{ Information = Comprehension × Extension } Revisited • Selection 4

Selection 3 showed how it was possible to combine symbols in such a way as to end up with species of representation outside the class of genuine symbols and introduced the concepts of conjunctive terms and disjunctive terms to describe two ways of doing this.  The essence of wit being quickly grasping the middle term, Peirce’s wit fastens on these terms to highlight the links between manners of representation and modes of inference.

Selection 4 finds Peirce in the middle of articulating the connection between indexical reference and inductive inference, using examples of disjunctive terms as pivotal cases.

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.

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{ Information = Comprehension × Extension } • Discussion 10

Re: Ontolog ForumAzamat Abdoullaev

As I see it, the semiotics or theory of signs handed down by Aristotle, Augustine, the Medieval Scholastics, Locke, and others took a significant leap toward a truly scientific theory with the work of C.S. Peirce.  This became possible, I believe, not so much driven by any mutation in the taxonomy of signs as catalyzed by Peirce’s concurrent development of the logic of relatives and the mathematics of relations, especially triadic relations.

The task I’ve set for myself under this heading is threefold —

  1. There is the scholarly task of figuring out what Peirce meant by the formula:  “Information = Comprehension × Extension”.
  2. There is the scientific task of finding out whether Peirce’s theory of information tells us anything useful about empirical realities.
  3. There is the theory-engineering task that bridges tasks 1 and 2.  It looks for ways to repair incomplete or inconsistent theories in order to give them a better grasp of the empirical realities.

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{ Information = Comprehension × Extension } Revisited • Selection 3

Selection 3 opens with Peirce remarking a critical property of genuine symbols — the class of symbols is not closed under combinations.  In particular, there are logical conjunctions of symbols and logical disjunctions of symbols which are not themselves genuine symbols.

Applying this paradigm to terms, Peirce introduces two sets of examples under the headings of conjunctive terms and disjunctive terms designed to illustrate the correspondence between manners of representation and modes of inference.

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.

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{ Information = Comprehension × Extension } Revisited • Selection 2

Over the course of Selection 1 Peirce introduces the ideas he needs to answer stubborn questions about the validity of scientific inference.  Briefly put, the validity of scientific inference depends on the ability of symbols to express superfluous comprehension, the measure of which Peirce calls information.

Selection 2 sharpens our picture of symbols as general representations, contrasting them with two species of representation whose characters fall short of genuine symbols.

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.

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{ Information = Comprehension × Extension } Revisited • Selection 1

Our first text comes from Peirce’s Lowell Lectures of 1866, titled “The Logic of Science, or, Induction and Hypothesis”.  I still remember the first time I read these words and the light that lit up the page and my mind.

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.

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{ Information = Comprehension × Extension } • Discussion 9

Re: Ontolog ForumMihai Nadin

Here’s an article of mine bearing on the present topic, namely, the relationship between Peirce’s theory of triadic sign relations and his theory of inquiry.  It also gives a smattering of historical context, bracketing Peirce between Aristotle and Dewey.

  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  ArchiveJournalOnline.

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{ Information = Comprehension × Extension } • Discussion 8

Re: Structural ModelingJoseph Simpson

Links to previous Selections, Comments, and Discussions on this topic can be found here.  See especially the section on I = C × E.

My plan going forward is to review this material in a systematic manner, redeeming the benefit of what second and third thoughts have surfaced in the meantime.  I’ve studied Peirce for just over 50 years now and I’m still seeing new facets of his work each time I return to what I thought was familiar business.

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