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PEIRCE-L Digest 1297 -- February 14-15, 1998
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From PEIRCE-L Forum, Jan 5, 1998, [name of author of message],
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Topics covered in this issue include:
1) Re: Does Language Determine Our Scientific Ideas?
by piat@juno.com (Jim L Piat)
2) Re: What is zero? What is number?
by piat@juno.com (Jim L Piat)
3) Re: Hypostatic Symbols
by piat@juno.com (Jim L Piat)
4) Re:Hookway --Chapter I Introspection
by piat@juno.com (Jim L Piat)
5) Re: a question about LISP and recursion
by piat@juno.com (Jim L Piat)
6) Re: Porphyry: On Aristotle's Categories/The New List (4)
by Thomas.Riese@t-online.de (Thomas Riese)
7) quotes: definition & logical & analysis
by joseph.ransdell@yahoo.com (ransdell, joseph m.)
----------------------------------------------------------------------
Date: Sat, 14 Feb 1998 20:54:52 -0500
From: piat@juno.com (Jim L Piat)
To: peirce-l@ttacs6.ttu.edu
Subject: Re: Does Language Determine Our Scientific Ideas?
Message-ID: <19980215.004355.9118.9.piat@juno.com>
On Sat, 14 Feb 1998 16:53:41 -0600 (CST) KWells2382@aol.com writes:
>Dear Howard,
>
>I'd like to read it.
>
>Kelley
>
Me too, Jim Piat
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------------------------------
Date: Sun, 15 Feb 1998 00:19:53 -0500
From: piat@juno.com (Jim L Piat)
To: peirce-l@ttacs6.ttu.edu
Subject: Re: What is zero? What is number?
Message-ID: <19980215.004355.9118.12.piat@juno.com>
>Best...
>
>Benoit
>Favreault
>Programme
>de
>doctorat
>en
>semiologie
>Universite
>du Quebec
>a Montreal
>
Voir
c'est
croire!
Actionsspeaklouderthanwordsjimpiat.
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------------------------------
Date: Sun, 15 Feb 1998 00:19:20 -0500
From: piat@juno.com (Jim L Piat)
To: peirce-l@ttacs6.ttu.edu
Subject: Re: Hypostatic Symbols
Message-ID: <19980215.004355.9118.11.piat@juno.com>
On Mon, 9 Feb 1998 05:58:54 -0600 (CST) Cathy Legg
writes:
>
>"Come sit down beside me", I said to myself,
>And although it doesn't make sense,
>I held my own hand in a small sign of trust,
>And together I sat on the fence.
> (Michael Leunig, a classic work from the mid-'80s).
>
>Now *that's* bootstrapping.
>
>Cheers,
>Cathy.
You bet your boots it is! A fine occurrence of recurrence right on the
fence. I'd like the name of the Leunig work if you have it handy.
Jim Piat
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Date: Sun, 15 Feb 1998 00:10:12 -0500
From: piat@juno.com (Jim L Piat)
To: peirce-l@ttacs6.ttu.edu
Subject: Re:Hookway --Chapter I Introspection
Message-ID: <19980215.004355.9118.10.piat@juno.com>
Tom Anderson asked me why I think public private means about the same as
external internal.
I guess it's just my behaviorist bias in equating internal with sub-vocal
or self talk. Perhaps this is also an example of Hookways Quinean bias
that Dennis Knepp warned us about. Last week I read John Murphy's
_Pragmatism: From Peirce to Davidson_ in which I found this quote of
Quine:
"Peirce scored a major point for naturalism, moreover, in envisioning a
behavioristic semantics. Naturalism in psychology and semantics is
behaviorism; and Peirce declared for such a semantics when he declared
that beliefs consist in dispositions to action."
But most importantly, Tom, the whole issue would have slipped by me
entirely if you hadn't brought it to my attention. Seems to me Peirce
was an early behaviorist even if anti behaviorists (Chomsky for example)
often cite him as their champion as well. Peirce was attempting a
synthesis that has perhaps yet to be fully appreciated by either side.
It is interesting, don't you think, how perfectly legitimate but limited
insights first get over generalized and then dismissed as completely
misguided when the pendulum swings the other way.
I see this in philosophy as an outsider but equally or even more so in my
own field. Such self satisfaction with which we dismiss the the
primitive notions of the preceding generation and wonder whatever could
they have been thinking. The most annoying part is that surely we must
all realize at some level that our own current beliefs (as is perhaps
necessarily always the case) are largely based upon authority, tenacity
and taste. To act on well founded beliefs - (which also excludes
authoritarian scientism) that is the challenge as Peirce the philospher
and Dostoevsky the novelist have pointed out
Lastly, Tom, I just wanted to mention (for whatever reaction it might
elicit) that My reaction to Rorty's (who edited the Murphy book)
divorcing pragmaticism from objective reality (my interpretation) seems
to me (at first blush at least) to be going completely off the tracks.
My question to Rorty is, what in the world is it that he supposes we are
all talking about?
I guess I'm drifting into chapter II if you'll want to discuss that now.
Jim Piat
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Date: Sun, 15 Feb 1998 00:22:20 -0500
From: piat@juno.com (Jim L Piat)
To: peirce-l@ttacs6.ttu.edu
Subject: Re: a question about LISP and recursion
Message-ID: <19980215.004355.9118.13.piat@juno.com>
On Thu, 12 Feb 1998 14:33:05 -0600 (CST) joseph.ransdell@yahoo.com (ransdell,
joseph m.) writes:
>Doug:
>
>I had a further question I forgot to include in the earlier batch. I
>wondered if you could say more exactly what is meant in speaking of
>LISP
>as involving "naturally recursive control flow"? Mathematicians,
>logicians, and computer scientists sometimes seem to have somewhat
>different ideas of what recursion is, and I have never been clear on
>exactly what everyone agrees on as fundamental in it.
