Limited Mark Universes

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Inquiry Into Symbolization

SYM. Note 1


I think that it is time for me to take a little break --
to try and pose my current guess as to what Peirce is
going on about with all that guff about symbolization.

I find our pupallary Peirce trying to wriggle his way
out of the Cartesian/Kantian cocoon in which he finds
himself encased at his phase of metamorphosis in view. 

Remember that for Peirce "concepts are a species of symbols", and so
to talk about "symbolization" and "symbolizability" is tantamount to
invoking a generalization of "conceptualization" and "conceivability".

So the whole scene in question is taking place on the stage set by Kant,
whose depiction of the Creation, Development, and Elimination operators
that work on concept-ions Peirce has already intoned in his prologue to
the entire drama:

| The essential of a thing -- the character of it --
| is the unity of the manifold therein contained.
| 'Id est', the logical principle, from which as
| major premiss the facts thereof can be deduced.
|
| What are called a man's principles however
| are only certain beliefs of his that he may or
| may not carry out.  They therefore do not compose
| his character, but the general expression of the facts --
| the ACTS OF HIS SOUL -- does.
|
| What he does is important.
| How he feels is incidental.
|
| CSP, CE 1, page 6.
|
| Charles Sanders Peirce,
|"Private Thoughts, Principally On The Conduct Of Life" (Number 37, August 1860),
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

If that is a bit too oracular, then he will echo it again in the interlude to come.

| This paper is based upon the theory already established, that the function of
| conceptions is to reduce the manifold of sensuous impressions to unity, and that
| the validity of a conception consists in the impossibility of reducing the content
| of consciousness to unity without the introduction of it.  (CSP, CP 1.545, CE 2.49).

Before I can say any more about this business I will have to dig up some old
essays of mine on the relationship between artificial sets and natural kinds.

SYM. Note 2


I am going to try and explain one of the conceptual schemes
that I use to interpret what Peirce says about symbolization
and symbolizability in his First "Logic Of Science" Lectures.

To do this I need to discuss the relations between two lattices
or partial orders, one being a lattice of "arbitrary sets" (SET),
the other being a lattice of "natural kinds" (NAT).  For the time
being, I limit myself to concrete, discrete, even finite universes
of discourse, where all of the sets in view are subsets of a set X.

Here is a little essay in which I first broached this subject to the
Peirce Forum last year, in what I once thought was an amusing manner.

Subj:  HOPE's & FEAR's
Date:  Tue, 23 May 2000 12:20:58 -0400
From:  Jon Awbrey <jawbrey@oakland.edu>
  To:  Peirce Discussion Forum <peirce-l@lyris.acs.ttu.edu>

Day 1

Bright and early Monday morning I woke up with the
idea of returning to a peculiar theme in Peirce's
work, one that persists in nagging me, often while
I am reading something else, or working on another
topic to which it seems fairly incidental and all
too tangential, but a theme, nevertheless, that I
can barely catch a glimmer of here and there, not
even get so much as a firm handle on it, though it
continues to plague me with that not-to-be-denied
sense of its vague imports and general importance.
Moreover, it seems like every attempt that I have
made to raise this sunken ship of a topic, whether
in conversations here or elsewhere, has soon gone
down in flames or subsided with nary a whimper --
I cannot decide which is worse, but either way it
has been a fruitless, a frustrating, and a vastly
unsatisfactory experience just to try to express it.

Now, with that sort of build-up I think that you all
must be terribly -- yes, "terribly" is precisely the
word that fits -- excited about the prospects of my
bringing it up -- whatever the heck it is -- again,
but, in spite of this all too painful suspense, on
both our parts, I woke up, as I said, on that day,
rather enthusiastic about my prospects this time.

Without further ado -- well, thank you! -- let me
just say that this is the topic of Peirce's Notes
on "A Limited Universe of Marks" (ALUOM), appearing
for the first time in the volume by Peirce and his
students entitled 'Studies in Logic', published by
Little, Brown, and Company, of Boston, MA, in 1883,
(CE 4, 405-466, specifically for this note, 450-453),
and reworked again for the "Grand Logic", in 1893,
(CP 2.517-531).  The original book of studies was
republished as a kind of Centenary Edition in 1983,
but I have apparently misplaced my copy of this work
in one of my last few of many geographic relocations.

Since these two versions of the remarks on ALUOM,
as I will call it, may enjoy widely varying levels
of accessibility across the breadth of this Forum,
I think that it might benefit discussion to copy as
much of them as I think necessary into the Sources
thread.  The reason why I prefer to do things this
way is to separate my own remarks and speculations
as much as possible from the texts themselves, and
further, to leave the texts available in a maximally
uncluttered fashion for use in the light of what may
well turn out to be a variety of different purposes.

To this task I will now turn, but before I leave this
bit of preamble behind for what for may well turn out,
or maybe not so well turn out, to be a lengthy interval
of time, I probably ought to say a few more words in
explanation of my current tag-line for this thread.

On what do I feel that the theme of ALUOM has a bearing?
Well, on many things, easy to collect but difficult to
classify, at least, in any thoroughly rationalized and
schematized way.  Natural kinds, the logic of inquiry,
especially the teasing apart of abduction and induction,
"giving a rule to abduction" (GARTA), constraints, and
thus information, innate or acquired, on admssible hype
and on permissible hypotheses, the nature of the human,
the Pragmatic Cosmos that orders the normative sciences
into a concentrically focusing and yet climbing spiral
of ascent, aspiration, reflection, and regardedness,
from aesthetics via ethics to logic in their turns --
you will no doubt begin to think that I am merely
free-associating or spinning out topics at random
if I even begin to unroll my rigamarole shopping
list of things that I think are encompassed here.

Nevertheless, what little form of organization that I can e-spy
in this seeming chaos and this teeming cornucopia of pragmatic
commonplaces is enough to get me cranking, at least, on some
sort of exposition, no matter how chancy and risky it may
start out being at first.

In this vein, I find what appears to be two distinct ways,
perhaps a couple of dual ways to approach the instigatory
question of it all, the question of "GARTA", of what sort
of limits may exist on our admission, creation, generation,
and imagination of propositions to describe our worlds of
experience, including as a special case the propositions
that we may choose to employ as explanations of striking
phenomena.

