Re: [PEIRCE-L] Re: Stjernfelt Revisited • Propositions and Information

2017-06-29 Thread John F Sowa

On 6/29/2017 2:34 PM, Gary Richmond wrote:
Jon--The Peirce list is a forum, not a kind of personal 'storage' site. 
Gary R


I second that motion.

My email handler (Thunderbird) has a place to store "drafts".
Other email handlers I've used or seen also have such storage
sites.  I suggest them as places to store "email in progress".

John

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Re: [PEIRCE-L] Re: Stjernfelt Revisited • Propositions and Information

2017-06-29 Thread Gary Richmond
Jon--The Peirce list is a forum, not a kind of personal 'storage' site.
Gary R

[image: Gary Richmond]

*Gary Richmond*
*Philosophy and Critical Thinking*
*Communication Studies*
*LaGuardia College of the City University of New York*
*C 745*
*718 482-5690*

On Thu, Jun 29, 2017 at 2:25 PM, Jon Awbrey  wrote:

> Storing this here for later discussion ...
>
> On 6/28/2017 11:53 AM, John F Sowa wrote:
> > Jon,
> >
> > That's an important topic to explore:
> >
> > JA
> >> we can take up the issue of propositions in more detail
> >> as it arises in the relevant context.
> >
> > For a good analysis of the issues, I recommend the following book:
> > Stjernfelt, Frederik (2014) Natural Propositions: The Actuality
> > of Peirce’s Doctrine of Dicisigns, Boston: Docent Press.
> >
> > I wrote a 5-page article on propositions from a Peircean perspective:
> > http://www.jfsowa.com/logic/proposit.pdf
> >
> > That article is based on Peirce's notion of equivalence (CP 5.569):
> >> A sign is only a sign in actu by virtue of its receiving an
> >> interpretation, that is, by virtue of its determining another sign
> >> of the same object. This is as true of mental judgments as it is of
> >> external signs. To say that a proposition is true is to say that
> >> every interpretation of it is true. Two propositions are equivalent
> >> when either might have been an interpretant of the other. This
> >> equivalence, like others, is by an act of abstraction (in the sense
> >> in which forming an abstract noun is abstraction) conceived as identity.
> >>
> >> And we speak of believing in a proposition, having in mind an entire
> >> collection of equivalent propositions with their partial interpretants.
> >> Thus, two persons are said to have the same proposition in mind. The
> >> interpretant of a proposition is itself a proposition. Any necessary
> >> inference from a proposition is an interpretant of it.
> >>
> >> When we speak of truth and falsity, we refer to the possibility of the
> >> proposition being refuted; and this refutation (roughly speaking) takes
> >> place in but one way. Namely, an interpretant of the proposition would,
> >> if believed, produce the expectation of a certain description of percept
> >> on a certain occasion. The occasion arrives: the percept forced upon
> >> us is different. This constitutes the falsity of every proposition of
> >> which the disappointing prediction was the interpretant. Thus, a false
> >> proposition is a proposition of which some interpretant represents
> >> that, on an occasion which it indicates, a percept will have a certain
> >> character, while the immediate perceptual judgment on that occasion is
> >> that the percept has not that character.
> >>
> >> A true proposition is a proposition belief in which would never lead
> >> to such disappointment so long as the proposition is not understood
> >> otherwise than it was intended.
> >
> > In the article, I formalize Peirce's notion of equivalence in terms
> > of *meaning-preserving translations* (MPTs), which specify a class
> > of equivalent sentences in some language or languages.  It's easy to
> > define MPTs for formal logics, but much harder for natural languages.
> >
> > John
> >
>
> --
>
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>
>
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[PEIRCE-L] Re: Stjernfelt Revisited • Propositions and Information

2017-06-29 Thread Jon Awbrey

Storing this here for later discussion ...

