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User:Jon Awbrey/Tone, Token, Type
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Tone, Token, Type
Note 1
| A Sign may 'itself' have a "possible" Mode of Being, | e.g., a hexagon inscribed in or circumscribed about | a conic. It is a Sign, in that the collinearity of | the intersections of opposite sides shows the curve | to be a conic, if the hexagon is inscribed; but if | it be circumscribed the co-punctuality of its three | diameters (joining opposite vertices). Its Mode of | Being may be Actuality: as with any barometer. Or | Necessitant: as the word "the" or any other in the | dictionary. For a "possible" Sign I have no better | designation than a 'Tone', though I am considering | replacing this by "Mark". Can you suggest a really | good name? An Actual Sign I call a 'Token'; | a Necessitant Sign, a 'Type'. | | Charles S. Peirce, "Letters to Lady Welby", 31 Jan 1909, page 406 in: |'Charles S. Peirce: Selected Writings (Values in a Universe of Chance)', | Edited with an Introduction by Philip P. Wiener, Dover, New York, NY, 1966.
Note 2
| A common mode of estimating the amount of matter in | a MS. or printed book is to count the number of words. | There will ordinarily be about twenty 'the's on a page, | and of course they count as twenty words. In another sense | of the word "word", however, there is but one word "the" in | the English language; and it is impossible that this word | should lie visibly on a page or be heard in any voice, for | the reason that it is not a Single thing or Single event. | It does not exist; it only determines things that do | exist. Such a definitely significant Form, I propose | to term a 'Type'. | | A Single event which happens once and whose identity is | limited to that one happening or a Single object or thing | which is in some single place at any one instant of time, | such event or thing being significant only as occurring | just when and where it does, such as this or that word | on a single line of a single page of a single copy of | a book, I will venture to call a 'Token'. | | An indefinite significant character such as | a tone of voice can neither be called a Type | nor a Token. I propose to call such a Sign | a 'Tone'. | | In order that a Type may be used, it has to be embodied | in a Token which shall be a sign of the Type, and thereby | of the object the Type signifies. I propose to call such | a Token of a Type an 'Instance' of the Type. Thus, there | may be twenty Instances of the type "the" on a page. | | Charles Sanders Peirce, 'Collected Papers', CP 4.537.
Note 3
| Questioner: Well, if you choose so to make Doing the Be-all | and the End-all of human life, why do you not make meaning to | consist simply in doing? Doing has to be done at a certain time | upon a certain object. Individual objects and single events cover | all reality, as everybody knows, and as a practicalist ought to be | the first to insist. Yet, your meaning, as you have described it, | is 'general'. Thus, it is of the nature of a mere word and not | a reality. You say yourself that your meaning of a proposition | is only the same proposition in another dress. But a practical | man's meaning is the very thing he means. What do you make to | be the meaning of "George Washington"? | | Pragmaticist: Forcibly put! A good half dozen of your points must certainly be | admitted. It must be admitted, in the first place, that if pragmaticism really | made Doing to be the Be-all and the End-all of life, that would be its death. | For to say that we live for the mere sake of action, as action, regardless of | the thought it carries out, would be to say that there is no such thing as | rational purport. Secondly, it must be admitted that every proposition | professes to be true of a certain real individual object, often the | environing universe. Thirdly, it must be admitted that pragmaticism | fails to furnish any translation or meaning of a proper name, or other | designation of an individual object. Fourthly, the pragmatistic meaning | is undoubtedly general; and it is equally undisputable that the general | is of the nature of a word or sign. Fifthly, it must be admitted that | individuals alone exist; and sixthly, it may be admitted that the | very meaning of a word or significant object ought to be the very | essence of reality of what it signifies. | | But when those admissions have been unreservedly made, you find the pragmaticist | still constrained most earnestly to deny the force of your objection, you ought | to infer that there is some consideration that has escaped you. Putting the | admissions together, you will perceive that the pragmaticist grants that a | proper name (although it is not customary to say that it has a 'meaning'), | has a certain denotative function peculiar, in each case, to that name and | its equivalents; and that he grants that every assertion contains such a | denotative or pointing-out function. In its peculiar individuality, the | pragmaticist excludes this from the rational purport of the assertion, | although 'the like' of it, being common to all assertions, and so, | being general and not individual, may enter into the pragmaticistic | purport. Whatever exists, 'ex-sists', that is really acts upon other | existents, so obtains a self-identity, and is definitely individual. | | As to the general, it will be a help to thought to notice that there | are two ways of being general. A statue of a soldier on some village | monument, in his overcoat and with his musket, is for each of a hundred | families the image of its uncle, its sacrifice to the Union. That statue, | then, though it is itself single, represents any one man of whom a certain | predicate may be true. It is 'objectively' general. The word "soldier", | whether spoken or written, is general in the same way; while the name, | "George Washington", is not so. But each of these two terms remains | one and the same noun, whether it be spoken or written, and whenever | and wherever it be spoken or written. This noun is not an existent | thing: it is a 'type', or 'form', to which objects, both those that | are externally existent and those which are imagined, may 'conform', | but which none of them can exactly be. This is subjective generality. | The pragmaticistic purport is general in both ways. | | Charles Sanders Peirce, 'Collected Papers', CP 5.429.
Note 4
| All general, or definable, Words, whether in the sense of | Types or of Tokens, are certainly Symbols. That is to say, | they denote the objects that they do by virtue only of there | being a habit that associates their signification with them. | As to Proper Names, there might perhaps be a difference of | opinion, especially if the Tokens are meant. But they should | probably be regarded as Indices, since the actual connection | (as we listen to talk) of Instances of the same typical words | with the same Objects, alone causes them to be interpreted as | denoting those Objects. Excepting, if necessary, propositions | in which all the subjects are such signs as these, no proposition | can be expressed without the use of Indices. If, for example, a man | remarks, "Why, it is raining!" it is only by some such 'circumstances' | as that he is now standing here looking out at a window as he speaks, | which would serve as an Index (not, however, as a Symbol) that he is | speaking of this place at this time, whereby we can be assured that | he cannot be speaking of the weather on the satellite of Procyon, | fifty centuries ago. | | Charles Sanders Peirce, 'Collected Papers', CP 4.544.
Note 5
| So then, a natural class being a family whose members are the sole | offspring and vehicles of one idea, from which they derive their | peculiar faculty, to classify by abstract definitions is simply | a sure means of avoiding a natural classification. I am not | decrying definitions. I have a lively sense of their great | value in science. I only say that it should not be by means | of definitions that one should seek to find natural classes. | When the classes have been found, then it is proper to try to | define them; and one may even, with great caution and reserve, | allow the definitions to lead us to turn back and see whether | our classes ought not to have their boundaries differently | drawn. After all, boundary lines in some cases can only be | artificial, although the classes are natural, as we saw in | the case of the 'kets'. When one can lay one's finger upon | the purpose to which a class of things owes its origin, then | indeed abstract definition may formulate that purpose. But | when one cannot do that, but one can trace the genesis of a | class and ascertain how several have been derived by different | lines of descent from one less specialized form, this is the | best route toward an understanding of what the natural classes | are. This is true even in biology; it is much more clearly so | when the objects generated are, like sciences, themselves of the | nature of ideas. | | Charles Sanders Peirce, 'Collected Papers', CP 1.222.