One "way of thinking" (WOT) is the one that I will dub as
the way of "Higher Order Propositional Expressions" (HOPE's).

The other WOT that I can see, at least, in so far as I can see any
other way at all, is the one that I will dub, fittingly enough, as
the way of "Framed Extensions And Restrictions" (FEAR's).

With apologies to Pandora, I will choose to introduce
the HOPE's first, and put off the FEAR's until later.

With gratitude to Shirazad, I will choose to make these HOPE's
the story of yet another day, as who knows what any day brings?

Day 2

If one treats hypotheses as any other propositions,
as so many simple closed circles in a venn diagram,
as so many logical variables in a truth table, then
one way of talking about constraints on hypotheses
is by making use of propositions about propositions,
or "higher order propositions" (HO propositions),
which are naturally expressed in the formulas of
"higher order propositional expressions" (HOPE's),
telling what propositions, in general, hypotheses
or not, are admitted to the universe of discourse,
curtailing discussion to "a limited number of marks",
as Peirce had a habit of putting it, in his studied
and exquisitely classical way.

The dizzying hypes of these orders of abstraction
makes it advisable to begin with a concrete and
a memorable case, if a rather ridiculous example.
Here, understand that we are only concerned with
the purified form, and not the ignoble content,
of this artificially simple example.

Don't bother to try and stop me if you have already
heard this one, as I think that it is likely to be
one of Aristotle's most outlandish jokes, because
I already have in mind another end altogether that
I hope will eventually serve to redeem the evident
absurdity of it.

Consider the humorous definition of a Human Being as
a Featherless Two-legged Critter, to schematize it,
if not utterly to traumatize it, let me express the
subject matter in the following way:

|  A  =  Apterous  (featherless)
|  B  =  Bipedal   (two-legged)
|  C  =  Critter   (animal)
|  H  =  Human     (human being)

Now, I had been planning to introduce some venn diagrams
at this point, but after wasting two days of trying, and
trying the patience of all concerned, and not concerned,
I am afraid that will have to forego, for now, that brand
of diagram -- what conceivable significance could iconic
diagrams have in philosophy, anyway? -- at least, until
I can figure out a way to arrive at a non-distorted form
of representation without the occasional experiment or two,
indeed, short of a persistent, persevering, indefinite series
of experiments.  Of course, if I could figure out how to do that,
in full generality and without loss of geniality, as they will say,
what need would there be for any inquiry at all, much less any brand
of theory concerned with the logic and the practice of actual inquiry?
But never mind all that.  Where there is a will, there will be a way,
whether the pathways of the requistite varieties of reactions have
all of their semiotic catalysts in all of the most optimal places
or not.  There is always one way or another to go forward, even
if one's active duty status as an exponent of Peirce and one's
interim role as an interpretant of Peirce must abdicate a
few of the iconic attributes that remain most fitting to
these tasks, and even if we must relegate ourselves to
symbolically talking about the kinds of diagrams that
Peirce regarded as important to actually, brutally,
crudely, deliberately, existentially, faithfully
take a modicum of trouble to draw, for all that
one can learn from the concrete and practical
process of going through the exercise to do it.
In short, short of the facilities of the graphic
medium, I will have to require you to exercise your
imagination to a somewhat greater extent than you otherwise
might have to, and I will count myself fortunate in the circumstance,
that when it comes to imagination, you folks have no shortage of that!

But I did have a bit more luck with a somewhat simpler class of diagrams --
What double-edged luck, indeed, that it should have encouraged me to go
on ahead and rush in blindly where even angles and anglers fear to leave
the marks of their treads! -- but never mind all that.  These rather more
tractable diagrams, although they lack that one critically important and
crucially iconic property of continuously reminding the viewer, not only
of the conceivably-continuously-supporting extensions of human concepts,
but also of the arbitrariness of the heraldic distinctions that humans
are wont to mark upon the underlying fields of existential experience,
as if the divisions we impose in our own conceits and in our images
were capable of placing any brand of demand on Nature at its joints
that Nature could not cast off as quick as Nature can dispose of us.
But never mind all that.  The sort of diagram that I have been able
to draw on these walls, at least, so far, are none other than the
"logical lattice", the "propositional partial order", or, with
a tip of the hat to Tom Gollier, the "implicative food chain"
type of diagram.  And so I will satisfice with these for now.

So let me try to draw you a picture of the situation that I want to discuss,
if it must be one that requires the viewer to "connect the dots" just a bit.
Figure 1 outlines the subject matter, to wit, the category "human being" (H)
here defined as falling under the head of an "apterous biped" (G  =  A |^| B),
hence bound by the set-theoretic intersection G of the respective extensions
of the two concepts, "apterous" (= featherless) and "bipedal" (= two-legged).
Now the wise-cracking sort of person, that everyone among us has encountered
before, will naturally be compelled to say, ignoring the natural and implicit
constraints of the discussion to what are often described as "natural kinds",
"But what of the plucked chicken? -- a two-legged critter without feathers? --
Is that your model of a genuine Mensch?"  (To get the full effect, you have
to imagine this being said in a Woody Allen voice.)  And so you patiently go
about explaining, as if your interrupreter did not already know this -- such
is the role of a straightman in this genre of commodious eristic, as you know
that you'll get your turn sooner or later, hopefully in the very next bit --
all about how a "plucked chicken", along with many other hype-o-thetical and
hi-pathetical creatures that might be abduced, construed, confabulated, and
otherwise plucked from thin air, is what one calls an "artificial kind", in
"essence", or the lack thereof, not really a "kind" (Greek 'genus') at all.

|      A             B
|      o             o
|      |\           /|
|      |.\         /.|
|      |  \       /  |
|      | . \     / . |
|      |    \   /    |
|      |  .  \ /  .  |
|      |      G      |
|      |   . / \ .   |
|      |    /   \    |
|      |   / . . \   |
|      |  /  . .  \  |
|      | / .     . \ |
|      |/.         .\|
|      o             o
|      H             P
|
| Figure 1.  On Being Human
|
| A  =  Apterous (featherless animal)
| B  =  Bipedal  (two-legged being)
| C  =  Critter  (creature, creation)
| G  =  GLB  =  Intersection of A and B
| H  =  HB   =  Human Being
| P  =  PC   =  Plucked Chicken

Okay, I think that will do to set up the joke.
I will save the explanation and the resolution
of it -- not nearly so fun a task -- till next
we meet in this space.