On 6/28/2017 11:53 AM, John F Sowa wrote:
> Jon,
>
> That's an important topic to explore:
>
> JA
>> we can take up the issue of propositions in more detail
>> as it arises in the relevant context.
>
> For a good analysis of the issues, I recommend the following book:
> Stjernfelt, Frederik (2014) Natural Propositions: The Actuality
> of Peirce’s Doctrine of Dicisigns, Boston: Docent Press.
>
> I wrote a 5-page article on propositions from a Peircean perspective:
> http://www.jfsowa.com/logic/proposit.pdf
>
> That article is based on Peirce's notion of equivalence (CP 5.569):
>> A sign is only a sign in actu by virtue of its receiving an
>> interpretation, that is, by virtue of its determining another sign
>> of the same object. This is as true of mental judgments as it is of
>> external signs. To say that a proposition is true is to say that
>> every interpretation of it is true. Two propositions are equivalent
>> when either might have been an interpretant of the other. This
>> equivalence, like others, is by an act of abstraction (in the sense
>> in which forming an abstract noun is abstraction) conceived as identity.
>>
>> And we speak of believing in a proposition, having in mind an entire
>> collection of equivalent propositions with their partial interpretants.
>> Thus, two persons are said to have the same proposition in mind. The
>> interpretant of a proposition is itself a proposition. Any necessary
>> inference from a proposition is an interpretant of it.
>>
>> When we speak of truth and falsity, we refer to the possibility of the
>> proposition being refuted; and this refutation (roughly speaking) takes
>> place in but one way. Namely, an interpretant of the proposition would,
>> if believed, produce the expectation of a certain description of percept
>> on a certain occasion. The occasion arrives: the percept forced upon
>> us is different. This constitutes the falsity of every proposition of
>> which the disappointing prediction was the interpretant. Thus, a false
>> proposition is a proposition of which some interpretant represents
>> that, on an occasion which it indicates, a percept will have a certain
>> character, while the immediate perceptual judgment on that occasion is
>> that the percept has not that character.
>>
>> A true proposition is a proposition belief in which would never lead
>> to such disappointment so long as the proposition is not understood
>> otherwise than it was intended.
>
> In the article, I formalize Peirce's notion of equivalence in terms
> of *meaning-preserving translations* (MPTs), which specify a class
> of equivalent sentences in some language or languages.  It's easy to
> define MPTs for formal logics, but much harder for natural languages.
>
> John
>

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[PEIRCE-L] Re: Peirce's own definition of 'information'

2017-06-29 Thread John F Sowa

Jon A, Charles Pyle, and John C.,

The main point I was trying to make is that the term 'proposition'
is basic, and that information is knowledge (propositional content)
that is being communicated or derived in some way.

Jon

there is simply no reason why we should not take seriously the
lectures where Peirce develops in breadth, depth, and length his
conception of information


I take Peirce's Lowell lectures very seriously.  But in order
to understand what Peirce was trying to say, his Century definition
should also be taken seriously:

CSP, from the Century Dictionary for 'information':

3. Knowledge inculcated or derived; known facts or principles,
however communicated or acquired, as from reading, instruction,
or observation 


That knowledge may be stated or expressed in propositions or
in some form from which propositions can be derived.  The static
view emphasizes the propositions.  Information is the dynamic
process of communicating or transmitting those propositions.

Jon

I don't think anyone can fairly encounter his definition of
information as “superfluous comprehension” without being
downright shocked at its novelty.


That is an interesting comment, but it is *not* a definition of
information.  His comment notes that sometimes the propositions
that are communicated will narrow the extension, and sometimes
they will have no effect on the extension.  Therefore, the
property of being superfluous cannot be part of the definition.

CSP from Writings, p. 1.467

The information of a term is the measure of its superfluous
comprehension.  That is to say that the proper office of the
comprehension is to determine the extension of the term...


When those additional propositions have no effect on extension,
Peirce said that they are "superfluous" to the task of narrowing
the extension.  But that same information might be useful for
some other task, such as providing a more detailed description.

This example shows how a definition of the word 'information' --
either from the OED or by Peirce himself -- can clarify what he meant.
The word 'superfluous' is part of the comment, not of the definition.

C Pyle

I have always been concerned about the implications of false
information for the definition of information.


Good question.  If you define information as the communication of
raw bits (as Shannon did), the question of meaning is irrelevant.

If you define it as the communication of propositions (which may be
true or false) then meaning is required, but truth is irrelevant.

If you define it as the communication of knowledge (as Peirce did),
then meaning and truth are both required.

John C

On the nature of information flow (transfer), I recommend the book
by that name by Barwise and Seligman. It is far superior to anything
written by or about Shannon...


Yes.  And note that B & P emphasize the transmission.

