer time arriving than others.
But the question stays the night, no matter the number of live-long days.

We return to the question of what sort of formal or informational process
makes a circumstance different from what it might have been otherwise, as
analogous to the causal or material process in Peirce's Rain-Stone system.

To address this question let us think of the form of determination in
the "two points determine a line" example as proceeding by a stepwise
information process, as if we were to be given one point, asking what
sort of constraint it places on the lines that must be incident to it,
and then we are given a second point, which finally determines one of
the lines among the set of lines that are incident to the first point.
Then the circumstance that is determined to be different from what it
might have been otherwise is just our state of information before and
after the information is actually given about the first point and the
second point.

To make this concrete, but also to keep it simple,
let's focus on a small sample of points and lines
that we pick from a larger geometry, specifically,
the six points and the three lines in this Figure:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` 2 o ` ` ` o 6 ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` o-------o-------o ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `3` ` ` ` 4 ` ` ` `5` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

    1.  The set of points is !P! = {p_1, p_2, p_3, p_4, p_5, p_6},
        it being usually enough to use the indices for the points.

    2.  The set of lines is !L! = {L_1, L_2, L_3}, with the data:

        a.  L_1  =  {1, 2, 3}

        b.  L_2  =  {3, 4, 5}

        c.  L_3  =  {5, 6, 1}

Earlier I defined a "dyad" as a pair of distinct points.
Notice that I did not say "ordered pair", so we have as
many dyads in the overall geometry as there are choices
of two different things from a collection of six things,
in short, (6 * 5)/2 = 15 dyads.  But only nine of these
dyads lie on the three lines of our sample, leaving the
following picture of the incidence relations among this
particular sample of geometric points, dyads, and lines:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
1 ` 2 ` 2 ` 3 ` 3 ` 1 ` 3 ` 4 ` 4 ` 5 ` 5 ` 3 ` 5 ` 6 ` 6 ` 1 ` 1 ` 5
o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o
`\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/`
`D_1` ` `D_2` ` `D_3` ` `D_4` ` `D_5` ` `D_6` ` `D_7` ` `D_8` ` `D_9`
` `\` ` ` | ` ` `/` ` ` ` `\` ` ` | ` ` `/` ` ` ` `\` ` ` | ` ` `/` `
` ` \ ` ` | ` ` / ` ` ` ` ` \ ` ` | ` ` / ` ` ` ` ` \ ` ` | ` ` / ` `
` ` `\` ` | ` `/` ` ` ` ` ` `\` ` | ` `/` ` ` ` ` ` `\` ` | ` `/` ` `
` ` ` \ ` | ` / ` ` ` ` ` ` ` \ ` | ` / ` ` ` ` ` ` ` \ ` | ` / ` ` `
` ` ` `\` | `/` ` ` ` ` ` ` ` `\` | `/` ` ` ` ` ` ` ` `\` | `/` ` ` `
` ` ` ` \ | / ` ` ` ` ` ` ` ` ` \ | / ` ` ` ` ` ` ` ` ` \ | / ` ` ` `
` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` `
` ` ` ` `L_1` ` ` ` ` ` ` ` ` ` `L_2` ` ` ` ` ` ` ` ` ` `L_3` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Considered within the context of our geometric sample, then, we can
say that picking the point p_1 by itself would pin down the dyads to
the set {D_1, D_3, D_8, D_9}, and that picking the point p_3 by itself
would pin down the dyads to the set {D_2, D_3, D_4, D_6}, so picking the
points p_1 and p_3 determines the dyad D_3, which determines the line L_1.

SR. Comment 36


Arisbe List, Inquiry List,

Suppose we now absorb into our "sample of geometry" (SOG)
all of the remaining dyads on six points, thus taking up
the full set of "6 choose 2" = 15 dyads, for convenience
re-indexed in lexicographic order, drawn and detailed so:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/|\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / | \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` | `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` 2 o---|---o 6 ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\`.|,`/`\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` , | . ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/`,` `\|/` `.`\` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` o-------o-------o ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `3` ` ` ` 4 ` ` ` `5` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` D_1 = {1, 2}, D_2 = {1, 3}, D_3 = {1, 4}, D_4 = {1, 5}, D_5 = {1, 6},
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` D_6 = {2, 3}, D_7 = {2, 4}, D_8 = {2, 5}, D_9 = {2, 6},
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` D_a = {3, 4}, D_b = {3, 5}, D_c = {3, 6},
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` D_d = {4, 5}, D_e = {4, 6},
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` D_f = {5, 6}.
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The lexicographic re-indexing requires us to re-label our old sample so:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
1 ` 2 ` 1 ` 3 ` 2 ` 3 ` 1 ` 5 ` 1 ` 6 ` 5 ` 6 ` 3 ` 4 ` 3 ` 5 ` 4 ` 5
o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o
`\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/`
`D_1` ` `D_2` ` `D_6` ` `D_4` ` `D_5` ` `D_f` ` `D_a` ` `D_b` ` `D_d`
` `\` ` ` | ` ` `/` ` ` ` `\` ` ` | ` ` `/` ` ` ` `\` ` ` | ` ` `/` `
` ` \ ` ` | ` ` / ` ` ` ` ` \ ` ` | ` ` / ` ` ` ` ` \ ` ` | ` ` / ` `
` ` `\` ` | ` `/` ` ` ` ` ` `\` ` | ` `/` ` ` ` ` ` `\` ` | ` `/` ` `
` ` ` \ ` | ` / ` ` ` ` ` ` ` \ ` | ` / ` ` ` ` ` ` ` \ ` | ` / ` ` `
` ` ` `\` | `/` ` ` ` ` ` ` ` `\` | `/` ` ` ` ` ` ` ` `\` | `/` ` ` `
` ` ` ` \ | / ` ` ` ` ` ` ` ` ` \ | / ` ` ` ` ` ` ` ` ` \ | / ` ` ` `
` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` `
` ` ` ` `L_1` ` ` ` ` ` ` ` ` ` `L_2` ` ` ` ` ` ` ` ` ` `L_3` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

SR. Comment 37


Arisbe List, Inquiry List,

| We return to the question of what sort of formal or informational process
| makes a circumstance different from what it might have been otherwise, in
| abstraction from or analogy to the causal or material process in Peirce's
| Rain-Stone example.

I think that we have now attained a suitable outlook from which to review
the salient features of the just now passing geometric topography, and to
recapitulate the lessons that we ought to extract from our side-trip into
extremely partial and special 2-adic relations of geometric determination.

Within the context of our particular sample of geometry, which involves
a relation of sampling that is itself a form of attention, bias, choice,
constraint, determination, election, information, partiality, selection,
or representation -- as far as information goes it's all the same stuff --
we may find a number of further relations of determination, whereby the
showcasing of particular subsets among the available geometric elements
acts to showcase further subsets among the available geometric elements.
And though we may stroll through a gallery of showcases in a particular
order on a particular show occasion, there is nothing that qualifies as
a causal, material, or physical order in this promenade sequence, as we
could just as easily have conducted our peripatetics in many other ways.

SR. Comment Work Area 1


SOP.  Discussion Note 1

Different readers of Peirce seem to favor different "definitions"
or "characterizations" of the sign relation.  Robert Marty, et al.
have collected 76 + 12 of them at his site in Perpignan, and these
are also available at Joe Ransdell's "Arisbe" website:

http://www.univ-perp.fr/see/rch/lts/marty/
http://www.univ-perp.fr/see/rch/lts/marty/76defeng.htm

http://members.door.net/arisbe/
http://members.door.net/arisbe/menu/library/rsources/76defs/76defs.htm

My personal favorite definition would probably be Number 14 from the above lists.
The current definition is Number 15, and I will confess that I find some aspects
of it confusing to me, still, but I like it especially because of the "sunflower"
illustration, which was critical to piquing my interest in the subject many years
ago, partly because it made a connection in my mind to the way that I used to view
group theory in those days.

To represent a 3-tuple <x, y, z> such that xy = z in a group G,
I used to draw a picture of something like the following shape:

          x       y
           \    //
            \  //
             \//
             |||
             |||
             |||
              z

Then I would chain these together in trees and cycles to represent the
more complex constructions that I had to think about in a given setting.

When I used this tactic to represent the elementary sign relations among
the sunflowers and the sun, I ended up with pictures something like this:

          s_1   s_2
          o o   o o
          |  \ /  |
   s_8 o--o   o   o--o s_3
       o   \  |  /   o 
        \   o o o   /
         o--o O o--o
        /   o o o   \
       o   /  |  \   o 
   s_7 o--o   o   o--o s_4
          |  / \  |
          o o   o o
          s_6   s_5
 
Of course, it could be any multitude of sunflowers, not just eight.
When it comes to the placemattes |, ||, ||, however, what assignment
makes the most sense?  Does it really matter all that much if we place
the Sun, the object O, in first or second of third place, so long as we
dub it the object, isn't that enough?  Well, that depends.  Some people
will read this story as Peirce saying something significant about the
relationship between the three sign roles and his three "Categories",
which he often gave the maximally abstract names First, Second, Third.
And here you will find a lot of fights break out.  Where have all the
flowers gone?  And so it goes.  I personally do not choose to take it
that way, mostly because it is apparently possible to rationalize so
many different correspondences that none of them wins out as unique.
Let a thousand flowers bloom.

SOP.  Discussion Note 2

HT = Hugh Trenchard

Re: SOP 1.  http://suo.ieee.org/ontology/msg05410.html

HT: I find this quite confusing.  It looks like Peirce is saying that
    the "triadic" relationship of the sign, object, and interpretant
    cannot exist in any other combination except in the triad.

HT: If I can paraphrase this a bit, it sounds like an object (the "second")
    may be perceived by a person, but it is meaningless as any particular
    object until that person interprets the object by relating it to certain
    properties (i.e. conceptions and extensions (?)) she knows that can help
    to identify and give meaning to the particular object (these properties
    being the sign or "representamen").  The triadic relationship, if I am
    anything close to the mark, is about gleaning meaning from perception.

The first thing to do is to realize that a "sign relation" is a certain type
of formal structure that one is trying to characterize in a formal way, much
like the way that we define mathematical objects like graphs and groups and
so on ad infinitum.  Whether a particular family of formal objects is good
for any particular purpose is a whole separate question.  So quite a few
of the words that we necessarily have to use to talk about this familiy
of formal objects, for instance, "object" or "sign", will have to be
shorn of their ordinary connotations, and perhaps get to keep only
some of their former associations.  In this situation, all of the
designations Object, Sign, Interpretant refer to the relational
roles that particular elements play in a particular 3-tuple of
a particular set of 3-tuples that we call a "sign relation".

To make it short but rough, "object" is used more in the
sense of "object of discussion" or "object of thought".
It can be, but doesn't have to be an ordinary physical
or even a presently existing object.  Indeed, very
often the word "objective", in the sense of "goal"
or "intentional object" is a better paraphrase of
what's being intended in a particular application.

So let us go back to "the" definition of a sign relation --
I'd recommend No. 14 as being the clearest on this score:

http://suo.ieee.org/email/msg11187.html

There you can see that whatever we are talking about has a lot to do
with a special relationship of "correspondence" and "determination",
and to find out what Peirce meant by those words you have to do some
further reading, where you'd find that he is talking about a triple
correspondence, no kind of 2-adic mirror correspondence need apply,
and a type of determination that is informational and partial in
general, not of necessity causal or absolutely deterministic.

SOP.  Discussion Note 3

HT = Hugh Trenchard

Re: SOP 1.  http://suo.ieee.org/ontology/msg05410.html

HT: I get confused (or more confused) when he talks about a sunflower being
    a representamen of the sun.  Is the sunflower in this case a property
    of the sun -- a thing which triggers an image of the sun;  something
    which implies the sun?  If so, I can't help but think again of Borges'
    story in which the jaguar implies the deer the jaguar ate, the grass
    the deer ate ... the universe (or that idea), and I question the
    proximity of the relationship.  A sunflower may conjure images
    of the sun, but isn't a closer representamen of the sun
    a circle and bright skies?

I will go ahead and use the word "sign" for the general case,
and if I want to talk specifically about a "mental sign",
I will just call it that.

I used to worry a lot about whether a sign of something
is a property of something, and vice versa, but this is
really a separate issue, and discussing it would depend
on picking a particular theory of what a "property" is,
which we don't really have to get into, just yet, not
in order to understand the bare essentials of what
a sign relation is.

Peirce says that a sunflower is a sign of the sun, to another sunflower,
so the speak -- more exactly, as interpreted in the inspired performance
of another sunflower -- and only with the proviso that you can imagine
a certain very unlikely condition as holding true, just because the
3-place relation of the sun O, sunflower s_1, and sunflower s_2
fits the definition of a sign relation that he more or less
gives in various places.

HT: Also, if a person is involved in the perception of the object, isn't
    the person himself part of the relationship?  It sounds like we have
    object, sign, and interpretation of sign=meaning.  But what about the
    physical system (i.e. the brain) which facilitates this transformative
    process?  Isn't this a quadratic relationship? (he-he-heaven forbid --
    anything but a triad!  Bring out the crucifixes and the wooden stakes!
    Sweep up the stars, dismantle the sun!)