>
>
>Joe
Joe, I'm none of the above but your persistent queries and comments
elsewhere about mathematicians gave me a chuckle and sent me to the
dictionary. I recall once asking a mathematician to explain exactly what
a Markov chain was (in connection with Chomsky's claim that language
contrary to Skinner's approach could not be accounted for by a Markov
chain) His answer was short, no doubt correct and altogether
unsatisfactory to me at the time. Sometimes the application of the math
is more profound (not to be pretentious) than the mathematical concept
itself. I didn't have the credibility then to keep asking- but in the
spirit of not wishing to be a damn fool all my life I've decided to share
with you what I found in the dictionary, realizing full well that you
know the dictionary definition but at the same time remain unsettled
about something at the core of what you are talking about. Turns out
cursive (as in handwriting) comes from the latin "currere" to run or
flow. So we have incursive, discursive and recursive. Also, "recursion
formula" in math a formula for determining the next term of a sequence
from one or more of the preceding terms. And, "recursive definition" in
logic a definition consisting of a set of rules such that by repeated
application of the rules the meaning of the definiendum is uniquely
determined in terms of ideas that are already familiar.
Seems to me it's this logical definition (relating to continuums,
bootstrapping and semiosis) that's the interesting one worth unpacking as
you philosophers sometimes say. Just as a wild speculation I wonder if
in the case of bootstrapping we have an example of a syllogism in which
the rule is applied to the rule itself yielding the rule itself as the
conclusion. For example: Repeat this sentence. Or am I just confusing
this with the story about pete and repete sitting on the corner.
Jim Piat
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------------------------------
Date: Sun, 15 Feb 1998 13:07:47 +0100
From: Thomas.Riese@t-online.de (Thomas Riese)
To: peirce-l@ttacs6.ttu.edu
Subject: Re: Porphyry: On Aristotle's Categories/The New List (4)
Message-ID:
In response to Bill Overcamp (BugDaddy, 13 Feb 98 - 22:33):
Bill, I think your important message on Porphyry and Aristotle might
bring us a step further to the decisive things.
With Peirce there is the distinction
ICONICITY vs AXIOMATICS
and the parallel distinction between theorematic and demonstrative
reasoning. Goedel and recursive function theory etc. fall into the
second class. They have nothing systematically to say about how to
insert new elements, except that, at a certain point we indeed do have
to insert new elements -- that's the content of Goedel's famous
theorems. They are indeed 'restrictive' relative to any 'powerful
enough' axiomatic basis. 'Powerful enough' here means: of the type of
the system of the natural numbers and that finally means: of the
Dedekindian kind. What special axiom system we base our argumentation
on, e.g. Peano etc., doesn't matter here.
But that's finally an assertion on the class of admissible transitive
relations (i.e., philosophically speaking, the interpretation of
Aristotle's nota notae).
And here Peirce's 'expansion by restriction' comes in.
I think it is decisive for a deeper understanding of Peirce's logic to
see the systematic connection. If this seems far fetched I could give
it additional plausibility by the following: Peirce got a decisive
impulse for his logic based on categories by considering the
_geometry_ of Aristotle's syllogistic system and Peirce's categories
are akin to bases in geometry. Concerning Goedel it has often been
remarked that the extension of incomplete systems by Goedelian
propositions is very similar to the branching into different types of
non-Euclidean geometries.
And I do indeed think that concepts in the 'Principia
Mathematica'-type of systems are the logical equivalent of the rigid
bodies in Euclidean geometry. It even seems to me to be quite obvious
once one hits upon the idea.
In the axiomatic logical approach concepts can only be compared (or:
are so to speak 'commensurable'), when they are _similar_ exactly in
the sense in which Euclidean geometry is based on similarity (and
congruence) of triangles etc. There is nothing like expansion and
contraction of reference etc.: such effects as that concepts can
'acquire meaning', Peirce once gave 'electricity' as an example, are
systematically excluded -- except if we _take_serious_ that axioms are
'implicit definitions' of the involved objects. But this is obviously
not the case for then nobody would dream of calling Goedel's results
limitative in a destructive sense, as an ultimate barrier etc.
So even if what I say might seem near the bone, it nevertheless has
some plausibility, I think, prior to the close scrutiny of the nice
mathematical details. More: I do indeed think that our presently
accepted 'logical atomism' is even dangerously rigid, practically
speaking.
I think what is psychologically difficult to accept is the fact that
things can be IRREDUCIBLE WITHOUT BEING ULTIMATE. That's how I would
put the idea of relativity in a nutshell. (Just another form of the
Peircean reduction thesis;-))
All this seems to promise abhorrent and totally counterintuitive
mathematical nightmares. In one sense this indeed is so. In another
sense there seems to me to be a new level of simplicity which is even
more intuitive than 'rigid body logic' -- according to my experience
especially for children who didn't yet have too much 'education'.
The key is simply to say what we do and take that 'as a concept'.
In the case of recursion and mathematical induction that means to take
the whole structure of induction, together with the 'initial element'
into one relation. This is exactly what happens when we expand the
general relation of transitivity (respectively Aristotles nota notae)
_by_restricting_it_.
So we can make the transition from the (potentially) purely
self-referential form of general transitivity, i.e.
is lover whatever is loved by
to
is both lover of and lover of everything loved by
and take this _as_an_irreducible_form_.
One of the merits of this idea is that we regain the idea of truth
without having to deny relativity. Simple truth is an astonishingly
rich concept;-) -- even socially.
