{ Information = Comprehension × Extension } • Selection 4

Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol.  But suppose we know of no symbol for neat, swine, sheep, and deer except cloven-hoofed animals.  There is but one objection to substituting this for the disjunctive term;  it is that we should, then, say more than we have observed.  In short, it has a superfluous information.  But we have already seen that this is an objection which must always stand in the way of taking symbols.  If therefore we are to use symbols at all we must use them notwithstanding that.  Now all thinking is a process of symbolization, for the conceptions of the understanding are symbols in the strict sense.  Unless, therefore, we are to give up thinking altogeher we must admit the validity of induction.  But even to doubt is to think.  So we cannot give up thinking and the validity of induction must be admitted.

(Peirce 1866, p. 469)

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

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{ Information = Comprehension × Extension } • Selection 3

Yet there are combinations of words and combinations of conceptions which are not strictly speaking symbols.  These are of two kinds of which I will give you instances.  We have first cases like:

man and horse and kangaroo and whale,

and secondly, cases like:

spherical bright fragrant juicy tropical fruit.

The first of these terms has no comprehension which is adequate to the limitation of the extension.  In fact, men, horses, kangaroos, and whales have no attributes in common which are not possessed by the entire class of mammals.  For this reason, this disjunctive term, man and horse and kangaroo and whale, is of no use whatever.  For suppose it is the subject of a sentence;  suppose we know that men and horses and kangaroos and whales have some common character.  Since they have no common character which does not belong to the whole class of mammals, it is plain that mammals may be substituted for this term.  Suppose it is the predicate of a sentence, and that we know that something is either a man or a horse or a kangaroo or a whale;  then, the person who has found out this, knows more about this thing than that it is a mammal;  he therefore knows which of these four it is for these four have nothing in common except what belongs to all other mammals.  Hence in this case the particular one may be substituted for the disjunctive term.  A disjunctive term, then, — one which aggregates the extension of several symbols, — may always be replaced by a simple term.

Hence if we find out that neat are herbivorous, swine are herbivorous, sheep are herbivorous, and deer are herbivorous;  we may be sure that there is some class of animals which covers all these, all the members of which are herbivorous.  Now a disjunctive term — such as neat swine sheep and deer, or man, horse, kangaroo, and whale — is not a true symbol.  It does not denote what it does in consequence of its connotation, as a symbol does;  on the contrary, no part of its connotation goes at all to determine what it denotes — it is in that respect a mere accident if it denote anything.  Its sphere is determined by the concurrence of the four members, man, horse, kangaroo, and whale, or neat swine sheep and deer as the case may be.

Now those who are not accustomed to the homologies of the conceptions of men and words, will think it very fanciful if I say that this concurrence of four terms to determine the sphere of a disjunctive term resembles the arbitrary convention by which men agree that a certain sign shall stand for a certain thing.  And yet how is such a convention made?  The men all look upon or think of the thing and each gets a certain conception and then they agree that whatever calls up or becomes an object of that conception in either of them shall be denoted by the sign.  In the one case, then, we have several different words and the disjunctive term denotes whatever is the object of either of them.  In the other case, we have several different conceptions — the conceptions of different men — and the conventional sign stands for whatever is an object of either of them.  It is plain the two cases are essentially the same, and that a disjunctive term is to be regarded as a conventional sign or index.  And we find both agree in having a determinate extension but an inadequate comprehension.

(Peirce 1866, pp. 468–469)

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

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{ Information = Comprehension × Extension } • Selection 2

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

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{ Information = Comprehension × Extension } • Selection 1

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 redRed colour has considerably less extension than colour;  add to this the comprehension of darkdark red colour has still less [extension].  Add to this the comprehension of non-bluenon-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

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{ Information = Comprehension × Extension }

Another angle from which to approach the incidence of signs and inquiry is by way of C.S. Peirce’s “laws of information” and the corresponding theory of information that 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

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Peirce’s Categories • 7

Re: Peirce List DiscussionBen 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

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Peirce’s Categories • 6

Re: Peirce List DiscussionBen 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

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