Note 6
| There are cases where we are quite in the dark, alike concerning the creating | purpose and concerning the genesis of things, but where we find a system of | classes connected with a system of abstract ideas -- most frequently numbers -- | and that in such a manner as to give us reason to guess that those ideas in | some way, usually obscure, determine the possibilities of things. For example, | chemical compounds, generally -- or at least the more decidedly characterized | of them, including, it would seem, the so-called elements -- seem to belong | to types, so that, to take a single example, chlorates KClO3, manganates | KMnO3, bromates KBrO3, rutheniates KRuO3, iodates KIO3, behave chemically | in strikingly analogous ways. That this sort of argument for the existence | of natural classes -- I mean the argument drawn from types, that is, from | a connection between the things and a system of formal ideas -- may be much | stronger and more direct than one might expect to find it, is shown by the | circumstance that ideas themselves -- and are they not the easiest of all | things to classify naturally, with assured truth? -- can be classified | on no other grounds than this, except in a few exceptional cases. Even | in these few cases, this method would seem to be the safest. For example, | in pure mathematics, almost all the classification reposes on the relations | of the forms classified to numbers or other multitudes. Thus, in topical | geometry, figures are classified according to the whole numbers attached | to their 'choresis', 'cyclosis', 'periphraxis', 'apeiresis', etc. As for | the exceptions, such as the classes of hessians, jacobians, invariants, | vectors, etc., they all depend upon types, too, although upon types of | a different kind. It is plain that it must be so; and all the natural | classes of logic will be found to have the same character. | | Charles Sanders Peirce, 'Collected Papers', CP 1.223.
Discussion
I think that the ideas are important here, but the terminology
is probably going to stay hopelessly confused from here on out.
There are different sorts of type/token issues in mathematics,
computer science, Peirce's logic, and lately the "ontology via
formal concept analysis" crowd has invented a whole new way of
using these words, so I will have to pick my preferences and
forge ahead.
What interests me in Peirce's Comprehension/Extension/Information,
Quality/Reaction/Symbolization, Tone/Token/Type analogies is that
he started out with trying to understand how inquiry is possible --
the conditions for the possibility of scientific thinking -- and
developed the theory of signs in a supporting role to that effort.
Though I started out with a healthy dose of "pattern recognition AI",
like most folks were doing 20-30 years ago, the way that this issue
comes up in my current applications is more like this, certainly not
a problem about the identity of objects, but about the various kinds
of "partitions", "quotients", "equivalence relations", or "equivalence
class structures" that can be overlaid on a space of signs. Remember,
too, that "signs" here can mean "data of the senses". Accordingly, the
strong theme for me is that these clusterings are always "interpretive"
or "perspective-&-purpose-relative", at least, initially. For instance,
certain species of sign relations lead to various sorts of equiv. classes
that I call "referential", "semiotic", or "logical" equivalence classes,
REC's, SEC's, LEC's, respectively. The generic picture looks like this:
| Object Domain Syntactic Domain
|
| o-----------o
| o~~~~~~~~~~~~~~~~~~/| s s s ... |\
| / \ / o-----------o \
| / \ / \
| / \ o-----------o \
| o~~~~~~~\~~~~~~~~~~~| s s s ... | \
| \ \ o-----------o \
| \ \ \ o-----------o
| \ o~~~~~~~~~~\~~~~~~~~| s s s ... |
| \ / \ o-----------o
| \ / \ /
| \ / \ o-----------o /
| o~~~~~~~~~~~~~~~~~~\| s s s ... |/
| o-----------o
|
| Figure 1. Objects Inducing A Sign Partition
In a sense, one "reconstructs" the structure of the Object domain O
within the equivalence class structure of the Syntactic domain S |_| I.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
These are important issues, and I will return to your note next week
when I have some chance of better focus, but let me say a few things
by way of trying to clear a mental working space.
First, I must respond under a Peircean, if not a Cerberean heading,
as I earnestly believe that any attempt to deal with this issue in
dichotomous terms is doomed to end up two-thirds baked at best.
Second, this is an issue that has occupied me since my
first post-pubescent identity crisis, some time before
I ran into Peirce -- my very first undergrad essay on
Peirce was titled "Distinction and Coincidence" (1972),
in which I compared the various calculi of Peirce with
those of George Spencer Brown.
From my point of view, the critical treatment, the culmination
of five years intense and groundbreaking work, is the 1870
"Description of a Notation for the Logic of Relatives",
(CP 3.45-149; CE 2, 359-429). The critical passage
is what he says about the "doctrine of individuals".
Understanding the implications of that critique
would be a big part of grasping Peirce's whole
subsequent temporal evolution.
Third, it is crucial to recognize Aristotle's dimension,
the one that stretches between the things that are closer
to Nature and the things that are closer to us. Others have
called the opposing directions "Reality" and "Representation" --
for reading Peirce, "Objects" and "Signs" will do well enough.
So, we have to sort out from moment to moment whether we are talking about
relations -- for example, difference and indifference -- among Objects or
among Signs, and then to say what the relationships among these separate
realms of relations are or ought to be. All very obvious, of course,
but what comes out of Peirce's way of doing this will be very different
from what comes out of, say, Frege's way of doing this.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
SS: This is a bit of a nuisance, but I wonder if I could ask you to include,
so far as you are able, the dates of the passages you will be quoting.
This is not meant as a criticism, but we know that Peirce changed his
theory of identity at least once, and I think we shall have better luck
understanding the evolution of his views if we keep track of the dates.
The passage you quote in this post came from an 1870 publication.
(I am not suggesting that he changed his views on the ideality of
absolute individuality after 1870, but the significance of that
claim has to be assessed in context.)
No problem at all. I included a full reference
on my first citation of this passage, which for
ease of reference can be found here:
http://suo.ieee.org/ontology/msg04332.html
When you say "we know that Peirce changed his theory of identity at least once",
please list me as agnostic on that point. So far, "we" have heard little more
than Murphey's opinion and two opinions on Murphey's law, all of which I used
to buy right up until the 'Chronological Edition" started coming out, when I
discovered, much to my initial shock, that many of the things that I thought
were Peirce's last and best ideas were actually his first ideas out of the
starting blocks. About the only the difference is that he initially wrote
it all out in far more detail than he later apparently tired of repeating.
Of course, there were the usual assortments of minor dead ends and
back tracks, but nothing all that radical, so far as I can yet see.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
I notice the risk of a certain confusion as to what exactly
we are talking about when we talk about a "law of identity",
and I only have time for a short sermon this morning, so ...
The reason that the so-called "law of identity" -- in one of the ways of
laying it down taking the form "A is A", where, of course, its meaning
already depends a little bit, though not all, on what the meaning of
the word "is" is -- is called a law of "logic", a normative science,
instead of a law of "ontology", a descriptive science, is that it
tells us how we ought to use signs if we earnestly desire those
signs to function as they ought to in scientific reasoning,
and thus this law of sign design puts no constraint at all
on the being-in-relation-to-itself of any being, per se,
but only on the beings that would be signs for our sake.
