# Information = Comprehension × Extension

Another angle from which to approach the incidence of signs and inquiry is by way of 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, in part, 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 this formula:

 $$\operatorname{Information} = \operatorname{Comprehension} \times \operatorname{Extension}$$

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 the author'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.

## Selections from Peirce's "Logic of Science" (1865–1866)

### 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 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, Lowell Lecture 7, CE 1, 467).

### 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, Lowell Lecture 7, CE 1, 467–468).

### 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, "Lowell Lecture 7", CE 1, 468–469).

### 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, "Lowell Lecture 7", CE 1, 469).

### Selection 5

 A similar line of thought may be gone through in reference to hypothesis. In this case we must start with the consideration of the term: spherical, bright, fragrant, juicy, tropical fruit. Such a term, formed by the sum of the comprehensions of several terms, is called a conjunctive term. A conjunctive term has no extension adequate to its comprehension. Thus the only spherical bright fragrant juicy tropical fruit we know is the orange and that has many other characters besides these. Hence, such a term is of no use whatever. If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once. On the other hand, if the conjunctive term is subject and we know that every spherical bright fragrant juicy tropical fruit necessarily has certain properties, it must be that we know more than that and can simplify the subject. Thus a conjunctive term may always be replaced by a simple one. So if we find that light is capable of producing certain phenomena which could only be enumerated by a long conjunction of terms, we may be sure that this compound predicate may be replaced by a simple one. And if only one simple one is known in which the conjunctive term is contained, this must be provisionally adopted. (Peirce 1866, "Lowell Lecture 7", CE 1, 470).

### Selection 6

 We have now seen how the mind is forced by the very nature of inference itself to make use of induction and hypothesis. But the question arises how these conclusions come to receive their justification by the event. Why are most inductions and hypotheses true? I reply that they are not true. On the contrary, experience shows that of the most rigid and careful inductions and hypotheses only an infinitesimal proportion are never found to be in any respect false. And yet it is a fact that all careful inductions are nearly true and all well-grounded hypotheses resemble the truth; why is that? If we put our hand in a bag of beans the sample we take out has perhaps not quite but about the same proportion of the different colours as the whole bag. Why is that? The answer is that which I gave a week ago. Namely, that there is a certain vague tendency for the whole to be like any of its parts taken at random because it is composed of its parts. And, therefore, there must be some slight preponderance of true over false scientific inferences. Now the falsity in conclusions is eliminated and neutralized by opposing falsity while the slight tendency to the truth is always one way and is accumulated by experience. The same principle of balancing of errors holds alike in observation and in reasoning. (Peirce 1866, "Lowell Lecture 7", CE 1, 470–471.

### Discussion

At this point in his discussion, Peirce is relating the nature of inference, inquiry, and information to the character of the signs that are invoked in support of the overall process in question, a process that he is presently describing as symbolization.

In the interests of the maximum possible clarity I would like to pause for a while and try to extract from Peirce's account a couple of quick sketches, designed to show how the examples that he gives of a conjunctive term and a disjunctive term might look if they were cast within a lattice-theoretic frame.

Let's examine Peirce's example of a conjunctive term, "spherical, bright, fragrant, juicy, tropical fruit", within a lattice framework. We have these six terms:

t1 = spherical
t2 = bright
t3 = fragrant
t4 = juicy
t5 = tropical
t6 = fruit

Suppose that z is the logical conjunction of these six terms:

z = t1 t2 t3 t4 t5 t6

What on earth could Peirce mean by saying that such a term is "not a true symbol", or that it is of "no use whatever"?

In particular, let us consider the following statement:

 If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once.

That is to say, if something x is said to be z, then we may guess fairly surely that x is really an orange, in other words, that x has all of the additional features that would be summed up quite succinctly in the much more constrained term y = an orange.