Back to the question of 'recursive definition' and iconicity:
In CP 3.523 ("Professor Schroeder's Iconic Solution of x -< PHI x")
Peirce analyses the form
x -< PHI x
He then continues:
[...]
"Suppose for example, that we desire the general form of a
'transitive' relative, that is, such a one, x, that xx -< x"
further continues to use what today would be called a 'generating
function' approach ending up in finding the formal equivalent of the
above 'lover of and lover of everything loved by' saying:
"This is a truly iconic result; that is it shows us what the
constitution of a transitive relative really is."
He then amplifies on this form saying that the first factor has the
effect to exclude in certain cases universal identity (which would be
Dedekind's totally self-refential system) and thus to extend the class
of relatives represented by the other factor so as to include those of
which it is not true that 1 -< x (i.e. tautologies).
He continues:
"Here we have an instance of restriction having the effect of
extension, that is, restriction of special relatives extends the class
of relatives represented. This does not take place in all cases, but
only where certain relatives can be represented in more than one way."
Finally he says:
[...]
"For the chief end of formal logic is the representation of the
syllogism.";-)
.., eh, my smiley!
If we keep in mind that for the system of natural numbers there is
indeed, mildly speaking, a relative, that can be represented in more
than one way, for this is Dedekind's chain automorphism, then this
means that we get a more complete representation of mathematical
induction and thus the number system, if we represent the form of
mathematical induction in one form, i.e. we, so to speak "pull in" the
initial element of the induction ("Induktionsanfang" in German) to the
form of transitivity.
This is indeed a truly *iconic* result, since the relative finally
says that something has to be valid for something to be named
concretely and -- going down and coming up again --, well, it is a bit
difficult to describe:-)
This is what one could call a 'recursive definition'.
There are some strange effects: there are then forms that are as much
operators as they are operands! I.e. we can throw any mathematical
description into an 'active form', can apprehend an object as a
transformation, a process. Well, of course this should indeed be so if
mathematics works by icons, as Peirce contends. (In the other
direction we can revert the process to more solid 'objects' via
hypostatic abstraction).
This at first sounds mystical, but has a sound algebraic basis! It is
algebraically speaking simply that any linear associative algebra can
be put into matrix form, i.e. into a transformation algebra.
And this is exactly what Peirce develops in a letter to William James,
26 Feb 1909 (NEM, III/2, p.837ff.) where he again starts with the nota
notae, the fact that in order to get a form that holds good of *all*
transitive relations that form must be *restricted* (have a look into
the Goedel papers!), talks about the barycentric calculus and then
proves a precursor and essential part of what later became the
celebrated Wedderburn structure theorems for abstract algebra (which,
by the way, indeed show that abstract algebra in a certain sense has a
tri-partite structure), finally saying (on page 861):
"It is thus proved that any linear associative system of units may be
put into matricular, or dyadic relative form by simply regarding the
coefficients of its multiplication table as coefficients of
capital-pairs in the precise manner in which the formula [...] ...,
well, here we are again at that 'truely iconic' thing! (see formula on
top of page 860)
And now please compare what Peirce said before in the same letter on
page 858:
"Any two *operations* whatsoever (using the word "operation" not for
the individual deed but for whatever deed may be done under a fully
explicit *rule* or *prescription*, covering a multitude of possible
situations) are *equal*, i.e. the same *en*effet*, if and only if, the
*result* of either is, in *every*conceivable*case*, indistinguishable
from that of the other. They are identical, that is, the same in
essence, if and only if, whatever is prescribed in the defining
explicit *rule* for doing the one is prescribed in that of the
other."(emphasis by Peirce)
Please compare this with NEM IV, MS 441, pp.175ff (or "Logic of Things
pp.131ff., "Types of Reasoning") where you'll find the above again and
something on the geometry of the syllogism and then with MS 293,
Prolegomena to an Apology..." where there is a passage beginning with:
"... a diagram is the icon of a set of rationally related
objects.[...]" and further: "The scheme _sees_, as we can say, that
the transformed diagram is in substance contained in the diagram that
is to be transformed...", ending with "And evidence belongs to every
necessary conclusion."
Email is not particularly appropriate for algebraic notation and so I
don't want to go through the argumentation in detail here. It would be
probably only confusing; the important thing being that these things
*can* be shown with rigidity(;-)). Peirce himself has already done so.
So logic has necessarily more than just only one dimension. Seems as
if there were three, as if ideas had sort of 'spatial' extension
(Peirce's idea, by the way).
And thus things can be coordinated without being necessarily similar!
And further Peirce's discussion concerning J.S.Mill that 'uniformity
of nature' is _not_ necessary for lawfulness and reasoning to be
possible is, I believe, only on this background fully understandable.
In fact it can be shown that three-dimensionality is a prerequisite
for the fact that an electromagnetic signal with a sharp onset is not
hopelessly blurred, i.e. that information is possible. So, if we could
show the converse to be true,... In fact I believe that Peirce's
discussion of Mill can only be really understood on these lines or,
and perhaps similar, on Ramsey's theorems that logically 'complete
disorder is impossible'. Perhaps there is even a connection.
That's, by the way, why Peirce, being a realist, could simply decide
for _Existential_ Graphs, which is much more than 'nominalism' could
ever dare. Pretty paradoxical, isn't it?
There is still more: In Dedekinds system the rational numbers in a
sense just simply 'fall down from the sky' and have no important
meaning except as means to approximate irrational numbers.
Not so for Peirce.
There is a very little known construction in mathematics called
"Stern-Peirce-Tree" or "Stern-Brocot-Tree". Peirce mentions the
details in a letter to Henry B. Fine, dated July 17, 1903 (New
Elements of Mathematics, vol.III/2, pp.781ff.) and in other places.
