{ Information = Comprehension × Extension } • Selection 2

Posted on May 19, 2016 by Jon Awbrey

For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.

In the first place there are likenesses or copies — such as statues, pictures, emblems, hieroglyphics, and the like.  Such representations stand for their objects only so far as they have an actual resemblance to them — that is agree with them in some characters.  The peculiarity of such representations is that they do not determine their objects — they stand for anything more or less;  for they stand for whatever they resemble and they resemble everything more or less.

The second kind of representations are such as are set up by a convention of men or a decree of God.  Such are tallies, proper names, &c.  The peculiarity of these conventional signs is that they represent no character of their objects.

Likenesses denote nothing in particular;  conventional signs connote nothing in particular.

The third and last kind of representations are symbols or general representations.  They connote attributes and so connote them as to determine what they denote.  To this class belong all words and all conceptions.  Most combinations of words are also symbols.  A proposition, an argument, even a whole book may be, and should be, a single symbol.

(Peirce 1866, pp. 467–468)

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

Posted in Abduction, Belief Fixation, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , | 13 Comments

{ Information = Comprehension × Extension } • Selection 1

Posted on May 19, 2016 by Jon Awbrey

Let us now return to the information.  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.  For instance, you and I are men because we possess those attributes — having two legs, being rational, &c. — which make up the comprehension of man.  Every addition to the comprehension of a term lessens its extension up to a certain point, after that further additions increase the information instead.

Thus, let us commence with the term colour;  add to the comprehension of this term, that of red.  Red colour has considerably less extension than colour;  add to this the comprehension of dark;  dark red colour has still less [extension].  Add to this the comprehension of non-blue — non-blue dark red colour has the same extension as dark red colour, so that the non-blue here performs a work of supererogation;  it tells us that no dark red colour is blue, but does none of the proper business of connotation, that of diminishing the extension at all.  Thus information measures the superfluous comprehension.  And, hence, whenever we make a symbol to express any thing or any attribute we cannot make it so empty that it shall have no superfluous comprehension.

I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of information.

(Peirce 1866, p. 467)

Reference

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

Resources

Posted in Abduction, Belief Fixation, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , | 16 Comments

{ Information = Comprehension × Extension }

Posted on May 18, 2016 by Jon Awbrey

Another angle from which to approach the incidence of signs and inquiry is by way of Charles S. Peirce’s “laws of information” and the corresponding theory of information he developed from the time of his lectures on the “Logic of Science” at Harvard University (1865) and the Lowell Institute (1866).

When it comes to the supposed reciprocity between extensions and intensions, Peirce, of course, has another idea, and I would say a better idea, partly because it forms the occasion for him to bring in his new-fangled notion of “information” to mediate the otherwise static dualism between the other two.  The development of this novel idea brings Peirce to enunciate the formula:

\mathrm{Information} = \mathrm{Comprehension} \times \mathrm{Extension}

But comprehending what in the world that might mean is a much longer story, the end of which your present teller has yet to reach.  So, this time around, I will take up the story near the end of the beginning of Peirce’s own telling of it, for no better reason than that’s where I myself initially came in, or, at least, where it all started making any kind of sense to me.  And from this point we will find it easy enough to flash both backward and forward, to and fro, as the occasions arise for doing so.

Readings

Posted in Abduction, Belief Fixation, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , | 17 Comments

Peirce’s Categories • 7

Posted on May 13, 2016 by Jon Awbrey

Re: Peirce List Discussion • Ben Udell

The week before last my home office got tossed like a salad into the middle of our bedroom floor while workmen worked on various things that needed re-working.  There’s probably a metaphor of brute secondness there, I don’t know.  One of the unintended but beneficial (in the long run) side-effects of all that uproar in the Awbrey household is that books and notes and papers at the bottom of their respective categories of stacks all got flipped to the tops of their heaps with the causal consequence that I am now busy re-acquainting myself with the unfinished business of a decade or more ago.  So it may be a while before I can manage to get any sort of concentration again.

By way of interlusory comments then …

Earlier on I said a little about what I mean by charitable interpretation and just now I said a little more about how I understand critical interpretation.  Before you can agree or disagree with someone you have to figure out what he or she is intending to say.  That is the question we have to ask with respect to the corpus of Peirce’s texts.

Most readers of Peirce have their pet correspondences among any budget of threesomes he happens to mention and they all have their favorite snippets to support their choices.  In 50 years of following these animadversions I have seen no total agreement among the various parties, though some do agree on some.  My reading of Peirce over the years leaves me with no certainty on these scores and certainly nothing approaching the orders of axiomatic definitions and formal proofs that would privilege any one-to-one correspondences among the trios that might be fixed and unique in all contexts for all intents and purposes and times.  I find Peirce making suggestive correlations in various contexts of application and others in others.  But when he is casting the most critical reflection on the alignment of the moment I see him expressing a duly requisite doubt and then begging off with a conclusion more apology than logical proof.