John

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RE: [PEIRCE-L] Peirce's own definition of 'information'

2017-06-29 Thread John Collier
In the technical sense (algorithmic information theory, Shannon, various 
others), information is understood syntactically only, so there is no content 
involved. Content is required for truth or falsity. So the technical notion of 
information has nothing to say about truth or falsity of the information. 
Information is either transferred, or it is not.

On the nature of information flow (transfer), I recommend the book by that name 
by Barwise and Seligman. It is far superior to anything written by or about 
Shannon, but it is based on pre-Shannon work on networks by electrical 
engineers in the 1930s. It is a difficult book, but you can find the basics 
summarised in several of my articles on my web page.

John Collier
Emeritus Professor and Senior Research Associate
Philosophy, University of KwaZulu-Natal
http://web.ncf.ca/collier

From: Charles Pyle [mailto:charlesp...@comcast.net]
Sent: Thursday, 29 June 2017 4:35 PM
To: peirce-l@list.iupui.edu
Subject: Re: [PEIRCE-L] Peirce's own definition of 'information'


I have always been concerned about the implications of false information for 
the definition of information. Is false information information? Is false 
knowledge knowledge? I should think the answer must certainly be "No" for 
knowledge, because to know is a factive verb, meaning that it presupposes the 
truth of its object. I believe in common usage of 'information' in the 
technical sense, as in information theory, false information would be 
information even if it was false,  but information in the ordinary sense of the 
word would not be information if it is false.



If, as I have argued, all signs are of a duplicitous nature, then this would be 
a moot question, or at least a very different question.

On June 29, 2017 at 7:59 AM John F Sowa 
> wrote:

Jon A, Jeff D, and Gary F,

JA

Why don't we put this on hold for later discussion?

I was about to send the following when your note appeared in
my inbox. It should be sufficient for the word 'information',
but we can discuss other issues later.

JD

I take the following passage to indicate that Peirce changed his use of
"depth" and "breadth" in some respects some time between 1867 and 1896.
The change was a broadening of the use of both terms.

GF

What Peirce wrote in 1893 is that he had broadened the application
of the terms, i.e. the breadth of the propositions involving them.
That does not mean that their depth, or “signification” as Peirce
often called it, changed in any way;

I agree. One example I use is the broadening of the word 'number'
from integers to rational numbers to irrational numbers to complex
numbers to quaternions... That broadens the application of the word,
but it does not make the definitions for its earlier uses obsolete.

For any particular application, the definition can be narrowed
by adding an adjective, such as real, complex, hypercomplex...

JA

BTW, is it really necessary to point out once again that the job
of a lexicographer presenting a survey of significant usages in
common or technical is very different from the role of a philosopher
expounding his or her own conception?

Many of Peirce's definitions for the Century Dictionary or Baldwin's
dictionary include short philosophical essays. They are as significant
for his Opera Omnia as any other publications.

And note his Ethics of Terminology. From EP 2.265:

The first rule of good taste in writing is to use words whose
meanings will not be misunderstood

Implication: For a common word such as 'information', a dictionary
that cites dates for the word senses, such as the OED, would be
sufficient to determine what Peirce had intended. But when he wrote
the definition himself, that's even better: I'm sure he would not
use a word in a sense that was inconsistent with his own definition.

John

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Re: [PEIRCE-L] Peirce's own definition of 'information'

2017-06-29 Thread Charles Pyle
I have always been concerned about the implications of false information for 
the definition of information. Is false information information? Is false 
knowledge knowledge? I should think the answer must certainly be "No" for 
knowledge, because to know is a factive verb, meaning that it presupposes the 
truth of its object. I believe in common usage of 'information' in the 
technical sense, as in information theory, false information would be 
information even if it was false,  but information in the ordinary sense of the 
word would not be information if it is false. 


If, as I have argued, all signs are of a duplicitous nature, then this would be 
a moot question, or at least a very different question. 