The moral of the sermon being this, that though we use signs to
describe things, that does not make logic a descriptive science,
since not every description is, or even ought to be, logical.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| Identity, Law of.
|
| Given by traditional logicians as "A is A".
| Because of the various possible meanings of
| the 'copula' (q.v.) and the uncertainty as to
| the range of the variable A, this formulation is
| ambiguous. The traditional law is perhaps best
| identified with the theorem x = x, either of the
| functional calculus of first order with equality,
| or in the theory of types (with equality defined),
| or in the algebra of classes, etc. It has been, or
| may be, also identified with either of the theorems
| of the propositional calculus, p => p, p = p, or with
| the theorem of the functional calculus of first order,
| F(x) =>_x F(x).
|
| Alonzo Church, in:
| Dagobert D. Runes (ed.),
|'Dictionary of Philosophy',
| Littlefield, Adams, & Co.,
| Totowa, NJ, 1972.
Now, which of these, if any, comes closest
to helping us understand what Peirce meant
by "identity"?
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
WS = William Thomas Sherman
Explaining a sermon is about like explaining a joke.
I will make one try and then confess my sins in the
appropriate place, returning to the more significant
text in relation to which it was intended as no more
than an incidental sidelight.
WS: You wrote:
JA: "since not every description is, or even ought to be, logical."
WS: Imagine someone reading this statement of yours, by what standard,
criteria or authority should someone believe what you say? If for
example, you might say, experience tells me or us that this is so.
If experience tells that this is so, how do you derive your "ought",
and how can experience give us an "ought"? Indeed, what do you mean
by "ought?"
Both the "is" and the "ought" occurred in a negative or limitative context:
The "is" part is my empirical summary of descriptions that I have known.
If your experience is terribly different, then I would be surprised, and
most likely try to explain the difference by the hypothesis that we are
using some of the words differently.
The "ought" part is simply my statement that I would like to try
and avoid telling someone that a description ought to be logical
when I do not know the purpose of making it. If the purpose is
logical, then I would strive to make my advice logical.
I almost added the qualifier "at least, when taken literally" to the
end of the sentence, but I judged that it was most likely redundant.
What I am puzzling over here is simply the fact that we
use terms to describe things, sometimes in a way that the
terms obey "laws of logic" and sometimes not, and this does
not always bear on the goodness of the description unless our
purpose is of a very special sort, namely, logical or oriented
toward a scientific use. I am merely turning over in my own mind
the issue of Peirce's criterion "as to what 'must be' the characters
of all signs used by a 'scientific' intelligence, that is to say, by an
intelligence capable of learning by experience".
| Logic, in its general sense, is, as I believe I have shown, only another
| name for 'semiotic' [Greek 'semeiotike'], the quasi-necessary, or formal,
| doctrine of signs. By describing the doctrine as "quasi-necessary", or
| formal, I mean that we observe the characters of such signs as we know,
| and from such an observation, by a process which I will not object to
| naming Abstraction, we are led to statements, eminently fallible, and
| therefore in one sense by no means necessary, as to what 'must be' the
| characters of all signs used by a "scientific" intelligence, that is to say,
| by an intelligence capable of learning by experience. As to that process of
| abstraction, it is itself a sort of observation. The faculty which I call
| abstractive observation is one which ordinary people perfectly recognize,
| but for which the theories of philosophers sometimes hardly leave room.
| It is a familiar experience to every human being to wish for something
| quite beyond his present means, and to follow that wish by the question,
| "Should I wish for that thing just the same, if I had ample means to gratify it?"
| To answer that question, he searches his heart, and in doing so makes what I term
| an abstractive observation. He makes in his imagination a sort of skeleton diagram,
| or outline sketch, of himself, considers what modifications the hypothetical state
| of things would require to be made in that picture, and then examines it, that is,
| 'observes' what he has imagined, to see whether the same ardent desire is there to
| be discerned. By such a process, which is at bottom very much like mathematical
| reasoning, we can reach conclusions as to what 'would be' true of signs in all
| cases, so long as the intelligence using them was scientific. (CP 2.227).
|
| Charles Sanders Peirce, 'Collected Papers', CP 2.227,
| Editor Data: From An Unidentified Fragment, c. 1897.
WS: Can you give us an example of a description which cannot be made
consistent with logic? Or do you mean merely to say that people
sometimes make descriptions which are illogical but which are,
nevertheless, accepted as being somehow consistent with reality?
WS: Again, picture someone discovering your statement (above) for
the first time, and they ask themselves, "is what Jon Awbrey
saying really true or is it only his opinion? If really true,
how will I know it is really true?"
WS: If what you are saying is "really" true, as opposed to an opinion,
how would you answer these questions so as to command belief?
WS: Asking these questions for a better understanding.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
The best I remember, you asked a question about Peirce's "theory of identity"
in the context of a discussion about types and tokens, and where we eventually
adduced source materials on the related issues of general names, individuals,
instances, intensions, laws, natural kinds, proper names, universals, etc.
I am aware that the "theory of identity" can mean many things, from the
ontological to the logical, and who knows what else, but I tried to give
you the best information that I have with regard to what is distinctive
about Peirce's thought on this question, as relevant to the context of
issues that were being discussed.
As far as Leibniz's principle goes, aside from the fact that it means
different things when viewed logically vs. ontologically, I know lots
of serious thinkers who just plain treat it as a parameter, like the
parallel axiom, exploring the consequences of A on even days and ~A
on odd days, so I would never count a vacillation here as a serious
revolution in anyone's thinking.
The statements about the various identity relations, taking "identity"
in the sense that it has within the logic of relatives, not in, say,
meteorology, you are just plain misunderstanding, by dint of removing
these statements from the distinctive contexts in which each is true.
There is a definition of "decomposable" that has to be observed here.
This definition is invoked when one says that no 3-adic relation is
"composed" of 2-adic relations. It is not invoked in the statement
that one fact is "contained" in several other facts, because the
form of that style of "containment" involves the application of
several other 3-adic relations, including 3-identity relations.
In general, the fact that there exist k-identity relations
I_k c X^k for each k = 2, 3, 4, ..., is one thing. Which
of them can be defined in terms of which others in which
ways -- that is a whole manifold of different questions.
As far as realism versus nominalism goes --
all mathematicians are pythagorean realists.
What age any one of them decides to come out of
the closet about it is a whole different question.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Let me recall the weather advisories
that I posted for your consideration
when first we set out on this course,
in search of Peirce's "identity".
| It is crucial to recognize Aristotle's dimension, the one that
| stretches between the things that are closer to Nature and the
| things that are closer to us. Others have called the opposing
| directions "Reality" and "Representation" -- for reading Peirce,
| "Objects" and "Signs" will do well enough.
| So, we have to sort out from moment to moment whether we are talking
| about relations -- for example, difference and indifference -- among
| Objects or among Signs, and then to say what the relationships among
| these separate realms of relations are or ought to be.
Given the experience of the discussion since then, I think that I can
clarify things a little better at this point by adding the contrast
between "Ontology" and "Logic" to the other pairs of comparisons.
In these terms, the points that I have been trying to make are,
first, that Peirce's theory of identity will vary as we pass
in its bearing from descriptive ontology to normative logic,
but more importantly, it is really the relationship between
the similarities of Objects and the similarities of Signs
that is of fundamental interest here.
But it took me about two hours to write that paragraph,
which is a symptom that it's way past my dormitive hour.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
JA = Jon Awbrey
WS = William Thomas Sherman
JA: It is crucial to recognize Aristotle's dimension, the one that
stretches between the things that are closer to Nature and the
things that are closer to us. Others have called the opposing
directions "Reality" and "Representation" -- for reading Peirce,
"Objects" and "Signs" will do well enough.
WS: Let us be careful about taking for granted assumptions.