The important thing in this connection here is that Peirce filled the
gap in a way consistent with what I have described above. The
rationals can then be interpreted as paths in a binary tree and the
remarkable thing is that each rational is at the same time a sequence
of transformation matrices such that if M is a functiom mapping
strings S to matrices we have
M(S) = S
so that strings again have a dual role as transformations (operations,
matrices) _and_ numbers.
The system in itself forms a number system and some irrational
numbers, as e.g. Euler's 'e', which in the decimal system are
irregular have in this system a regular pattern of digits!
If we keep in mind that Zeno's "Achilles and the Tortoise" to Peirce
not even was not paradoxical at all, but far more important: he
claimed the the case is soluble in the rationals -- here you have the
key! It's indeed a question of the representation of the numbers and
we tend to forget that we don't know numbers directly. What we know
are representations of the numbers: decimal, polynomial etc..
I think that's what is behind what we might call a 'recursive
definition' (as Conway in his 'On Numbers and Games' calls it, where,
by the way, numbers are at the same time 'games'. -- That's why the
book has this title).
I think Peirce's representation relation is exactly the prototype of a
recursive definition. And this works, to repeat it, because this
structure is as much an object as it is a relation (operation). So we
evade the necessity to assume the initial _existence_ of anything (as
you would in any Dedekindian approach). There are many beginnings then
-- a process of beginning, if you like that form of speech.
What ever conforms to such a definition behaves as if it were the real
thing. And if _everything_ behaves "as if it were so" then it is so.
What else?
I am very hesitant to say this, but logically expansion_by_restriction
is very similar to the basic idea of Heisenberg's principle in his
'matrix mechanics': the _idea_ of it (not the content of the physical
theory). But what Peirce has discovered is of course much more
general.
By way of this sort of restriction we are able to know _more_ than
without this restriction and not less! That's the point that is so
often overlooked: it's limitative and _thereby_ expands our possible
knowledge enormously. And I am constantly amazed that physicist were
much more prepared to draw the logical consequences than logicians.
But even in recursive function theory proper it can be clearly seen,
just by looking at the center piece, Cantor's diagonal argument, that
undecidability and universality (generality) in fact "meet". What
else? A general system as e.g. projective geometry is of course
"undecidable" -- what else. That is exactly why it is general. This is
as plain as anything can be. Though most people would of course say
that this a 'false' way to look at things. Well, in order to make a
step we have to do something 'false', i.e. beyond what is accustomed.
But then it is not a question of complexity, it is the problem to
_see_ what is apparent! -- Again: if a theory were 'decidable' it
simply couldn't be a theory, since it could not be applied to
something 'foreign' to it -- it couldn't function as a sign. (I
express it as strange and 'sarcastic' as possible for the sake of
contrast. We are too much used to these things, so sunken into it,
and so we are unable to perceive such platitudes). 'Undecidedness'
and utility are two sides of the same medal in these things. Of
course!
The price we have to pay is 'uncertainty' which means nothing else
than that we have based our definition on a relation -- "relational
generality".
And again: we are then in a position to regard a determination
(definition) as a restriction of the continuum. That's the 'trick', so
to speak. It is virtually _all_there_already_. In a sense we just only
make it visible though at the same time it is a constructive process,
too. The initial element for the definition process is a limitation of
our knowledge, i.e. an act of directed attention which excludes other
possible knowledge.
In still another sense what one initially does is nothing but 'making
a mistake'. But one can correct that afterwards. It's like that
conception of 'force' in Newton's physics: Newton never says what a
force 'really, ultimately, is'. But if you use this conception
consistently, things will come out right. It's like a scaffolding
which you remove afterwards.
Similarly in logic some problems are never "really" solved. It's just
only that the question disappears..., in a sense.
The best "toy-model" of the problem is perhaps the relation of a point
to a line. If you have a point and a line then you never get a line by
piling up points. The "glue" is simply missing. And if you try to
"glue", then things become even worse, in a sense, for then you have
points, lines and "glue". And what the hell is "glue" then...!?
So another approach is to generate a point from a line. If you turn
the line so that you see it from the 'small end', then your line will
appear as a point. But that doesn't help either, for if you see a line
as a point before you, then you can thereby not anymore see it as
line. The line has disappeared.
And if you move a point "fast enough" before your eyes, it will be
"smeared", it will appear as a line. But then your point is so "fuzzy"
that you can't see it any more (well, that is what you intended in
this experiment, though then, not quite...).
The method of 'degenerate cases'. But the decisive thing then is that
we do never ultimately "really" know what a point or a line is. But we
can answer any possible question concerning points and lines. But what
a point or a line is we do not really know. We have only discovered
projective geometry which allows us to answer many questions where we
before were not even able to dream up the question. Well, that's the
"tree of knowledge". No question escapes it's answer.
This is exactly what physicists did with quantum mechanics and what
Peirce did with the Logic of Relatives. And thereby, though you
haven't "solved" the problem, you can get a lot of more answers to
your problem, even new problems. Problems and answers which before
were simply unimaginable.
You can see a point as an aspect of a line "at infinitude" and you can
see a line "as points", i.e. as a "moving" point, i.e. you even get
such concepts as "velocity", "time" etc. into logic, i.e. you can
treat them from a logical point of view. (I think we can get at what
Peirce meant with the splitting of a point if we consider, as in
projective geometry, a point as a pencil of lines and then lines
dually as points again. Well, my guess...)
And you are not even in the position that you can generate answers
without end, without ever stopping (though in another sense this is
so), you get even "complete" theories. As for example "projective
geometry" or classical physics or quantum physics etc. etc.. I.e. in
your generation process there are indeed "natural" stopping points,
i.e.information is possible. Though you can go on and on and on ...