SYM. Note 3


| Note A.  On A Limited Universe Of Marks
|
| Boole, De Morgan, and their followers, frequently speak of
| a "limited universe of discourse" in logic.  An unlimited universe
| would comprise the whole realm of the logically possible.  In such
| a universe, every universal proposition, not tautologous, is false;
| every particular proposition, not absurd, is true.  Our discourse
| seldom relates to this universe:  we are either thinking of the
| physically possible, or of the historically existent, or of
| the world of some romance, or of some other limited universe.
|
| But besides its universe of objects, our discourse also refers to
| a universe of characters.  Thus, we might naturally say that virtue
| and an orange have nothing in common.  It is true that the English
| word for each is spelt with six letters, but this is not one of the
| marks of the universe of our discourse.
|
| A universe of things is unlimited in which every combination of characters,
| short of the whole universe of characters, occurs in some object.  In like
| manner, the universe of characters is unlimited in case every aggregate
| of things short of the whole universe of things possesses in common one
| of the characters of the universe of characters.  The conception of
| ordinary syllogistic is so unclear that it would hardly be accurate
| to say that it supposes an unlimited universe of characters;  but
| it comes nearer to that than to any other consistent view.  The
| non-possession of any character is regarded as implying the
| possession of another character the negative of the first.
|
| In our ordinary discourse, on the other hand, not only are both universes limited, but,
| further than that, we have nothing to do with individual objects nor simple marks;
| so that we have simply the two distinct universes of things and marks related to
| one another, in general, in a perfectly indeterminate manner.  The consequence
| is, that a proposition concerning the relations of two groups of marks is not
| necessarily equivalent to any proposition concerning classes of things;  so
| that the distinction between propositions in extension and propositions in
| comprehension is a real one, separating two kinds of facts, whereas in the
| view of ordinary syllogistic the distinction only relates to two modes of
| considering any fact.  To say that every object of the class S is included
| among the class of P's, of course must imply that every common character of
| the P's is a common character of the S's.  But the converse implication is by
| no means necessary, except with an unlimited universe of marks.  The reasonings
| in depth of which I have spoken, suppose, of course, the absence of any general
| regularity about the relations of marks and things.  (CSP, SIL, 182-183).
|
| CSP, SIL, pages 182-186.  (Cf. CE 4, pages 450-453, CP 2.517-531).
|
| Charles Sanders Peirce, "Note A.  On A Limited Universe Of Marks" (1883),
| CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University',
| Reprinted with an Introduction by Max H. Fisch & a Preface by Achim Eschbach,
| in 'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983.
|
|'Writings of Charles S. Peirce: A Chronological Edition, Volume 4, 1879-1884',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986.

SYM. Note 4


This is one of those puzzles that I have been puzzling away at for almost as long
as I can remember.  I have gotten fairly well acquainted with the various pieces
of the puzzle, but haven't quite figured out yet how they ought to fit together.

It all seems to have something to do with an intricate relationship
among concepts, kinds (of the natural kind, naturally), and symbols.
I know that I am always reminded of it when I read what Peirce says
on the issues of "symbolization" and "symbolizability".  And I have
the impression that there is a vast order of generalization in the
works  here, taking the topics of "observation" and "observables",
along with "computation" and "computables", and even "conception"
and "conceivables" under its wing with plenty of room left over.

Still, the best that I seem able to do at this juncture in time
is just to keep assembling the pieces together and just to keep
staring at them till the right sorts of connections occur to me.

The pieces of the puzzle are these:

1.  Remember that for Peirce "concepts are a species of symbols", and so
    to talk about "symbolization" and "symbolizability" is tantamount to
    invoking a generalization of "conceptualization" and "conceivability".

    So the whole scene in question is taking place on the stage set by Kant,
    whose depiction of the creation, development, and elimination operators
    that work on conceptions Peirce has already intoned in his prologue to
    the entire drama:

    | The essential of a thing -- the character of it --
    | is the unity of the manifold therein contained.
    | 'Id est', the logical principle, from which as
    | major premiss the facts thereof can be deduced.
    |
    | CSP, CE 1, page 6.
    |
    | Charles Sanders Peirce,
    |"Private Thoughts, Principally On The Conduct Of Life" (Number 37, August 1860),
    |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866',
    | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

2.  The idea about the function of conceptions that Peirce obtained from Kant:

    | This paper is based upon the theory already established, that the function of
    | conceptions is to reduce the manifold of sensuous impressions to unity, and that
    | the validity of a conception consists in the impossibility of reducing the content
    | of consciousness to unity without the introduction of it.  (CSP, CP 1.545, CE 2.49).
    |
    | http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm

3.  Peirce's "Note On A Limited Universe Of Marks" (NOALUOM).

    | http://suo.ieee.org/ontology/msg03204.html
    |
    | CSP, SIL, pages 182-186.  (Cf. CE 4, pages 450-453, CP 2.517-531).
    |
    | Charles Sanders Peirce, "Note A.  On A Limited Universe Of Marks" (1883),
    | CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University',
    | Reprinted with an Introduction by Max H. Fisch & a Preface by Achim Eschbach,
    | in 'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983.
    |
    |'Writings of Charles S. Peirce: A Chronological Edition, Volume 4, 1879-1884',
    | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986.

4.  An image that I have about the relationship between artificial kinds
    and natural kinds in terms of a mapping, a morphism, a restriction,
    or a quotient relation between two lattices.  I worked this out
    once in application to the "apterous biped" definition of the
    human, but I Kant quite recall the punchline.

Let me begin again with this last bit to see if I
can get a little bit further this time through it.