After all, there are those who would argue that language
(or signhood) and logic are what is most real, and objects
only derive their reality in our higher cognitive higher
understandings as they are known through language and logic.
A physical object may be quite real say to our unreflective
feelings, but unless mind is present to identify and catalog
the object felt as something "real" we are not even aware of
the object as such, only the feeling or sensation of it.
I am only indicating a line of orientation for understanding
what Peirce was about here. Peirce echoed Aristotle's insight
in pointing out that we must begin from what is closer to us,
in this case, the world of sense-signs and thought-signs, the
latter reaches of which we take part in shaping, and that we
must use these givens and tokens to work toward what is closer
to "Nature", "Reality", "the world beyond the cave", whatever
you wish to call it. In doing this we hammer out concepts and
construct conceptual architectures from the raw data of sense,
and we use all of this as so much instrumentation to arrive at
a better sense of the things that are slightly more permanent.
How we do this, how science works, is the longstanding question that
Peirce takes up. In the process of trying to answer this question,
he finds it necessary to reflect on how we use signs, better said,
on the "formal" functions of signs in inquiry, along with the way
that signs come to embody and bear information. In this pursuit,
Peirce had to develop the theory of sign relations, a beginning
theory of information, and this in turn required him to develop
the logic of relative terms, just to deal with the complexities
that arose in the work.
Probably I should continue to point out that the distinction
between Object and Sign is a distinction of relational roles,
not of absolute essences. There is no dichotomy or dualism
of that sort being set up here. It is perfectly possible
to have a sign relation L contained in the product OxSxI
where the object domain O, the sign domain S, and the
interpretant domain I are all the same set, indeed,
we tend to like working toward such cases.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Working on the philosophy that any external landmark is better than none,
I will repost this gloss from Church, in spite of the anachronisms that it
is bound to contain with respect to Peirce's thought. My own sense is that
Peirce started out with something much like the traditional meaning, while
the propositional meanings are quite prevalent throughout his work, from
the beginning up through the logical graphs, but you can only read him
as using the "functional calculus" interpretations if you understand
the distinctive point of view that comes out of their mathematical
provenance, where one is thinking of "aggregates" and "composites"
as Sums and Products and using Sigma's and Pi's to signify them.
| Identity, Law of.
|
| Given by traditional logicians as "A is A".
| Because of the various possible meanings of
| the 'copula' (q.v.) and the uncertainty as to
| the range of the variable A, this formulation is
| ambiguous. The traditional law is perhaps best
| identified with the theorem x = x, either of the
| functional calculus of first order with equality,
| or in the theory of types (with equality defined),
| or in the algebra of classes, etc. It has been, or
| may be, also identified with either of the theorems
| of the propositional calculus, p => p, p = p, or with
| the theorem of the functional calculus of first order,
| F(x) =>_x F(x).
|
| Alonzo Church, in:
| Dagobert D. Runes (ed.),
|'Dictionary of Philosophy',
| Littlefield, Adams, & Co.,
| Totowa, NJ, 1972.
Nota Bene. It gets muffed a bit in ascii, but Church
is using subscripts on infix connectives to signify
the same thing as prefixing universal quantifiers,
that is, "F(x) =>_x F(x)" = "`A`x (F(x) => F(x))".
???
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| I shall follow Boole in taking the sign of equality to signify identity.
| Thus, if v denotes the Vice-President of the United States, and p the
| President of the Senate of the United States,
|
| v = p
|
| means that every Vice-President of the United States is President of the
| Senate, and every President of the United States Senate is Vice-President.
|
| Charles Sanders Peirce, "Description of a Notation ...", CP 3.66 (1870).
Now, it's not the end of the story, of course, but it's a start.
The significant thing is what is usually the significant thing,
in mathematics, at least, that two distinct descriptions refer
to the same things. Incidentally, Peirce is not really being
as indifferent to the distinctions between signs and things,
mention and use, as this ascii text makes him look, but he
uses a host of other type-faces to distinguish the types
and the uses of signs.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| Ismism. The tendency to make inferences of the following forms:
|
| X is good.
| -------------------
| X is the only good.
|
| X is good for something.
| -------------------------
| X is good for everything.
SS: With respect to "theory of identity" meaning different things in different
contexts in the Peirce papers, that may be true. A search for the expression
"theory of identity" in the Collected Papers yielded no examples. I note, though,
that in 1873, Peirce wrote that "Logic may be considered as the science of identity"
(MS229). In that paper, he also gave a version of Leibniz's law, as one of three
fundamental principles in the "science of identity", calling it "the principle
of the singleness of the same", and holding that this principle is the only
thing that distinguishes the relation expressed by the logical copula from
other relations of a similar kind. Maybe he later changed his mind about
this?
I have told you how most serious thinkers I know of regard Leibniz's Law.
It's a parameter of a formal system, analogous to the Parallel Axiom in
geometry. As stated, any reference to "all predicates" leaves a lot to
be desired in terms of what you mean by that, and it only makes sense in
a context where the statement can be regarded as well-posed. Outside of
such a frame, it simply has no meaning at all. Peirce's thinking on this
score is very nuanced, and I have repeatedly referred people to his note
"On A Limited Universe Of Marks" for a sample of his best thinking on it:
http://suo.ieee.org/ontology/msg03204.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
It seems to me that the sources
of your confusion are quite clear:
1. You consistently ignore what Peirce said in his 1870 remark on
the "doctrine of undividuals", that attributions of identity are
relative to a context of discourse, an insight that he did not
abandon but developed to its finest degree in his 1883 remark
"On a Limited Universe of Marks":
http://suo.ieee.org/ontology/msg03204.html
2. You consistently ignore the distinctions and the meanings
of technical terms in mathematics, as they were used by
Peirce and as they have been used for at least 200 years,
for instance, the meanings of "composition" and "reduction"
as they are used in the algebra of relations and in the
logic of relative terms.
I have pointed out all of this to you before.
It is all pretty clear to anybody who does
not have a prior theory of what can qualify
as a theory of identity, or a philosophy,
for that matter, that they keep trying
shove Peirce's more general theory,
and more general philosophy, into.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
I started to read your comments on Peirce's 1883 remark, and I'm
afraid to say that they are wholly off-base almost from the very
outset. Anyone who accorded even the most casual notice to the
extended discussion that we had earlier this year on Peirce's
"Extension x Comprehension = Information" equation must have
noted that Peirce has a very different idea about the whole
relationship between extensions and intensions than are
dreamt of in your stock (and pillory) dichotomies.
The mere recognition of the question as to which predicates are "admissible"
to a given context of discussion and stage of play strikes what we normally
regard as a very "modern" chord. And when one says that two objects are
identical iff they have all "admitted" predicates in common, then it has
an obvious bearing on yielding a "relativized indiscernibility principle".
Sung another way, this is just the question of which hypotheses are
admissible, which is the problem of "giving a rule to abduction",
which rule is none other than the "pragmatic maxim", so I think
it is clear why Peirce emphasizes this identity question as yet
another variation on the main motive of pragmatism. The Upshot.