In fact this process of expansion by restriction seems itself to be of
the nature of perception -- the focusing of attention to the neglect
of other possibilities. I don't think that logically much more is
necessary. It has the form of reasoning. So we can have ICONICITY. And
mathematics, as a system, is *not* just a huge tautology.
Sort of back at the beginning.
Well, these are my observations and that's my guess at this riddle;-)
Return-path:
Date: Sun, 15 Feb 1998 08:12:44 -0600 (CST)
From: peirce-l@ttacs6.ttu.edu
Subject: PEIRCE-L digest 1297
Sender: peirce-l@ttacs6.ttu.edu
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What I have to say is deliberately 'large scale' and risky. I think it
sometimes helps to have a look at the territory from the air in order
to make one's maps.
Thomas Riese.
P.S. I certainly remember to have read that Peirce somewhere says
"definition, i.e. logical analysis". Must have been on a left side
page, first quarter of it. I am unable to recover the reference.
Perhaps someone on the list can help me?!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Thomas Riese * Hedwigstr.24 * 45130 Essen * Germany
Tel.: +49 201 77 94 45 * Fax: +49 201 77 94 81
email: thomas.riese@t-online.de
0201779445-0001@t-online.de
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
------------------------------
Date: Sun, 15 Feb 1998 08:09:41 -0600
From: joseph.ransdell@yahoo.com (ransdell, joseph m.)
To:
Subject: quotes: definition & logical & analysis
Message-ID: <002401bd3a1b$5e875f60$43a432ce@ransdell.door.net>
In view of something Thomas Riese said in his most recent post, I did a
string search of the Collected Papers, with a Boolean conjunction of
"definition" and "logical" and "analysis", and it seems to me to have
come up with results that might be worth posting in that a number of
them are relevant to what Thomas said in connection with the idea of
recursive definition and are interesting, in any case. Again, I have
done nothing toward refining the data other than to eliminate some
impertinent verbiage, and also to omit a few which don't seem closely
enough related to the present context (but I leave the reference so you
can check for yourself), so let the user beware! -- meaning be sure and
check the context yourself in the Collected Papers. This type of
material has to be regarded as suggestive only.
Joe Ransdell
===========result of string search described above============
Peirce: CP 1.443
443. But accepting this amendment, which to his customary way of
thinking is microscopic, the mathematician will be inclined to say, here
is a perfect definition; and excepting a few little corollaries, there
is nothing more to be said of the dyad. It behooves me, then, to clearly
state what the inquiry is which I propose to institute. It is not a
mathematical inquiry; because the business of the mathematician is to
frame an arbitrary hypothesis, which must be perfectly distinct at the
outset, so far, at least, as concerns those features of it upon which
mathematical reasoning can turn, and then to deduce from this hypothesis
such necessary consequences as can be drawn by diagrammatical reasoning.
The present problem is one of logical analysis. Instead of setting out
with a distinct hypothesis of a diagrammatic kind, we have the confused
fact that a dyad is a conception of the highest utility, though we are
not prepared to say exactly what its nature is, nor even, in all cases,
whether a given case should properly be reckoned as a duality or not. We
are somewhat in the position of a naturalist who knows that whales are
large swimming animals, which spout water, and yield blubber,
spermaceti, and whalebone, but knows little else about them, and who
proposes to himself to examine the anatomy and physiology of whales so
as to assign them their place in the system of the animal kingdom. He
does not intend to preserve the popular description nor delimitation of
the class of whales. He will perhaps see reason to extend the name to
some animals not popularly called whales and to refuse it to others that
are so called. He will also subdivide the group, and classify it
according to the facts. So far as our inquiry is a logical analysis, the
greatest difference between it and that of a taxonomic biologist
consists in the circumstance that we are not forced to institute special
observations, because all the facts are either well known or can be
ascertained by careful reflection upon those that are known.
Peirce: CP 3.432
432. The introduction, which relates to first principles, while
containing many excellent observations, is somewhat fragmentary and
wanting in a unifying idea; and it makes logic too much a matter of
feeling.
It cannot be said to belong to exact logic in any sense. Thus, under á
(Vol. I., p. 2) the reader is told that the sciences have to suppose,
not only that their objects really exist, but also that they are
knowable and that for every question there is a true answer and but one.
But, in the first place, it seems more exact to say that in the
discussion of one question nothing at all concerning a wholly unrelated
question can be implied. And, in the second place, as to an inquiry
presupposing that there is some one truth, what can this possibly mean
except it be that there is one destined upshot to inquiry with reference
to the question in hand --one result, which when reached will never be
overthrown? Undoubtedly, we hope that this, or something approximating
to this, is so, or we should not trouble ourselves to make the inquiry.
But we do not necessarily have much confidence that it is so. Still less
need we think it is so about the majority of the questions with which we
concern ourselves. But in so exaggerating the presupposition, both in
regard to its universality, its precision, and the amount of belief
there need be in it, Schroeder merely falls into an error common to
almost all philosophers about all sorts of "presuppositions." Schroeder
(under {e}, p. 5) undertakes to define a contradiction in terms without
having first made an ultimate analysis of the proposition. The result is
a definition of the usual peripatetic type; that is, it affords no
analysis of the conception whatever. It amounts to making the
contradiction in terms an ultimate unanalysable relation between two
propositions -- a sort of blind reaction between them. He goes on (under
{z}, p. 9) to define, after Sigwart, logical consequentiality, as a
compulsion of thought. Of course, he at once endeavors to avoid the
dangerous consequences of this theory, by various qualifications. But
all that is to no purpose. Exact logic will say that C's following
logically from A is a state of things which no impotence of thought can
alone bring about, unless there is also an impotence of existence for A
to be a fact without C being a fact. Indeed, as long as this latter
impotence exists and can be ascertained, it makes little or no odds
whether the former impotence exists or not. And the last anchor-hold of
logic he makes (under {i}) to lie in the correctness of a feeling! If
the reader asks why so subjective a view of logic is adopted, the answer
seems to be (under á, p. 2), that in this way Sigwart escapes the
necessity of founding logic upon the theory of cognition. By the theory
of cognition is usually meant an explanation of the possibility of
knowledge drawn from principles of psychology. Now, the only sound
psychology being a special science, which ought itself to be based upon
a wellgrounded logic, it is indeed a vicious circle to make logic rest
upon a theory of cognition so understood. But there is a much more
general doctrine to which the name theory of cognition might be applied.