Figure 1 depicts the situation that is being contemplated here.

o---------------------------------------------------------------------o
|                                   |
| t_1 t_2    t_5 t_6                    |
|  o   o  ...  o   o                     |
|   *  *    *  *                      |
|    *  *    *  *                       |
|     * *  * *                        |
|      * *  * *                         |
|       ** **                          |
|        o z = spherical bright fragrant juicy tropical fruit |
|        * *                          |
|        *  *  Rule                     |
|        *   * y=>z                     |
|        *    *                       |
|        *     *                      |
|     Fact *      o y = orange               |
|     x=>z *     *                      |
|        *    *                       |
|        *   * Case                     |
|        *  *  x=>y                     |
|        * *                          |
|        o                           |
|        x = subject                      |
|                                   |
o---------------------------------------------------------------------o
Figure 1.  Conjunctive Term z, Taken as Predicate


What Peirce is saying about z not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction z, in lattice terms, the greatest lower bound (glb) of the conjoined terms, z = glb{tj : j = 1 to 6}, and what we might regard as the "natural conjunction" or the "natural glb" of these terms, namely, y = an orange. That is to say, there is an extra measure of constraint that goes into forming the natural kinds lattice from the free lattice that logic and set theory would otherwise impose. The local manifestations of this global information are meted out over the structure of the natural lattice by just such abductive gaps as the one between z and y.

### Discussion

Let us now consider Peirce's alternate example of a disjunctive term, "neat, swine, sheep, deer".

 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. 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.

This is apparently a stock example of inductive reasoning that Peiece borrows from traditional discussions, so let us pass over the circumstance that modern taxonomies may classify swine as omniverous.

In view of the analogical symmetries that the disjunctive term shares with the conjunctive case, I think that we can run through this example in fairly short order. We have an aggregation over four terms:

s1 = neat
s2 = swine
s3 = sheep
s4 = deer

Suppose that u is the logical disjunction of these four terms:

u = ((s1)(s2)(s3)(s4)).

Figure 2 depicts the situation that we have before us.

o---------------------------------------------------------------------o
|                                   |
|        w = herbivorous                    |
|        o                           |
|        * *   Rule                     |
|        *  *  v=>w                     |
|        *   *                        |
|        *    *                       |
|        *     *                      |
|     Fact *      o v = cloven-hoofed            |
|     u=>w *     *                      |
|        *    *                       |
|        *   * Case                     |
|        *  *  u=>v                     |
|        * *                          |
|        o u = ((neat)(swine)(sheep)(deer))          |
|       ** **                          |
|      * *  * *                         |
|     * *  * *                        |
|    *  *    *  *                       |
|   *  *    *  *                      |
|  o   o      o   o                     |
| s_1 s_2    s_3 s_4                    |
|                                   |
o---------------------------------------------------------------------o
Figure 2.  Disjunctive Term u, Taken as Subject


In a similar but dual fashion to the preceding consideration, there is a gap between the the logical disjunction u, in lattice terminology, the least upper bound (lub) of the disjoined terms, u = lub{sj : j = 1 to 4}, and what we might regard as the "natural disjunction" or the "natural lub", namely, v = cloven-hoofed.

Once again, the sheer implausibility of imagining that the disjunctive term u would ever be embedded exactly per se in a lattice of natural kinds, leads to the evident naturalness of the induction to vw, namely, the rule that cloven-hoofed animals are herbivorous.

### Discussion

I continue with the out lay of my incidental musings on the theme of approximate inference rules.

 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, "Lowell Lecture 7", CE 1, 467–468).

Aside from Aristotle, the influence of Kant on Peirce is very strongly marked in these earliest expositions. The invocations of "conceptions of the understanding", the "use" of concepts and thus of symbols in reducing the manifold of extension, and the not so subtle hint of the synthetic à priori in Peirce's discussion, not only of natural kinds, but of the kinds of signs that lead up to genuine symbols, can all be recognized as being reprises of dominant, pervasive Kantian themes.

In order to draw out these themes, and to see how Peirce was led and often inspired to develop their main motives, let us bring together our previous Figures, abstracting away from all of those distractingly ephemeral details about defunct stockyards full of imaginary beasts, and see if we can see what is really going to go on here.

Figure 3 shows an abductive step of inquiry, as it is taken on the cue of an iconic sign.

o-----------------------------------------------------------o
|                              |
| t_1 t_2    t_3 t_4               |
|  o   o      o   o                |
|   *  *    *  *                 |
|    *  *    *  *                  |
|     * *  * *                   |
|      * *  * *                    |
|       ** **                     |
|        o z = icon?                 |
|        * *                     |
|        *  *  Rule                |
|        *   * y=>z                |
|        *    *                  |
|        *     *                 |
|     Fact *      o y = object?          |
|     x=>z *     *                 |
|        *    *                  |
|        *   * Case                |
|        *  *  x=>y                |
|        * *                     |
|        o                      |
|        x = subject                 |
|                              |
o-----------------------------------------------------------o
Figure 3.  Conjunctive Predicate z, Abduction of Case (x (y))