My first ten years of reading Peirce were quite a struggle.  I came to college as a math and physics major.  I couldn’t say Peirce is wholly responsible for my wandering years through fields and majors as diverse as communication and computer science to psychology and philosophy, but my efforts to understand what he was saying are decidedly one of the main forces that drove me back to graduate school, first mathematics, then adding psychology again along a parallel track, then more computer science and systems engineering as I worked to program a theorem prover for his logical graphs and then broadened that into my long-running work on Inquiry Driven Systems.  But the way I read his scientific work stabilized fairly well after that first decade, and I know I have done little on the Peirce List over the last ten years but rehash what I said during the first five.

To be continued …

Resources

Posted in Abstraction, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatism, Relation Theory, Semiotics, Thirdness, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 8 Comments

Peirce’s Categories • 6

Posted on May 12, 2016 by Jon Awbrey

Re: Peirce List Discussion • Ben Udell

I read Peirce primarily for his insights into logic, mathematics, and science, which are considerable enough to occupy several lifetimes, and I read him the same way I read other thinkers in those areas.  Maybe some people read Peirce as Charles the Revelator, applying the principles of scriptural interpretation and chasing his tale around hermeneutic circles in hopes of cornering a sublime truth.  Scientific texts are read a different way.  There we have a line between two kinds of statements, those that serve as conjectures, heuristics, or suggestions and those that are proved (or proven).

To be continued …

Resources

Posted in Abstraction, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatism, Relation Theory, Semiotics, Thirdness, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 8 Comments

Peirce’s Categories • 5

Posted on May 12, 2016 by Jon Awbrey

Re: Peirce List Discussion • Ben Udell

For my part, I see a distinctive paradigm of thought and practice immanent in Peirce’s work and all I’ve been trying to do for many years now has been to nudge it a little further from immanence to implementation.  It is precisely the engagement with applications that brings these criticisms to the fore.

Applications to empirical situations force one to view sign relations as large collections of triples, you might even say clouds of triples.  A sign relation, in this extended sense, is more like the environment in which our discussion and thought takes place than any single focal triple of the all too fixed gaze.  Any attempt at significant application simply never gets off square one, or rather triangle one, if it fails to step back and take in the big picture of an extended sign relation.

Resources

Posted in Abstraction, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatism, Relation Theory, Semiotics, Thirdness, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | 8 Comments

Peirce’s Categories • 4

Posted on May 11, 2016 by Jon Awbrey

Re: Peirce List Discussions • (1) • (2) • (3) • (4) • (5)

Let me state a few principles that have guided me in my efforts to read and understand Peirce for the past fifty years.

There is a long-running strain of Peirce commentary that sees radical modifications in his thinking over the years.  I do not belong to that tradition.  I see more continuity than radical re-thinking in his thought through the years.  But seeing things that way is due to a certain perspective.

I apply the same principles of charitable and critical interpretation to Peirce that I do to any other writer.

Charity entails a search for a consistent interpretation if one is possible at all.  Charity goes only so far with some writers and some styles of writing;  contradictions of a sort that cannot be glossed over develop almost immediately and about all one can do is read things emotively or impressionistically after that point.  In Peirce’s case I almost always find that a little extra charity repays itself in the long run.  That is not to deny the apparent inconsistencies that we find in Peirce’s work, taken whole cloth, as careful readers have noted many, but it does imply a particular strategy for dealing with the wrinkles that do appear.

To be continued …

Resources

Posted in Abstraction, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Peirce's Categories, Phenomenology, Philosophy, Pragmatism, Relation Theory, Semiotics, Thirdness, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , , | 8 Comments

Definition and Determination • 14

Posted on May 5, 2016 by Jon Awbrey

Re: Peirce List (1) (2)

I’ve been trying to sort through the explosion of topics and tangents that have arisen over the past month — disruptions in my actual and virtual office spaces have made it hard for me to keep track — I still have the general impression that many things once relatively well understood among Peircean pilgrims are no longer as commonly consensual.

I can almost hear Galadriel’s prologue echoing in my ears …

I’m still not sure how we jumped from relatively simple questions about the meaning of correspondence and determination in Peirce’s definition of triadic sign relations to all the other issues that flared up, and I’m not seeing anything in the late Peirce manuscripts he didn’t say many times before, perhaps less subtly but rather more concretely and thus more clearly for all that.

By way of trying to get organized again, or maybe just recapping my part before moving on, here are the blog editions of my main comments on these subjects over the past month.