Consider the joke definition of a Human Being as a Featherless Two-legged Critter.
By way of a schematic formalization, I set out the matter in the following manner:

|      A             B
|      o             o
|      |\.         ./|
|      | \ .     . / |
|      |  \  . .  /  |
|      |   \ . . /   |
|      |    \   /    |
|      |   . \ / .   |
|      |      G      |
|      |  .  / \  .  |
|      |    /   \    |
|      | . /     \ . |
|      |  /       \  |
|      |./         \.|
|      |/           \|
|      o             o
|      H             P
|
| Figure 1.  On Being Human
|
| A  =  Apterous   =  featherless animal
| B  =  Bipedal    =  two-legged being
| C  =  Critter    =  creature, creation
| G  =  glb(A, B)  =  A |^| B
| H  =  Human Being
| P  =  Plucked Chicken

Figure 1 outlines the subject matter, to wit, the category "human being" (H)
here defined as falling under the head of an "apterous biped" (G  =  A |^| B),
hence bound by the set-theoretic intersection G of the respective extensions
of the two concepts, "apterous" (= featherless) and "bipedal" (= two-legged).
Now the wise-cracking sort of person, one who ignores the naturally implicit
constraints of the discussion to what are often described as "natural kinds",
will naturally be compelled to pipe up, "But what of the plucked chicken? --
a two-legged critter without feathers? -- is that your idea of human being?"

Now, we know that the response to this witlesscism must invoke the distinction
between what one calls an "artificial kind" and a "natural kind", respectively,
even though it is difficult to say just how this difference makes a difference.

Here is one possible way to view the situation:

|            SET                        NAT                        NAT
|
|      A             B            A             B            A             B
|      o             o            o             o            o             o
|      |\.         ./|            |            /              \           /
|      | \ .     . / |            |           /                \         /
|      |  \  . .  /  |            |          /                  \       /
|      |   \ . . /   |            |         /                    \     /
|      |    \   /    |            |        /                      \   /
|      |   . \ / .   |            |       /                        \ /
|      |      G      |            |      /                          G
|      |  .  / \  .  |            |     /                           =
|      |    /   \    |            |    /                            =
|      | . /     \ . |            |   /                             =
|      |  /       \  |            |  /                              =
|      |./         \.|            | /                               =
|      |/           \|            |/                                =
|      o             o            o                                 o
|      H             P            H                                 H
|
| Figure 2.  On Being Human, All Too Human

Think of the initial set-up as being cast in a lattice of arbitrary sets.
Within that setting, the "greatest lower bound" (glb) of the extensions
of A and B is their set-theoretic intersection, G = glb(A, B) = A |^| B.
This G covers the desired class H but also admits the risible category P.

Now, suppose that we are clued into the fact that not all sets in SET
are admissible, allowable, natural, pertinent, relevant, or whatever,
to the aims of the discussion in view, and that only some mysterious
'je ne sais quoi' subset of "natural kinds", NAT c SET, is at stake,
a limitation that, whatever else it does, excludes the set P and all
of that ilk from beneath glb(A, B).  Though we cannot quite say how
we apply this information, we know it by its effects to give us the
lattice structure in the next frame, where H = glb(A, B), and thus
in this more natural setting the proposed definition works okay.

An alternative way to look at the transformation of our views
from the arbitrary lattice SET to the natural lattice NAT,
is illustrated in the last frame, where the equal signs
indicate that the nodes for G and H are identified.
In this picture, the measure of the interval that
once existed between G and H, now shrunk to nil,
gives a rough indication of the quantity of
information that went into forming the
natural end result.

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

Limited Mark Universes

LMU 1. Comment on Peirce “On a Limited Universe of Marks”

Whenever we read what writers have written, we should try to figure out what they were trying to say, within their own frame of reference, and with an eye to their own chief motives, and put off till later, maybe, the task of staking them out on the dissecting trays of our chosen isms and anti-isms.  It is just barely conceivable, after all, that the pre-dissected writer is trying to tell us something of import about the limitations of our taxonomic labels.

In that spirit, I will return to Peirce's remark “On A Limited Universe of Marks” and try to give it an independent reading.


| On A Limited Universe Of Marks
|
| Boole, De Morgan, and their followers, frequently speak of
| a "limited universe of discourse" in logic.  An unlimited universe
| would comprise the whole realm of the logically possible.  In such
| a universe, every universal proposition, not tautologous, is false;
| every particular proposition, not absurd, is true.  Our discourse
| seldom relates to this universe:  we are either thinking of the
| physically possible, or of the historically existent, or of
| the world of some romance, or of some other limited universe.
|
| But besides its universe of objects, our discourse also refers to
| a universe of characters.  Thus, we might naturally say that virtue
| and an orange have nothing in common.  It is true that the English
| word for each is spelt with six letters, but this is not one of the
| marks of the universe of our discourse.
|
| A universe of things is unlimited in which every combination of characters,
| short of the whole universe of characters, occurs in some object.  In like
| manner, the universe of characters is unlimited in case every aggregate
| of things short of the whole universe of things possesses in common one
| of the characters of the universe of characters.  The conception of
| ordinary syllogistic is so unclear that it would hardly be accurate
| to say that it supposes an unlimited universe of characters;  but
| it comes nearer to that than to any other consistent view.  The
| non-possession of any character is regarded as implying the
| possession of another character the negative of the first.
|
| In our ordinary discourse, on the other hand, not only are both universes limited, but,
| further than that, we have nothing to do with individual objects nor simple marks;
| so that we have simply the two distinct universes of things and marks related to
| one another, in general, in a perfectly indeterminate manner.  The consequence
| is, that a proposition concerning the relations of two groups of marks is not
| necessarily equivalent to any proposition concerning classes of things;  so
| that the distinction between propositions in extension and propositions in
| comprehension is a real one, separating two kinds of facts, whereas in the
| view of ordinary syllogistic the distinction only relates to two modes of
| considering any fact.  To say that every object of the class S is included
| among the class of P's, of course must imply that every common character of
| the P's is a common character of the S's.  But the converse implication is by
| no means necessary, except with an unlimited universe of marks.  The reasonings
| in depth of which I have spoken, suppose, of course, the absence of any general
| regularity about the relations of marks and things.  (CSP, SIL, 182-183).
|
| CSP, SIL, pp. 182-186.  (CP 2.517-531;  CE 4, 450-453).
|
| Charles Sanders Peirce, "On A Limited Universe Of Marks" (1883), in:
| CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University',
| Reprinted with an Introduction by Max H. Fisch and a Preface by Achim Eschbach,
|'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983.
|
|'Writings of Charles S. Peirce:  A Chronological Edition, Volume 4, 1879-1884',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986.