Figure 4 depicts an inductive step of inquiry, as it is taken on the cue of an indicial sign.

o-----------------------------------------------------------o
|                              |
|        w = predicate                |
|        o                      |
|        * *   Rule                |
|        *  *  v=>w                |
|        *   *                   |
|        *    *                  |
|        *     *                 |
|     Fact *      o v = object?          |
|     u=>w *     *                 |
|        *    *                  |
|        *   * Case                |
|        *  *  u=>v                |
|        * *                     |
|        o u = index?                |
|       ** **                     |
|      * *  * *                    |
|     * *  * *                   |
|    *  *    *  *                  |
|   *  *    *  *                 |
|  o   o      o   o                |
| s_1 s_2    s_3 s_4               |
|                              |
o-----------------------------------------------------------o
Figure 4.  Disjunctive Subject u, Induction of Rule (v (w))


I have up to this point followed Peirce's suggestions somewhat unthinkingly, but I can tell you now that previous unfortunate experience has led me concurrently to remain suspicious of all attempts to conflate the types of signs and the roles of terms in arguments quite so facilely, so I will keep that as a topic for future inquiry.

### Selection 7

 It is obvious that all deductive reasoning has a common property unshared by the other kinds — in being purely explicatory. Buffier mentions a definition of logic as the art of confessing in the conclusion what we have avowed in the premisses. This bit of satire translated into the language of sobriety — amounts to charging that the logicians confine their attention exclusively to deductive reasoning. A charge which against the logicians of other days, was quite just. All deductive reasoning is merely explicatory. That is to say, that which appears in the conclusion explicitly was contained in the premisses implicitly. All explication is of one of two kinds — direct or indirect. Explication direct consists in simply substituting for a word what is implied in that word. A statement therefore in order to imply something not expressed must either say that a word denotes something or else that something is meant by a word. Then the direct explication consists in saying that that what a word denotes is what is meant by the word. Indirect explication consists in saying that what is not what is meant by the word is not denoted by the word or else in saying that which what a word denotes is not is not meant by the word. Explication in general, then, may be said to be the application of the maxim that what a word denotes is what is meant by the word. (Peirce 1866, "Lowell Lecture 7", CE 1, 458–459).

### Selection 8

 It is important to distinguish between the two functions of a word: 1st to denote something — to stand for something, and 2nd to mean something — or as Mr. Mill phrases it — to connote something. What it denotes is called its Sphere. What it connotes is called its Content. Thus the sphere of the word man is for me every man I know; and for each of you it is every man you know. The content of man is all that we know of all men, as being two-legged, having souls, having language, &c., &c. It is plain that both the sphere and the content admit of more and less. … Now the sphere considered as a quantity is called the Extension; and the content considered as quantity is called the Comprehension. Extension and Comprehension are also termed Breadth and Depth. So that a wider term is one which has a greater extension; a narrower one is one which has a less extension. A higher term is one which has a less Comprehension and a lower one has more. The narrower term is said to be contained under the wider one; and the higher term to be contained in the lower one. We have then: o-----------------------------o-----------------------------o |               |               | | What is 'denoted'     | What is 'connoted'     | |               |               | | Sphere           | Content          | |               |               | | Extension         | Comprehension       | |               |               | |      ( wider      |     ( lower       | | Breadth <         | Depth <          | |      ( narrower    |     ( higher      | |               |               | | What is contained 'under' | What is contained 'in'   | |               |               | o-----------------------------o-----------------------------o  The principle of explicatory or deductive reasoning then is that: Every part of a word's Content belongs to every part of its Sphere, or: Whatever is contained in a word belongs to whatever is contained under it. Now this maxim would not be true if the Extension and Comprehension were directly proportional to one another; this is to say if the Greater the one the greater the other. For in that case, though the whole Content would belong to the whole Sphere; yet only a particular part of it would belong to a part of that Sphere and not every part to every part. On the other hand if the Comprehension and Extension were not in some way proportional to one another, that is if terms of different spheres could have the same content or terms of the same content different spheres; then there would be no such fact as a content's belonging to a sphere and hence again the maxim would fail. For the maxim to be true, then, it is absolutely necessary that the comprehension and extension should be inversely proportional to one another. That is that the greater the sphere, the less the content. Now this evidently true. If we take the term man and increase its comprehension by the addition of black, we have black man and this has less extension than man. So if we take black man and add non-black man to its sphere, we have man again, and so have decreased the comprehension. So that whenever the extension is increased the comprehension is diminished and vice versa. (Peirce 1866, "Lowell Lecture 7", CE 1, 459–460).