Namely, it is that speculative grammar, or analysis of the nature of
assertion, which rests upon observations, indeed, but upon observations
of the rudest kind, open to the eye of every attentive person who is
familiar with the use of language, and which, we may be sure, no
rational being, able to converse at all with his fellows, and so to
express a doubt of anything, will ever have any doubt. Now, proof does
not consist in giving superfluous and superpossible certainty to that
which nobody ever did or ever will doubt, but in removing doubts which
do, or at least might at some time, arise. A man first comes to the
study of logic with an immense multitude of opinions upon a vast variety
of topics; and they are held with a degree of confidence, upon which,
after he has studied logic, he comes to look back with no little
amusement. There remains, however, a small minority of opinions that
logic never shakes; and among these are certain observations about
assertions. The student would never have had a desire to learn logic if
he had not paid some little attention to assertion, so as at least to
attach a definite signification to assertion. So that, if he has not
thought more accurately about assertions, he must at least be conscious,
in some out-of-focus fashion, of certain properties of assertion. When
he comes to the study, if he has a good teacher, these already dimly
recognised facts will be placed before him in accurate formulation, and
will be accepted as soon as he can clearly apprehend their statements.
Peirce: CP 4.233
233. Mathematics is the study of what is true of hypothetical states of
things. That is its essence and definition. Everything in it, therefore,
beyond the first precepts for the construction of the hypotheses, has to
be of the nature of apodictic inference. No doubt, we may reason
imperfectly and jump at a conclusion; still, the conclusion so guessed
at is, after all, that in a certain supposed state of things something
would necessarily be true. Conversely, too, every apodictic inference
is, strictly speaking, mathematics. But mathematics, as a serious
science, has, over and above its essential character of being
hypothetical, an accidental characteristic peculiarity -- a proprium, as
the Aristotelians used to say -- which is of the greatest logical
interest. Namely, while all the "philosophers" follow Aristotle in
holding no demonstration to be thoroughly satisfactory except what they
call a "direct" demonstration, or a "demonstration why" -- by which they
mean a demonstration which employs only general concepts and concludes
nothing but what would be an item of a definition if all its terms were
themselves distinctly defined -- the mathematicians, on the contrary,
entertain a contempt for that style of reasoning, and glory in what the
philosophers stigmatize as "mere" indirect demonstrations, or
"demonstrations that." Those propositions which can be deduced from
others by reasoning of the kind that the philosophers extol are set down
by mathematicians as "corollaries." That is to say, they are like those
geometrical truths which Euclid did not deem worthy of particular
mention, and which his editors inserted with a garland, or corolla,
against each in the margin, implying perhaps that it was to them that
such honor as might attach to these insignificant remarks was due. In
the theorems, or at least in all the major theorems, a different kind of
reasoning is demanded. Here, it will not do to confine oneself to
general terms. It is necessary to set down, or to imagine, some
individual and definite schema, or diagram -- in geometry, a figure
composed of lines with letters attached; in algebra an array of letters
of which some are repeated. This schema is constructed so as to conform
to a hypothesis set forth in general terms in the thesis of the theorem.
Pains are taken so to construct it that there would be something closely
similar in every possible state of things to which the hypothetical
description in the thesis would be applicable, and furthermore to
construct it so that it shall have no other characters which could
influence the reasoning. How it can be that, although the reasoning is
based upon the study of an individual schema, it is nevertheless
necessary, that is, applicable, to all possible cases, is one of the
questions we shall have to consider. Just now, I wish to point out that
after the schema has been constructed according to the precept virtually
contained in the thesis, the assertion of the theorem is not evidently
true, even for the individual schema; nor will any amount of hard
thinking of the philosophers' corollarial kind ever render it evident.
Thinking in general terms is not enough. It is necessary that something
should be DONE. In geometry, subsidiary lines are drawn. In algebra
permissible transformations are made. Thereupon, the faculty of
observation is called into play. Some relation between the parts of the
schema is remarked. But would this relation subsist in every possible
case? Mere corollarial reasoning will sometimes assure us of this. But,
generally speaking, it may be necessary to draw distinct schemata to
represent alternative possibilities. Theorematic reasoning invariably
depends upon experimentation with individual schemata. We shall find
that, in the last analysis, the same thing is true of the corollarial
reasoning, too; even the Aristotelian "demonstration why." Only in this
case, the very words serve as schemata. Accordingly, we may say that
corollarial, or "philosophical" reasoning is reasoning with words; while
theorematic, or mathematical reasoning proper, is reasoning with
specially constructed schemata.
Peirce: CP 4.325
325. By a 'postulate,' Dr. Huntington seems to understand any one of a
body of propositions such that nothing can be deduced from one that
could equally be deduced from another, while, from them all, every
proposition of a given branch of mathematics might be deduced. The
utility of such a body of premisses for the logical analysis of the
branch of mathematics in question is beyond dispute. But I think we
ought to distinguish between postulates and definitions. As for axioms,
or propositions already well-known to the student who takes up the
branch of mathematics in question, the ancients themselves admitted that
they might be omitted without detriment to the course of deduction of
the theorems. Indeed, an axiom could only be a maxim of logical nature.