I think that most of the things that you are talking about are covered by
the concept of an "interpreter", and we can go ahead and use that word in
a broad enough sense to cover any sort of interpretive system, whether an
individual agent, a whole community of interpretation, Gaia, Global Brain,
Over Soul, or whatever.  Again, this is just the standard system-theoretic
attitude with regard to the agent, "representative point", "test particle",
its names are legion.  It does not really matter much what imagery you find
most catalytic of fruitful ideas, since we are really only talking about the
forms that we see from a particular perspective.  From that outlook, what we
care about is the effect on the system, and that effect is the interpretant.

SOP.  Discussion Note 5

JA = Jon Awbrey
RX = Reader X

Re: SOP 2.  http://suo.ieee.org/ontology/msg05411.html
In: SOP.    http://suo.ieee.org/ontology/thrd1.html#05410

JA: Another style of graphical picture can be given by letting the fact that
    interpreter J employs a triple of the form <x, y, z> be depicted like so:

            y
           /
     x o--J
           \
            z

RX: The triple here denotes the triad of <sign, object, interpretant>
    all relating to Interpreter J.  What is the conceptual distinction
    to be made between the Object A and the Interpreter A, or is this
    what you are driving at with the special case of O and Interpreter
    being the same with the use of demonstratives, indexicals, etc.?

There are depths of complexity here that I will try to avoid falling into --
somehow the phrase "Run, you fools!" comes to mind -- at any rate, let us
try to approach the abyss more gingerly than your average hobbit will do.

Interpretive agents, in so far as we formally consider them -- since we
do not concern ourselves with what they had for breakfast except insofar
as we conceive it to have a bearing on their sign relations -- are really
just personifications of these sign relations themselves.  In this formal
regard, we can replace Interpreter J with a particular sign relation L(J),
and oftentimes it will be safe enough in context to use just "J" for L(J).

In that light, the picture above is just another syntax for saying
that <x, y, z> is an element of L(J), more briefly, <x, y, z> in J.

In our eavesdropping on the discussion between A and B, we take them
as having names for A and B as unanalyzed objects of discussion, and
nowhere near enough vocabulary yet to talk about their own discourse
in sign relational terms.  It would take the development of language
about "higher order sign relations" of various sorts in order for us
to be able to talk about this intelligently and without getting lost
in near-hopeless confusion.  So let's leave that quest to the sequel.

SR. Comment Work Area 2


Cf: DEF 33. 'Semiotics & Significs : Letters to Lady Welby', p. 196 (1906) 
At: http://members.door.net/arisbe/menu/library/rsources/76defs/76defs.htm

| The same form of distinction extends to the interpretant;
| but as applied to the interpretant, it is complicated by
| the circumstance that the sign not only determines the
| interpretant to represent (or to take the form of) the
| object, but also determines the interpretant to represent
| the sign.  Indeed in what we may, from one point of view,
| regard as the principal kind of signs, there is one distinct
| part appropriated to representing the object, and another to
| representing how this very sign itself represents that object.
| The class of signs I refer to are the dicisigns.  In "John is
| in love with Helen" the object signified is the pair, John and
| Helen.  But the "is in love with" signifies the form this sign
| represents itself to represent John and Helen's Form to be.
| That this is so, is shown by the precise equivalence between
| any verb in the indicative and the same made the object of
| "I tell you". "Jesus wept" = "I tell you that Jesus wept". 

SR. Sign Relations • Discussion

SR. Discussion Note 1


BM = Bernard Morand
JA = Jon Awbrey

Re: SR-COM 3.  http://stderr.org/pipermail/inquiry/2005-January/002244.html
In: SR-COM.    http://stderr.org/pipermail/inquiry/2005-January/thread.html#2242

BM: This is the good point Jon!  I split your commentary into its two components:

JA: 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, ...

BM: Yes, this is the problem with samples (individual signs):  they can only 
    stand for the generality of the sign relation to some extend.  However, it 
    is often interesting to inquire whether we manage to retrieve the general 
    features through the sample.  And it is clear that we need the knowledge of 
    such general structure to do this job, one kind of inductive method I think.

We have in fact two kinds of sampling:  the elementary sign relation <o, s, i>,
a particular moment in which something (s) means something (o) to somebody (i),
is a sample from the sign relation, L c O x S x I, that we have in mind at the
moment, and the sign relation L is just one example under the definition of a
sign relation.  Indeed, we may consider the set L itself to be a sample from
a larger sign relation L' c O' x S' x I', of a similar type or otherwise.

At this point, though, at least so far as the most general concept
of a sign relation, I am content to think that Peirce has already
done sufficient work of induction to find a fruitful definition,
and the work of necessary reasoning could potentially proceed
from that alone, though, as a practical matter it is nearly
indispensable to prop up one's resoning with the assistance
of concrete examples, so long as one recognizes the extra
properties that they are bound have as concrete examples.
Moreover, in doing so, one advances the work of what we
may call "species reasoning", or the classification of
natural subtypes.  And then, there's always the chance
that we'll hit upon a more fruitful concept by keeping
in touch with the empirical phenomena that continue to
turn up on their own, quite unbidden by our deductions.

JA: ... 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.

BM: Quite agreed on the principle.  I am not sure as regards to "the same effect".
    In fact thinking through the structure invariance will not be enough for
    understanding the transformation at work within individual signs semiosis;
    nevertheless I agree that this structure has to be preserved there.

Yes, we can water the hedges around "same effect" as needed over time.
And you are also right about the second point.  As I have occasionally
pointed out, the process "semiosis" exists at a more specific level than
the structure "sign relation", because it requires the specification of
a temporal sequence as an additional parameter, one that is not required
by the bare definition of a sign relation.  There is something analogous
to a type-token relationship here, in which the time-free types of signs
are cast at a level more abstract than their time-bound tokens in a given
example of semiosis, which is in general just one of the many instances
of process that are possible to generate from out of the same abstract
sign relation.

SR. Discussion Note 2


GR = Gary Richmond
JA = Jon Awbrey

Re: SR-COM 4.  http://stderr.org/pipermail/inquiry/2005-January/002248.html
In: SR-COM.    http://stderr.org/pipermail/inquiry/2005-January/thread.html#2242

Amended here:

JA: | 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.
    |
    | All in all, sign relations are not limited to purely linguistic types of systems.
    | They encompass the data of the senses, natural signs, and plastic representation,
    | just to name some randomly-chosen species of this very widely disseminated genus.

GR, quoting JA:

JA: All in all, sign relations are not limited to purely linguistic types of systems.
    They encompass the data of the senses, natural signs, and plastic representation,
    just to name some randomly-chosen species of this very widely disseminated genus.

GR: A profound truth (in my opinion), perfectly well stated.  Your analysis 
    here brings to my mind the great chasm existing between dyadic semiology
    and triadic semeiotic, just for example.

The scheme above is an organization of the subject matter between
signs relations and inquiry processes that I frequently recur to,
and it does contain some remedies against the tenacious tendency
to regress semiotics back to the level of 2-adic relations.  For
a naturalist like myself, who thinks that everything that keeps on
happening is natural, it is a complementary consideration to think
how everything in nature may be causally related to everything else,
and to that extent everything is potentially an index of everything
else.  In dual consideration, we may think how everything in nature
shares at least one quality with everything else in nature, and so
everything is potentially an icon of everything else in some way.
But these considerations are based on purely 2-adic relations and
their virtues as icons and indices must still be remitted in the
receipt of an interpretant sign before we have a full-fledged
sign relation.

SR. Discussion Note 3


BB = Bill Bailey

Re: SR-COM 13.  http://stderr.org/pipermail/inquiry/2005-January/002260.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-January/thread.html#2242

BB: Perhaps the kicker is this -- when you really begin to look at
    how people actually use language, you'll find that they don't.

If you find another paraphrase more useful, feel free to use it instead.
I think most people know what I mean when I say I use "I" to refer to me.
At any rate, I don't think that my use of the word "use" affects the very
schematic sorts of examples that I chose to begin with here, more or less
extracted from ordinary langauge use, but intended more to illustrate some
of the more generic and salient properties of sign relations in a not quite
trivial couple of examples.

BB: Earlier, someone  posted a William James witticism that its "turtles all
    the way down".  An anthropologist professor of mine frequently said, in
    essence, "It's artifacts all the way down".  In communication it is not
    very useful to speak of using language, unless we define language as
    something other than the relationships of words.  People communicate
    to get work done.  They are often not very good at the work because
    they typically learned their communication behavior from their parents,
    who, like their own parents, didn't know how to do the work either.
    We assimilate what ever communication behavior is available in our
    families.  The traditional conception is that we are acquiring language.
    But we are not.  We are learning how to act when your wife criticizes
    your drinking, or your husband criticizes your spending -- yes, I know
    those are cliches, but many of the families I encountered were living
    them.  Starting with that principle, within a few minutes of working
    with chronically conflicted couples, you can tell them a great deal
    about the nature of their respective parents.  They learn communication
    behavior just like they learn how to fix a plugged sink or shingle a roof --
    by participation.

Of course you know it's a very good question whether we "learn" language at all,
or merely undergo the natural maturation process of an innate linguistic organ,
but it's hardly within my scope to tackle such a bedeviled hen or egg questions
at this stage of the game.

BB: When you say "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", perhaps the idiosyncrasy is in the approach?
    I'm not sure what you mean by "initial idiosyncrasy", but, if you mean in
    the sampled behavior, I'm pretty certain that much of the communication
    behavior that seems idiosyncratic from the outside is solid convention
    within the given culture of the communicators.  In fact, I'm willing to
    bet you that any actual communication behavior you want to present from
    a reasonably normal population of people can be shown to have a base in
    cultural convention.  If so, how can it be described as idiosyncratic?

I was referring to the curious property of indexicals, that they
are to designed to denote different objects when used by different
people or when used at different times.  I chose this example with
a larger purpose in mind, as it reflects certain issues that arise
whenever observers in different "reference frames" have to reconcile
the frame relative observations that they make with regard to what we
generally assume is actually the very same world.

BB: If you are going to make sense of situated communication, I think you
    have to look at the presumed "work to be done", and, yes, that often
    seems stupid and it often ends up in stupid, confused outcomes.
    Communication transactions are often sad, ugly failures because
    one person wants to use a given artifact (metaphorically) as
    a paperweight while the other wishes to use it as a weapon.

That was kinda my point.

BB: Or put it differently, if you you want to use my handling of money as a 
    sign of my stinginess, and I want to use it as a sign of my thriftiness, 
    one of us in our universe of two is functionally insane, and neither of 
    us has  information that functions in the relationship.  But neither of 
    use is exactly "idiosyncratic" in our communication behavior.

BB: I view my comments as seconding your first two paragraph below,
    and most of the third.  It is that last clause I object to,
    and suggest the madness is in the method, not the data.

Well, I hope I explained what I meant a little better.

SR. Discussion Note 4


BB = Bill Bailey

BB: Jon, I was apparently the one who was unclear;  I think I understood your post.
    Let me see if I can respond without the usual cut and paste job.  First, as I
    tried to convey, I have no problem with any of your analytics and schemata
    regarding sign relations, and if I did, given my lack of sophistication in
    semiotics, it wouldn't mean much.  Mainly, I think, my comments about not
    learning and using language (I didn't get the relevance of the hypothetical
    biological bases of language, by the way) went astray.  My only issue was
    what seems to me paradoxical in your summary statement.  You say people's
    actual use of words and language are problematic as regards extracting
    sense from all the idiosyncratic usage, but when you reference/anchor
    those sign relations relative to the abstracted conventions of words
    in language, wouldn't you expect that?  The behavior may be far less
    idiosyncratic when assessed relative to the pragmatics of context,
    where people do not compose their discourse word by word according
    to linguistic function, but rather use the learned communication
    artifacts of their culture/community/family.

Maybe the word "idiosyncractic" is too red a flag.
When you say that signs need to be "assessed relative
to the pragmatics of context", that is pretty much all
that I am trying to flag, if you consider the additional
circumstance that context and the person speaking are not
constants but extremely variable.  The fact that we learn
to use, or otherwise grow into, languages that have these
qualities is not disputed, it simply goes beyond what many
people with no sensitivity to the pragmatic dimension take
for granted as being adequate to semantics and ontology.

SR. Discussion Note 5


JA = Jon Awbrey
KM = Kirsti Määttänen

Re: SR-COM 12.  http://stderr.org/pipermail/inquiry/2005-January/002259.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-January/thread.html#2242

KM: Some questions on what you write below -- one to check
    whether my feeling of understanding is predominantly
    correct, one to get more informed on the subject. 

JA: 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. 

JA: 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. 

KM: Who are the "some folks" who read Frege this way? (I sincerely hope you
    do not answer with fifty-one references to your admirably arranged network
    of your (net)work.  That would not help me in this.  Although I'm learning
    a lot from musing from time to time your method of arranging the threads.
    Seven plus/minus two references -- if that will be the way how you choose
    to answer -- would a digestible amount.  Not that I'm complaining, the
    dialogue has been most interesting and enlightening.  Your focus of
    interest on Peirce's writings is exactly where mine is not.)