### Selection 9

 The highest terms are therefore broadest and the lowest terms the narrowest. We can take a term so broad that it contains all other spheres under it. Then it will have no content whatever. There is but one such term — with its synonyms — it is Being. We can also take a term so low that it contains all other content within it. Then it will have no sphere whatever. There is but one such term — it is Nothing. o------------------------o------------------------o |            |            | | Being        | Nothing       | |            |            | | All breadth     | All depth      | |            |            | | No depth       | No breadth      | |            |            | o------------------------o------------------------o  We can conceive of terms so narrow that they are next to nothing, that is have an absolutely individual sphere. Such terms would be innumerable in number. We can also conceive of terms so high that they are next to being, that is have an entirely simple content. Such terms would also be innumerable. o------------------------o------------------------o |            |            | | Simple terms     | Individual terms   | |            |            | o------------------------o------------------------o  (Peirce 1866, "Lowell Lecture 7", CE 1, 460).

### Selection 10

 But such terms though conceivable in one sense — that is intelligible in their conditions — are yet impossible. You never can narrow down to an individual. Do you say Daniel Webster is an individual? He is so in common parlance, but in logical strictness he is not. We think of certain images in our memory — a platform and a noble form uttering convincing and patriotic words — a statue — certain printed matter — and we say that which that speaker and the man whom that statue was taken for and the writer of this speech — that which these are in common is Daniel Webster. Thus, even the proper name of a man is a general term or the name of a class, for it names a class of sensations and thoughts. The true individual term the absolutely singular this & that cannot be reached. Whatever has comprehension must be general. (Peirce 1866, "Lowell Lecture 7", CE 1, 461).

### Selection 11

 In like manner, it is impossible to find any simple term. This is obvious from this consideration. If there is any simple term, simple terms are innumerable for in that case all attributes which are not simple are made up of simple attributes. Now none of these attributes can be affirmed or denied universally of whatever has any one. For let A be one simple term and B be another. Now suppose we can say All A is B; then B is contained in A. If, therefore, A contains anything but B it is a compound term, but A is different from B, and is simple; hence it cannot be that All A is B. Suppose No A is B, then not-B is contained in A; if therefore A contains anything besides not-B it is not a simple term; but if it is the same as not-B, it is not a simple term but is a term relative to B. Now it is a simple term and therefore Some A is B. Hence if we take any two simple terms and call one A and the other B we have: Some A is B and Some A is not B or in other words the universe will contain every possible kind of thing afforded by the permutation of simple qualities. Now the universe does not contain all these things; it contains no well-known green horse. Hence the consequence of supposing a simple term to exist is an error of fact. There are several other ways of showing this besides the one that I have adopted. They all concur to show that whatever has extension must be composite. (Peirce 1866, "Lowell Lecture 7", CE 1, 461).

### Selection 12

 The moment, then, that we pass from nothing and the vacuity of being to any content or sphere, we come at once to a composite content and sphere. In fact, extension and comprehension — like space and time — are quantities which are not composed of ultimate elements; but every part however small is divisible. The consequence of this fact is that when we wish to enumerate the sphere of a term — a process termed division — or when we wish to run over the content of a term — a process called definition — since we cannot take the elements of our enumeration singly but must take them in groups, there is danger that we shall take some element twice over, or that we shall omit some. Hence the extension and comprehension which we know will be somewhat indeterminate. But we must distinguish two kinds of these quantities. If we were to subtilize we might make other distinctions but I shall be content with two. They are the extension and comprehension relatively to our actual knowledge, and what these would be were our knowledge perfect. Logicians have hitherto left the doctrine of extension and comprehension in a very imperfect state owing to the blinding influence of a psychological treatment of the matter. They have, therefore, not made this distinction and have reduced the comprehension of a term to what it would be if we had no knowledge of fact at all. I mention this because if you should come across the matter I am now discussing in any book, you would find the matter left in quite a different state. (Peirce 1866, "Lowell Lecture 7", CE 1, 462).