> 
> On June 29, 2017 at 7:59 AM John F Sowa  wrote:
> 
> Jon A, Jeff D, and Gary F,
> 
> JA
> 
> > > 
> > Why don't we put this on hold for later discussion?
> > 
> > > 
> I was about to send the following when your note appeared in
> my inbox. It should be sufficient for the word 'information',
> but we can discuss other issues later.
> 
> JD
> 
> > > 
> > I take the following passage to indicate that Peirce changed his 
> > use of
> > "depth" and "breadth" in some respects some time between 1867 and 
> > 1896.
> > The change was a broadening of the use of both terms.
> > 
> > > 
> GF
> 
> > > 
> > What Peirce wrote in 1893 is that he had broadened the application
> > of the terms, i.e. the breadth of the propositions involving them.
> > That does not mean that their depth, or “signification” as Peirce
> > often called it, changed in any way;
> > 
> > > 
> I agree. One example I use is the broadening of the word 'number'
> from integers to rational numbers to irrational numbers to complex
> numbers to quaternions... That broadens the application of the word,
> but it does not make the definitions for its earlier uses obsolete.
> 
> For any particular application, the definition can be narrowed
> by adding an adjective, such as real, complex, hypercomplex...
> 
> JA
> 
> > > 
> > BTW, is it really necessary to point out once again that the job
> > of a lexicographer presenting a survey of significant usages in
> > common or technical is very different from the role of a philosopher
> > expounding his or her own conception?
> > 
> > > 
> Many of Peirce's definitions for the Century Dictionary or Baldwin's
> dictionary include short philosophical essays. They are as significant
> for his Opera Omnia as any other publications.
> 
> And note his Ethics of Terminology. From EP 2.265:
> 
> > > 
> > The first rule of good taste in writing is to use words whose
> > meanings will not be misunderstood
> > 
> > > 
> Implication: For a common word such as 'information', a dictionary
> that cites dates for the word senses, such as the OED, would be
> sufficient to determine what Peirce had intended. But when he wrote
> the definition himself, that's even better: I'm sure he would not
> use a word in a sense that was inconsistent with his own definition.
> 
> John
> 
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[PEIRCE-L] Re: Peirce's own definition of 'information'

2017-06-29 Thread Jon Awbrey

John, List ...

Yes, those are all the reasons that all of us who
have been reading Peirce for 50 or 40 or 30 years
have always read the Baldwin and Century Dictionary
entries that Peirce had a hand in with avid interest
for what we can glean of his own point of view.  And
you remind us of all the sensible considerations that
every sensible reader eventually observes every time
this comes up.  So we're all copacetic on that score.

Now I don't imagine those dictionary editors greeted Peirce's
contributions with quite the same screams of NOR! and NPOV!
that the WikiPeanut Gallery of “Editors” are duty bound to
hue and cry against any scent of original perspective or
creative thought, but still, the job of a lexicographer
will always have its constraints, in any place or time.
In view of all that, then, there is simply no reason
why we should not take seriously the lectures where
Peirce develops in breadth, depth, and length his
conception of information, however concise that
may fail to be.

What interests me so much about Peirce's first legislation of
the “laws of information” in his 1865-66 “Logic of Science” is
that the primal twins of Semiotics and Inquiry nestle so closely
in their first nest that we can see their kinship far better and
more easily than ever again.  (I am cautiously optimistic their
further development won't go the same way it did for Rome.)

More than that, whatever disclaimers Peirce may issue about his own
originality, I don't think anyone can fairly encounter his definition
of information as “superfluous comprehension” without being downright
shocked at its novelty.

Reference
https://inquiryintoinquiry.com/2016/05/19/information-comprehension-x-extension-%e2%80%a2-selection-1/

Regards,

Jon

On 6/29/2017 7:59 AM, John F Sowa wrote:

Jon A, Jeff D, and Gary F,

JA

Why don't we put this on hold for later discussion?


I was about to send the following when your note appeared in
my inbox.  It should be sufficient for the word 'information',
but we can discuss other issues later.

JD

I take the following passage to indicate that Peirce changed his use of
"depth" and "breadth" in some respects some time between 1867 and 1896.
The change was a broadening of the use of both terms.


GF

What Peirce wrote in 1893 is that he had broadened the application
of the terms, i.e. the breadth of the propositions involving them.
That does not mean that their depth, or “signification” as Peirce
often called it, changed in any way;


I agree.  One example I use is the broadening of the word 'number'
from integers to rational numbers to irrational numbers to complex
numbers to quaternions...  That broadens the application of the word,
but it does not make the definitions for its earlier uses obsolete.

For any particular application, the definition can be narrowed
by adding an adjective, such as real, complex, hypercomplex...

JA

BTW, is it really necessary to point out once again that the job
of a lexicographer presenting a survey of significant usages in
common or technical [use] is very different from the role of
a philosopher expounding his or her own conception?