I had in mind Dummett's account of Tugendhat's approach
to Frege's theory of reference, found on pp. 199-203 of
Michael Dummett, 'Frege:  Philosophy of Language', 2nd,
Harvard University Press, Cambridge, MA, 1981.  I will
copy out some relevant excerpts after dinner, but have
to run for now.

JA: 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.

KM: Here I have a feeling of understanding -- or rather:
    a feeling of mutual agreement.  But I'm not sure.
    "Partition into equivalence classes"?

Will get to this later, also.

JA: This is a very general phenomenon, betraying a certain
    "point of view relativity" in that way that the structure
    of the objective world is represented in the structure of
    languages.

KM: "Betraying"?

Here just meaning "to show, indicate, disclose, or reveal".
Maybe the connotative connection is through "informing on"?

SR. Discussion Note 6


JA = Jon Awbrey
KM = Kirsti Määttänen

Re: SR-COM 12.  http://stderr.org/pipermail/inquiry/2005-January/002259.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-January/thread.html#2242

Amended here:

JA: 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.

JA: 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.

KM: Here I have a feeling of understanding -- or rather:
    a feeling of mutual agreement.  But I'm not sure.
    "Partition into equivalence classes"?

An equivalence relation E on a set S is a 2-adic relation E c S x S that is
reflexive, symmetric, and transitive.  A partition of a non-empty set S is
a set of mutually disjoint non-empty subsets of S whose union is all of S.
Any equivalence relation E on a set S induces a partition of S into subsets
that are called the "equivalence classes" under E.  The members of a given
equivalence class are all pairwise equivalent under E, while elements of S
from different equivalence classes are inequivalent under E.  Equivalence
relations on S can be represented as directed graphs or "digraphs" on the
point set S, drawing an arc from x to y in S if and only if the ordered
pair <x, y> belongs to E.  Since E is reflexive there is a loop at each
point.  Since E is symmetric, each arc x->y is paired with an arc y->x.

In the case of the sign relations L(A) and L(B), their projections
on the SI plane, L(A)_SI and L(B)_SI, respectively, yield equivalence
relations  on the set of signs !S! = {"A", "B", "i", "u"} that I call
"semiotic equivalence relations" (SER's).  The corresponding partitions
I call "semiotic partitions" (SEP's).

The SEP for interpreter A consists of the sets {"A", "i"} and {"B", "u"}.
The SEP for interpreter B consists of the sets {"A", "u"} and {"B", "i"}.

SR. Discussion Note 7


JA = Jon Awbrey
KM = Kirsti Määttänen

Re: SR-COM 13.  http://stderr.org/pipermail/inquiry/2005-January/002260.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-January/thread.html#2242

In part:

JA: 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. 

KM: Well, here I do not agree.  This follows from an adultomorphic view
    of human understanding and qualities (=> classes) of human experience.

I'm not talking here about the acquisition of the core language,
but about the more generally analogous problems that we have to
achieve without the help of several million years of evolution.

KM: By the way, you seem to have noticed that
    there is a system in my use of ( ).
    Have you?

You always deploy them in pairs?
(Minimal committment hypothesis.)

SR. Discussion Note 8


JA = Jon Awbrey
KM = Kirsti Määttänen

Re: SR-COM 12.  http://stderr.org/pipermail/inquiry/2005-January/002259.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-January/thread.html#2242

JA: 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. 

KM: Who are the "some folks" who read Frege this way?

JA: I had in mind Dummett's account of Tugendhat's approach
    to Frege's theory of reference, found on pp. 199-203 of
    Michael Dummett, 'Frege:  Philosophy of Language', 2nd,
    Harvard University Press, Cambridge, MA, 1981.

Two excerpts:

| An attempt has recently been made by Tugendhat [E. Tugendhat, in M. Schirn (ed.),
| 'Studien zu Frege / Studies on Frege', vol. 3, Stuttgart and Bad Canstatt, 1976,
| pp. 51-69] to approach Frege's notion of reference from a new direction.  This
| involves, in effect, casting off the use of the name/bearer relation as prototype,
| and presenting reference wholly as semantic role.  It is instructive to study the
| results of this attempt.
|
| Tugendhat proposes that reference should be understood as what he calls
| "truth-value potential".  The truth-value potential of an expression is,
| in effect, its semantic role, its contribution to the determination of the
| truth-value of a sentence in which it occurs.  Tugendhat does not explain
| this notion so much by apppeal to the semantics which are to be given for
| the language, a semantics which will lay down what the semantic role of
| each primitive expression is:  rather, what he actually defines is the
| relation between two expressions of having the same truth-value potential,
| in terms of the truth-values, assumed as already known, of the sentences
| in which they occur.  Two expressions will then be said to have the same
| truth-value potential just in case, whenever each is supplemented by the
| same expression to form a sentence, the two resulting sentences have the
| same truth-value.  Tugendhat does not say what truth-value potential
| itself is:  it could, in accordance with a device introduced by Frege,
| and compatibly with all that Tugendhat says, be identified with an
| equivalence class of expressions under the equivalence relation
| of having the same truth-value potential.  (Dummett, p. 199).

| Tugendhat has stripped the notion of reference of the character of being
| a 'relation' to something extra-linguistic:  it has become, in his hands,
| essentially an equivalence relation between expressions.  (Dummett, p. 200).

Dummett then goes on to criticize the consequences of this interpretation.

SR. Discussion Note 9


KM = Kirsti Määttänen

Re: SR-DIS 8.  http://stderr.org/pipermail/inquiry/2005-January/002267.html
In: SR-DIS.    http://stderr.org/pipermail/inquiry/2005-January/thread.html#2247

KM: Thanks a lot for this.  I'll read it later more carefully, in detail.

KM: Dummett, once I happened to read something of him, I dropped after a few sentences.
    Thinking:  Why on earth, having Peirce to read, would I read Dummett.

At any rate, the idea itself, of relating semantics to equivalence classes
of syntactic entities, is pretty generic, often enlightening, and seems to
occur to most folks sooner or later.  The big difference in attitude comes
in whether one thinks that this "reduces" semantics to syntax or not.
Weighing against the reduction, aside from the non-uniqueness of the
construction, already mentioned, is the "generative" character of
a living language, and the fact that a set of syntactic entities
is not in general a syntactic entity itself, but an abstract
object over and above the syntax.  These factors compell
an approach to semantics, and even the syntactic theory
itself, that involves rational concepts and categories,
not merely empirical ones.

SR. Discussion Note 10


Strictly speaking, Peirce advises a non-psychological approach
to logic, which he defines as formal semiotic, using "formal"
to mean "quasi-necessary", which is the moral equivalent of
"normative" to us.  I have mentioned before that the prefix,
"non" frequently serves as a generalizing functor in math,
as in the study of non-associative algebras, which includes
those algebras that do satisfy the associative axiom along
with those that do not.  It is just as if "non" was really
an acronym for "not of necessity".  I have also argued that
semiotics in general has room for a descriptive semiotics,
under which would fall many applications to the descriptive,
or non-therapeutic, side of psychology, in which Peirce was
evidently rather interested, of course.

But there is nothing about cardinality, causality, cognition, or continuity
in the barest unpsychological definitions of sign relations, and so if we
find those considerations coming into our discussions of sign relations,
it is either because we have explicitly added some additional axioms and
definitions, or else because we are treading on unexpressed assumptions,
which being non-conscious, are likely to vary widely from participant to
participant in the discussion.  Of course, much diversion lies that way.

SR. Discussion Note 11


Cf: Jon Awbrey & Susan Awbrey, "Interpretation as Action: The Risk of Inquiry"
    http://www.chss.montclair.edu/inquiry/fall95/awbrey.html

As Susan Awbrey and I pointed out in the paper cited above,
brute expereience and the outward clash are already present
in Aristotle's account of the sign relation, under the heading
of 'pathemata' or "affective impressions".  These are actually
the closest things to the objects of experience, of which signs
and concepts are fragmentary and shadowy derivatives.  Fudging
a little, it is possible to relate these affective impressions
to interpretant signs, with the understanding that less direct
signs, of the sort that we speak and write and cognize in,
are derived by a process of reflection on these impresses,
and that one mode of interpreting words and thoughts is to
return to the more concrete impressions as interpretants.
Thus there is a back and forth play between pathemata
and their derivatives that we may call "reflection"
and "interpretation", in some senses, respectively.

I do not know if there are any creatures who experience nothing but
the 2-adic clash in a stimulus-response fashion, but I'm pretty sure
that intelligence depends on reflection thereof, thus inserting the
mediating parameter of signs into the flow of experience.  In this
connection, we may note that, far from the passive determination
that marches from objects to signs in a stimulus-response fashion,
the pragmatic maxim is stated in terms of the reverse direction:
"if you do this, you will experience this".  This is critical
to transforming 2-adic reactions into 3-adic reflections.

SR. Discussion Note 12


JA = Jon Awbrey
JP = Jim Piat
VA = Victoria N. Alexander

Re: SR-COM 14.  http://stderr.org/pipermail/inquiry/2005-September/003028.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-September/thread.html#3028

In part:

JA: For a pivotal instance, when we say that an icon is a sign
    that receives its interpretant by virtue of a property that
    it shares with its object, what exactly do we mean by that?
    And how can we tell, in practical and operational terms, when
    we have such a thing and when we do not?  In the sequel, I'll
    stake out an approach to answering these and related questions.

In reply:

JP: I'm up way too late but just want to tell you I look forward
    to your discussion below.  That's the sort of nuts and bolts
    examination of a key Peircean notion that I think can be very
    helpful and productive -- at least for me.  Maybe I'm just a
    sucker for promises of exact and operational meanings.  I'm
    forever promising myself to at least seek if not provide them.
    All said with a smile, Jon -- I really am looking forward to
    your comments.

VA: I look forward to your discussion of icons.

VA: I've been wondering whether or not it's possible to think about
    "real" icons and "accidental" icons.  Here are some excerpts from
    a paper I'm currently working on (mentioned previously on this list).

VA: There are patterns (such as resemblances) that are generated
    according to physical laws (e.g., various spiral galaxies)
    and there are patterns that are merely coincidental (e.g.,
    the big and the little dippers).  According to Peirce, a
    "real" law governing a resemblance indicates intelligible
    regularities (habits, if not laws) of nature.  When a
    pattern is found in an "accidental" law, a "synthetising"
    subject "introduces an idea not contained in the data,
    which gives connections, which they would not otherwise
    have had" (Peirce, 1890, 261).

VA: If I write a "to do" list, then I notice that it happens
    to have fourteen lines, as sonnets do, I can interpret
    it as if it were a sonnet.  I can ignore the "to do"
    list's essential features and utilizes a coincidental
    pattern instead -- the fact that it has fourteen lines.
    But this is a special kind of iconic interpretation.
    Making use of a side effect entails not seeing what
    is essential (pertinent to the creation/development
    of the object in question) and seeing what is not
    essential.

VA: Curiously the symbol seems to be able to take special advantage of what I will
    call (following Peirce in the context of distinguishing between types of laws,
    see above) "accidental", as opposed to "real", icons and indices, that is,
    icons that only coincidentally seem similar to an object (because of
    a side-effect) and indices that only seem contiguous with an object
    (because of a coincidental association between an object and a
    context).  A "real" icon, for example, would be a photograph
    of a person. The likeness is not coincidental.

Just a quick note, partly to myself, as the leading edge of my catching-up process
is about 20,000 fathoms nearer the bottom of my inbox, but a quick sounding leads
me to believe that these are some of the same issues that I tried to tackle with
my theory of "frameworks, genres, and motifs", some of which I sketched beginning
at Subsection 1.3.4.12 "Objective Plans and Levels" in my "Inquiry Driven Systems":

http://members.door.net/arisbe/menu/library/aboutcsp/awbrey/inquiry.htm

There are some corrections and extensions here:

IDS.  http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434
IDS.  http://stderr.org/pipermail/inquiry/2004-June/thread.html#1574

There is also the beginning of an attempt at a fuller exposition here:

OPAL-May.  http://stderr.org/pipermail/inquiry/2005-May/thread.html#2755
OPAL-Jun.  http://stderr.org/pipermail/inquiry/2005-June/thread.html#2767

We should also recall that Peirce's analysis of photographs and portraits
makes them more akin to indices than first impressions usually take them.

SR. Discussion Note 13


JA = Jon Awbrey
JP = Jim Piat

Re: SR-COM 15.  http://stderr.org/pipermail/inquiry/2005-September/003031.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-September/thread.html#3028

Comments interlaced below ...

JA: We take up the question of virtue in the case of an icon,
    where we say that an icon is a sign that gets or begets
    its interpretant sign by virtue of a property that it
    bears in common with its object.

