Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction

[Jon Awbrey]

Ben, Steven, & All ...

I may have missed a few posts but I don't understand the fuss about indices.
The types of signs not in one-to-one correspondence with the types of objects.
You can refer to the same object by means of a pronoun or some other index --
for example, "Looky there!", "Voila!", or "I don't know what it is, but there
it goes again" -- or you can refer to it by means of a noun, or some figure of
speech with iconic properties.  It is simply a matter of convenience in certain
cases that we use an index or icon when a more definitive symbol might take a 
lot
of work to fashion.

Regards,

Jon

Steven Ericsson-Zenith wrote:
Ben and I appear to be speaking across each other and, possibly, agreeing fiercely. 

Recall that in the 1906 dialectic Peirce is drawing a distinction between the wider usage of "Category" at the time, i.e., Aristotle's Categories considered by "you" in the dialog, and saying that he prefers to call these "Predicaments." Having made this distinction he then speaks about the indices that are his categories. 


As I said earlier, the index in this case does not point to the elements of the category but the 
category itself. "There is Firstness" as opposed to "x is a first." The 
confusion may be that Ben thinks I am saying that a category is some set of indices to its members. 
That is not the case, a category stands alone and we can point to it (index). Icons are the 
selection mechanisms of properties of classes, not indices.

Predicaments are higher order, assertions about assertions, predicates of predicates, I prefer to 
say "predicated predicates" or "assertions about assertions" which is more 
generally understood today.

Being as careful as he is, I see no evidence to cause us to suppose that the 
categories that Peirce attributes to himself in 1906 are different than those 
he identifies as early as 1866.

With respect,
Steven

--
Dr. Steven Ericsson-Zenith
Institute for Advanced Science & Engineering
http://iase.info


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[peirce-l] Peirce and Neitzsche Precursors of the Future

[Stephen C. Rose]

Peirce and Neitzsche Precursors of the Future http://ping.fm/fnwjR

fyi, Cheers, S
*ShortFormContent at Blogger* 

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Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction

[Steven Ericsson-Zenith]

Ben and I appear to be speaking across each other and, possibly, agreeing 
fiercely. 

Recall that in the 1906 dialectic Peirce is drawing a distinction between the 
wider usage of "Category" at the time, i.e., Aristotle's Categories considered 
by "you" in the dialog, and saying that he prefers to call these 
"Predicaments." Having made this distinction he then speaks about the indices 
that are his categories. 

As I said earlier, the index in this case does not point to the elements of the 
category but the category itself. "There is Firstness" as opposed to "x is a 
first." The confusion may be that Ben thinks I am saying that a category is 
some set of indices to its members. That is not the case, a category stands 
alone and we can point to it (index). Icons are the selection mechanisms of 
properties of classes, not indices.

Predicaments are higher order, assertions about assertions, predicates of 
predicates, I prefer to say "predicated predicates" or "assertions about 
assertions" which is more generally understood today.

Being as careful as he is, I see no evidence to cause us to suppose that the 
categories that Peirce attributes to himself in 1906 are different than those 
he identifies as early as 1866.

With respect,
Steven

--
Dr. Steven Ericsson-Zenith
Institute for Advanced Science & Engineering
http://iase.info







On Mar 12, 2012, at 7:26 AM, Gary Fuhrman wrote:

> Steven, in addition to what Ben said ...
>  
> Your sense of chronology (and therefore of context) is completely askew here. 
> You wrote,
> [[ Again:
> 
> "... of superior importance in Logic is the use of Indices to denote 
> Categories and Universes, which are classes that, being enormously large, 
> very promiscuous, and known but in small part, cannot be satisfactorily 
> defined, and therefore can only be denoted by Indices."
> 
> A year earlier, in 1866, Peirce wrote "On A Method Of Searching For The 
> Categories" ... ]]
>  
> That quote is from the “Prolegomena” of 1906. Your next quote is not from “a 
> year earlier” but from 40 years earlier. The Prolegomena clearly does not use 
> the word “Categories” in reference to Firstness, Secondness and Thirdness, 
> but in reference to classes of predicates (as opposed to subjects). Although 
> Peirce calls those classes “Categories” in the 1906 Prolegomena, they do not 
> map onto his usual categorial triad, nor vice versa. Also note that in 1866-7 
> Peirce was not yet using the term “index” for the kind of sign that is 
> capable of genuine denotation; but in 1906, “indices” has its precise 
> semiotic meaning in Peirce, which your analysis does not reflect.
>  
> Gary F.
>  
> From: C S Peirce discussion list [mailto:PEIRCE-L@LISTSERV.IUPUI.EDU] On 
> Behalf Of Benjamin Udell
> Sent: March-12-12 9:28 AM
> To: PEIRCE-L@LISTSERV.IUPUI.EDU
> Subject: Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction
>  
> Dear Steven,
> 
> Okay, 1866 instead of 1867. Indeed he regularly said that his categories are 
> indecomposible into more basic elements; they _are_ his basic elements. 
> That's why your indecomposability argument fails in the case of the 
> Prolegomena-categories (which he says he prefers to call "Predicaments") - 
> it's because there he does _not_ call them indecomposable; instead he says 
> that they are "classes that, being enormously large, very promiscuous, and 
> known but in small part, cannot be satisfactorily defined, and therefore can 
> only be denoted by Indices."
> 
> Note that he did not say that the classes in question are indices of their 
> elements or members. Instead he said that they are classes denotable 
> (actually) only by indices and not by satisfactory definitions (since there 
> actually are none). You imply that he could not have meant that because it 
> would have led to an infinite regress. Yet it is in fact what he did say, and 
> we are not entitled to silently revise him as if it were a mere typographical 
> error. I'm not sure why you think it leads to infinite regress but, supposing 
> that it does, it is not necessarily a problem for Peirce. Peirce believed in 
> infinite series of signs in semiosis that has nevertheless a beginning and an 
> end (at least by interruption) in time, since he was a synechist. In fact he 
> based his synechism on the four incapacities, for example the incapacity for 
> intuition, that is, the incapacity for a cognition devoid of inferential 
> relation to a previous cognition. From final paragraph (CP 5.263) of 
> "Questions concerning certain Faculties claimed for Man":
> 
> So that it is not true that there must be a first. Explicate the logical 
> difficulties of this paradox (they are identical with those of the Achilles) 
> in whatever way you may. I am content with the result, as long as your 
> principles are fully applied to the particular case of cognitions determining 
> one another. Deny motion, if it seems proper to do so; only then deny the 