  • Definition and Determination
    (11) • (12) • (13)
  • Readings On Determination
    (0) • (1)
  • Readings On Determination • Discussion
    (1) • (2) • (3)
  • Types of Reasoning in C.S. Peirce and Aristotle
    (1) • (2)
  • Abduction, Deduction, Induction, Analogy, Inquiry
    (23) • (24)

cc: Inquiry List • Peirce List (1) (2) (3) (4) (5)

Posted in C.S. Peirce, Comprehension, Constraint, Definition, Determination, Extension, Form, Indication, Information = Comprehension × Extension, Inquiry, Intension, Logic, Logic of Science, Mathematics, Peirce, Semiotics, Structure | Tagged , , , , , , , , , , , , , , , , | 5 Comments

Abduction, Deduction, Induction, Analogy, Inquiry • 24

Posted on May 5, 2016 by Jon Awbrey

Re: Peirce List • Jon Alan Schmidt

Peirce’s categories are best viewed as categories of relations.  To a first approximation, firstness, secondness, thirdness are simply what all monadic, dyadic, triadic relations, respectively, have in common.  At a second approximation, we may take up the issues of generic versus degenerate cases of 1-, 2-, 3-adicity, but it is critical to address the first approximation first before attempting to deal with the second.

In that light, thirdness is a global property of the whole triadic relation in question and it is a category error to attribute thirdness to any one relational domain or role, much less any of the elements belonging to those domains or filling the roles of the triadic relation.

As it happens, we often approach a complex relation by picking one of its elements, that is, a single tuple as exemplary of the whole set of tuples making up the relation, and then we take up the components of that tuple in one convenient order or another.  That method lends itself to the impression k-ness abides in the k-th component we happen to take up, but that impression begs the question of whether that order is a property of the relation itself or merely an artifact of our choice.

Failing to examine that question puts us at risk for a type of error I’ve previously dubbed the Fallacy Of Misplaced Abstraction (FOMA).  As I see it, there is a lot of that going on in the present discussion, arising from a tendency to assign Peircean categories to everything in sight, despite the fact that Peirce’s categories apply only to certain levels of structure.

cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

Posted in Abduction, Analogy, Aristotle, Artificial Intelligence, C.S. Peirce, Deduction, Induction, Inquiry, Inquiry Driven Systems, Intelligent Systems Engineering, Logic, Mental Models, Peirce, Scientific Method, Semiotics, Systems | Tagged , , , , , , , , , , , , , , , | 8 Comments

Abduction, Deduction, Induction, Analogy, Inquiry • 23

Posted on May 1, 2016 by Jon Awbrey

Re: Peirce List • Benjamin Udell • Jon Awbrey • Gary Richmond (1) (2) • Jon Alan Schmidt

These days it takes me a web search to discover what I was thinking and writing the month before.  I went looking for the passage in McCulloch where he uses Case, Fact, Rule and it led me through hill and dale and back to my own post here on March 11.  See McCulloch’s paper with the Bardic title “What’s In The Brain That Ink May Character?”, but watch out for a few typos in the online copy.

Abductive reasoning was one of the first topics that pulled me into the briar patch of AI many years ago.  There were early papers by Harry Pople that I recall, partly because they came up again when I was working at the University of Texas Medical Branch and folks there were just beginning to explore computer-aided medical diagnosis.  At any rate, my search did turn up a copy of one of Pople’s early papers that references both Peirce and McCulloch.

  • Pople, H., and Werner, G. (1972), “An Information Processing Approach to Theory Formation in Biomedical Research”.  Online.

There have been four or five distinct waves of AI literature on so-called “abduction” since that time but almost all of it takes off from the same syntactic over-simplification of Peirce’s more complex model of abductive reasoning as it performs its role within the process of inquiry, so I have largely lost interest in that departure from the fons et origo.

References

  • McCulloch, W.S. (1964), “What’s in the Brain That Ink May Character?”, International Congress for Logic, Methodology, and Philosophy of Science, Israel, August 28, 1964.  Reprinted in Embodiments of Mind, pp. 387–397.
  • McCulloch, W.S. (1965), Embodiments of Mind, MIT Press, Cambridge, MA.
  • Pople, H., and Werner, G. (1972), “An Information Processing Approach to Theory Formation in Biomedical Research”, International Workshop on Managing Requirements Knowledge, December 5–7, 1972, Anaheim, CA, American Federation of Information Processing Societies (AFIPS), 1972 Proceedings of the Spring Joint Computer Conference, pp. 1125–1138.

cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Posted in Abduction, Analogy, Aristotle, Artificial Intelligence, C.S. Peirce, Deduction, Induction, Inquiry, Inquiry Driven Systems, Intelligent Systems Engineering, Logic, Mental Models, Peirce, Scientific Method, Semiotics, Systems | Tagged , , , , , , , , , , , , , , , | 8 Comments