JP: Yes that seems right to me.  So in my view the icon in effect serves
    as a virtual proxie for the object.  It is for some purposes taken
    as the object and indeed with respect to the property they have in
    common it is quite proper to do so.  It's the properties that the
    object and the icon don't have in common that make one an icon
    and the other an object.  And further (in my view) it's a safe
    bet that the icon is less physically burdernsome to manipulate
    or move about than the object it represents -- otherwise it
    would not be particularly useful as a means of representing
    an object in thought.

For the moment, I'm trying to stay focused on the more definitive properties
of iconic sign relations, leaving to later phases of inquiry the additional
properties that we might find various and sundry special sorts exhibiting.
But as a general rule I'll be interested in the ways that signs convey
information and the way that they figure in inquiry.  In this regard,
many of the features that you mention above, for all their typical
prevalence, are not in fact definitive or even necessary features
of icons, as Peirce often notes in many places.  For instance,
it doesn't have to be that the object and icon are distinct.

Will have to break here, as it's getting past the twitching hour ...

SR. Discussion Note 14


JA = Jon Awbrey
JP = Jim Piat

Re: SR-COM 15.  http://stderr.org/pipermail/inquiry/2005-September/003031.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-September/thread.html#3028

I continue from where I left off ...

JP: So operationally what is similarity?  My candidate is as follows:
    any two or more objects (or properties of objects) that evoke the
    same response in some third object are similar.  Yes -- similarity
    itself can not be defined except triadically.  There is no inherent
    absolute qualites that can be established empirically without reference
    to a triadic relationship.  We can in theory posit their mode of being --
    but even to do this requires triadic thought.  And my further guess is
    that we can not establish a triadic relationship without reference to
    the iconic and or indexical.  They are all inextricable parts of the
    whole -- conceptually and otherwise.

My intent is to pragmatize -- to clarify by way of the pragmatic maxim --
some of the fuzziness that still persists in our collective impressions
in-about the dints of reason and virtue, as they're taught in the schools
of hard and soft knocks that we all know and love.  The way that "virtue"
figures in the various definitions of signs and sign species is, of course,
of special interest, and we are currently attending to the case of icons,
but the question of virtue and its kin is much more general than that.

With that broader view in mind, I'm content for the moment to accept
any respectable definitions of similarity, at its simplest being no
more than the bearing in common of identifiable predicates.  Later,
of course, we will want to cover more complex notions, like those
of arrows in mathematical category theory, to wit, homomorphisms,
isomorphisms, and morphisms in general.  But it's best to begin
simply, and so I will start with the simple idea of sharing one
or more attributes, characters, features, marks, qualities,
predicates, properties, or whatever you want to call them.

JP: So where I'm going is that a symbol (however pure) is
    an icon or index stripped of some non essential purely
    actual features (as opposed to essential conceptual or
    imputed features that serve the same purpose).  It's the
    effect of similarity or iconicity (a sort of equivalence)
    that is essential.  Not the actual physical equivalence or
    spatial temporal correlation.  In a way a symbol is less not
    more than an icon or index -- physically symbols are less.
    Conceptually they are equivalent to icons and indexes.
    In the sense I am speaking of above -- as standing for
    to by virtue of either actual or imputed similarity
    or spatial temporal correlation.

JP: Just thinking out loud (perhaps in error) as I'm following along
    and trying to anticipate the development of your discussion, Jon.

Well, you 'are' racing ahead of me just a bit,
so I'll have to restrain myself to proceeding
in my own plodding way.

SR. Discussion Note 15


JA = Jon Awbrey
JR = Joe Ransdell

Re: SR-COM 15.  http://stderr.org/pipermail/inquiry/2005-September/003031.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-September/thread.html#3028

In part:

JA: We take up the question of virtue in the case of an icon,
    where we say that an icon is a sign that gets or begets
    its interpretant sign by virtue of a property that it
    bears in common with its object.

JR responds:

JR: I don't recall any place where Peirce defines or characterizes
    an icon in that way, as producing its interpretant in virtue of
    a property that it bears in common with its object.  It impresses
    me initially as being a sloppy rewording that may well be senseless:
    intrinsic (monadic) properties do not, as such, affect production.
    Why not start off from one of his own statements, and then if you
    want to argue that from that you can derive immediately the
    formulation above by mere rewording, it will be possible to
    see whether there is any basis for attributing that to him
    or not.  Otherwise we may well be wasting our time with
    something that has to do only with you but which is not
    attributable to Peirce.

JR: Also, it is not at all clear to me that what you take as obvious
    in the rest of your message is as obvious as you take it to be
    inasmuch as it seems to imply a denial that existential pairs
    or semiotical triplets can iconize other such pairs or triplets.
    That is a worthwhile question to pursue but it would be foolish
    to try to do so on the basis of the apparently idiosyncratic
    forumulation you supply above.

I believe that the characterizations of iconic sign relations
that I've been using are fair paraphrases of everything that
I've read Peirce say in definition or description of their
generic properties, within the constraints of what can be
said in vernacular précis.  Naturally, any paraphrase in
ordinary language is bound to be limited in exactitude,
up to the point where every concept invoked in it can
either be defined in formal fashion or identified as
a primitive undefined term.  The circumstance that
words and phrases like "virtue" and "by virtue of"
are yet to be clarified in a moderately pragmatic
way is precisely the task of the present effort
to address.  One of the ways that this can be
achieved is to examine a variety of concrete
examples, constructed to exemplify the ideas
and conjectures that arise from the ongoing
formal description, in terms of axioms and
formal definitions, of the subject matter
that is currently under consideration.
That is just what I am doing here.

The tenor of the rest of your remarks does not strike me as forming
a promising basis for dialogue, so I'll just let the rest of it go.

SR. Discussion Note 16


JP = Jim Piat

JP: Continuing to think out loud as I read  your interesting
    inquiry into the nature of icons, indexes, and symbols.

JP: It seems that similarity comes in two forms:  intrinsic-iconic
    (inherent quality that is connoted) and extrinsic-indexical
    (temporal-spatial quality that is denoted*).  Objects can
    share either of theses sorts of similarity, and the crux
    of a triadic relationship is that two objects (ie objects
    that lack at least one form of similarity -- elsewise we
    would take them to be the same object) can be interpreted
    triadically "as if" similar in one of these two ways.
    A fully triadic symbol functions (ie is interpreted)
    "as if" it were similar to the object it connotes
    (iconizes) and denotes (indexes).  All tools employ
    this triadic "as if" function.  For example flat
    sticks can be employed "as if" shovels or levers.
    But the particular tools we call signs, function
    "as if" they were icons and indexes.  Even those
    that actually are icons and indexes.  That's the
    way in which all triadic signs incorprate icons
    and indexes.  Maybe.

JP: * Objects can have similar locations in time and space just
      as they can have similar inherent qualities such as color.
      These are the two broad classes of information that specify
      an object:  Its essence and its location.

JP: A friend reminds me that "function" is also a common third way of specifying
    an object's nature but in my view function is not a property of the object
    per se but of what man thinks about the object.  Thus not a real property
    but an imaginary, interpreted, imputed or "as if" property.  In short a
    property of thought.  So we have three possible types of similarity:
    iconic, indexical, and functional -- of the first, second, and
    third kind.

I can appreciate the way that you stick with it.
I can only recommend that that you not be in such
a hurry to identify so many things that are really
different, like the topics of connotation/denotation
icons/indices, intrinsic/extrinsic, etc. -- these are
related but not usually by straightforward parallelisms.

It is a continuing mistake of many people to think that
they can construct 3-adic relations by mushing together
the structures of 2-adic relations, making a silk 3-ad
out of sow's 2-ads.  All I can say is:  "When pigs fly".
A symbol is not in general made by cramming icons and
indices togther, not unless you consider the cramming
to refer to a genuinely 3-adic or symbolic relation.

Here it might help to think about the sense in which
Peirce names icons and indices as "degenerate" signs.
He recalls the example of conic sections in geometry.
Suppose I list examples of conic sections as follows:

   1.  a point   "."
   2.  a line    "|"
   3.  a cross   "X"
   4.  a circle  "O"
   5.  ellipses
   6.  hyperbolas
   7.  parabolas

The first 3 or 4 would be described as "atypical" or "degenerate" cases,
and if we interrupted the list of examples there, a listener who didn't
already know better would be likely to form a concept of conic sections
that overgeneralizes from overly special cases.  One might even get the
notion that one can construct the generic cases by mushing together X's
and O's in some fashion.  A lot of the current litter on sign relations
is just about that misconceived.  The proper approach to the subject is
by going the distance to tackle the generic cases first, and then going
back to see the simpler special cases as degenerate or derivative forms.

In this business there is no substitute for
looking at a whole lot of concrete examples.

SR. Discussion Note 17


JP = Jim Piat

Re: SR-DIS 16.  http://stderr.org/pipermail/inquiry/2005-September/003048.html
In: SR-DIS.     http://stderr.org/pipermail/inquiry/2005-September/thread.html#3029

Let me go back and pick up some of the implications that
I missed at first reading in your following two remarks:

JP: A fully triadic symbol functions (ie is interpreted)
    "as if" it were similar to the object it connotes
    (iconizes) and denotes (indexes).  All tools employ
    this triadic "as if" function.  For example flat
    sticks can be employed "as if" shovels or levers.
    But the particular tools we call signs, function
    "as if" they were icons and indexes.  Even those
    that actually are icons and indexes.  That's the
    way in which all triadic signs incorprate icons
    and indexes.  Maybe.

JP: A friend reminds me that "function" is also a common third way of specifying
    an object's nature but in my view function is not a property of the object
    per se but of what man thinks about the object.  Thus not a real property
    but an imaginary, interpreted, imputed or "as if" property.  In short a
    property of thought.  So we have three possible types of similarity:
    iconic, indexical, and functional -- of the first, second, and
    third kind.

Now, I am mainly using the word "function" in its mathematical sense,
namely, to denote a special sort of 2-adic relation between two sets
such that:  each element of the first set is related to some element
of the second set -- this property by itself is referenced by saying
that the relation is "total" on the first domain -- and each element
of the first set is related to one element at most of the second set --
this property by itself is referenced by saying that the relation is
"tubular" on the first domain.  In short, a "function" f from X to Y,
that we write as f : X -> Y, is a relation f c X x Y that is "total"
and "tubular" on X.  (It's the Che Valley definition of a function.)

Still, there is a connection with the sense of the word "function"
that we use in speaking of how devices like instruments and tools
function, at least, when they are functioning according to specs,
and this bears on the functioning of some types of sign systems.

For instance, consider the weather-cock, where the fact that
the rooster's tail presents a larger surface area to the wind
than the rooster's head means that our rooster robot will face
into the wind, in tandem constituting, as Peirce describes it,
"an index which forces something to be an icon".  The type of
functional determination that is involved in the functioning
of this instrument is one of those "whole system" properties
that cannot be defined solely at the level of single tuples.
In other words, the domain of wind directions is related to
the range of instrument pointings in a functional manner,
such that changes in the first correspond to changes in
the second.  You cannot tell if x determines y simply
by inspecting a single ordered pair <x, y>.  If the
weather-vane froze up in last night's ice-storm,
it's ostentatious indices will remain in vain,
and will not function as informative indices
until Chanticleer chances to thaw again.

As it were ...

SR. Discussion Note 18


CP = Charles Pyle

Charles Pyle wrote:
> 
> Eugene Halton says, "You need to state how direct unmediated
> knowledge is possible, given Peirce's devastating criticism
> of it."
> 
> You.  Meaning me.  I need to.  Which is quite surprising.
> 
> I can rely on previous quotes entirely.
> 
> Since this is not a new debate. This debate has been going on for
> many years in this very forum. And Eugene Halton has taken part in it.
> And in the context of this debate, it is all the more notable that his
> assertion makes so many debated presuppositions, ignoring this context,
> that it is all the more difficult to reply.
> 
> To speak to one point, I deny, as others have denied in this forum, that
> Peirce made a criticism of unmediated knowledge, let alone a devastating
> criticism. On the contrary. I cite, for example, the following quotes from
> Peirce, which appear on the face of it to assert the validity of direct,
> unmediated knowledge, from an email of April 17, 2001 to this forum sent
> by Jean-Marc Orliaguet [jmo@medialab.chalmers.se] as a turn in an extensive
> debate. (I extract the essential point following the quote):
> 
> Peirce: "our knowledge of things in themselves is entirely relative, it is true;
> but all experience and all knowledge is knowledge of that which is, independently
> of being represented" 6.95
> 
> knowledge of that which is, independently of being represented
> 
> I paraphrase:  knowledge independent of representation

That is not a good paraphrase.

The "independently" modifies the "is", not the "knowledge".
More precisely, "independently"  modifies the relation
between "is" and "represented", not the relation
between "knowledge" and "represented".

Thus we have:

"knowledge" <--- dependent ---> "representation"
           ^                   ^
            \   independent   /
             \               /
              "that which is"

There is something of a good puzzle that remains here --
having to do with whether there is an asymmetric
sort of independence relation that we have to
imagine, or whether that is not necessary.

None of this can be understood without knowing
what Peirce meant by "independent", which is
a form of relationship that is not at all
the same thing as "disconnected from",
"exclusive of", or "unmediated by",
indeed, all sorts of things that
are connected by long chains of
mediation can nevertheless be
independent, both of those
chains and of each other.