LMU 2. Upon the Logic of Mathematics (CP 3.43)


It will be necessary to do a little bit of background reading.
For ease of study, I'll break up long paragraphs as I see fit.

| Still it may be well to consider the matter a little further.
| Imagine, then, a particular case under Boole's calculus, in
| which the letters are no longer terms of first intention,
| but terms of second intention, and that of a special kind.
| Genus, species, difference, property, and accident, are
| the well-known terms of second intention.  These relate
| particularly to the 'comprehension' of first intentions;
| that is, they refer to different sorts of predication.
| Genus and species, however, have at least a secondary
| reference to the 'extension' of first intentions.
|
| Now let the letters, in the particular application of
| Boole's calculus now supposed, be terms of second intention
| which relate exclusively to the extension of first intentions.
| Let the differences of the characters of things and events be
| disregarded, and let the letters signify only the differences
| of classes as wider or narrower.  In other words, the only
| logical comprehension which the letters considered as terms
| will have is the greater or less divisibility of the classes.
|
| Thus, 'n' in another case of Boole's calculus might,
| for example, denote "New England States";  but in the
| case now supposed, all the characters which make these
| States what they are being neglected, it would signify
| only what essentially belongs to a class which has the
| same relations to higher and lower classes which the
| class of New England States has, -- that is,
| a collection of 'six'.
|
| C.S. Peirce, 'Collected Papers', CP 3.43
|
| Charles Sanders Peirce, "Upon the Logic of Mathematics",
|'Proceedings of the American Academy of Arts and Sciences',
| Volume 7, pp. 402-412, September 1867.

LMU 3. Upon the Logic of Mathematics (CP 3.44)


| In this case, the sign of identity will receive a special meaning.
| For, if 'm' denotes what essentially belongs to a class of the
| rank of "sides of a cube", then 'm =, n' will imply, not that
| every New England state is a side of a cube, and conversely,
| but that whatever essentially belongs to a class of the
| numerical rank of "New England States" essentially belongs
| to a class of the rank of "sides of a cube", and conversely.
|
| 'Identity' of this particular sort may be termed 'equality',
| and be denoted by the sign "=".  Moreover, since the numerical
| rank of a 'logical sum' depends on the identity or diversity
| (in first intention) of the integrant parts, and since the
| numerical rank of a 'logical product' depends on the identity
| or diversity (in first intention) of parts of the factors,
| logical addition and multiplication can have no place in
| this system.
|
| Arithmetical addition and multiplication, however, will not be destroyed.
|
| 'ab = c' will imply that whatever essentially belongs at once to a class
| of the rank of 'a', and to another independent class of the rank of 'b'
| belongs essentially to a class of the rank of 'c', and conversely.
|
| 'a + b = c' implies that whatever belongs essentially to a class
| which is the logical sum of two mutually exclusive classes of
| the ranks of 'a' and 'b' belongs essentially to a class of
| the rank of 'c', and conversely.
|
| It is plain that from these definitions the
| same theorems follow as from those given above.
|
| 'Zero' and 'unity' will, as before, denote the classes which have respectively
| no extension and no comprehension;  only the comprehension here spoken of is,
| of course, that comprehension which alone belongs to letters in the system now
| considered, that is, this or that degree of divisibility;  and therefore 'unity'
| will be what belongs essentially to a class of any rank independently of its
| divisibility.  These two classes alone are common to the two systems, because
| the first intentions of these alone determine, and are determined by, their
| second intentions.
|
| Finally, the laws of the Boolian calculus, in its ordinary form,
| are identical with those of this other so far as the latter apply
| to 'zero' and 'unity', because every class, in its first intention,
| is either without any extension (that is, is nothing), or belongs
| essentially to that rank to which every class belongs, whether
| divisible or not.
|
| These considerations, together with those advanced [in CP 1.556] will,
| I hope, put the relations of logic and arithmetic in a somewhat clearer
| light that heretofore.
|
| C.S. Peirce, 'Collected Papers', CP 3.44
|
| Charles Sanders Peirce, "Upon the Logic of Mathematics",
|'Proceedings of the American Academy of Arts and Sciences',
| Volume 7, pp. 402-412, September 1867.

NB.  A symbol that the editors transcribe as an equal sign
     with a subtended comma is here transcribed as "=,".

LMU 4. Comment


There is a little more background material that I ought to fill in,
but I am going to make an initial attempt to paint the Big Picture,
and try to explain in broad terms why I think that Peirce's remarks
on "Limited Mark Universes" (LMU's) are so significant, and how they
anticipate many ideas that I personally did not encounter until the
mid 1980's, in two different fields, cognitive psychology and the
area that is known as "category theory applied to computation".

I am beginning to get a better understanding of the ways that different
thinkers differ in their thinking processes.  Peirce was what I think of
as an "exploratory heuristic" (EH) and a "3-adic relational" (3R) thinker.
Thinkers of this sort, a category to which I aspire to aggrandize myself
one day, think in very different ways from those that I am coming to
recognize as "absolute dichotomous" (AD) thinkers.

For example, you can forget all that guff about classifying
languages into "extensional" versus "intensional" brands,
and making some 'auto da fe' to one or the other article
of faith.  All the real languages that anybody really
uses in reality have names for things that can be
instances or properties in relation to suitable
other things, not to mention, but they do,
names for these relations themselves.
This is so whether we are talking
logic, math, or normal people.