Re: [peirce-l] [Inquiry] Categorical Aspects of Abduction, Deduction, Induction

[Jon Awbrey]

Peircers,

There is a continuity of purpose that unites all the various category systems,
from Aristotle through the present day.  Clearly, the categories of Aristotle,
Kant, Peirce, and contemporary mathematics are the same in neither number nor
content, but the logical function and semiotic utility they serve is the same.

The problem for which categories are proposed as a solution by Aristotle is
that signs are equivocal, perhaps inherently and even necessarily, at least,
for creatures like us, and so we have need of additional signs for reducing
their ambiguities to the point where logic can begin to apply, as it cannot
apply to words that are used in incompatible categories of meaning or sense.

Here is the that inaugural passage from Aristotle again —

| Things are equivocally named, when they have the name only in common,
| the definition (or statement of essence) corresponding with the name
| being different. For instance, while a man and a portrait can properly
| both be called animals (ζωον), these are equivocally named. For they
| have the name only in common, the definitions (or statements of essence)
| corresponding with the name being different. For if you are asked to
| define what the being an animal means in the case of the man and the
| portrait, you give in either case a definition appropriate to that
| case alone.
|
| Things are univocally named, when not only they bear the same name but the
| name means the same in each case — has the same definition corresponding.
| Thus a man and an ox are called animals. The name is the same in both cases;
| so also the statement of essence. For if you are asked what is meant by their
| both of them being called animals, you give that particular name in both cases
| the same definition.
|
| Aristotle, Categories, 1.1a1–12.
|
| Translator's Note. “Ζωον in Greek had two meanings, that is to say, living 
creature, and,
| secondly, a figure or image in painting, embroidery, sculpture.  We have no 
ambiguous noun.
| However, we use the word ‘living’ of portraits to mean ‘true to life’.”  
(H.P. Cooke).
|
| http://mywikibiz.com/Directory:Jon_Awbrey/Notes/Precursors#Aristotle

As I commented —

In the logic of Aristotle categories are adjuncts to reasoning that are 
designed to resolve ambiguities
and thus to prepare equivocal signs, that are otherwise recalcitrant to being 
ruled by logic, for the
application of logical laws.  The example of ζωον illustrates the fact that we 
don't need categories
to make generalizations so much as we need them to control generalizations, to 
reign in abstractions
and analogies that are stretched too far.

Regards,

Jon

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Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

[Benjamin Udell]

Jason, all,

If I had bothered to search on "computational mathematics" I would have found 
that the potential ambiguity that worried me is already actual, as you clearly 
show.  Do you think that the phrase "computative mathematics" is too close to 
the phrase "computational mathematics" for comfort?  I hope not, but please say 
so if it is.

Problem is, the "applied" in "applied mathematics" is used in various ways 
that, as Dieudonné of the Bourbaki group pointed out in his Britannica article 
(15th edition I think), jumbles trivial and nontrivial areas of math together, 
and has all too many, umm, applications. One area of pure math X may be 
_applied_ in another area of math Y, whih is to say that Y is the guiding 
research interest. If on the other hand Y is applied in X, then that's to say 
that X is the guiding research interest. And both X and Y remain areas of 
'pure' math. Then there are areas of so-called 'applied' but often nontrivial 
math like probability theory. Then there are applications in statistics and in 
the special sciences. Then there applications in practical/productive 
sciences/arts. And of course, sometimes theoretical or 'pure' math is developed 
specifically for a particular application. (All in all, we won't be able to get 
rid of the term "applied," but in some cases we may be find an alternate term 
with the same denotation in the given context).

Best, Ben

- Original Message - 
From: Khadimir 
To: Benjamin Udell 
Cc: PEIRCE-L@listserv.iupui.edu 
Sent: Monday, March 12, 2012 2:14 PM
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition

This latest post caught my attention.

Since my first degree was a B.S. in "computational mathematics," I thought that 
I would weigh-in.  