> Peirce: "Effort is effort by virtue of its being opposed; and no third
> element enters. Note that I speak of the experience, not of the feeling
> of effort. " 8. 330.
> 
> no third element enters
> 
> I paraphrase: experience independent of signs
> 
> Peirce: "Secondness is the most prominent of the three. This is not a
> conception, nor is it a peculiar quality. It is an experience. It comes out
> most fully in the shock of reaction between ego and non-ego. It is there the
> double consciousness of effort and resistance. That is something which
> cannot properly be conceived. For to conceive it is to generalize it; and to
> generalize it is to miss altogether the hereness and nowness which is its
> essence." 8.267
> 
> something which cannot properly be conceived
> something which is not a conception
> 
> I paraphrase: something prior to signs
> 
> So, I don't think it is necessary that _I_ need to state how unmediated
> knowledge is possible. I am convinced that this is already one of Peirce's
> premises. Firstness and secondness are prior to thirdness, which is the
> realm of signs.
> 
> Charles Pyle
> 
> -----Original Message-----
> From: Eugene Halton [mailto:Eugene.W.Halton.2@nd.edu]
> Sent: Monday, September 26, 2005 1:19 PM
> To: Peirce Discussion Forum
> Subject: [peirce-l] Re: Sign Relations
> 
> Charles Pyle stated:  The crux of Zen and Tao is that one can only truly
> know by direct unmediated experience. Whatever knowledge is mediated is not
> true knowledge. As Huang Po said, "All signs are no signs."
> 
> Dear Charles et al,
> 
>              Huang Po, as quoted here, had a faulty semiotic, too
> nominalistic. Zen and Tao would also be false, as you characterize them, in
> my view. c, as Peirce showed so clearly in
> 1868 papers. You need to state how direct unmediated knowledge is possible,
> given Peirce's devastating criticism of it. You also need to state why a
> Zen or Taoist master would be interested in unmediated knowledge, or any
> kind of knowledge, as something worth more than the being in and of the
> moment. Being in and of the moment is aesthetic, not knowledge. Awareness
> is not reducible to knowledge or even self-consciousness.
> 
>              Descartes also attempted to meditate the world and all signs
> away, and thought he had succeeded with "I think, therefore I am." But he
> could not have been thinking without a medium of thought, which was,
> unfortunately, already doubted away by him. Peirce's thought: The very idea
> that one is thinking is an inference occurring in time, not an immediate
> intuition of the very present in which it takes place.
> 
>              Losing the conceptualization of an activity does not mean that
> the activity is not a fluid flow of signs, only that they are not
> conceptual signs. All activity involves a medium of the activity.
> Perception is inferential, all inferences are signs, but not all signs are
> concerned with knowing.
> 
>           In my view, the Buddhist idea of Enlightenment is a becoming
> iconic with experience. Perhaps a Zen master might say, with an invisible
> wink, that "All signs are Noh signs."
> 
> Gene Halton

SR. Discussion Note 19


CSP: | But we have direct experience of things in themselves.
     | Nothing can be more completely false than that we can
     | experience only our own ideas.

The way I read this, Peirce is making a point that
is a familiar one from the stance of phenomenology.
It is just that we have experience of things that
are independent of us.  Our experience constitutes
a phenomenological fact.  But that is not to say
that we infallibly experience things in themselves
'as' they are in themselves -- there is still that
shade of a difference there.  We know this from the
the comparison of conflicting experiences, not all
of which can be reconciled.  Peirce makes this point
in order to avoid the trap of solipsistic idealism,
but it does not say that experience is identical
to the thing in itself.

Also, I don't think Peirce identifies experience and knowledge.
Yes, we sometimes use "knowledge" to mean "acquaintance", but
I think that Peirce uses it more in the sense of "episteme",
or "underatanding".  We have many experiences that we do
not understand.

Charles Pyle wrote:
> 
> Yes, perhaps I parsed those statements by Peirce incorrectly.
> 
> What about this quote:
> 
> "But we have direct experience of things in themselves."
> 
> I paraphrase: We have direct experience of things.
> = Experience that is not mediated by signs.
> 
> Here is the whole paragraph to provide context.
> 
> 95. The first thing to be taken into consideration is the general upshot of
> Kant's Critic of the Pure Reason. The first step of Kant's thought -- the
> first moment of it, if you like that phraseology -- is to recognize that all
> our knowledge is, and forever must be, relative to human experience and to
> the nature of the human mind. That conception being well digested, the
> second moment of the reasoning becomes evident, namely, that as soon as it
> has been shown concerning any conception that it is essentially involved in
> the very forms of logic or other forms of knowing, from that moment there
> can no longer be any rational hesitation about fully accepting that
> conception as valid for the universe of our possible experience. To repeat
> an example I have given before, you look at an object and say "That is red."
> I ask you how you prove that. You tell me you see it. Yes, you see
> something; but you do not see that it is red; because that it is red is a
> proposition; and you do not see a proposition. What you see is an image and
> has no resemblance to a proposition, and there is no logic in saying that
> your proposition is proved by the image. For a proposition can only be
> logically based on a premiss and a premiss is a proposition. To this you
> very properly reply, with Kant's aid, that my objections allege what is
> perfectly true, but that instead of showing that you have no right to say
> the thing is red they conclusively prove that you are logically justified in
> doing so. At this point, the idealist appears before the tribunal of your
> reason with the suggestion that since these metaphysical conceptions, that
> repose upon their being involved in the forms of logic, are only valid for
> experience and since all our knowledge is relative to the human mind, they
> are not valid for things as they objectively are; and since the conception
> of existence is preeminently a conception of that description, it is a mere
> fairy tale to say that outward objects exist, the only objects of possible
> experience being our own ideas. Hereupon comes the third moment of Kant's
> thought, which was only made prominent in the second edition, not, as Kant
> truly says, that it was not already in the book, but that it was an idea in
> which Kant's mind was so completely immersed that he failed to see the
> necessity of making an explicit statement of it, until Fichte misinterpreted
> him. It is really a most luminous and central element of Kant's thought. I
> may say that it is the very sun round which all the rest revolves. This
> third moment consists in the flat denial that the metaphysical conceptions
> do not apply to things in themselves. Kant never said that. What he said is
> that these conceptions do not apply beyond the limits of possible
> experience. But we have direct experience of things in themselves.†1 Nothing
> can be more completely false than that we can experience only our own ideas.
> That is indeed without exaggeration the very epitome of all falsity. Our
> knowledge of things in themselves is entirely relative, it is true; but all
> experience and all knowledge is knowledge of that which is, independently of
> being represented. Even lies invariably contain this much truth, that they
> represent themselves to be referring to something whose mode of being is
> independent of its being represented.†2 This is true even if the proposition
> relates to an object of representation as such. At the same time, no
> proposition can relate, or even thoroughly pretend to relate, to any object
> otherwise than as that object is represented. These things are utterly
> unintelligible as long as your thoughts are mere dreams. But as soon as you
> take into account that Secondness that jabs you perpetually in the ribs, you
> become awake to their truth. Duns Scotus and Kant are the great assertors of
> this doctrine, for which Thomas Reid deserves some credit too. But Kant
> failed to work out all the consequences of this third moment of thought and
> considerable retractions are called for, accordingly, from some of the
> positions of his Transcendental Dialectic. Nor in other respects must it be
> supposed that I assent to everything either in Scotus or in Kant. We all
> commit our blunders.
> 
> -----Original Message-----
> From: Jim Piat [mailto:piat325@charter.net]
> Sent: Saturday, October 01, 2005 3:04 PM
> To: Peirce Discussion Forum
> Subject: [peirce-l] Re: Sign Relations
> 
> Dear Charles, Jon, Folks--
> >
> > Peirce, which appear on the face of it to assert the validity of direct,
> > unmediated knowledge, from an email of April 17, 2001 to this forum sent
> > by
> > Jean-Marc Orliaguet [jmo@medialab.chalmers.se] as a turn in an extensive
> > debate. (I extract the essential point following the quote):
> >
> > Peirce: "our knowledge of things in themselves is entirely relative, it is
> > true; but all experience and all knowledge is knowledge of that which is,
> > independently of being represented" 6.95
> >
> > knowledge of that which is, independently of being represented
> >
> > I paraphrase: knowledge independent of representation
> 
> Dear Charles, Jon, Folks--
> 
> I favor what I understand to be Jon's interpretation of this issue.
> What I think Peirce is intending to convey in the quote above is
> the following:
> 
> 1.  We can only know things in themselves be means of medaition or
> representation.  In this sense knowledge depends upon or is relative to
> mediation.
> 
> 2.  However, notwithstanding the above, what we know of various things in
> themselves is not distorted by the mediation.  What we know of things in
> themselves (albeit mediated by representation) is truly the things in
> themselves as they actually exist in reality and not, for example,  some
> figment of our imaginations.
> 
> If one can not hear without a hearing aide, one is dependent upon a hearing
> aid to hear; but this does not mean that what one hears (the sound itself)
> is other than what it is.    If one requires a dollar to ride a bus this
> does not necessarily mean the bus ride in itself is other than what it would
> be if it a dollar fee were not required.  Of course, the meaning consequence
> and perhaps the very essence of riding a bus might depend upon whether or
> not one has to pay a dollar or not  -- but this is not a logical
> requirement.   Being able to ride the bus depends upon paying the fee but
> the nature of the bus ride in iself is independent of the fee.  Accesss to
> knowledge of things in themself depends upon mediation but the knowledge of
> the thing in itself thus accessed is in not otherwise dependent upon (or
> distorted by) mediation.
> 
> I paraphrase:  The only knowledge we have of things is mediated by
> representation, but this knowledge is of those things in themselves
> as they truly are.
> 
> Not to be confused with drawing false conclusions based upon applying faulty
> reasoning to to our mediated but true knowledge of things in themselves as
> they actually are.
> 
> Course I could be misunderstanding Jon, Peirce or you Charles --
> and I comment mostly to gain a better understanding myself.
> 
> Not sure this helps to advance the discussion but it might help
> to point out where I've misunderstood.
> 
> Cheers,
> Jim Piat

SR. Discussion Note 20


CP = Charles Pyle

CP: How bout this then:

CSP: | every correlate of an existential relation is a single object
     | which may be indefinite, or may be distributed;  that is, may
     | be chosen from a class by the interpreter of the assertion of
     | which the relation or relationship is the predicate, or may be
     | designated by a proper name, but in itself, though in some guise
     | or under some mask, it can always be perceived, yet never can it
     | be unmistakably identified by any sign whatever, without collateral
     | observation.  Far less can it be defined.  It is 'existent', in that
     | its being does not consist in any 'qualities', but in its effects --
     | in its actually acting and being acted on, so long as this action
     | and suffering endures.  Those who experience its effects perceive
     | and know it in that action;  and just that constitutes its very
     | being.  It is not in perceiving its qualities that they know it,
     | but in hefting its insistency then and there, which Duns called
     | its 'haecceitas' -- or, if he didn't, it was this that he was
     | groping after.
     |
     | C.S. Peirce, 'Collected Papers', CP 6.318

CP, emphasizing CSP:

CSP: | Those who experience its effects perceive and know it ...
     | It is not in perceiving its qualities that they know it ..."

CP: I [read] Peirce as asserting that there is a knowing that
    comes from direct experience, not indirect experience as
    mediated by signs.  This is not to deny that there is
    also a kind of knowledge that is mediated by signs.

This is the sense of the word "knowledge" that I previously
acknowledged as falling under the heading of "acquaintance".
It is the sense in which we can say we know a thing without
really knowing anything about it.  I have to regard that as
very peculiar sense, most likely generated by a process of
"idealizing to the limit", and like most limiting concepts,
for instance, "haecceitas" or "individual", nobody really
uses them in any literal fashion, but only approximately.
Abstract idealizations, like Peirce's categories, in this
case secondness, are useful to the extent that we can use
them to describe actual experiences, becoming a nuisance
when reified and geneticized into metaphysical substance.
Peirce himself usually avoided this extreme, but was not
always so careful as to avoid endangering his readers,
no doubt underestimating their precipitous natures.

SR. Discussion Note 21


JA = Jon Awbrey
KM = Kirsti Määttänen

Re: SR-COM 26.  http://stderr.org/pipermail/inquiry/2005-October/003083.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-October/thread.html#3083

JA: One of the fonder illusions of nominal thinking is that there is 
    an absolute distinction between generals and individuals, beyond 
    any that are relative to the conventions of a particular context. 
    On the contrary, the relativity of this distinction to a context 
    of interpretation, in other words, to a particular sign relation, 
    means that 3-adic relations are the minimal context in which one 
    can talk about qualities and reactions. 

KM: Yes, I've always thought so.  But how do you
    relate what you have written on idiosyncracy
    to this?