I will give it another try tomorrow. In the meantime you might glance at
a few of the following excerpts that I shared with the Peirce List earlier
this spring in connection with the "Question Regarding Indexicality" and
the "Semiosis & Inference" threads, and that I recall Tom and Fernando
and I discussing quite a bit:
Extension x Comprehension = Information -- Selected Links
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
SS: To put the whole story in a nutshell (which necessarily distorts it a bit),
recall that in the history of philosophy, there have been two main contenders
as criteria of identity for existents: essential properties and spatio-temporal
continuity. Peirce's 1870 view clearly falls into the former group, identity of
existents always being in respect of some property. Peirce's later view appears
to fall in the latter group, allowing entities that are completely indiscernable
at one time, even occupying the same space at that time, to be distinct, because
they are in spatio-temporal continuity with entities occupying different places
at another time.
SS: Recall how this discussion of identity started. I said that
I would have expected Peirce to take a "realists" position:
SS: quoting from SS post of 11/9/02, 10:39 AM:
| namely, that identity is always in respect to some universal
| or type. For example, a is the same as b with respect to color,
| or size, or personhood (the same "person" as), etc. Then what has
| been called (since Aristotle) "numerical identity" could be regarded
| as a "degenerate" form of relative identity, in which a is the same as b
| in 'every' respect. This would make Leibniz's law of indiscernables true
| for numerical identity but not for identity simpliciter, since a could be
| identical to b in one respect and different in another.
SS: Only I found, on reading Peirce, seemingly conflicted statements that
prevented me from confirming this hypothesis and indeed, prevented me
from forming a clear idea of Peirce's theory of identity.
I will continue with the reading from Leibniz, which I began for two reasons:
one, to introduce some of the terminology that Peirce was taking for granted
in his writing about such concepts as "composite", "individual", "primitive",
"simple", and so on, two, in order to give an account of Leibniz's principle
as Leibniz was given to write about it.
As far as what you have been writing on this thread,
I find it at the present time to be incommensurable
with any of the meanings that I know for words like
"identity", "realist", "relative", "degenerate", etc.
So let me ask the following questions:
Why do you call the conflating of identity with similarity a "realist" position?
For that matter, why not call your "relative identity" by the name "similarity"?
The use of "relative" in this way, to refer to a universal or an absolute term,
seems to be just begging for trouble. Moreover, it introduces a confound with
all of the other sorts of relativity that might be involved in predication.
Why do you call "numerical identity" the "degenerate" form of "relative identity",
and why do you call your "relative identity" by the name "identity simpliciter"?
You obviously understand that any statement involving a phrase
like "all predicates", "all properties", or "every respect" is
to be regarded with extreme circumspection. Why can you not
accord to Peirce the right that we all assume for ourselves,
to wit, of having to look at it from many different angles?
As I see it, there is/are a host of ambiguities lurking in
all of these concepts, one that cannot be addressed short
of saying what one means by "all", "every", "predicate",
"property", and "respect".
If one finds even the simplest question, for instance,
whether mass is a "property" of a physical "entity",
one whereof one must be silent, then does it not
appear that the issue of Leibniz's principle is
not so much whether it is true, just yet, but
what in the heceity it means?
Finally, I will just point out that your continuing projection of the
3-fold (tone, token, type) upon the 2-some (particular, universal) is
causing more than a bit of distortion in the texts of Peirce you read.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
JA, citing Peirce's "On a Limited Universe of Marks",
in 'Studies in Logic' (1883, 1983), pp. 182-186,
CP 2.517-531; CE 4, 450-453):
http://suo.ieee.org/ontology/msg03204.html
SS: This passage is opposed to the kind of extensionalism advocated by Quine.
Quine's extensional languages are ones in which classes substitute for
properties and relations, two classes being identical if they have the
same members. In an intensional language, admitting properties as well
as classes, different properties may belong to exactly the same things.
In an intensional language, "a proposition concerning the relations of
two groups of marks is not necessarily equivalent to any proposition
concerning classes of things". Extensional languages, such as the
first-order predicate calculus, or set theory, are adequate for
mathematics, but it is controversial whether the sentences of
ordinary language or the sciences in general can be translated
into such an extensional language sentence for sentence.
I wish that you would try every now and then reading what Peirce writes
without trying to atomize each and every remark, if not the man himself,
according to your true-false checklist of dichotomies, especially since
the most casual reader of Peirce would know that he would consider your
attempt to pit extensions versus intensions (properly "comprehensions")
to be an utterly false and misleading antagonism.
Just reaching into the bean bag of all possible quotations:
| The moment, then, that we pass from nothing and the vacuity of being
| to any content or sphere, we come at once to a composite content and
| sphere. In fact, extension and comprehension -- like space and time --
| are quantities which are not composed of ultimate elements; but
| every part however small is divisible.
|
| The consequence of this fact is that when we wish to enumerate the
| sphere of a term -- a process termed 'division' -- or when we wish
| to run over the content of a term -- a process called 'definition' --
| since we cannot take the elements of our enumeration singly but must
| take them in groups, there is danger that we shall take some element
| twice over, or that we shall omit some. Hence the extension and
| comprehension which we know will be somewhat indeterminate. But
| we must distinguish two kinds of these quantities. If we were to
| subtilize we might make other distinctions but I shall be content
| with two. They are the extension and comprehension relatively to
| our actual knowledge, and what these would be were our knowledge
| perfect.
|
| Logicians have hitherto left the doctrine of extension
| and comprehension in a very imperfect state owing to the
| blinding influence of a psychological treatment of the
| matter. They have, therefore, not made this distinction
| and have reduced the comprehension of a term to what it
| would be if we had no knowledge of fact at all. I mention
| this because if you should come across the matter I am now
| discussing in any book, you would find the matter left in
| quite a different state.
|
| CSP, CE 1, page 462.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
T^3. Note 28
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CL = Cathy Legg
SS = Seth Sharpless
CL: And the long 1885 quote is about the indiscernibility of identicals only.
SS: Are you sure? Notice in the 1885 quote:
| But this relation of identity has peculiar properties.
| The first is that if i and j are identical, whatever
| is true of i is true of j. The other property is that
| if everything which is true of i is true of j, then
| i and j are identical.
SS: Isn't the "other property" the identity of indiscernibles? (Well, almost.
He obviously slipped, intending the protasis to be "if everything which is
true of i is true of j 'and' everything which is true of j is true of i,
then i and j are identical".)
If "i" and "j" are individual terms, then they are determinate
on all of the properties that are available in the discussion.
If j has all of the properties of i, and if i is determinate
on all of the available properties, then, as an atom, j = i.
This assumes, as taken for granted in this context, that the "universe of marks"
is closed under negation, that is, if A is a property then ~A is a property, and,
of course, if A is false of x then ~A is true of x. Hence, the peculiarity.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Work 1
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
SS = Seth Sharpless
SS: Yes, the fact that there can be different dimensions along
which "identity" can be assessed is brought home forcibly
in lexicography, where, for example, we have to deal with:
phonetic identity and typographic identity, identity of
reference, identity of meaning, homonymy, synonymy, etc.
Philosophic and logical theories of identity often do not
seem to do justice to this problem of criterial identity
(i.e., "sameness in respect to"), though it does seem that
in most commonplace judgements of identity (e.g., same man,
same word, same river, same word, same meaning, etc.), it
is always sameness in respect of some criterion or property
that is at issue. The hoary problem of indiscernability of
identicals vs identity of indiscernables is with us yet in
philosophy of logic. And for this reason, perhaps, Harris
has a point in focusing on the type-token distinction as
a problem area. However, it seems to me that this problem
is most acute for the nominalist, who in judging identity,
has always to come back to "same particular". As a student
of Quine, you must have thought a good deal about this problem.