One can make the distinctions as follows, beginning with pure vs. applied 
mathematics.  I will give a negative definition, since I am not so skilled with 
the Peircean terminology used so far; applied mathematics is the use of 
mathematics as a formal, ideal system to specific problems of existence.  For 
instance, consider the use of statistical confidence intervals to solve 
problems in manufactoring relating to the rate of production of defective vs. 
non-defective goods.  Pure mathematics is not bound by existent conditions, but 
"pure" becomes "applied" when used in that context.  Hence, I am treated 
applied mathematics as an informal, existential constraint that alters the 
purpose and use of pure mathematics.

Computational mathematics is for the most part a subset of applied mathematics, 
which focuses on how to adapt computational formulas so that they may be run or 
run more efficiently on a given computation system, e.g., a binary computer.  
Computational mathematics, then, is primarily focused on formulas and 
computation of said formulas, which is to be more specific about the limits 
that make it an applied mathematic.

I offer this as a different viewpoint, one coming from where the distinction 
has practical effects.

Jason H.

On Mon, Mar 12, 2012 at 12:47 PM, Benjamin Udell  wrote:

  Malgosia, Irving, Gary, list,

  I should add that this whole line of discussion began because I put the cart 
in front of the horse. The adjectives bothered me. "Theoretical math" vs. 
"computational math" - the latter sounds like of math about computation. And 
"creative math" vs. what - "consumptive math"? "consumptorial math"?  Then I 
thought of theorematic vs. corollarial, thought it was an interesting idea and 
gave it a try. The comparison is interesting and there is some likeness between 
the distinctions.  However I now think that trying to align it to Irving's and 
Pratt's distinctions just stretches it too far.  And it's occurred to me that 
I'd be happy with the adjective "computative" - hence, theoretical math versus 
computative math.

  However, I don't think that we've thoroughly replaced the terms "pure" and 
"applied" as affirmed of math areas until we find some way to justly 
distinguish between so-called 'pure' maths as opposed to so-called 'applied' 
yet often (if not absolutely always) mathematically nontrivial areas such as 
maths of optimization (linear and nonlinear programming), probability theory, 
the maths of information (with laws of information corresponding to 
group-theoretical principles), etc.

  Best, Ben


  - Original Message - 
  From: Benjamin Udell 
  To: PEIRCE-L@LISTSERV.IUPUI.EDU 

  Sent: Monday, March 12, 2012 1:14 PM 
  Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's 
Peirce edition

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Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

[Khadimir]

This latest post caught my attention.

Since my first degree was a B.S. in "computational mathematics," I thought
that I would weigh-in.

One can make the distinctions as follows, beginning with pure vs. applied
mathematics.  I will give a negative definition, since I am not so skilled
with the Peircean terminology used so far; applied mathematics is the use
of mathematics as a formal, ideal system to specific problems of existence.
 For instance, consider the use of statistical confidence intervals to
solve problems in manufactoring relating to the rate of production of
defective vs. non-defective goods.  Pure mathematics is not bound by
existent conditions, but "pure" becomes "applied" when used in that
context.  Hence, I am treated applied mathematics as an informal,
existential constraint that alters the purpose and use of pure mathematics.

Computational mathematics is for the most part a subset of applied
mathematics, which focuses on how to adapt computational formulas so that
they may be run or run more efficiently on a given computation system,
e.g., a binary computer.  Computational mathematics, then, is primarily
focused on formulas and computation of said formulas, which is to be more
specific about the limits that make it an applied mathematic.

I offer this as a different viewpoint, one coming from where the
distinction has practical effects.

Jason H.

On Mon, Mar 12, 2012 at 12:47 PM, Benjamin Udell  wrote:

> **
>
> Malgosia, Irving, Gary, list,
>
> I should add that this whole line of discussion began because I put the
> cart in front of the horse. The adjectives bothered me. "Theoretical math"
> vs. "computational math" - the latter sounds like of math about
> computation. And "creative math" vs. what - "consumptive math"?
> "consumptorial math"?  Then I thought of theorematic vs. corollarial,
> thought it was an interesting idea and gave it a try. The comparison is
> interesting and there is some likeness between the distinctions.  However I
> now think that trying to align it to Irving's and Pratt's distinctions just
> stretches it too far.  And it's occurred to me that I'd be happy with the
> adjective "computative" - hence, theoretical math versus computative math.
>
> However, I don't think that we've thoroughly replaced the terms "pure" and
> "applied" as affirmed of math areas until we find some way to justly
> distinguish between so-called 'pure' maths as opposed to so-called
> 'applied' yet often (if not absolutely always) mathematically nontrivial
> areas such as maths of optimization (linear and nonlinear programming),
> probability theory, the maths of information (with laws of information
> corresponding to group-theoretical principles), etc.
>
> Best, Ben
>
> - Original Message -
> *From:* Benjamin Udell  *
> **To:* PEIRCE-L@LISTSERV.IUPUI.EDU *
> *
> *Sent:* Monday, March 12, 2012 1:14 PM
> *Subject:* Re: [peirce-l] Mathematical terminology, was, review of
> Moore's Peirce edition
>
> Malgosia, list,
>
> Responses interleaved.
>
> - Original Message -
> From: malgosia askanas
> To: PEIRCE-L@LISTSERV.IUPUI.EDU
> Sent: Monday, March 12, 2012 12:31 PM
> Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's
> Peirce edition
>
> >>[BU] Yes, the theorematic-vs.-corollarial distinction does not appear in
> the Peirce quote to depend on whether the premisses - _up until some lemma_
> - already warrant presumption.
> >>BUT, but, but, the theorematic deduction does involve the introdution of
> that lemma, and the lemma needs to be proven (in terms of some postulate
> system), or at least include a definition (in remarkable cases supported by
> a "proper postulate") in order to stand as a premiss, and that is what
> Irving is referring to.
>
> >[MA] OK, but how does this connect to the corollarial/theorematic
> distinction?  On the basis purely of the quote from Peirce that Irving was
> discussing, the theorem, again, could follow from the lemma either
> corollarially (by virtue purely of "logical form") or theorematically
> (requiring additional work with the actual mathematical objects of which
> the theorem speaks).
>
> [BU] So far, so good.
>
> >[MA] And the lemma, too, could have been obtained either corollarially (a
> rather needless lemma, in that case)
>
> [BU] Only if it comes from another area of math, otherwise it is
> corollarially drawn from what's already on the table and isn't a lemma.
>
> >[MA] or theorematically.   Doesn't this particular distinction, in either
> case, refer to the nature of the _deduction_ that is required in order to
> pass from the premisses to the conclusion, rather than referring to the
> warrant (or lack of it) of presuming the premisses?
>
> [BU] It's both, to the extent that the nature of that deduction depends on
> whether the premisses require a lemma, a lemma that either gets something
> from elsewhere (i.e., the lemma must refer to where its content is
> established elsewhere), or needs to be proven on t

Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction

[Jon Awbrey]

GF: Good point, Jon -- we should not neglect the element of performance art in 
philosophy! :-)

GF: However I'm not sure it's right to say that the metaphysical order is more 
fundamental than the
phenomenological. It doesn't seem to jibe with Peirce's classification of 
the sciences, either.

JA: Yes, we always have the choice between "first in nature" and "first for us".
I have no strong feelings about which first comes first -- I was just going
by Peirce's statement:

CSP: Besides, it would be illogical to rely upon the categories to decide so 
fundamental a question.

JA: But you are right, one could just as well say that independent foundations 
are both fundamental
without one foundation being more fundamental than the other.

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Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

[Benjamin Udell]

Malgosia, Irving, Gary, list,

I should add that this whole line of discussion began because I put the cart in 
front of the horse. The adjectives bothered me. "Theoretical math" vs. 
"computational math" - the latter sounds like of math about computation. And 
"creative math" vs. what - "consumptive math"? "consumptorial math"?  Then I 
thought of theorematic vs. corollarial, thought it was an interesting idea and 
gave it a try. The comparison is interesting and there is some likeness between 
the distinctions.  However I now think that trying to align it to Irving's and 
Pratt's distinctions just stretches it too far.  And it's occurred to me that 
I'd be happy with the adjective "computative" - hence, theoretical math versus 
computative math.

However, I don't think that we've thoroughly replaced the terms "pure" and 
"applied" as affirmed of math areas until we find some way to justly 
distinguish between so-called 'pure' maths as opposed to so-called 'applied' 
yet often (if not absolutely always) mathematically nontrivial areas such as 
maths of optimization (linear and nonlinear programming), probability theory, 
the maths of information (with laws of information corresponding to 
group-theoretical principles), etc.

Best, Ben

- Original Message - 
From: Benjamin Udell 
To: PEIRCE-L@LISTSERV.IUPUI.EDU 
Sent: Monday, March 12, 2012 1:14 PM 
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition

Malgosia, list,

Responses interleaved.

- Original Message - 
From: malgosia askanas 
To: PEIRCE-L@LISTSERV.IUPUI.EDU 
Sent: Monday, March 12, 2012 12:31 PM 
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition

>>[BU] Yes, the theorematic-vs.-corollarial distinction does not appear in the 
>>Peirce quote to depend on whether the premisses - _up until some lemma_ - 
>>already warrant presumption.
>>BUT, but, but, the theorematic deduction does involve the introdution of that 
>>lemma, and the lemma needs to be proven (in terms of some postulate system), 
>>or at least include a definition (in remarkable cases supported by a "proper 
>>postulate") in order to stand as a premiss, and that is what Irving is 
>>referring to.

>[MA] OK, but how does this connect to the corollarial/theorematic distinction? 
> On the basis purely of the quote from Peirce that Irving was discussing, the 
>theorem, again, could follow from the lemma either corollarially (by virtue 
>purely of "logical form") or theorematically (requiring additional work with 
>the actual mathematical objects of which the theorem speaks).  

[BU] So far, so good.

>[MA] And the lemma, too, could have been obtained either corollarially (a 
>rather needless lemma, in that case) 

[BU] Only if it comes from another area of math, otherwise it is corollarially 
drawn from what's already on the table and isn't a lemma.

>[MA] or theorematically.   Doesn't this particular distinction, in either 
>case, refer to the nature of the _deduction_ that is required in order to pass 
>from the premisses to the conclusion, rather than referring to the warrant (or 
>lack of it) of presuming the premisses?  