A postulate is a statement that might be questioned or denied without
absurdity. A definition, or rather, one of the pair, or larger number,
of propositions that constitute a definition, cannot be questioned,
because it merely states the logical relation of a conception thereby
introduced to conceptions already in use. It is quite true, on the one
hand, that a postulate, after all, is merely a part of the definition of
the underlying conception of the branch of mathematics to which it
refers, (Euclid's celebrated fifth postulate, for example, being merely
a part of the definition of Euclidean space); while on the other hand, a
definition is a statement of positive fact about the use of the word
defined, and may thus be regarded as a sort of postulate. But to argue
from these truths that there is no important difference between a
postulate and a definition would be to fall into a fallacy of a very
common kind, that of denying all important difference between two things
because they are in an important respect alike, or of denying all
important likeness between two things because they are in an important
respect unlike. There is a vast difference between the logical relations
to a branch of mathematics of propositions defining its very purpose in
defining its fundamental hypothesis, and those of propositions that
merely define conceptions which it is convenient or even which it is
necessary to introduce in order to develop that branch.
Peirce: CP 4.370
370. Any broad mathematical hypothesis, like that of a system of
values, will attract three classes of students by three different
interests that attach to it. The first is the special interest in the
circumstance that that hypothesis necessarily involves certain relations
among the things supposed, over and above those that were supposed in
the definition of it. This is the mathematical interest proper. The
second is the methodeutic interest in the devices which have to be
employed to bring those new relations to light. This is a matter of
supreme interest to the mathematician and of considerable, though
subordinate, interest to the logician. The third is the analytical
interest in the essential elements of the hypothesis and of the
deductive processes of the second study, in their intellectual pedigrees
and in their conceptual affiliations with ideas met with elsewhere. This
is the logical interest, par excellence. In the case of non-relative
deductive logic, that is, the doctrine of the relations of truth and
falsity between combinations of non-relative terms, the methodeutic
interest is slight owing to the extreme simplicity of the methods. The
logical interest, on the other hand, limited as the subject is when
relative terms are excluded, is very considerable, not to say great. In
the inquiries which it prompts, it is the simplest cases which will
chiefly attract attention, and therefore the circumstance, that the
system of Eulerian diagrams becomes too cumbrous and laborious in
complicated problems, is no objection to it. While the student cannot be
counselled to confine himself to any single method of representation,
the system of Eulerian diagrams is probably the best of any single one
for the purely non-relative analysis of thought. Thus, it at once
directs attention to the circumstance that the syllogism may be
considered as a special case of the inference from Fig. 63 to Fig. 64,
where the blots may either be zero or crosses or one a zero and the
other a cross. Another example of the analytical interest of the system
lies in the higher particular propositions, where we see an evolution of
the conception of multitude. Multitude, or maniness, is a property of
collections. Now a collection is an ens rationis, or abstraction; and
abstraction appears as the highest product of the development of the
logic
of relatives. The student is thus directed to the deeply interesting and
important problem of just how it is that the conception of multitude
merges in the Eulerian diagrams.
Peirce: CP 4.435 [omitted]
Peirce: CP 4.619
619. My reason for expressing the definition of a cyclic system in
Existential Graphs is that if one learns to think of relations in the
forms of those graphs, one gets the most distinct and ecthetically as
well as otherwise intellectually, iconic conception of them likely to
suggest circumstances of theoric utility, that one can obtain in any
way. The aid that the system of graphs thus affords to the process of
logical analysis, by virtue of its own analytical purity, is
surprisingly great, and reaches further than one would dream. Taught to
boys and girls before grammar, to the point of thorough familiarization,
it would aid them through all their lives. For there are few important
questions that the analysis of ideas does not help to answer. The
theoretical value of the graphs, too, depends on this.
Peirce: CP 5.491
491. In every case, after some preliminaries, the activity takes the
form of experimentation in the inner world; and the conclusion (if it
comes to a definite conclusion), is that under given conditions, the
interpreter will have formed the habit of acting in a given way whenever
he may desire a given kind of result. The real and living logical
conclusion is that habit; the verbal formulation merely expresses it. I
do not deny that a concept, proposition, or argument may be a logical
interpretant. I only insist that it cannot be the final logical
interpretant, for the reason that it is itself a sign of that very kind
that has itself a logical interpretant. The habit alone, which though it
may be a sign in some other way, is not a sign in that way in which that
sign of which it is the logical interpretant is the sign. The habit
conjoined with the motive and the conditions has the action for its
energetic interpretant; but action cannot be a logical interpretant,
because it lacks generality. The concept which is a logical interpretant
is only imperfectly so. It somewhat partakes of the nature of a verbal
definition, and is as inferior to the habit, and much in the same way,
as a verbal definition is inferior to the real definition. The
deliberately formed, self-analyzing habit -- self-analyzing because
formed by the aid of analysis of the exercises that nourished it -- is
the living definition, the veritable and final logical interpretant.
Consequently, the most perfect account of a concept that words can
convey will consist in a description of the habit which that concept is
calculated to produce. But how otherwise can a habit be described than
by a description of the kind of action to which it gives rise, with the
specification of the conditions and of the motive?