Many of Peirce's definitions for the Century Dictionary or Baldwin's
dictionary include short philosophical essays.  They are as significant
for his Opera Omnia as any other publications.

And note his Ethics of Terminology.  From EP 2.265:

The first rule of good taste in writing is to use words whose
meanings will not be misunderstood


Implication:  For a common word such as 'information', a dictionary
that cites dates for the word senses, such as the OED, would be
sufficient to determine what Peirce had intended.  But when he wrote
the definition himself, that's even better:  I'm sure he would not
use a word in a sense that was inconsistent with his own definition.

John



--

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Re: [PEIRCE-L] Peirce's own definition of 'information'

2017-06-29 Thread John F Sowa

Jon A, Jeff D, and Gary F,

JA

Why don't we put this on hold for later discussion?


I was about to send the following when your note appeared in
my inbox.  It should be sufficient for the word 'information',
but we can discuss other issues later.

JD

I take the following passage to indicate that Peirce changed his use of
"depth" and "breadth" in some respects some time between 1867 and 1896.
The change was a broadening of the use of both terms.


GF

What Peirce wrote in 1893 is that he had broadened the application
of the terms, i.e. the breadth of the propositions involving them.
That does not mean that their depth, or “signification” as Peirce
often called it, changed in any way;


I agree.  One example I use is the broadening of the word 'number'
from integers to rational numbers to irrational numbers to complex
numbers to quaternions...  That broadens the application of the word,
but it does not make the definitions for its earlier uses obsolete.

For any particular application, the definition can be narrowed
by adding an adjective, such as real, complex, hypercomplex...

JA

BTW, is it really necessary to point out once again that the job
of a lexicographer presenting a survey of significant usages in
common or technical is very different from the role of a philosopher
expounding his or her own conception?


Many of Peirce's definitions for the Century Dictionary or Baldwin's
dictionary include short philosophical essays.  They are as significant
for his Opera Omnia as any other publications.

And note his Ethics of Terminology.  From EP 2.265:

The first rule of good taste in writing is to use words whose
meanings will not be misunderstood


Implication:  For a common word such as 'information', a dictionary
that cites dates for the word senses, such as the OED, would be
sufficient to determine what Peirce had intended.  But when he wrote
the definition himself, that's even better:  I'm sure he would not
use a word in a sense that was inconsistent with his own definition.

John

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[PEIRCE-L] Re: { Information = Comprehension × Extension }

2017-06-29 Thread Jon Awbrey

John,

Why don't we put this on hold for later discussion?
What you say goes to the heart of a problem I saw in
Natural Propositions, whether it was Peirce's account
or Stjernfelt's analysis I did not have time to decide
as the schedule of the slow reading went too fast for me
to take it up on the List.  I marked the critical passages
and my copy of Natural Propositions is around here someplace
but I am trying to stay focused on the subject matter and the
set of problems that I introduced under the above subject line.

There are many issues here about cross-disciplinary communication,
the varieties of quasi-religious belief about the uses of words in
the whole proposition/sentence/statement complex, the various uses
Peirce and even sub-genius people use across contexts, disciplines,
historical time, and even within the same discussion.  But I think
it's best to hold the forte on that for now.

Regards,

Jon

On 6/28/2017 3:19 PM, John F Sowa wrote:> On 6/28/2017 1:44 PM, Jon Awbrey 
wrote:
>>
>> The short shrift for now is that neither Peirce nor I is talking about
>> propositions in the sense of dicisigns or dicent symbols at this juncture
>> but rather the simpler sorts of propositions that fall under the heading of
>> the Propositional Calculus in current usage, adequately and most felicitously
>> dealt with of course by means of Peirce's own Alpha Graphs.
>
> There is no difference.  A proposition (or dicisign or dicent symbol)
> is a proposition, no matter how it is used.  In propositional calculus
> or Peirce's Alpha, the letter p can represent any proposition or dicisign
> or dicent symbol of any kind.
>
>> The concept of information that comes up in this context is rather distinct.
>
> Yes.  Information is propositional content that is being communicated
> to someone in some way.  As an employee of AT, Claude Shannon focused
> on the transmission methods.  But he did not reject the fact that the
> people who use a telephone communicate propositional content.
>
> The terms 'comprehension' and 'extension' address the propositional content,
> not the method of communication.
>
> John
>

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[PEIRCE-L] Re: Stjernfelt Revisited (was some other topic)

2017-06-29 Thread Jon Awbrey

Peircers,

Storing this here for later discussion.