[BU] It's both, to the extent that the nature of that deduction depends on 
whether the premisses require a lemma, a lemma that either gets something from 
elsewhere (i.e., the lemma must refer to where its content is established 
elsewhere), or needs to be proven on the spot. But - in some cases there's no 
lemma but merely a definition that is uncontemplated in the thesis, and is not 
demanded by the premisses or postulates but is still consistent with them, and 
so Irving and I, as it seems to me now, are wrong to say that it's _always_ a 
matter of whether some premiss requires special proof. Not always, then, but 
merely often. In some cases said definition needs to be supported by a new 
postulate, so there the proof-need revives but is solved by recognizing the 
need and "conceding" a new postulate to its account.

>[MA] If the premisses are presumed without warrant, that - it seems to me - 
>does not make the deduction more corollarial or more theorematic; it just 
>makes it uncompleted, and perhaps uncompletable.

[BU] That sounds right.

Best, Ben

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Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

[Benjamin Udell]

Malgosia, list,

Responses interleaved.

- Original Message - 
From: malgosia askanas 
To: PEIRCE-L@LISTSERV.IUPUI.EDU 
Sent: Monday, March 12, 2012 12:31 PM 
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition

>>[BU] Yes, the theorematic-vs.-corollarial distinction does not appear in the 
>>Peirce quote to depend on whether the premisses - _up until some lemma_ - 
>>already warrant presumption.
>>BUT, but, but, the theorematic deduction does involve the introdution of that 
>>lemma, and the lemma needs to be proven (in terms of some postulate system), 
>>or at least include a definition (in remarkable cases supported by a "proper 
>>postulate") in order to stand as a premiss, and that is what Irving is 
>>referring to.

>[MA] OK, but how does this connect to the corollarial/theorematic distinction? 
> On the basis purely of the quote from Peirce that Irving was discussing, the 
>theorem, again, could follow from the lemma either corollarially (by virtue 
>purely of "logical form") or theorematically (requiring additional work with 
>the actual mathematical objects of which the theorem speaks).  

[BU] So far, so good.

>[MA] And the lemma, too, could have been obtained either corollarially (a 
>rather needless lemma, in that case) 

[BU] Only if it comes from another area of math, otherwise it is corollarially 
drawn from what's already on the table and isn't a lemma.

>[MA] or theorematically.   Doesn't this particular distinction, in either 
>case, refer to the nature of the _deduction_ that is required in order to pass 
>from the premisses to the conclusion, rather than referring to the warrant (or 
>lack of it) of presuming the premisses?  

[BU] It's both, to the extent that the nature of that deduction depends on 
whether the premisses require a lemma, a lemma that either gets something from 
elsewhere (i.e., the lemma must refer to where its content is established 
elsewhere), or needs to be proven on the spot. But - in some cases there's no 
lemma but merely a definition that is uncontemplated in the thesis, and is not 
demanded by the premisses or postulates but is still consistent with them, and 
so Irving and I, as it seems to me now, are wrong to say that it's _always_ a 
matter of whether some premiss requires special proof. Not always, then, but 
merely often. In some cases said definition needs to be supported by a new 
postulate, so there the proof-need revives but is solved by recognizing the 
need and "conceding" a new postulate to its account.

>[MA] If the premisses are presumed without warrant, that - it seems to me - 
>does not make the deduction more corollarial or more theorematic; it just 
>makes it uncompleted, and perhaps uncompletable.

[BU] That sounds right.

Best, Ben

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Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

[malgosia askanas]

Benjamin Udell wrote:

>Yes, the theorematic-vs.-corollarial distinction does not appear in the Peirce 
>quote to depend on whether the premisses - _up until some lemma_ - already 
>warrant presumption.
>
>BUT, but, but, the theorematic deduction does involve the introdution of that 
>lemma, and the lemma needs to be proven (in terms of some postulate system), 
>or at least include a definition (in remarkable cases supported by a "proper 
>postulate") in order to stand as a premiss, and that is what Irving is 
>referring to.

OK, but how does this connect to the corollarial/theorematic distinction?  On 
the basis purely of the quote from Peirce that Irving was discussing, the 
theorem, again, could follow from the lemma either corollarially (by virtue 
purely of "logical form") or theorematically (requiring additional work with 
the actual mathematical objects of which the theorem speaks).   And the lemma, 
too, could have been obtained either corollarially (a rather needless lemma, in 
that case) or theorematically.   Doesn't this particular distinction, in either 
case, refer to the nature of the _deduction_ that is required in order to pass 
from the premisses to the conclusion, rather than referring to the warrant (or 
lack of it) of presuming the premisses?  If the premisses are presumed without 
warrant, that - it seems to me - does not make the deduction more corollarial 
or more theorematic; it just makes it uncompleted, and perhaps uncompletable.

-malgosia

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Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

[Benjamin Udell]

Irving, Gary, Malgosia, list,

Yes, the theorematic-vs.-corollarial distinction does not appear in the Peirce 
quote to depend on whether the premisses - _up until some lemma_ - already 
warrant presumption. 

BUT, but, but, the theorematic deduction does involve the introdution of that 
lemma, and the lemma needs to be proven (in terms of some postulate system), or 
at least include a definition (in remarkable cases supported by a "proper 
postulate") in order to stand as a premiss, and that is what Irving is 
referring to. 