I've forgotten the occasion on which we last spoke of idiosyncrasy,
and so I'll just have to whip up a fresh batch of ideas on the spur
of the moment.  I always think of the word "idiosyncrasy" in linkage
with a couple of old Greek themes:  (1) there is the "krater", a kind
of mixing bowl or jar, and the "someone left the cake out in the rain"
idea that chance mixtures, without the benefit of their precise recipes,
tend to fall among those irreproducible results with which our real life
is so rife, and (2) there is the obsession of the Pythagorean sectarians
with the idea that irrational numbers would puncture their surest proof
of reincarnation, to wit, the circumstance that in a wholly rational
cosmos every state of the universe, no matter how crass or peculiar
the mix of its elements, would of necessity be eternally recurrent.

But this is another one of those places where the early Peirce
appears to have been even more prescient than the later Peirce,
where he found, if not fully following through, the middle way
that guides life between the gelid crystal and the vapid chaos.

SR. Discussion Note 22


Just in case it wasn't clear which themes from Peirce's early work
I was referring to, here are a few links to the pertinent passages:

ICE.  Information = Comprehension x Extension
00.   http://stderr.org/pipermail/inquiry/2004-November/thread.html#1913
12.   http://stderr.org/pipermail/inquiry/2004-November/001926.html
13.   http://stderr.org/pipermail/inquiry/2004-November/001927.html
14.   http://stderr.org/pipermail/inquiry/2004-November/001928.html
15.   http://stderr.org/pipermail/inquiry/2004-November/001929.html
16.   http://stderr.org/pipermail/inquiry/2004-November/001930.html
17.   http://stderr.org/pipermail/inquiry/2004-November/001931.html

DOI.  Doctrine of Individuals
00.   http://stderr.org/pipermail/inquiry/2005-January/thread.html#2320
01.   http://stderr.org/pipermail/inquiry/2005-January/002320.html
02.   http://stderr.org/pipermail/inquiry/2005-January/002321.html
03.   http://stderr.org/pipermail/inquiry/2005-January/002322.html
04.   http://stderr.org/pipermail/inquiry/2005-January/002323.html

SR. Discussion Note 23


AB = Auke van Breemen
JA = Jon Awbrey

Re: SR-COM 28.  http://stderr.org/pipermail/inquiry/2005-December/003325.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-December/thread.html#3325

JA: With all that in mind, let us return to the quest for something
    nigh unto the "smallest possible iconic sign relation" (SPICON).
    I think that many of the difficulties that we find in reasoning
    about the relations of signs in general to the assorted species
    of signs, like icons and indices, would be greatly eased by the
    examination of such concretely detailed examples, if we can but
    find a few.

AB: As I understand Peirce a pure icon only offers the possibility
    to partake in a sign relation.  The sign relation is established
    through the contribution of a symbolic element.  In which case the
    relation you have to look for may be of a symbolic, iconic nature,
    but not purely iconic. 

AB: Next, a question.  What do you mean by smallest possible?
    Is it according to the quantum of information it conveys or
    to the amount of possible different kinds of interpretants
    involved as for instance when only an idea is raised but
    no response asked for?  Or still something else?

I am planning to observe the model that is usual in
constructing concrete examples of abstract concepts
in mathematics and in the allied disciplines of the
formal sciences.  The way that this usually goes is
a bit as follows.

For definiteness, one picks a definition of the concept in question.
As far as the bare definition of a sign relation goes I like the one
in L75, so I will make that my base camp.  Folks who prefer different
definitions are encouraged to set out from those, and at some point the
re-assembled parties can compare their gleanings.  Whatever the outcomes,
much can be learned thereby.  A suitable definition of an icon is a little
more chancey.  I will most likely go with the moral equivalents of the idea
that an icon is a sign whose sharing of a property with its object determines
a sign, not of necessity distinct, that is usually dubbed its interpretant sign.

I haven't said anything about pure icons, and don't really plan to,
so I won't stop to inquire into the divirginces of its definitions.

As far as "smallest possible" goes, I said "nigh unto".
The typical strategy here is to construct the smallest
one that you can cook up, and to await another thinker
to underwhelm you.  When there's a chance of finding a
finite sign relation that satisfies the specifications,
then the cardinality of the set of triples will do for
the measure of size.  By "smallest" we mean "minimal",
not "minimum", as there may be multiple minimal ones.

SR. Discussion Note 24


AB = Auke van Breemen
JA = Jon Awbrey

On subsequent readings I 'think' I get a couple more of the
questions that you may be asking, so I will answer those now.

AB: As I understand Peirce a pure icon only offers the possibility
    to partake in a sign relation.  The sign relation is established
    through the contribution of a symbolic element.  In which case the
    relation you have to look for may be of a symbolic, iconic nature,
    but not purely iconic.

As a general tactic, the modality of possibility is taken care
of in these sorts of formal considerations by the fact that we
are contemplating "all possible structures" that satisfy given
specifications.  In a way, we deal with possible happenings by
treating them as actual happenings in a possible configuration.

I still don't get your second sentence -- though I think things
like that are precisely what we may hope to clarify in carrying
out this exercise.  The third sentence is also vague to me, but
maybe it will help to explain that we are looking for any brand
of sign relation that has at least one icon somewhere within it.

AB: Next, a question.  What do you mean by smallest possible?
    Is it according to the quantum of information it conveys or
    to the amount of possible different kinds of interpretants
    involved as for instance when only an idea is raised but
    no response asked for?  Or still something else?

The scale of rank was explained last time.
If you are using the notion of information
in a sense that is compatible with Peirce's,
that's an interesting question on independent
grounds, but cannot be addressed until we say
exactly how to measure it.  I'm not quite sure
of the last bit, but if you intend a notion of
"sign relations that have missing interpretants",
those are generalizations of sign relations that
actually come up in practical applications, being
dubbed (by me) "sign relational complexes", taking
the word "complex" from analogous uses in geometry.
But I think this is something to save for much later.

SR. Discussion Note 25


JA = Jon Awbrey
JP = Jim Piat

Re: SR-COM 31.  http://stderr.org/pipermail/inquiry/2005-December/003332.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-December/thread.html#3325

JP: Interesting.  Couple of questions.

JA: Our biologically evolved natural languages are apparently
    not yet evolved enough to handle relations in general with
    any facility.  They avail us with forms of syntax that can
    be patched together more or mostly less well to handle many
    particular situations, to be sure, else we would not have
    evolved far enough to reflect on the issue, but there is
    the tendency of these languages to focus our attention
    on the single case rather than the systems ruled, and,
    what amounts to their greatest aberrancy of continged
    coloring and distorted shaping, the liability to which
    they expose us of confounding the forms of objects with
    the forms of syntax.  

JP: My sense is that the syntax embedded in natural languages conforms
    (at a deep level) very closely to the grammar of the real world --
    but I'm not quite sure if this is the issue you are contesting
    or just what you mean by the "confounding the forms of objects
    with the forms of syntax".  Please tell me more, I'm interested.

Of course, you could hardly help but to think like that.
I said "hardly", I did not say "not possibly" -- that'd
be a thesis of linguistic entrapment that I do not make --
I'm just pointing out that it's a very hard task indeed
to push the comfort zone of our nurtural languages, our
humm'n'hallowed mutter tongues, far enough to peek over
the verge and beyond the pale of their fine confinement.
It is the brand of thing, to get a new tongue catalyzed,
that might get you branded a poet*, if not branded more
hot than that, for the sin of irrigating common fluency.
So it's likely to take, not just the help of others for
you to hardly help yourself, but helpings of protection
to help you help yourself through the helpings of farce.

Yikes, is it time for lunch already !?

SR. Discussion Note 26


JA = Jon Awbrey
JP = Jim Piat

Re: SR-DIS 25.  http://stderr.org/pipermail/inquiry/2005-December/003333.html
In: SR-DIS.     http://stderr.org/pipermail/inquiry/2005-December/thread.html#3326

Having to skip around --- as time intermittent today ...

JP: And lastly, Jon, a question that might interest you.  I was thinking the
    other day about forward and backward causation and got to wondering what,
    from a physical, semiotic and psychological standpoint, was the difference
    between pulling and pushing an object.  I'm curious what your quick take on
    this might yield.  Your broad take on the issue -- before you get down to
    the detailed examination.  I am trying to avoid one of those emoticon but
    want you to know I say all this with warm regards and hopefully a friendly,
    even if mischievious, twinkle in my eye.  Count of it's Christmas and all.
    Seriously, what do you think?

Pushing the reindeer ...
He endangers his elf ---
Pulling the reindeer ...
He endangers himself ---
Seeing reindeer play ...
Knowing reindeer way ---
DotDotDot TripleDash

SR. Discussion Note 27


JA = Jon Awbrey
JP = Jim Piat

Re: SR-COM 31.  http://stderr.org/pipermail/inquiry/2005-December/003332.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-December/thread.html#3325

There's a bit more to say on this point:

JP: My sense is that the syntax embedded in natural languages conforms
    (at a deep level) very closely to the grammar of the real world --
    but I'm not quite sure if this is the issue you are contesting
    or just what you mean by the "confounding the forms of objects
    with the forms of syntax".  Please tell me more, I'm interested.

Some of the most difficult advances in logic, math, and even the special
sciences have had to overcome the obstacles that are thrown in our way,
projected onto the scene by the apparent transparency of our language,
as it were.  A big part of overcoming the particle versus wave split
in physics was due to the iron grip that the subject-predicate or
noun-verb pattern of familiar languages had on our imaginations.
Peirce, along with other formal reasoners in the 19th Century,
struggled mightily with this lack of complementarity in the
infrastructure of Indo-European syntax, as it interferes
with the ability to treat k-adic relations in anything
but reductionary terms.  And that revolution has yet
to work its way through, even among many readers of
Peirce, as many of our misundertandings here show,
in particular, the one about icons being involved
in every symbol, not just complex propositions,
which is the implicit subject in every context
where Peirce speaks of symbols involving icons.

JA: As we have discussed on many occasions, determination is not a property
    of single cases, single pairs of elements, or single tuples in relation.
    It is a concept that only makes sense with regard to whole systems that
    are affected by or participate in systematic relations of the type that
    we call determinate.

JA: Determination is not just a property of 2-adic relations, as "x determines y",
    but is a concept that can be applied to arbitrary relations, even those which
    do not have a definable arity or adicity.  Without going to those extremes of
    generality just yet, however, we can say for the present that it also applies
    to k-adic relations, as "x_1 determines x_2, ..., x_k".  We're naturally very
    focused on the present application to sign relations, in the case where k = 3.

JP: OK, fine, determination in all cases (dyadic and triadic) but would it not be
    helpful (given the importance attached to the distinction between the two) to
    have a separate term for the sort of determination (or dare I say cause) that
    occurs in each case.

I think it's helpful to use a generic word for a generic idea,
and to use differential adjectives to distinguish the species.
We have to say what dimensions of relations we are discussing,
anyway, so there's no reason for confusion if we just do that.

JP: And on a related note -- I'm often confused as to when Peirce is using
    sign to refer to an index and when he is using sign as the generic term
    for iconic, indexical, and symbolic signs.  I think he may have switched
    his usage at some point to avoid confusion but left me confused.  Do you
    or others know if I am correct about this or just confused.  I think this
    "double" usage may account for some of the differing opinions as to whether
    or not Peirce fundamentally changed his view of signs over the course of his
    work.  Joe I'd be especially interested in your comments regarding this if you
    are reading this and have the time/interest.

My impression is that Peirce made this switch very early on,
so it's only in his earliest writings that we have to look
harder for contextual clues as to what he's talking about.

Signs Of Pragmata

SOP. Note 1


| A 'Sign', or 'Representamen', is a First which stands
| in such a genuine triadic relation to a Second, called
| its 'Object', as to be capable of determining a Third,
| called its 'Interpretant', to assume the same triadic
| relation to its Object in which it stands itself to
| the same Object.
|
| The triadic relation is 'genuine', that is, its three members are
| bound together by it in a way that does not consist in any complexus
| of dyadic relations.  That is the reason the Interpretant, or Third,
| cannot stand in a mere dyadic relation to the Object, but must stand
| in such a relation to it as the Representamen itself does.
|
| Nor can the triadic relation in which the Third stands be merely similar
| to that in which the First stands, for this would make the relation of the
| Third to the First a degenerate Secondness merely.  The Third must indeed
| stand in such a relation, and thus must be capable of determining a Third
| of its own;  but besides that, it must have a second triadic relation in
| which the Representamen, or rather the relation thereof to its Object,
| shall be its own (the Third's) Object, and must be capable of determining
| a Third to this relation.  All this must equally be true of the Third's
| Third and so on endlessly;  and this, and more, is involved in the familiar
| idea of a Sign;  and as the term Representamen is here used, nothing more
| is implied.
|
| A 'Sign' is a Representamen with a mental Interpretant.
|
| Possibly there may be Representamens that are not Signs.
|
| 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.
|
| But 'thought' is the chief, if not the only, mode of representation.
|
| C.S. Peirce, "Syllabus" (c. 1902), 'Collected Papers', CP 2.274

SOP. Note 2


What is the most primitive form in which this POV-relativity arises?
One guess would be with the understanding of so-called demonstratives,
indexicals, pronouns, words like that.  What is the "ontology" of words
like "I", "you", "they", "this", "that"?  Silly way to ask the question.
There is no way to understand these words by way of a simple fixed map
from signs to objects.