Regards,

Jon

On 6/28/2017 1:44 PM, Jon Awbrey wrote:
> John,
>
> Yes, I gave it a careful reading back when the List took it up:
>
> 
http://web.archive.org/web/20150116150400/http://comments.gmane.org/gmane.science.philosophy.peirce/13825
>
> I find some remnants of my comments here:
>
> https://inquiryintoinquiry.com/2014/10/12/semiotic-theory-of-information-3/
> https://inquiryintoinquiry.com/2014/10/13/semiotic-theory-of-information-4/
>
> 
https://inquiryintoinquiry.com/2014/08/24/c-s-peirce-%e2%80%a2-syllabus-%e2%80%a2-selection-1/
> 
https://inquiryintoinquiry.com/2014/10/04/c-s-peirce-%E2%80%A2-syllabus-%E2%80%A2-selection-2/
>
> I have in mind getting back to the issues raised by that reading someday
> but it would take me too far afield from my current focus to do that now.
>
> The short shrift for now is that neither Peirce nor I is/am talking about
> propositions in the sense of dicisigns or dicent symbols at this juncture
> but rather the simpler sorts of propositions that fall under the heading of
> the Propositional Calculus in current usage, adequately and most felicitously
> dealt with of course by means of Peirce's own Alpha Graphs.
>
> The concept of information that comes up in this context is rather distinct.
> To my way of thinking the earlier notion of information, however roughly cut,
> is superior in its basic principles, it being more realistic compared to the
> residual nominalism in the later concept, at least, as interpreted by others.
>
> Regards,
>
> Jon
>
> On 6/28/2017 11:53 AM, John F Sowa wrote:
>> Jon,
>>
>> That's an important topic to explore:
>>
>> JA
>>> we can take up the issue of propositions in more detail
>>> as it arises in the relevant context.
>>
>> For a good analysis of the issues, I recommend the following book:
>> Stjernfelt, Frederik (2014) Natural Propositions: The Actuality
>> of Peirce’s Doctrine of Dicisigns, Boston: Docent Press.
>>
>> I wrote a 5-page article on propositions from a Peircean perspective:
>> http://www.jfsowa.com/logic/proposit.pdf
>>
>> That article is based on Peirce's notion of equivalence (CP 5.569):
>>
>>> A sign is only a sign in actu by virtue of its receiving an
>>> interpretation, that is, by virtue of its determining another sign
>>> of the same object. This is as true of mental judgments as it is of
>>> external signs. To say that a proposition is true is to say that
>>> every interpretation of it is true. Two propositions are equivalent
>>> when either might have been an interpretant of the other. This
>>> equivalence, like others, is by an act of abstraction (in the sense
>>> in which forming an abstract noun is abstraction) conceived as identity.
>>>
>>> And we speak of believing in a proposition, having in mind an entire
>>> collection of equivalent propositions with their partial interpretants.
>>> Thus, two persons are said to have the same proposition in mind. The
>>> interpretant of a proposition is itself a proposition. Any necessary
>>> inference from a proposition is an interpretant of it.
>>>
>>> When we speak of truth and falsity, we refer to the possibility of the
>>> proposition being refuted; and this refutation (roughly speaking) takes
>>> place in but one way. Namely, an interpretant of the proposition would,
>>> if believed, produce the expectation of a certain description of percept
>>> on a certain occasion. The occasion arrives: the percept forced upon
>>> us is different. This constitutes the falsity of every proposition of
>>> which the disappointing prediction was the interpretant. Thus, a false
>>> proposition is a proposition of which some interpretant represents
>>> that, on an occasion which it indicates, a percept will have a certain
>>> character, while the immediate perceptual judgment on that occasion is
>>> that the percept has not that character.
>>>
>>> A true proposition is a proposition belief in which would never lead
>>> to such disappointment so long as the proposition is not understood
>>> otherwise than it was intended.
>>
>> In the article, I formalize Peirce's notion of equivalence in terms
>> of *meaning-preserving translations* (MPTs), which specify a class
>> of equivalent sentences in some language or languages.  It's easy to
>> define MPTs for formal logics, but much harder for natural languages.
>>
>> John
>>
>

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