Quine objects to properties on the ground that we do not have
criteria of identity for properties. For Quine, one would need
criteria of identity for classes (or properties, if one insists
on admitting them) but no criteria of identity for individuals;
individuals ARE criteria of identity for Quine. To know whether
class a = class b, one looks to the individuals that they (or
member classes) contain.
SS: As a realist, I take exactly the opposite view. I would say
that one needs criteria of identity for individuals but not
for properties (or at least not for all properties) since
properties ARE criteria of identity. (Of course, Quine --
like Russell in his nominalistic phase -- has a certain
amount of trouble in specifying what is an individual;
time-slices and all that.)
SS: From a logical or metaphysical point of view, this is
a bewildering and fundamental problem area. I don't
feel very confident about Peirce's theory of identity.
A computer search of 'Collected Papers' has left me
somewhat confused. Have you (or any lister) a good
idea of Peirce's philosophy of identity? Of course,
I would expect him to take something like what
I have called the "realist's" position above.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Work 2
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
SS = Seth Sharpless
SS: Thanks to Cathy, Jon, Joe, Howard, and Benjamin for responding to
my request for help in understanding Peirce's theory of identity.
SS: Let me try to make my quandary clear: I would have expected Peirce
to take a position on identity akin to that which I have described as
a "realist's" position, namely, that identity is always with respect to
some universal or type. For example, a is the same as b with respect to
color, or size, or personhood (the same "person" as), etc. Then, what has
been called (since Aristotle) "numerical identity" could be regarded as
a "degenerate" form of relative identity, in which a is the same as b
in 'every' respect. This would make Leibniz's law of indiscernables
true for numerical identity but not for identity simpliciter, since
a could be identical to b in one respect and different in another.
In the late 20th century, the classic defense of a "realist" theory
of identity of this kind is that of Peter Geach ("Identity", reprinted
in 'Logic Matters', Univ Cal. Press, 1972, 238-249). The opposing
"nominalistic" position is expounded by Quine and Wiggins in their
respective criticisms of Geach: Quine, Phil. Rev, 1964, p. 102 and
Wiggins, 'Identity and Spatio-Temporal Continuity', Blackwell, 1971.
SS: However, surveying passages on identity or sameness in 'Collected Papers' and
most of of the 'Writings' (I don't have access to Vol 4 of the latter at the
moment) has failed to support this hypothesis. Indeed, I find it difficult
to extract a coherent theory of identity from Peirce's writings, at least
up until 1903 or thereabouts. Just to give an example of the problem,
Cathy and Jon mentioned Peirce's concept of teridentity, which seems
to be essential to the proof of "Peirce's Theorem" about reducibility
to triads. Cathy recommends Burch's papers in Houser's book. (I have
looked at these papers in the past, but the book is not available to me
at the moment, so I'm relying on CP and 'Writings' and dating passages in
CP is difficult as you know.) The first reference I can find to teridentity
in CP is from "A Syllabus of Certain Topics of Logic" (1903). But here is the
problem: As late as 1896, Peirce wrote:
SS, quoting CSP:
| Now, identity is essentially a dual relation.
| That is, it requires two subjects and no more.
| If three objects are identical, this fact is
| entirely contained in the fact that the three
| pairs of objects are identical. CP1.446 (1896)
SS: So trying to make a coherent story out of Peirce's writings on identity,
most of which precede his development of the concept of teridentity, even
though the latter is essential to the proof of "Peirce's theorem" and must
somehow have played a role in his thinking, if not a conscious one, early
on, is exceedingly difficult. There are many interesting comments on
identity scattered through Peirce's papers, but I have not been able
to make a coherent story out of them, even of the ones preceding the
explicit development of the teridentity theory. I sometimes have the
impression in his earlier writings on this subject that he is all over
the map. In any case, I have to say that neither his late theory of
teridentity nor his earlier treatments of identity as an essentially
binary relation seem to be compatible with what I have called the
"realist" theory of identity. In discussing Peirce's theory of
identity, one has to remember his devotion of Scotus's haecceities,
but making things difficult for the interpreter, Peirce has put his
own spin on haecceities. This comes out in his observation that
"Even Duns Scotus is too nominalistic when he says that universals
are contracted to the mode of individuality in singulars, meaning,
as he does, by singulars, ordinary existing things" (8.208, 1905).
SS: Jon and Joe cite the pragmatic maxim, either Peirce's or James's version:
"A difference that makes no difference is no difference at all", as the
way to discover Peirce's views on identity. But one has to remember
that it took Peirce himself many years and the development of his
theory of synechism to show that the incommensurability of the
diagonal conforms to the requirements of the pragmatic maxim,
that it is a "difference that makes a difference."
SS: I admit to utter confusion over Peirce's theory of identity,
and further help would be welcome.
SS: Things are not helped by Peirce's devotion to coining new words.
Here is a classic bit of Peirceana touching on identity I think?
SS, quoting CSP:
| There is but one ambilative suilation. It is the juxtalation, or coëxistence.
| There is but one contrambilative suilation: it is the relation of individual
| identity, called numerical identity by the logicians. But the adjective seems
| needless. There is but one ambilative [contra] suilation: it is the relation
| of individual otherness, or negation. There is properly no contrambilative
| contrasuilation: it would be the absurd relation of incompossibility.
| These four relations are to be termed the Four Cardinal Dyadic Relations
| of Second Intention. It will be enough to call them the cardinilations,
| or cardinal relations. (CP 3.586).
|
| Any peneperlation or penereperlation is a juxtambilation;
| any perlation or reperlation is, in addition, a juxtasuilation.
| Any penecontraperlation or penecontrareperlation is an extrambilation:
| any contraperlation or contrareperlation is, in addition, an extrasuilation.
| Every ambilation is a penereperlative penereperlation: every contrambilation
| is a penecontrareperlative penecontraperlation. Every suilation is
| a juxtareperlative juxtaperlation: every contrasuilation is
| an extrareperlative extraperlation. (CP 3.587).
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Work 3
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
JA: It seems to me that the sources of your confusion are quite clear:
JA: 1. You consistently ignore what Peirce said in his 1870 remark on the
"doctrine of undividuals", that attributions of identity are relative to
a context of discourse, an insight that he did not abandon but developed
to its finest degree in his 1883 remarke "On a Limited Universe of Marks":
http://suo.ieee.org/ontology/msg03204.html
SS: For the benefit of those who may not be following Jon Awbrey's
arcane argument, I'm going to try to explain it, and to explain
why it is mostly nonsense. For convenience, I give here the
quote to which Awbrey is referring. Then I shall comment
on its bearing, if any, on logical criteria of identity.
| 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.
SS: I comment on this quote, showing that Jon Awbrey's contention that
it has some deep significance concerning Leibniz's law is nonsense.
SS: This passage is opposed to the kind of extensionalism advocated by Quine.
Quine's extensional languages are ones in which classes substitute for
properties and relations, two classes being identical if they have the
same members. In an intensional language, admitting properties as well
as classes, different properties may belong to exactly the same things.
In an intensional language, "a proposition concerning the relations of
two groups of marks is not necessarily equivalent to any proposition
concerning classes of things". Extensional languages, such as the
first-order predicate calculus, or set theory, are adequate for
mathematics, but it is controversial whether the sentences of
ordinary language or the sciences in general can be translated
into such an extensional language sentence for sentence.