I think that I will begin from what is closest to me, a problem that
I worked on all through the 1980's, one of the hot topics in AI and
cognitive science at the time, namely, "language acquisition" (LA).
You may remember the analogies that Chomsky pointed out, time and
again, between the problem of "giving a rule to abduction" and
the "poverty of the stimulus" argument for rational grammars,
that is to say, cartesian rationalism and innate grammars.

A large part of the work that I did on this problem reduced me
to working on computational models of formal language learning,
where the formal languages that I could handle from the outset
were very "impoverished" in comparison to natural languages,
but still not entirely trivial, and with many interesting
facets that would repay even a minimalist treatment.

To make a decade-long story as short as I can make it,
here are some of the ideas that gradually worked their
way into my probable density as I errored and trialed:

1.  Although it initially looks like a problem of classical induction,
    that is to say, forming rules from facts and cases, it turned out
    that I could not detach this from more abductive, anticipatory,
    or hypothetical forms of concept formation.

2.  Instead of just the extensions and the intensions of concepts
    (I had not yet clued into comprehensions at that time), there
    was another sort of relation between data and concepts that
    I was forced to consider, on account of what many call the
    "generative property" of any non-trivial language, or what
    is just about the same thing, the circumstance that the
    language learner, by the very nature of the task, does
    not have the whole extension of a language or any of
    its grammatical categories, but at every stage of
    the game has only a finite experiential record
    of the instances that fall under the putative,
    contingent, and ever hypothetical, concepts.

This last aspect of the problem led me to recognize the importance of the
sampling relation, which reminded me, via some vague or vagrant memory,
of Aristotle's "enumerative induction", and so I came to call this
relation between the data and the concept the "enumeration" of
the concept, and extension and intension makes three.

LMU 5. Comment


Peirce came into this arena with a question about "how science works",
and he took off from a standard sort of Kantian platform that permits
you to get started by just going ahead and accepting the evident fact,
the apparent phenomenon, or the provisional hypothesis that science,
as we do it, but not necessarily as we know it, does work, and then
to move on to the next question, to wit:  What are the conditions
for the possibility of science working?

I walked into this theatre with a problem about language learning, and
had very little acquaintance and a whole lot of wrong ideas about Kant.

But there is a natural analogy between the task of scientific knowing
and the task of language acquisition, as Newton clearly recognized in
the guise of his metaphor about science as the decoding of nature's
cryptographic laws.

So I will start out by explaining a very simple sort of language acquisition task.
One of the first obstacles that we run into is this huge gulf between all of the
realistic examples and all of the sorts of examples that one can discuss in the
beginning, the fact that all of the motive settings are very complex indeed
and all of the simple set-pieces are very simple indeed.

So I will beg you to use your imagination.

Okay, enough preamble.

An "alphabet" (or a "lexicon") is a finite set A.

The "kleene star" A* of the alphabet A
is the set of all finite sequences that
can be formed out of the elements of A.
We call these "strings" or "sequences".
Note that A* includes the empty string.

A "formal language" L over the alphabet A is an arbitrary subset of A*,
thus L c A*.  Depending on the setting, the strings or sequences of L
are called "L-words", "L-strands", or "L-sentences", in one locution,
or "words of L", "strands of L", or "sentences of L", in another.
Whenever there is only one language under discussion, or when
it is otherwise clear, the obvious abridgements may be used.

LMU 6. Comment


| By their fruit flies ye shall know them.
|
| ~~ Pragmatic definition of a geneticist.

To be a language, formally speaking, L c A*, is to embody
a distinction between the strings or sequences of A* that
are in L and those that aren't.  There are two exceptions,
or degenerate cases, if one prefers to view them that way,
two languages that draw no distinction in the space of A*.
These are the "empty language" L_0 = {}, so empty that it
fails to contain even so much as the empty string <>, and
the "total language" L_1 = A*.

Some would say that we are only doing syntax as this point,
but others say that the semantic and pragmatic dimensions
cannot be reduced to zero magnitudes, even in this frame.
For example it is possible to see in the present setting --
others would say "read into the present scene" -- a bit,
yes, exactly one bit, of semantic meaning and pragmatic
motive, namely the Peircean arrow from the nonsense of
non-L to the sense of L, except for the non-orientable
cases, of course.

Limited Mark Universes • Selections & Comments

Peirce • On A Limited Universe Of Marks (SIL 182–183)


| On A Limited Universe Of Marks
|
| Boole, De Morgan, and their followers, frequently speak of
| a "limited universe of discourse" in logic.  An unlimited universe
| would comprise the whole realm of the logically possible.  In such
| a universe, every universal proposition, not tautologous, is false;
| every particular proposition, not absurd, is true.  Our discourse
| seldom relates to this universe:  we are either thinking of the
| physically possible, or of the historically existent, or of
| the world of some romance, or of some other limited universe.
|
| But besides its universe of objects, our discourse also refers to
| a universe of characters.  Thus, we might naturally say that virtue
| and an orange have nothing in common.  It is true that the English
| word for each is spelt with six letters, but this is not one of the
| marks of the universe of our discourse.
|
| A universe of things is unlimited in which every combination of characters,
| short of the whole universe of characters, occurs in some object.  In like
| manner, the universe of characters is unlimited in case every aggregate
| of things short of the whole universe of things possesses in common one
| of the characters of the universe of characters.  The conception of
| ordinary syllogistic is so unclear that it would hardly be accurate
| to say that it supposes an unlimited universe of characters;  but
| it comes nearer to that than to any other consistent view.  The
| non-possession of any character is regarded as implying the
| possession of another character the negative of the first.
|
| In our ordinary discourse, on the other hand, not only are both universes limited, but,
| further than that, we have nothing to do with individual objects nor simple marks;
| so that we have simply the two distinct universes of things and marks related to
| one another, in general, in a perfectly indeterminate manner.  The consequence
| is, that a proposition concerning the relations of two groups of marks is not
| necessarily equivalent to any proposition concerning classes of things;  so
| that the distinction between propositions in extension and propositions in
| comprehension is a real one, separating two kinds of facts, whereas in the
| view of ordinary syllogistic the distinction only relates to two modes of
| considering any fact.  To say that every object of the class S is included
| among the class of P's, of course must imply that every common character of
| the P's is a common character of the S's.  But the converse implication is by
| no means necessary, except with an unlimited universe of marks.  The reasonings
| in depth of which I have spoken, suppose, of course, the absence of any general
| regularity about the relations of marks and things.
|
| C.S. Peirce, 'Studies in Logic', pp. 182-183.
|
| Charles Sanders Peirce, "Note A.  On A Limited Universe Of Marks" (1883),
| CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University',
| Reprinted with an Introduction by Max H. Fisch and a Preface by Achim Eschbach,
|'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983.
|
| C.S. Peirce, 'Studies in Logic', pp. 182-186.  (CP 2.517-531;  CE 4, 450-453).
|
|'Writings of Charles S. Peirce:  A Chronological Edition, Volume 4, 1879-1884',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986.