Peirce: CP 5.494 [omitted]
Peirce: CP 6.74 [omitted]
Peirce: CP 6.471
471. Deduction has two parts. For its first step must be by logical
analysis to Explicate the hypothesis, i.e. to render it as perfectly
distinct as possible. This process, like Retroduction, is Argument that
is not Argumentation. But unlike Retroduction, it cannot go wrong from
lack of experience, but so long as it proceeds rightly must reach a true
conclusion. Explication is followed by Demonstration, or Deductive
Argumentation. Its procedure is best learned from Book I of Euclid's
Elements, a masterpiece which in real insight is far superior to
Aristotle's Analytics; and its numerous fallacies render it all the more
instructive to a close student. It invariably requires something of the
nature of a diagram; that is, an "Icon," or Sign that represents its
Object in resembling it. It usually, too, needs "Indices," or Signs that
represent their Objects by being actually connected with them. But it is
mainly composed of "Symbols," or Signs that represent their Objects
essentially because they will be so interpreted. Demonstration should be
Corollarial when it can. An accurate definition of Corollarial
Demonstration would require a long explanation; but it will suffice to
say that it limits itself to considerations already introduced or else
involved in the Explication of its conclusion; while Theorematic
Demonstration resorts to a more complicated process of thought.
Peirce: CP 6.490 [omitted]
Peirce: CP 8.184
184. As to the Interpretant, i.e., the "signification," or
"interpretation" rather, of a sign, we must distinguish an Immediate and
a Dynamical, as we must the Immediate and Dynamical Objects. But we must
also note that there is certainly a third kind of Interpretant, which I
call the Final Interpretant, because it is that which would finally be
decided to be the true interpretation if consideration of the matter
were carried so far that an ultimate opinion were reached. My friend
Lady Welby has, she tells me, devoted her whole life to the study of
significs, which is what I should describe as the study of the relation
of signs to their interpretants; but it seems to me that she chiefly
occupies herself with the study of words. She also reaches the
conclusion that there are three senses in which words may be
interpreted. She calls them Sense, Meaning, and Significance.
Significance is the deepest and most lofty of these, and thus agrees
with my Final Interpretant; and Significance seems to be an excellent
name for it. Sense seems to be the logical analysis or definition, for
which I should prefer to stick to the old term Acception or Acceptation.
By Meaning she means the intention of the utterer.
Peirce: CP 8.302
302. [December 17, 1909] I was and had been long working as hard as I
could upon my "System of Logic, from the point of view of Semiotic,"
when Juliette was enabled to start up the repairs which may enable me to
finish that book, by keeping us alive. Then Carus having written me
already offering me $200 or $250 (I forget which) for the copyright of
my 6 articles in the Popular Science Monthly for 1877 and '78, I agreed
for $250 to allow him to print one edition, with a revision that I would
furnish together with an Article for the Monist on my method of
performing logical analyses. Owing to a lot of interruptions, . . . I
either forgot, or never comprehended that Carus particularly cared to
have that Article a separate one, and I had been working on it as a
chapter of my first essay, consisting of the two articles with which I
began in the Popular Science Monthly called, -- "The Settlement of
Opinion" and "How to make our Ideas clear." Of course Definition, which
is the end of Logical Analysis, is the first step, (after general
familiarity in use,) toward making Ideas clear. However, he has lately
written, remonstrating on my delay, and in consequence I am going
immediately to write that article, which I think will be a really
helpful one to many people.
Peirce: CP 8.305
305. Now third, I argue that there seem to be no other modes of
consciousness by taking up some of the most difficult and analyzing
them, which will at the same time illustrate my method of analysis. I
shall show that a Concept is a Sign and shall define a Sign and show its
triadic form. I shall define the Modality of a Sign and show that in
this respect every Object is either a Can-be, an Actual, or a Would-be.
I shall show (as generally recognized,) that an Actual cannot be defined
and that the Can-be's and Would-be's when accurately discriminated are
only definable in different senses. There is no use of going through
these headings, however, because they are unintelligible until they are
defined at length. I don't pretend that my argument that there are only
three kinds of consciousness does more than raise a presumption by the
precision with which I succeed in defining a great variety of terms
without calling in any fourth element. It will remain for those who
question the conclusion to find a term I cannot define with this
apparatus. After all this I shall undertake to show (still somewhat
imperfectly) that concepts are capable of such phaneroscopic analysis,
or in common parlance "logical analysis"; but there are only a few cases
in which I pretend as yet to carry the analysis so far as to resolve the
concept into its ultimate elements. After a few more such questions have
been discussed, I show how to go to work to perform the analysis; and
then I proceed to show that a definition constructed according to my
method at once clears up various puzzles relating to the concept.
Peirce: CP 8.343
343. It seems to me that one of the first useful steps toward a science
of semeiotic ({semeiotik‚}), or the cenoscopic science of signs, must be
the accurate definition, or logical analysis, of the concepts of the
science. I define a Sign as anything which on the one hand is so
determined by an Object and on the other hand so determines an idea in a
person's mind, that this latter determination, which I term the
Interpretant of the sign, is thereby mediately determined by that
Object. A sign, therefore, has a triadic relation to its Object and to
its Interpretant. But it is necessary to distinguish the Immediate
Object, or the Object as the Sign represents it, from the Dynamical
Object, or really efficient but not immediately present Object. It is
likewise requisite to distinguish the Immediate Interpretant, i.e. the
Interpretant represented or signified in the Sign, from the Dynamic
Interpretant, or effect actually produced on the mind by the Sign; and
both of these from the Normal Interpretant, or effect that would be
produced on the mind by the Sign after sufficient development of
thought. On these considerations I base a recognition of ten respects in
which Signs may be divided. I do not say that these divisions are
enough. But since every one of them turns out to be a trichotomy, it
follows that in order to decide what classes of signs result from them,
I have 3[to the 10th power] or 59049, difficult questions to carefully
consider; and therefore I will not undertake to carry my systematical
division of signs any further, but will leave that for future explorers.
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