### Selection 13

 With me — the Sphere of a term is all the things we know that it applies to, or the disjunctive sum of the subjects to which it can be predicate in an affirmative subsumptive proposition. The content of a term is all the attributes it tells us, or the conjunctive sum of the predicates to which it can be made subject in a universal necessary proposition. The maxim then which rules explicatory reasoning is that any part of the content of a term can be predicated of any part of its sphere. (Peirce 1866, "Lowell Lecture 7", CE 1, 462).

### Selection 14

 We come next to consider inductions. In inferences of this kind we proceed as if upon the principle that as is a sample of a class so is the whole class. The word class in this connection means nothing more than what is denoted by one term, — or in other words the sphere of a term. Whatever characters belong to the whole sphere of a term constitute the content of that term. Hence the principle of induction is that whatever can be predicated of a specimen of the sphere of a term is part of the content of that term. And what is a specimen? It is something taken from a class or the sphere of a term, at random — that is, not upon any further principle, not selected from a part of that sphere; in other words it is something taken from the sphere of a term and not taken as belonging to a narrower sphere. Hence the principle of induction is that whatever can be predicated of something taken as belonging to the sphere of a term is part of the content of that term. But this principle is not axiomatic by any means. Why then do we adopt it? (Peirce 1866, "Lowell Lecture 7", CE 1, 462–463).

### Selection 15

To explain this, we must remember that the process of induction is a process of adding to our knowledge; it differs therein from deduction — which merely explicates what we know — and is on this very account called scientific inference. Now deduction rests as we have seen upon the inverse proportionality of the extension and comprehension of every term; and this principle makes it impossible apparently to proceed in the direction of ascent to universals. But a little reflection will show that when our knowledge receives an addition this principle does not hold.

Thus suppose a blind man to be told that no red things are blue. He has previously known only that red is a color; and that certain things A, B, and C are red.

 The comprehension of red then has been for him color. Its extension has been A, B, C.

But when he learns that no red thing is blue, non-blue is added to the comprehension of red, without the least diminution of its extension.

 Its comprehension becomes non-blue color. Its extension remains A, B, C.

Suppose afterwards he learns that a fourth thing D is red. Then, the comprehension of red remains unchanged, non-blue color; while its extension becomes A, B, C, and D. Thus, the rule that the greater the extension of a term the less its comprehension and vice versa, holds good only so long as our knowledge is not added to; but as soon as our knowledge is increased, either the comprehension or extension of that term which the new information concerns is increased without a corresponding decrease of the other quantity.

The reason why this takes place is worthy of notice. Every addition to the information which is incased in a term, results in making some term equivalent to that term. Thus when the blind man learns that red is not-blue, red not-blue becomes for him equivalent to red. Before that, he might have thought that red not-blue was a little more restricted term than red, and therefore it was so to him, but the new information makes it the exact equivalent of red. In the same way, when he learns that D is red, the term D-like red becomes equivalent to red.

Thus, every addition to our information about a term is an addition to the number of equivalents which that term has. Now, in whatever way a term gets to have a new equivalent, whether by an increase in our knowledge, or by a change in the things it denotes, this always results in an increase either of extension or comprehension without a corresponding decrease in the other quantity.

For example we have here a number of circles dotted and undotted, crossed and uncrossed:

 (·X·) (···) (·X·) (···) ( X ) ( ) ( X ) ( ) 

Here it is evident that the greater the extension the less the comprehension:

 o-------------------o-------------------o |          |          | | dotted      | 4 circles     | |          |          | o-------------------o-------------------o |          |          | | dotted & crossed | 2 circles     | |          |          | o-------------------o-------------------o 

Now suppose we make these two terms dotted circle and crossed and dotted circle equivalent. This we can do by crossing our uncrossed dotted circles. In that way, we increase the comprehension of dotted circle and at the same time increase the extension of crossed and dotted circle since we now make it denote all dotted circles. (Peirce 1866, "Lowell Lecture 7", CE 1, 463–464).

### Selection 16

Thus every increase in the number of equivalents of any term increases either its extension or comprehension and conversely. It may be said that there are no equivalent terms in logic, since the only difference between such terms would be merely external and grammatical, while in logic terms which have the same meaning are identical. I fully admit that. Indeed, the process of getting an equivalent for a term is an identification of two terms previously diverse. It is, in fact, the process of nutrition of terms by which they get all their life and vigor and by which they put forth an energy almost creative — since it has the effect of reducing the chaos of ignorance to the cosmos of science. Each of these equivalents is the explication of what there is wrapt up in the primary — they are the surrogates, the interpreters of the original term. They are new bodies, animated by that same soul. I call them the interpretants of the term. And the quantity of these interpretants, I term the information or implication of the term. (Peirce 1866, "Lowell Lecture 7", CE 1, 464–465).