Peirce • On A Limited Universe Of Marks (SIL 183–184)


| I may mention here another respect in which this view differs from that of
| ordinary logic, although it is a point which has, so far as I am aware, no
| bearing upon the theory of probable inference.  It is that under this view
| there are propositions of which the subject is a class of things, while the
| predicate is a group of marks.  Of such propositions there are twelve species,
| distinct from one another in the sense that any fact capable of being expressed
| by a proposition of one of these species cannot be expressed by any proposition
| of another species.  The following are examples of six of the twelve species:--
|
|    1.  Every object of the class S possesses every character of the group !p!.
|
|    2.  Some object of the class S possesses all characters of the group !p!.
|
|    3.  Every character of the group !p! is possessed by some object of the class S.
|
|    4.  Some character of the group !p! is possessed by all the objects of the class S.
|
|    5.  Every object of the class S possesses some character of the group !p!.
|
|    6.  Some object of the class S possesses some character of the group !p!.
|
| The remaining six species of propositions are like the above,
| except that they speak of objects 'wanting' characters instead
| of 'possessing' characters.
|
| C.S. Peirce, 'Studies in Logic', pp. 183-184.
|
| Charles Sanders Peirce, "Note A.  On A Limited Universe Of Marks" (1883),
| CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University',
| Reprinted with an Introduction by Max H. Fisch and a Preface by Achim Eschbach,
|'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983.
|
| C.S. Peirce, 'Studies in Logic', pp. 182-186.  (CP 2.517-531;  CE 4, 450-453).
|
|'Writings of Charles S. Peirce:  A Chronological Edition, Volume 4, 1879-1884',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986.

Peirce • On A Limited Universe Of Marks (SIL 184–185)


| But the varieties of proposition do not end here;
| for we may have, for example, such a form as this:
| "Some object of the class S possesses every character
| not wanting to any object of the class P".  In short,
| the relative term "possessing as a character", or its
| negative, may enter into the proposition any number
| of times.  We may term this number the 'order' of
| the proposition.
|
| An important characteristic of this kind of logic is the part that
| immediate inference plays in it.  Thus, the proposition numbered 3,
| above, follows from No. 2, and No. 5 from No. 4.  It will be observed
| that in both cases a universal proposition (or one that states the
| non-existence of something) follows from a particular proposition
| (or one that states the existence of something).  All the immediate
| inferences are essentially of that nature.  A particular proposition
| is never immediately inferable from a universal one.  (It is true that
| from "no A exists" we can infer that "something not A exists";  but
| this is not properly an immediate inference, -- it really supposes
| the additional premiss that "something exists".)  There are also
| immediate inferences raising and reducing the 'order' of propositions.
| Thus, the proposition of the second order given in the last paragraph
| follows from "some S is a P".  On the other hand, the inference holds, --
|
|    Some common character of the S's is wanting to everything except P's;
|
|    .:  Every S is a P.
|
| C.S. Peirce, 'Studies in Logic', pp. 184-185.
|
| Charles Sanders Peirce, "Note A.  On A Limited Universe Of Marks" (1883),
| CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University',
| Reprinted with an Introduction by Max H. Fisch and a Preface by Achim Eschbach,
|'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983.
|
| C.S. Peirce, 'Studies in Logic', pp. 182-186.  (CP 2.517-531;  CE 4, 450-453).
|
|'Writings of Charles S. Peirce:  A Chronological Edition, Volume 4, 1879-1884',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986.

Peirce • On A Limited Universe Of Marks (SIL 185–186)


| The necessary and sufficient condition of the existence of
| a syllogistic conclusion from two premisses is simple enough.
| There is a conclusion if, and only if, there is a middle term
| distributed in one premiss and undistributed in the other.  But
| the conclusion is of the kind called 'spurious' by De Morgan,
| if, and only if, the middle term is affected by a "some" in
| both premisses.  For example, let the two premisses be, --
|
|    Every object of the class S wants some character of the group !m!;
|
|    Every object of the class P possesses some character not of the group !m!.
|
| The middle term !m! is distributed in the second premiss, but not in the first;
| so that a conclusion can be drawn.  But, though both propositions are universal,
| !m! is under a "some" in both;  hence only a spurious conclusion can be drawn,
| and in point of fact we can infer both of the following: --
|
|    Every object of the class S wants a character
|    other than some character common to the class P;
|
|    Every object of the class P possesses a character other
|    than some character wanting to every object of the class S.
|
| The order of the conclusion is always the sum of the orders of the
| premisses;  but to draw up a rule to determine precisely what the
| conclusion is, would be difficult.  It would at the same time be
| useless, because the problem is extremely simple when considered
| in the light of the logic of relatives.
|
| C.S. Peirce, 'Studies in Logic', pp. 185-186.
|
| Charles Sanders Peirce, "Note A.  On A Limited Universe Of Marks" (1883),
| CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University',
| Reprinted with an Introduction by Max H. Fisch and a Preface by Achim Eschbach,
|'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983.
|
| C.S. Peirce, 'Studies in Logic', pp. 182-186.  (CP 2.517-531;  CE 4, 450-453).
|
|'Writings of Charles S. Peirce:  A Chronological Edition, Volume 4, 1879-1884',
| Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986.