The confusion seems to be that, though Peirce says that deduction doesn't care 
whether the premisses are true (though it obviously cares that they be at least 
formally consistent), evidently it does matter whether they warrant presumption 
- are they supported by postulates already in place?, do they need another 
postulate?, etc.  This concern with the postulate system seems something native 
to 'pure' math, 

The rest of Irving's response seems to show that his 
theoretical-vs.-computational distinction and Pratt's creator-vs.-consumer 
distinction are made in terms of whether mathematical discovery and originality 
are involved, and are also, so to speak, at another level of scale. In other 
words, one could have merely computational-purposed and unoriginal definitions 
uncontemplated in a thesis. Moreover the computational math is not always out 
to prove a thesis, but to find a solution.

Another problem with calling computational/consumptorial math merely 
"corollarial" - Peirce said "Corollarial deduction is where it is only 
necessary to imagine any case in which the premisses are true in order to 
perceive immediately that the conclusion holds in that case." That's certainly 
untrue of a whole lot of computational deductions! - long calculations whose 
answers are anything but immediately evident. This may just be matter of 
allowing that a corollarial deduction, even if not involving an immediately 
evident conclusion, should at least be analyzable into steps each of which 
makes the next step immediately evident. The price is an alteration of Peirce's 
definition, and a less steep price would be instead to call such a deduction 
"multi-corollarial." or some such.

Best, Ben

- Original Message - 
From: "Gary Richmond" 
To: 
Sent: Monday, March 12, 2012 12:22 AM
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition

I *strongly* agree with your analysis, Malgosia.

Best,

Gary

On 3/11/12, malgosia askanas  wrote:
> Irving wrote, quoting Peirce MS L75:35-39:

>>"Deduction is only of value in tracing out the consequences of
>>hypotheses, which it regards as pure, or unfounded, hypotheses.
>>Deduction is divisible into sub-classes in various ways, of which the
>>most important is into corollarial and theorematic. Corollarial
>>deduction is where it is only necessary to imagine any case in which
>>the premisses are true in order to perceive immediately that the
>>conclusion holds in that case. Ordinary syllogisms and some deductions
>>in the logic of relatives belong to this class. Theorematic deduction
>>is deduction in which it is necessary to experiment in the imagination
>>upon the image of the premiss in order from the result of such
>>experiment to make corollarial deductions to the truth of the
>>conclusion. The subdivisions of theorematic deduction are of very high
>>theoretical importance. But I cannot go into them in this statement."

>>[...] Peirce's characterization of theorematic and corrolarial
>>deduction would seem, on the basis of this quote, to have to do with
>>whether the presumption that the premises of a deductive argument or
>>proof are true versus whether they require to be established to be
>>true [...]

> I would disagree with this reading of the Peirce passage.  It seems
> to me that the distinction he is making is, rather, between (1) the case
> where the conclusion can be seen to follow from the premisses
> by virtue of the "logical form" alone, as in "A function which is continuous
> on a closed interval is continuous on any subinterval of that interval"
> (whose truth is obvious without requiring us to imagine any continuous
> function or any interval), and (2) the case where the deduction of the
> conclusions from the premisses requires turning one's imagination
> upon, and experimenting with, the actual mathematical objects
> of which the theorem speaks, as in "A function which is continuous
> on a closed interval is bounded on that interval".

> -malgosia-- 

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Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction

[Gary Fuhrman]

Steven, in addition to what Ben said ...

 

Your sense of chronology (and therefore of context) is completely askew here. 
You wrote,

[[ Again:

"... of superior importance in Logic is the use of Indices to denote Categories 
and Universes, which are classes that, being enormously large, very 
promiscuous, and known but in small part, cannot be satisfactorily defined, and 
therefore can only be denoted by Indices."

A year earlier, in 1866, Peirce wrote "On A Method Of Searching For The 
Categories" ... ]]

 

That quote is from the “Prolegomena” of 1906. Your next quote is not from “a 
year earlier” but from 40 years earlier. The Prolegomena clearly does not use 
the word “Categories” in reference to Firstness, Secondness and Thirdness, but 
in reference to classes of predicates (as opposed to subjects). Although Peirce 
calls those classes “Categories” in the 1906 Prolegomena, they do not map onto 
his usual categorial triad, nor vice versa. Also note that in 1866-7 Peirce was 
not yet using the term “index” for the kind of sign that is capable of genuine 
denotation; but in 1906, “indices” has its precise semiotic meaning in Peirce, 
which your analysis does not reflect.

 

Gary F.

 

From: C S Peirce discussion list [mailto:PEIRCE-L@LISTSERV.IUPUI.EDU] On Behalf 
Of Benjamin Udell
Sent: March-12-12 9:28 AM
To: PEIRCE-L@LISTSERV.IUPUI.EDU
Subject: Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction

 

Dear Steven,

Okay, 1866 instead of 1867. Indeed he regularly said that his categories are 
indecomposible into more basic elements; they _are_ his basic elements. That's 
why your indecomposability argument fails in the case of the 
Prolegomena-categories (which he says he prefers to call "Predicaments") - it's 
because there he does _not_ call them indecomposable; instead he says that they 
are "classes that, being enormously large, very promiscuous, and known but in 
small part, cannot be satisfactorily defined, and therefore can only be denoted 
by Indices."