Consider the following two sign relations, reflecting, let us say,
the ways that Ann and Bob use the words "Ann", "Bob", "I", "you".

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"           |
o---------------o---------------o---------------o
| 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"           |
o---------------o---------------o---------------o
| B             | "B"           | "B"           |
| B             | "B"           | "i"           |
| B             | "i"           | "B"           |
| B             | "i"           | "i"           |
o---------------o---------------o---------------o

Normally, the structure of the Object domain
is reconstructed on the semiotic plane S x I,
between the Sign and Interpretant columns of
a sign relational database, by partitioning
signs into equivalence classes of some sort.

But here the partition is different for each interpreter:

   Interpreter A has the signs in {"i", "A"} denoting A,

                 and the signs in {"u", "B"} denoting B.

   Interpreter B has the signs in {"u", "A"} denoting A,

                 and the signs in {"i", "B"} denoting B.

The kicker 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 peculiar identities, and thus far more like
demonstratives and indexicals and pronouns, than most people would like to think
about, 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.

Another style of graphical picture can be given by letting the fact that
interpreter J employs a triple of the form <x, y, z> be depicted like so:

          y
         /
   x o--J
         \
          z

Then the 16 triples of the above two Tables can
be arranged around their shared objects like so:

          "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"
 
 
          "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"

SOP. Note 3


| If we do not now dare everything,
| the fulfillment of that prophecy,
| recreated from the Bible in song
| by a slave, is upon us:
|
| GOD GAVE NOAH THE RAINBOW SIGN,
| NO MORE WATER, THE FIRE NEXT TIME!
|
| James Baldwin, 'The Fire Next Time', 1963, taken as
| an epigraph by Cornel West, 'Race Matters', p. xiii.

SOP. Note 4


There is a point here that I am still struggling to express/explain on several
of my other lists, about the diference between the way that most mathematicians
think and the way that some logicians and some philosophers think (I probably
mean analytic not existentialist philosophers here).  Mat has this sense of
due proportion about Plato's heaven that tells him how inexhaustible it is,
and that he is darn lucky to grasp this or that tiny patch of it at a time.
So when Mat says "universe" he always means "universe of discourse", or
"object (= space) in a category of related objects (= spaces)", always
a thing well-bounded by the frame of some venn diagram or category.
So it's always local and relative to the discussion in progress.
Growing out of this there's a type of "polymorphism", where the
number 1, at first sight an instance of constant value, 1 : U,
is also the constant function 1 : U -> U, where 1(x) = 1.
When you are reading "1" as the logical value "true", then
1 : U -> B = {0, 1} is the proposition "it's all good",
that is, the predicate that is true of every-thing in U.
This 1 is what presides at the top of the lattice of
functions (propositions, predicates) on U, and so
some people call it T = Top = Thing, but we start
getting into Big Trouble here, as now we are
ontologizing it (= ossifying it) and losing
the good senses of its local, relative,
polymorphic meanings.  It led to big
trouble in set theory once to think
that what's a thing in one context
has to be a Thing in all contexts.

Phil, in contrast, seems to think that it's within his grasp
to say many useful things about the literal universe, if not,
indeed, all possible universes.  Mat thinks that sheer hubris.

Signs Of Pragmata • Discussion

SOP. Discussion Note 1


HT = Hugh Trenchard

Re:  Signs Of Pragmata 1.  http://suo.ieee.org/ontology/msg05410.html
NB.  I "think" I finally got all of the typos out of this latest copy.

I like your way of asking "all" the questions that come to mind,
and so I will take some care and some time in trying answer them.
I need to say at the outset, though, that I will answer from my
own peculiar point of view, and I really have no idea what may
be considered the "received view" on Peirce as of this moment.

Different readers of Peirce seem to favor different "definitions"
or "characterizations" of the sign relation.  Robert Marty, et al.
have collected 76 + 12 of them at his site in Perpignan, and these
are also available at Joe Ransdell's "Arisbe" website:

http://www.univ-perp.fr/see/rch/lts/marty/
http://www.univ-perp.fr/see/rch/lts/marty/76defeng.htm

http://members.door.net/arisbe/
http://members.door.net/arisbe/menu/library/rsources/76defs/76defs.htm

My personal favorite definition would probably be Number 14 from the above lists.
The current definition is Number 15, and I will confess that I find some aspects
of it confusing to me, still, but I like it especially because of the "sunflower"
illustration, which was critical to piquing my interest in the subject many years
ago, partly because it made a connection in my mind to the way that I used to view
group theory in those days.

To represent a 3-tuple <x, y, z> such that xy = z in a group G,
I used to draw a picture of something like the following shape:

          x       y
           \    //
            \  //
             \//
             |||
             |||
             |||
              z

Then I would chain these together in trees and cycles to represent the
more complex constructions that I had to think about in a given setting.

When I used this tactic to represent the elementary sign relations among
the sunflowers and the sun, I ended up with pictures something like this:

          s_1   s_2
          o o   o o
          |  \ /  |
   s_8 o--o   o   o--o s_3
       o   \  |  /   o 
        \   o o o   /
         o--o O o--o
        /   o o o   \
       o   /  |  \   o 
   s_7 o--o   o   o--o s_4
          |  / \  |
          o o   o o
          s_6   s_5
 
Of course, it could be any multitude of sunflowers, not just eight.
When it comes to the placemattes |, ||, ||, however, what assignment
makes the most sense?  Does it really matter all that much if we place
the Sun, the object O, in first or second of third place, so long as we
dub it the object, isn't that enough?  Well, that depends.  Some people
will read this story as Peirce saying something significant about the
relationship between the three sign roles and his three "Categories",
which he often gave the maximally abstract names First, Second, Third.
And here you will find a lot of fights break out.  Where have all the
flowers gone?  And so it goes.  I personally do not choose to take it
that way, mostly because it is apparently possible to rationalize so
many different correspondences that none of them wins out as unique.
Let 10^3 flowers bloom.  Or maybe it's just 3! -- yes, 3 factorial.

SOP. Discussion Note 2


HT = Hugh Trenchard

Re: SOP 1.  http://suo.ieee.org/ontology/msg05410.html

HT: I find this quite confusing.  It looks like Peirce is saying that
    the "triadic" relationship of the sign, object, and interpretant
    cannot exist in any other combination except in the triad.

HT: If I can paraphrase this a bit, it sounds like an object (the "second")
    may be perceived by a person, but it is meaningless as any particular
    object until that person interprets the object by relating it to certain
    properties (i.e. conceptions and extensions (?)) she knows that can help
    to identify and give meaning to the particular object (these properties
    being the sign or "representamen").  The triadic relationship, if I am
    anything close to the mark, is about gleaning meaning from perception.

The first thing to do is to realize that a "sign relation" is a certain type
of formal structure that one is trying to characterize in a formal way, much
like the way that we define mathematical objects like graphs and groups and
so on ad infinitum.  Whether a particular family of formal objects is good
for any particular purpose is a whole separate question.  So quite a few
of the words that we necessarily have to use to talk about this familiy
of formal objects, for instance, "object" or "sign", will have to be
shorn of their ordinary connotations, and perhaps get to keep only
some of their former associations.  In this situation, all of the
designations Object, Sign, Interpretant refer to the relational
roles that particular elements play in a particular 3-tuple of
a particular set of 3-tuples that we call a "sign relation".

To make it short but rough, "object" is used more in the
sense of "object of discussion" or "object of thought".
It can be, but doesn't have to be an ordinary physical
or even a presently existing object.  Indeed, very
often the word "objective", in the sense of "goal"
or "intentional object" is a better paraphrase of
what's being intended in a particular application.

So let us go back to "the" definition of a sign relation --
I'd recommend No. 14 as being the clearest on this score:

http://suo.ieee.org/email/msg11187.html

There you can see that whatever we are talking about has a lot to do
with a special relationship of "correspondence" and "determination",
and to find out what Peirce meant by those words you have to do some
further reading, where you'd find that he is talking about a triple
correspondence, no kind of 2-adic mirror correspondence need apply,
and a type of determination that is informational and partial in
general, not of necessity causal or absolutely deterministic.

SOP. Discussion Note 3


HT = Hugh Trenchard

Re: SOP 1.  http://suo.ieee.org/ontology/msg05410.html

HT: I get confused (or more confused) when he talks about a sunflower being
    a representamen of the sun.  Is the sunflower in this case a property
    of the sun -- a thing which triggers an image of the sun;  something
    which implies the sun?  If so, I can't help but think again of Borges'
    story in which the jaguar implies the deer the jaguar ate, the grass
    the deer ate ... the universe (or that idea), and I question the
    proximity of the relationship.  A sunflower may conjure images
    of the sun, but isn't a closer representamen of the sun
    a circle and bright skies?

I will go ahead and use the word "sign" for the general case,
and if I want to talk specifically about a "mental sign",
I will just call it that.

I used to worry a lot about whether a sign of something
is a property of something, and vice versa, but this is
really a separate issue, and discussing it would depend
on picking a particular theory of what a "property" is,
which we don't really have to get into, just yet, not
in order to understand the bare essentials of what
a sign relation is.

Peirce says that a sunflower is a sign of the sun, to another sunflower,
so the speak -- more exactly, as interpreted in the inspired performance
of another sunflower -- and only with the proviso that you can imagine
a certain very unlikely condition as holding true, just because the
3-place relation of the sun O, sunflower s_1, and sunflower s_2
fits the definition of a sign relation that he more or less
gives in various places.

HT: Also, if a person is involved in the perception of the object, isn't
    the person himself part of the relationship?  It sounds like we have
    object, sign, and interpretation of sign=meaning.  But what about the
    physical system (i.e. the brain) which facilitates this transformative
    process?  Isn't this a quadratic relationship? (he-he-heaven forbid --
    anything but a triad!  Bring out the crucifixes and the wooden stakes!
    Sweep up the stars, dismantle the sun!)

I think that most of the things that you are talking about are covered by
the concept of an "interpreter", and we can go ahead and use that word in
a broad enough sense to cover any sort of interpretive system, whether an
individual agent, a whole community of interpretation, Gaia, Global Brain,
Over Soul, or whatever.  Again, this is just the standard system-theoretic
attitude with regard to the agent, "representative point", "test particle",
its names are legion.  It does not really matter much what imagery you find
most catalytic of fruitful ideas, since we are really only talking about the
forms that we see from a particular perspective.  From that outlook, what we
care about is the effect on the system, and that effect is the interpretant.

SOP. Discussion Note 4


JA = Jon Awbrey
RX = Reader X

Re: SOP 2.  http://suo.ieee.org/ontology/msg05411.html
In: SOP.    http://suo.ieee.org/ontology/thrd1.html#05410

JA: Consider the following two sign relations, reflecting, let us say,
    the ways that Ann and Bob use the words "Ann", "Bob", "I", "you".

    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"           |
    o---------------o---------------o---------------o
    | B             | "B"           | "B"           |
    | B             | "B"           | "u"           |
    | B             | "u"           | "B"           |
    | B             | "u"           | "u"           |
    o---------------o---------------o---------------o

JA: Normally, the structure of the Object domain
    is reconstructed on the semiotic plane S x I,
    between the Sign and Interpretant columns of
    a sign relational database, by partitioning
    signs into equivalence classes of some sort.

RX: In the above table, I see the four relations resulting from S x I.
    I am not certain what one means by denoting these 4 objects as distinct.
    I also do not see how signs would be in equivalence classes, but rather,
    it seems to me, that the objects have been placed into such a class,
    hence the partitioning evident from the table.  This ties in to
    your "exception" following, but what would be a normal case?

In this extensional way of thinking, a "relation" L is a subset of
a cartesian product, that we write as L c X_1 x ... x X_k, at least,
so long as the number of factors X_j is finite.

In this particular case, the sign relation L(A) is a subset of O x S x I,
where O = {A, B}, S = {"A", "B", "i", "u"}, and I = {"A", "B", "i", "u"}.
Using the notation |X| for the cardinality of X, or the number of elements
in the set X, we have |O x S x I| = |O||S||I| = 2 * 4 * 4 = 32, out of which
we have selected the 8 triples listed in the Table to make up the relation L(A).

If we delete the object column, and don't count repeated pairs in what remains,
then we have what would be called the "projection" of L(A) on the S x I plane,
or some language to that effect, which we may write as Proj_SI (L(A)) or just
L(A)_SI, to be lazy I guess.  To be even more lazy, let L = L(A)_SI for now.
L is a 2-adic relation L c S x I that has the following 8 ordered pairs:

Table 1_SI.  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

L has the structure of an "equivalence relation".
That is, L is Reflexive, Symmetric, and Transitive.
First of all, S = I, so we can say that L c S x S.
L is reflexive, because <x, x> in L for all x in S.
L is symmetric, because <x, y> in L => <y, x> in L.
L is transitive, because <x, y> and <y, z> in L =>
<x, z> in L.  When you have an equivalence relation,
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 related.
In this case, the equivalence classes are {"A", "i"}
and {"B", "u"}, which codes up the fact that Agent A,
in the "No Exit" discourse that involves just A and B,
uses either "A" or "i" indifferently to denote A, and
uses either "B" or "u" indifferently to denote B.  Fin.