SS: Peirce evidently thinks (as I do) that such translation is not
always possible for ordinary "limited universes of discourse".
However, intensionality in itself neither supports nor defeats
Leibniz's Law. In spite of adopting an intensional language,
one could deny Leibniz's Law, saying, as Peirce did late in
his career, that two raindrops could have all their properties,
including position in space, in common (being merged together)
and yet not be numerically identical raindrops, owing to their
dynamic interactions with one another. Or one could affirm
Leibniz's Law in an intensional language, saying that two
things are numerically identical if and only if anything
true of one is true of the other, as Peirce did, early
in his career.
SS: The second paragraph of the quote bears on the law of "excluded middle,"
since, as Peirce says, in a universe of discourse in which the universe
of characters is limited, (x)(Px V ~Px) may not hold. [It may not be
the case that "the non-possession of any character is regarded as
implying the possession of another character the negative of the
first."] This paragraph is relevant to whether we require a sortal
logic or logic admitting of presuppositions to cope with natural
languages, but it has no bearing on Leibniz's law.
SS: To understand the third paragraph, we have to know what Peirce means by
an "unlimited universe of marks." Well, here is an example: A universe
consisting of only two "marks", namely the colors red and blue, and two
"objects," one red and the other blue. In this case, "the universe of
characters is unlimited [because] every aggregate of things short of
the whole universe of things possesses in common one of the characters
of the universe of characters". This is because there are only two
non-empty aggregates short of the whole universe, namely the aggregate
consisting of the one red thing and the aggregate consisting of the one
blue thing. Now, in such a universe as this, Peirce points out, when
the class S is included in the class P, every common character of the
S's must also belong to the P's. Why? Well, in this case, if class S
is included in class P, where P is an "aggregate of things short of the
whole universe," S and P must be the same class.
SS: But so what? Neither in this odd universe, nor in the more common
"limited" universes of ordinary discourse, described by an intensional
language, is there reason to deny Leibniz's Law. If Peirce chose to
deny Leibniz's law, as he did late in his career, he did so on grounds
not relevant to this passage, in spite of Jon Awbrey's handwaving.
SS: If I can muster up the patience, I will respond to Jon Awbrey's second
point (below) in another message. But to make a long story short, it is
just more self-aggrandizing handwaving, having no bearing on the
significance of the quotes in question.
JA: 2. You consistently ignore the distinctions and the meanings of technical
terms in mathematics, as they were used by Peirce and as they have been
used for at least 200 years, for instance, the meanings of "composition"
and "reduction" as they are used in the algebra of relations and in the
logic of relative terms.
JA: I have pointed out all of this to you before. It is all pretty clear to
anybody who does not have a prior theory of what can qualify as a theory
of identity, or a philosophy, for that matter, that they keep trying
shove Peirce's more general theory, and more general philosophy, into.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Subject: Re: Tone, Token, Type
From: "Seth Sharpless" <seth.sharpless@colorado.edu>
Date: Mon, 18 Nov 2002 13:35:39 -0700
X-Message-Number: 7
~~~~~~~~~Quote from Peirce, CP1.20, 1903~~~~~~~~~~
I have since [1871] very carefully and thoroughly revised my
philosophical opinions more than half a dozen times, and have
modified them more or less on most topics.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jon's publication project has finally yielded some passages directly
relevant to the problem I raised concerning Peirce's theory of identity.
I apologize for the length of this post, but the length is necessary to
exhibit the apparent conflict in Peirce's early and late theories of
identity.
SS: To put the whole story in a nutshell (which necessarily distorts it a bit),
recall that in the history of philosophy, there have been two main contenders
as criteria of identity for existents: essential properties and spatio-temporal
continuity. Peirce's 1870 view clearly falls into the former group, identity of
existents always being in respect of some property. Peirce's later view appears
to fall in the latter group, allowing entities that are completely indiscernable
at one time, even occupying the same space at that time, to be distinct, because
they are in spatio-temporal continuity with entities occupying different places
at another time.
SS: Recall how this discussion of identity started. I said that
I would have expected Peirce to take a "realists" position:
SS: quoting from SS post of 11/9/02, 10:39 AM:
| ... namely, that identity is always in respect to some universal
| or type. For example, a is the same as b with respect to color,
| or size, or personhood (the same "person" as), etc. Then what has
| been called (since Aristotle) "numerical identity" could be regarded
| as a "degenerate" form of relative identity, in which a is the same
| as b in 'every' respect. This would make Leibniz's law of indiscernables
| true for numerical identity but not for identity simpliciter, since a
| could be identical to b in one respect and different in another.
SS: Only I found, on reading Peirce, seemingly conflicted statements that
prevented me from confirming this hypothesis and indeed,prevented me
from forming a clear idea of Peirce's theory of identity.
The passages quoted by Jon that are relevant to my problem are:
CP 3.93, 1870 (Jon's 11/16/02, 5:30 PM: "Tone, Token, Type")
CP 3.613, 1911 (Jon's 11/17/02, 11:36PM: "Doctrine of Individuals")
Jon omitted a helpful footnote to the 1870 passage:
| The absolute individual can not only not be realized in sense or thought,
| but cannot exist, properly speaking. For whatever lasts for any time,
| however short, is capable of logical division, because in that time
| it will undergo some change in its relations. But what does not
| exist for any time, however short, does not exist at all.
| All, therefore, that we perceive or think, or that exists,
| is general. So far there is truth in the doctrine of
| scholastic realism. But all that exists is infinitely
| determinate, and the infinitely determinate is the
| absolutely individual. This seems paradoxical,
| but the contradiction is easily resolved. That
| which exists is the object of a true conception.
| This conception may be made more determinate than
| any assignable conception; and therefore it is
| never so determinate that it is capable of no
| further determination. (CP 3.93 n.)
The 1870 and 1911 passages seem conflicted. In 1870, Peirce held that
any logical "individual" would be "infinitely determinate". By this,
he meant infinitely specifiable, in the sense that however many
distinguishing properties we lay down, we should find that
the identity of the individual remains somewhat vague,
further specification always being possible.
This inescapable vagueness, according to Peirce, is not due simply to limited
cognitive abilities on our part -- that is, it is not that existing entities
are actually determinate in all their properties, only we cannot apprehend
them in their infinite complexity. It is not this; the indeterminacy of
character is a necessary feature of the world as its ontological potential
is realized. For Peirce, no genuinely determinate individual actually
exists at any given time. A wholly determinate individual is only
a possibility. If it existed, he says, it would have to last for
more than an instant, and it would have to undergo some change
from instant to instant; thus, the entity at instant one could
be said to be identical to the entity at instant two only
in some respects, not absolutely. Absolutely determinate
individuals belong only to the realm of the possible.
Normally, when we speak of "individuals", we are not speaking of
infinitely determinate individuals, but only what the scholastics
called "singulars", that is, things which undergo change but which
are identifiable because they remain the same in some respect.
Accordingly, so far as this 1870 passage quoted, I was right
in anticipating that Peirce would take a "realist's" view of
identity in which ordinary existents must be identified with
respect to some universal, so that though identical in one
respect, they remain discernable in other respects.
The only addition is that Peirce restricts what I called the "degenerate"
form of identity -- identity in all respects -- to the realm of the ideal
or possible (though it has to be remembered that for Peirce, the merely
possible is also real, just not existent). Actually existing entities,
which endure from instant to instant, cannot exhibit numerical identity,
identity in all respects, because there are always respects in which
they are vague or indeterminate.