Note that he did not say that the classes in question are indices of their 
elements or members. Instead he said that they are classes denotable (actually) 
only by indices and not by satisfactory definitions (since there actually are 
none). You imply that he could not have meant that because it would have led to 
an infinite regress. Yet it is in fact what he did say, and we are not entitled 
to silently revise him as if it were a mere typographical error. I'm not sure 
why you think it leads to infinite regress but, supposing that it does, it is 
not necessarily a problem for Peirce. Peirce believed in infinite series of 
signs in semiosis that has nevertheless a beginning and an end (at least by 
interruption) in time, since he was a synechist. In fact he based his synechism 
on the four incapacities, for example the incapacity for intuition, that is, 
the incapacity for a cognition devoid of inferential relation to a previous 
cognition. From final paragraph (CP 5.263) of "Questions concerning certain 
Faculties claimed for Man": 

So that it is not true that there must be a first. Explicate the logical 
difficulties of this paradox (they are identical with those of the Achilles) in 
whatever way you may. I am content with the result, as long as your principles 
are fully applied to the particular case of cognitions determining one another. 
Deny motion, if it seems proper to do so; only then deny the process of 
determination of one cognition by another. Say that instants and lines are 
fictions; only say, also, that states of cognition and judgments are fictions. 
The point here insisted on is not this or that logical solution of the 
difficulty, but merely that cognition arises by a _process_ of beginning, as 
any other change comes to pass.

In 1904 he still thought that the Four Incapacities lead to the establishment 
of synechism. From his brief intellectual autobiography*: "Upon these four 
propositions he based a doctrine of Synechism, or principle of the universality 
of the law of continuity, carrying with it a return to scholastic realism."

*(1904), Intellectual autobiography in draft letter L 107 (see the Robin 
Catalog  ) to Matthew Mattoon 
Curtis. Published 1983 in "A Brief Intellectual Autobiography by Charles 
Sanders Peirce" by Kenneth Laine Ketner in American Journal of Semiotics v. 2, 
nos. 1–2 (1983), 61–83. Some or all of it is in pp. 26–31 in Classical American 
Philosophy: Essential Readings and Interpretive Essays, John J. Stuhr, ed., 
Oxford University Press, USA, 1987. L 107 and MS 914 are in "Charles Sanders 
Peirce: Interdisciplinary Scientist" (first page at Oldenbourg 
 ) by Kenneth 
Laine Ketner in the 2009 Peirce collection Logic of Interdisciplinarity 
 .

As I said to Jon, I don't see why Peirce would refuse to call his own 
categori

Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction

[Jon Awbrey]

Peircers,

I retreated to my easy chair by the fireside with a stack of CPs, EPs, and NEMs
to refresh my memory of lost times in fond tomes, and it may take me a while to
bring those researches to any sort of satisfying, if provisional conclusion ...

Going by the lights of Peirce's detailed analysis of the question in the section
on Objective Logic, I think it is clear that Peirce does not simply identify the
Modes of Being and the Categories of Appearance, but treats them from the outset
as independent orders.  The first question is the more fundamental, 
metaphysical,
ontological matter and the second is a question of phenomenology, or 
appearances.

At the end of the section, Peirce brings the two orders into harmony with each 
other,
and that is very pleasing.  But arguments like that are a bit too reminiscent 
of the
old Cartesian prestidigitation, where he shows us the empty top hat at the 
beginning
of the trick, and then proceeds, quite rationally, to pull every last one of 
his own
pet rabbits out of it at the end.  Still, we cannot help but applaud the 
performance.

Voila!

Jon

GF = Gary Fuhrman
JA = Jon Awbrey

GF: I've been reading the section of the Minute Logic  that you've been posting
bits of (i don't think i've read it before) and i'm looking forward to your
way of connecting it to the category of categories ... if that's what you're
doing ... but i agree with Gary R. and Ben that it would be easier to follow
if you put it together into one message, or at least collect all the Peirce
quotes into one and your argument or comments into another one.

JA: Short on time till Monday, but I was able to redo the Objective Logic
excepts as a blog post, that may be easier to read all in one piece:

JA: 
http://inquiryintoinquiry.com/2012/03/09/c-s-peirce-%E2%80%A2-objective-logic/

--

academia: http://independent.academia.edu/JonAwbrey
inquiry list: http://stderr.org/pipermail/inquiry/
mwb: http://www.mywikibiz.com/Directory:Jon_Awbrey
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
word press blog 1: http://jonawbrey.wordpress.com/
word press blog 2: http://inquiryintoinquiry.com/

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Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction

[Benjamin Udell]

Dear Steven,

Okay, 1866 instead of 1867. Indeed he regularly said that his categories are 
indecomposible into more basic elements; they _are_ his basic elements. That's 
why your indecomposability argument fails in the case of the 
Prolegomena-categories (which he says he prefers to call "Predicaments") - it's 
because there he does _not_ call them indecomposable; instead he says that they 
are "classes that, being enormously large, very promiscuous, and known but in 
small part, cannot be satisfactorily defined, and therefore can only be denoted 
by Indices."