As it happens, the equivalence classes of L(A)_SI are
in correspondence with the objects, the elements in O.
The class of signs {"A", "i"} corresponds to object A.
The class of signs {"B", "u"} corresponds to 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.
And that is just what I was hinting at in the following observation:

JA: But here the partition is different for each interpreter:

    Interpreter A has the signs in {"i", "A"} denoting A,

                  and the signs in {"u", "B"} denoting B.

    Interpreter B has the signs in {"u", "A"} denoting A,

                  and the signs in {"i", "B"} denoting B.

Now, this is a slightly ad hoc example, but what I am
talking about here reflects a very general phenomenon.

SOP. Discussion Note 5


JA = Jon Awbrey
RX = Reader X

Re: SOP 2.  http://suo.ieee.org/ontology/msg05411.html
In: SOP.    http://suo.ieee.org/ontology/thrd1.html#05410

JA: Another style of graphical picture can be given by letting the fact that
    interpreter J employs a triple of the form <x, y, z> be depicted like so:

            y
           /
     x o--J
           \
            z

RX: The triple here denotes the triad of <sign, object, interpretant>
    all relating to Interpreter J.  What is the conceptual distinction
    to be made between the Object A and the Interpreter A, or is this
    what you are driving at with the special case of O and Interpreter
    being the same with the use of demonstratives, indexicals, etc.?

There are depths of complexity here that I will try to avoid falling into --
somehow the phrase "Run, you fools!" comes to mind -- at any rate, let us
try to approach the abyss more gingerly than your average hobbit will do.

Interpretive agents, in so far as we formally consider them -- since we
do not concern ourselves with what they had for breakfast except insofar
as we conceive it to have a bearing on their sign relations -- are really
just personifications of these sign relations themselves.  In this formal
regard, we can replace Interpreter J with a particular sign relation L(J),
and oftentimes it will be safe enough in context to use just "J" for L(J).

In that light, the picture above is just another syntax for saying
that <x, y, z> is an element of L(J), more briefly, <x, y, z> in J.

In our eavesdropping on the discussion between A and B, we take them
as having names for A and B as unanalyzed objects of discussion, and
nowhere near enough vocabulary yet to talk about their own discourse
in sign relational terms.  It would take the development of language
about "higher order sign relations" of various sorts in order for us
to be able to talk about this intelligently and without getting lost
in near-hopeless confusion.  So let's leave that quest to the sequel.

SOP. Discussion Note 6


JA = Jon Awbrey
RX = Reader X

Re: SOP 2.  http://suo.ieee.org/ontology/msg05411.html
In: SOP.    http://suo.ieee.org/ontology/thrd1.html#05410

What happened is that all of the data of the two Tables
went to make up the two Figures, but the Tables grouped
the triples by Interpreter, that is, according to their
presence in the same sign relation, whereas the Figures
regrouped the triples by common Object, that is, taking
their 3-tuples half from Table 1 and half from Table 2.

So we need to look at the whole dataset once again:

JA: Consider the following two sign relations, reflecting, let us say,
    the ways that Ann and Bob use the words "Ann", "Bob", "I", "you".

    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"           |
    o---------------o---------------o---------------o
    | 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"           |
    o---------------o---------------o---------------o
    | B             | "B"           | "B"           |
    | B             | "B"           | "i"           |
    | B             | "i"           | "B"           |
    | B             | "i"           | "i"           |
    o---------------o---------------o---------------o

JA: Then the 16 triples of the above two Tables can
    be arranged around their shared objects like so:

           "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"


           "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"

RX: This diagram does not parse for me, I am afraid.
    I am trying to reconstruct it locally, but I am
    unable to reconcile it with the simpler case of:

           y
          /
    x o--J
          \
           z

RX: Please remind me, in this ASCII notation,
    you are representing a graph, but the o--
    is a directed edge?

    Assume so, then I can see:

          "A"
          /
    A o--B
          \
          "A"

    from the RHS, for example, but cannot relate that
    to the relational table given for Interpreter A.

Yes, that is reading it right, and this is as it should be,
because that triple comes from the Table for Interpreter B.

Document History

Sign Relations 2004–2005

Sign Relations 2004–2005 • NKS Forum • History

  1. http://forum.wolframscience.com/showthread.php?postid=2298#post2298
  2. http://forum.wolframscience.com/showthread.php?postid=2393#post2393
  3. http://forum.wolframscience.com/showthread.php?postid=2395#post2395
  4. http://forum.wolframscience.com/showthread.php?postid=2420#post2420
  5. http://forum.wolframscience.com/showthread.php?postid=2426#post2426

Sign Relations 2004–2005 • Inquiry List • History

  1. http://stderr.org/pipermail/inquiry/2004-December/002134.html
  2. http://stderr.org/pipermail/inquiry/2004-December/002193.html
  3. http://stderr.org/pipermail/inquiry/2004-December/002194.html
  4. http://stderr.org/pipermail/inquiry/2005-January/002241.html
  5. http://stderr.org/pipermail/inquiry/2005-January/002246.html
  6. http://stderr.org/pipermail/inquiry/2005-September/003041.html
  7. http://stderr.org/pipermail/inquiry/2005-September/003042.html
  8. http://stderr.org/pipermail/inquiry/2005-September/003044.html
  9. http://stderr.org/pipermail/inquiry/2005-September/003046.html
  10. http://stderr.org/pipermail/inquiry/2005-September/003047.html

Sign Relations 2004–2005 Commentary • NKS Forum • History

  1. http://forum.wolframscience.com/showthread.php?postid=2443#post2443
  2. http://forum.wolframscience.com/showthread.php?postid=2444#post2444
  3. http://forum.wolframscience.com/showthread.php?postid=2445#post2445
  4. http://forum.wolframscience.com/showthread.php?postid=2446#post2446
  5. http://forum.wolframscience.com/showthread.php?postid=2457#post2457
  6. http://forum.wolframscience.com/showthread.php?postid=2459#post2459
  7. http://forum.wolframscience.com/showthread.php?postid=2462#post2462
  8. http://forum.wolframscience.com/showthread.php?postid=2464#post2464
  9. http://forum.wolframscience.com/showthread.php?postid=2466#post2466
  10. http://forum.wolframscience.com/showthread.php?postid=2471#post2471
  11. http://forum.wolframscience.com/showthread.php?postid=2472#post2472
  12. http://forum.wolframscience.com/showthread.php?postid=2479#post2479
  13. http://forum.wolframscience.com/showthread.php?postid=2481#post2481

Sign Relations 2005 Commentary • Inquiry List • History

  1. http://stderr.org/pipermail/inquiry/2005-January/002242.html
  2. http://stderr.org/pipermail/inquiry/2005-January/002243.html
  3. http://stderr.org/pipermail/inquiry/2005-January/002244.html
  4. http://stderr.org/pipermail/inquiry/2005-January/002248.html
  5. http://stderr.org/pipermail/inquiry/2005-January/002252.html
  6. http://stderr.org/pipermail/inquiry/2005-January/002253.html
  7. http://stderr.org/pipermail/inquiry/2005-January/002254.html
  8. http://stderr.org/pipermail/inquiry/2005-January/002255.html
  9. http://stderr.org/pipermail/inquiry/2005-January/002256.html
  10. http://stderr.org/pipermail/inquiry/2005-January/002257.html
  11. http://stderr.org/pipermail/inquiry/2005-January/002258.html
  12. http://stderr.org/pipermail/inquiry/2005-January/002259.html
  13. http://stderr.org/pipermail/inquiry/2005-January/002260.html
  14. http://stderr.org/pipermail/inquiry/2005-September/003028.html
  15. http://stderr.org/pipermail/inquiry/2005-September/003031.html
  16. http://stderr.org/pipermail/inquiry/2005-September/003032.html
  17. http://stderr.org/pipermail/inquiry/2005-September/003036.html
  18. http://stderr.org/pipermail/inquiry/2005-September/003037.html
  19. http://stderr.org/pipermail/inquiry/2005-September/003038.html
  20. http://stderr.org/pipermail/inquiry/2005-September/003039.html
  21. http://stderr.org/pipermail/inquiry/2005-September/003043.html
  22. http://stderr.org/pipermail/inquiry/2005-September/003045.html
  23. http://stderr.org/pipermail/inquiry/2005-September/003053.html
  24. http://stderr.org/pipermail/inquiry/2005-September/003059.html
  25. http://stderr.org/pipermail/inquiry/2005-September/003061.html
  26. http://stderr.org/pipermail/inquiry/2005-October/003083.html
  27. http://stderr.org/pipermail/inquiry/2005-October/003101.html
  28. http://stderr.org/pipermail/inquiry/2005-December/003325.html
  29. http://stderr.org/pipermail/inquiry/2005-December/003327.html
  30. http://stderr.org/pipermail/inquiry/2005-December/003328.html
  31. http://stderr.org/pipermail/inquiry/2005-December/003332.html
  32. http://stderr.org/pipermail/inquiry/2005-December/003336.html
  33. http://stderr.org/pipermail/inquiry/2005-December/003338.html
  34. http://stderr.org/pipermail/inquiry/2005-December/003339.html
  35. http://stderr.org/pipermail/inquiry/2005-December/003347.html
  36. http://stderr.org/pipermail/inquiry/2005-December/003348.html
  37. http://stderr.org/pipermail/inquiry/2005-December/003349.html

Sign Relations 2005 Discussion • Inquiry List • History

  1. http://stderr.org/pipermail/inquiry/2005-January/002247.html
  2. http://stderr.org/pipermail/inquiry/2005-January/002249.html
  3. http://stderr.org/pipermail/inquiry/2005-January/002261.html
  4. http://stderr.org/pipermail/inquiry/2005-January/002263.html
  5. http://stderr.org/pipermail/inquiry/2005-January/002264.html
  6. http://stderr.org/pipermail/inquiry/2005-January/002265.html
  7. http://stderr.org/pipermail/inquiry/2005-January/002266.html
  8. http://stderr.org/pipermail/inquiry/2005-January/002267.html
  9. http://stderr.org/pipermail/inquiry/2005-January/002302.html
  10. http://stderr.org/pipermail/inquiry/2005-January/002303.html
  11. http://stderr.org/pipermail/inquiry/2005-January/002311.html
  12. http://stderr.org/pipermail/inquiry/2005-September/003029.html
  13. http://stderr.org/pipermail/inquiry/2005-September/003033.html
  14. http://stderr.org/pipermail/inquiry/2005-September/003035.html
  15. http://stderr.org/pipermail/inquiry/2005-September/003040.html
  16. http://stderr.org/pipermail/inquiry/2005-September/003048.html
  17. http://stderr.org/pipermail/inquiry/2005-September/003054.html
  18. http://stderr.org/pipermail/inquiry/2005-October/003068.html
  19. http://stderr.org/pipermail/inquiry/2005-October/003069.html
  20. http://stderr.org/pipermail/inquiry/2005-October/003072.html
  21. http://stderr.org/pipermail/inquiry/2005-October/003093.html
  22. http://stderr.org/pipermail/inquiry/2005-October/003094.html
  23. http://stderr.org/pipermail/inquiry/2005-December/003326.html
  24. http://stderr.org/pipermail/inquiry/2005-December/003329.html
  25. http://stderr.org/pipermail/inquiry/2005-December/003333.html
  26. http://stderr.org/pipermail/inquiry/2005-December/003334.html
  27. http://stderr.org/pipermail/inquiry/2005-December/003335.html

Signs Of Pragmata 2004

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SOP.  Signs Of Pragmata

Ontology List

01.  http://suo.ieee.org/ontology/msg05410.html
02.  http://suo.ieee.org/ontology/msg05411.html
03.  http://suo.ieee.org/ontology/msg05412.html
04.  http://suo.ieee.org/ontology/msg05413.html

Inquiry List

01.  http://stderr.org/pipermail/inquiry/2004-February/001182.html
02.  http://stderr.org/pipermail/inquiry/2004-February/001183.html
03.  http://stderr.org/pipermail/inquiry/2004-February/001184.html
04.  http://stderr.org/pipermail/inquiry/2004-February/001185.html

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

SOP.  Signs Of Pragmata -- Discussion

Ontology List

01.  http://suo.ieee.org/ontology/msg05414.html
02.  http://suo.ieee.org/ontology/msg05415.html
03.  http://suo.ieee.org/ontology/msg05416.html
04.  http://suo.ieee.org/ontology/msg05417.html
05.  http://suo.ieee.org/ontology/msg05418.html
06.  http://suo.ieee.org/ontology/msg05419.html

Inquiry List

01.  http://stderr.org/pipermail/inquiry/2004-February/001186.html
02.  http://stderr.org/pipermail/inquiry/2004-February/001187.html
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05.  http://stderr.org/pipermail/inquiry/2004-February/001190.html
06.  http://stderr.org/pipermail/inquiry/2004-February/001191.html

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