Fine! That was 1870. Now, let us skip to Peirce's later period,
starting with a quote dated 1911, which must be nearly Peirce's
last word on the matter.
~~~~~Excerpts from CP3.613 (1911)~~~~~~~~~~~~~~
Another definition which avoids the above difficulties is that an
individual is something which reacts. That is to say, it does react
against some things, and is of such a nature that it might react, or
have reacted, against my will...
According to this definition, that which alone immediately presents
itself as an individual is a reaction against the will. But everything
whose identity consists in a continuity of reactions will be a single
logical individual. Thus any portion of space, so far as it can be
regarded as reacting, is for logic a single individual; its spatial
extension is no objection. With this definition there is no difficulty
about the truth that whatever exists is individual, since existence (not
reality) and individuality are essentially the same thing; and whatever
fulfills the present definition equally fulfills the former definition
by virtue of the principles of contradiction and excluded middle,
regarded as mere definitions of the relation expressed by "not".
As for the principle of indiscernibles, if two individual things
are exactly alike in all other respects, they must, according to
this definition, differ in their spatial relations, since space
is nothing but the intuitional presentation of the conditions
of reaction, or of some of them. But there will be no logical
hindrance to two things being exactly alike in all other respects;
and if they are never so, that is a physical law, not a necessity
of logic. This second definition, therefore, seems to be the
preferable one.
~~~~End of Peirce quote~~~~~~~~~~~~
The new theory was apparently developed by 1896,
as illustrated by this quote, from 1896.
~~~~~~~Peirce quote from CP1.458, 1896~~~~~~~~~~~
'Hic et nunc' is the phrase perpetually in the mouth of Duns Scotus,
who first elucidated individual existence. It is a forcible phrase
if understood as Duns did understand it, not as describing individual
existence, but as suggesting it by an example of the attributes found
in this world to accompany it. Two drops of water retain each its
identity and opposition to the other no matter in what or in how
many respects they are alike. Even could they interpenetrate one
another like optical images (which are also individual), they would
nevertheless react, though perhaps not at that moment, and by virtue
of that reaction would retain their identities. The point to be
remarked is that the qualities of the individual thing, however
permanent they may be, neither help nor hinder its identical
existence. However permanent and peculiar those qualities
may be, they are but accidents; that is to say, they are
not involved in the mode of being of the thing; for the
mode of being of the individual thing is existence;
and existence lies in opposition merely.
~~~~~~~~~~End of Peirce quote~~~~~~~~~~~
Notice that two drops of water completely merged and indiscernable
at time 't' nevertheless retain their distinct identities at time 't'
because they are continuous with spatially distinct drops interacting
with one another at a different time. The 1896 quote seems even more
radical than the 1911 quote, in that it was not evident from the latter
that two entities, indiscernable in all respects, 'even with respect to
spatial position', at one time, could nevertheless be distinct, owing to
being in spatio-temporal continuity with entities which are spatially
distinct at another time.
A problem with this theory of identity is in understanding how spatiotemporal
continuity is to be shown. If the two drops, a and b, are completely merged
and indiscernable in all qualities at time t, and apart and interacting at
time t+1, how can one know which of the interacting drops at time t+1 is a
and which is b?
But that is an aside. The exegetical problem is that of either
rendering the seemingly different theories of identity consistent,
or of coming to terms with the idea that Peirce might have changed
his mind on this, as on other things:
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Work Area
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CP 3.398 (1885)
CP 1.456 (1896)
CP 1.458 (1896)
CP 4.311 (1897)
authoritarian dogmatic vs. exploratory hypothetical
personal identity vs. atomic believer
| A_1 vs. ~A_1
|
| ...
|
| A_k vs. ~A_k
"reality is real" vs. "no it isn't"
"hermeneutic equivalence class" (HEC)
atomic philosopher, determinate on every pro-ism vs. con-ism
The difference between a realist and a personal infallibilist
is like the relation between a monotheist and a theomaniac.
It is the difference between
one who thinks that God is one
and one who thinks that one is God.
Let me put my last remarks -- about the false opposition between
any two such positions as might be dedicated solely to extensions
or to intensions, admitting neither a tertium quid nor the chance
of solid integration between these two aspects of a sign relation --
in the context of a contemporary problem that has been discussed
in many circles, namely, that of "language acquisition" (LA).
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I found this sample of passages on the Tone, Token, Type theme
that I collected pursuant to an earlier discussion on the List:
00. http://stderr.org/pipermail/inquiry/2003-April/thread.html#429
01. http://stderr.org/pipermail/inquiry/2003-April/000429.html
02. http://stderr.org/pipermail/inquiry/2003-April/000430.html
03. http://stderr.org/pipermail/inquiry/2003-April/000431.html
04. http://stderr.org/pipermail/inquiry/2003-April/000432.html
05. http://stderr.org/pipermail/inquiry/2003-April/000433.html
06. http://stderr.org/pipermail/inquiry/2003-April/000434.html
From back in days when a few of us were weaving interlacing threads
through several different interest groups, I find in addition these:
00. http://suo.ieee.org/ontology/thrd28.html#04053
01. http://suo.ieee.org/ontology/msg04053.html
02. http://suo.ieee.org/ontology/msg04325.html
03. http://suo.ieee.org/ontology/msg04326.html
04. http://suo.ieee.org/ontology/msg04327.html
05. http://suo.ieee.org/ontology/msg04328.html
06. http://suo.ieee.org/ontology/msg04329.html
07. http://suo.ieee.org/ontology/msg04330.html
08. http://suo.ieee.org/ontology/msg04331.html
09. http://suo.ieee.org/ontology/msg04334.html
10. http://suo.ieee.org/ontology/msg04347.html
It appears that we had quite a lot of discussion about this before,
but I am finding little more than fragments off the top of the web,
for example, this:
• http://www.cspeirce.com/peirce-l/11-19-02.htm
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JA: http://permalink.gmane.org/gmane.science.philosophy.peirce/8646
MF: http://permalink.gmane.org/gmane.science.philosophy.peirce/8647
Matt,
Just rummaging around my files and the web. Found a few bits
and bytes of other discussions that might be worth revisiting.
I tend to be skeptical about the whole "explaining consciousness" industry,
partly because I draw a distinction between describing and explaining, and
not everything that calls for description calls for explanation, at least,
not from the standpoint of a Peircean or Aristotelian perspective on the
role of abductive inference in the play of inquiry. That says nothing
about the possible use of the T3 framework in describing the contents
and modifications of consciousness, however, so I think that may be
worth another look.
Document History
- http://web.archive.org/web/20150206001000/http://stderr.org/pipermail/inquiry/2003-April/000429.html
- http://web.archive.org/web/20070310205736/http://stderr.org/pipermail/inquiry/2003-April/000430.html
- http://web.archive.org/web/20070311190954/http://stderr.org/pipermail/inquiry/2003-April/000431.html
- http://web.archive.org/web/20070311191006/http://stderr.org/pipermail/inquiry/2003-April/000432.html
- http://web.archive.org/web/20070309203706/http://stderr.org/pipermail/inquiry/2003-April/000433.html
- http://web.archive.org/web/20070311191016/http://stderr.org/pipermail/inquiry/2003-April/000434.html