Note that he did not say that the classes in question are indices of their 
elements or members. Instead he said that they are classes denotable (actually) 
only by indices and not by satisfactory definitions (since there actually are 
none). You imply that he could not have meant that because it would have led to 
an infinite regress. Yet it is in fact what he did say, and we are not entitled 
to silently revise him as if it were a mere typographical error. I'm not sure 
why you think it leads to infinite regress but, supposing that it does, it is 
not necessarily a problem for Peirce. Peirce believed in infinite series of 
signs in semiosis that has nevertheless a beginning and an end (at least by 
interruption) in time, since he was a synechist. In fact he based his synechism 
on the four incapacities, for example the incapacity for intuition, that is, 
the incapacity for a cognition devoid of inferential relation to a previous 
cognition. From final paragraph (CP 5.263) of "Questions concerning certain 
Faculties claimed for Man": 

  So that it is not true that there must be a first. Explicate the logical 
difficulties of this paradox (they are identical with those of the Achilles) in 
whatever way you may. I am content with the result, as long as your principles 
are fully applied to the particular case of cognitions determining one another. 
Deny motion, if it seems proper to do so; only then deny the process of 
determination of one cognition by another. Say that instants and lines are 
fictions; only say, also, that states of cognition and judgments are fictions. 
The point here insisted on is not this or that logical solution of the 
difficulty, but merely that cognition arises by a _process_ of beginning, as 
any other change comes to pass.
In 1904 he still thought that the Four Incapacities lead to the establishment 
of synechism. From his brief intellectual autobiography*: "Upon these four 
propositions he based a doctrine of Synechism, or principle of the universality 
of the law of continuity, carrying with it a return to scholastic realism."

*(1904), Intellectual autobiography in draft letter L 107 (see the Robin 
Catalog) to Matthew Mattoon Curtis. Published 1983 in "A Brief Intellectual 
Autobiography by Charles Sanders Peirce" by Kenneth Laine Ketner in American 
Journal of Semiotics v. 2, nos. 1-2 (1983), 61-83. Some or all of it is in pp. 
26-31 in Classical American Philosophy: Essential Readings and Interpretive 
Essays, John J. Stuhr, ed., Oxford University Press, USA, 1987. L 107 and MS 
914 are in "Charles Sanders Peirce: Interdisciplinary Scientist" (first page at 
Oldenbourg) by Kenneth Laine Ketner in the 2009 Peirce collection Logic of 
Interdisciplinarity.

As I said to Jon, I don't see why Peirce would refuse to call his own 
categories predicate of predicates, and maybe indeed he wouldn't refuse. I 
agree with Jon that "There is nothing very exotic about predicates of 
predicates." But it doesn't follow that the Prolegomena-categories are Peirce's 
own, and the other reasons given above, and below in my previous post, stand 
against such a consequence.

Best, Ben

- Original Message - 
From: "Steven Ericsson-Zenith" 
To: PEIRCE-L@LISTSERV.IUPUI.EDU 
Sent: Sunday, March 11, 2012 11:43 PM 
Subject: Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction

Dear Ben,

There is no inconsistency as I see it, though I may not have stated the case 
clearly enough. In the first I said Peirce is not referring to his categories 
AS "predicates of predicates," not that he is not referring to his categories.

As index I am referring to the category itself, not its elements. A category 
stands apart from the elements that it may select by virtue of its properties. 
Apprehended, denoted, the category is indexed; 1st, 2nd, 3rd. You object to my 
saying that a category IS an index, by which I mean that it has the properties 
of an index. You appear to suggest that indices has another level of being, 
that will lead to an infinite recurse. 

Again:

"... of superior importance in Logic is the use of Indices to denote Categories 
and Universes, which are classes that, being enormously large, very 
promiscuous, and known but in small part, cannot be satisfactorily defined, and 
therefore can only be denoted by Indices."

A year earlier, in 1866, Peirce wrote "On A Method Of Searching For The 
Categories" in which

Re: [peirce-l] Proemial: On The Origin Of Experience

[John Collier]

 
 
Professor John Collier  
Philosophy, University of KwaZulu-Natal
Durban 4041 South Africa
T: +27 (31) 260 3248 / 260 2292
F: +27 (31) 260 3031
email: colli...@ukzn.ac.za>>> On 2012/03/06 at 11:03 PM, in message 
<4a39e6c5-939f-49ba-bc6b-8af976028...@iase.us>, Steven Ericsson-Zenith 
 wrote:


I'm not sure I would say that the Mars lander computational analysis of data is 
"interpretation." It seems to me to be a further representation, although one 
filtered by a machine imbued with our intelligence. Interpretation would be the 
thing done by scientists on earth.

As a former planetary scientist, I would agree in general with this, but I also 
experienced new data that pretty much implied directly (along with other 
well-known principles) that lunar differentiation had occurred. (Even then, 
scientists had to interpret the results, but they were clear as crystal 
relative to the question.) I relied on much less direct data (gravity evidence 
and some general principles of physics and geochemistry) to argue for the same 
conclusion. My potential paper was scooped, and I hadn't even graduated yet. 
Both Harvard and MIT people in the field found my paper "very interesting" but 
lost complete interest when I was retrospectively scooped by firmer evidence. 
The moral is that nothing in science beats direct evidence, even the most 
appealing hypothesis. Nonetheless, your book sound interesting.
 
Regards,
John

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