Comment 4 Variant


There is a little more background material that I ought to fill in,
but I am going to make an initial attempt to paint the Big Picture,
and try to explain in broad terms why I think that Peirce's remarks
on "Limited Mark Universes" (LMU's) are so significant, and how they
anticipate many ideas that I personally did not encounter until the
mid 1980's, in two different fields, cognitive psychology and the
area that is known as "category theory applied to computation".

I am beginning to get a better understanding of the ways that different
thinkers differ in their thinking processes.  Peirce was what I think of
as an "exploratory heuristic" (EH) and a "3-adic relational" (3R) thinker.
Thinkers of this sort think in very different ways from those that I am
coming to recognize as "absolute dichotomous" (AD) thinkers.

For example, you can forget all that guff about classifying
languages into "extensional" versus "intensional" brands.
All the real languages that anybody really uses in reality
have names for things that can be instances or properties
in relation to suitable other things, not to mention, but
they do, names for these relations themselves.  This is
so whether we are talking logic, math, or normal people.

I think that I will begin from what is closest to me, a problem that
I worked on all through the 1980's, one of the hot topics in AI and
cognitive science at the time, namely, "language acquisition" (LA).
You may remember the analogy that Chomsky pointed out, time and
again, between the problem of "giving a rule to abduction" and
the "poverty of the stimulus" argument for rational grammars,
that is to say, cartesian rationalism and innate grammars.

A large part of the work that I did on this problem reduced me
to working on computational models of formal language learning,
where the formal languages that I could handle from the outset
were very "impoverished" in comparison to natural languages,
but still not entirely trivial, and with many interesting
facets that would repay even a minimalist treatment.

To make a decade-long story as short as I can make it,
here are some of the ideas that gradually worked their
way into my probable density as I errored and trialed:

1.  Although it initially looks like a problem of classical induction,
    that is to say, forming rules from facts and cases, it turned out
    that I couldn't detach this from the more abductive, anticipatory,
    and hypothetical forms of concept formation.

2.  Instead of just the extensions and the intensions of concepts
    (I had not yet clued into comprehensions at that time), there
    was another sort of relation between data and concepts that
    I was forced to consider, on account of what many call the
    "generative property" of any non-trivial language, or what
    is just about the same thing, the circumstance that the
    language learner, by the very nature of the task, does
    not have the whole extension of a language or any of
    its grammatical categories, but at every stage of
    the game has only a finite experiential record
    of the instances that fall under the putative,
    contingent, and ever hypothetical concepts.

This last aspect of the problem gradually led me to appreciate the importance
of the sampling relation, which reminded me, via some vague or vagrant memory,
of Aristotle's "enumerative induction", and so I eventually came to call this
sort of relation between the data and the concept the "enumeration" of the
concept, and extension and intension makes three.

Comment 5 Variant


Peirce came into this arena with a question about "how science works",
and he took off from a standard sort of Kantian platform that permits
you to get started by just going ahead and accepting the evident fact,
the apparent phenomenon, or the provisional hypothesis that science,
as we do it, but not necessarily as we know it, does work, and then
to move on to the next question, to wit:  What are the conditions
for the possibility of science working?

I walked into this fun-house with a question about language learning, and I
had very little acquaintance with and a whole lot of wrong ideas about Kant.

But there is a natural analogy between the task of scientific knowing
and the task of language acquisition, as Newton clearly recognized in
the guise of his metaphor about science as the decoding of nature's
cryptographic laws.

So I will start out by explaining a very simple sort of language acquisition task.
One of the first obstacles that we run into is this huge gulf between all of the
realistic examples and all of the sorts of examples that we can discuss in the
beginning, is sum, the fact that all of the motive settings are very complex
indeed and all of the rudimentary set-pieces are very simple indeed.

So I will beg you to use your imagination.

Okay, enough preamble.

An "alphabet" (or a "lexicon") is a finite set A.

The "kleene star" A* of the alphabet A
is the set of all finite sequences that
can be formed out of the elements of A.
We call these "strings" or "sequences".
Mark that A* includes the empty string.

A "formal language" L over the alphabet A is an arbitrary subset of A*,
thus L c A*.  Depending on the setting, the strings or sequences of L
are called "L-words", "L-strands", or "L-sentences", in one locution,
or "words of L", "strands of L", or "sentences of L", in another.
Whenever there is only one language under discussion, or when
it is otherwise clear, the obvious abridgements may be used.

Work Area

  • atom 3.93
  • individual 3.92f, 214ff
  • simple 3.216, 217, 220
  • singular 3.93, 216, 252, 602, 611

Document History

2001 • Ontology List • Inquiry Into Symbolization

  1. http://web.archive.org/web/20081204194050/http://suo.ieee.org/ontology/msg03201.html
  2. http://web.archive.org/web/20081204194631/http://suo.ieee.org/ontology/msg03202.html
  3. http://web.archive.org/web/20080906121059/http://suo.ieee.org/ontology/msg03204.html
  4. http://web.archive.org/web/20081011051100/http://suo.ieee.org/ontology/msg03234.html

2002 • Ontology List • Limited Mark Universes

  1. http://web.archive.org/web/20140429004255/http://suo.ieee.org/ontology/msg04349.html
  2. http://web.archive.org/web/20140429004359/http://suo.ieee.org/ontology/msg04350.html
  3. http://web.archive.org/web/20140429004130/http://suo.ieee.org/ontology/msg04351.html
  4. http://web.archive.org/web/20070304204618/http://suo.ieee.org/ontology/msg04364.html
  5. http://web.archive.org/web/20070304204628/http://suo.ieee.org/ontology/msg04365.html
  6. http://web.archive.org/web/20070302152005/http://suo.ieee.org/ontology/msg04368.html

2003 • Inquiry List • Limited Mark Universes

  1. http://web.archive.org/web/20140902230000/http://stderr.org/pipermail/inquiry/2003-April/000403.html
  2. http://web.archive.org/web/20140902230001/http://stderr.org/pipermail/inquiry/2003-April/000404.html
  3. http://web.archive.org/web/20070316093327/http://stderr.org/pipermail/inquiry/2003-April/000406.html
  4. http://web.archive.org/web/20070306142455/http://stderr.org/pipermail/inquiry/2003-April/000407.html