### Selection 17

 We must therefore modify the law of the inverse proportionality of extension and comprehension and instead of writing: Extension × Comprehension = Constant, which crudely expresses the fact that the greater the extension the less the comprehension, we must write: Extension × Comprehension = Information, which means that when the information is increased there is an increase of either extension or comprehension without any diminution of the other of these quantities. Now, ladies and gentlemen, as it is true that every increase of our knowledge is an increase in the information of a term — that is, is an addition to the number of terms equivalent to that term — so it is also true that the first step in the knowledge of a thing, the first framing of a term, is also the origin of the information of that term because it gives the first term equivalent to that term. I here announce the great and fundamental secret of the logic of science. There is no term, properly so called, which is entirely destitute of information, of equivalent terms. The moment an expression acquires sufficient comprehension to determine its extension, it already has more than enough to do so. (Peirce 1866, "Lowell Lecture 7", CE 1, 465).

### Discussion

If you dreamed that this inquiry had come full circle then I inform you of what you already know, that there are always greater circles. I revert to Peirce's Harvard University Lectures of the year before, to pick up additional background material and a bit more motivation.

 We are already familiar with the distinction between the extension and comprehension of terms. A term has comprehension in virtue of having a meaning and has extension in virtue of being applicable to objects. The meaning of a term is called its connotation; its applicability to things its denotation. Every symbol denotes by connoting. A representation which denotes without connoting is a mere sign. If it connotes without thereby denoting, it is a mere copy. It is universally held that extension and comprehension are in reciprocal relation; thus if horse be divided into black horse and non-black horse, black horse has more intension and therefore less extension than horse. It behooves me to say what the distinction between extension and comprehension is upon my view of logic. Before doing so, however, I must remark that the distinction extends to propositions; there are extensive and intensive propositions. An extensive proposition is defined to be one which states the relation between the extension of two terms. An intensive proposition is one which states the relation between the intension or comprehension of two terms. Subordination in extension is expressed by the term contained under. Subordination in intension is expressed by the term contained in. Hence in the case of affirmatives; an extensive judgment is expressed by the formula: A is contained under B, an equivalent intensive proposition by the formula: B is contained in A. Thus black horse is contained under horse, and horse [is contained in black horse]. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 272).

### Selection 19

Nota Bene. In the Table below a label of the form XY indicates a premiss of a classical syllogism in which X is the subject and Y is the predicate. Also, I suspect that the Third Figure syllogism ought to be XY & XZ.

 What we have to distinguish, therefore, is not so much the quantity of extension from the quantity of intension as it is the object of connotation from the object of denotation. In analytical judgments there is no denotation at all. In a synthetical judgment the subject is an object of denotation. o---------------------o-----------------------o-----------------o |           |            |         | |           | (  Subject: O of C |      ( XY | | Analytic      | <           | 2nd Fig. <   | |           | ( Predicate: O of C |      ( ZY | |           |            |         | o---------------------o-----------------------o-----------------o |           |            |         | |           | (  Subject: O of D |      ( YX | | Synthetic Intensive | <           | 1st Fig. <   | |           | ( Predicate: O of C |      ( ZY | |           |            |         | o---------------------o-----------------------o-----------------o |           |            |         | |           | (  Subject: O of D |      ( YX | | Extensive      | <           | 3rd Fig. <   | |           | ( Predicate: O of D |      ( ZX | |           |            |         | o---------------------o-----------------------o-----------------o  There cannot be a judgment whose subject is an object of connotation and whose predicate is an object of denotation. For a symbol denotes by virtue of connoting and not 'vice versa', hence the object of connotation determines the object of denotation and not 'vice versa', in the sense in which the subject of a proposition is the term determined and the predicate is the determining term. Whence if one of the terms is an object of connotation and the other is an object of denotation, the latter is the subject and not the former. In the other two cases, there is no difference between subject and predicate; except that one may be regarded as taken first. Thus these cases in which both terms are of the same kind are two kinds of twists of the first kind, just as the 2nd and 3rd Figures of Syllogism are right-handed and left-handed twists of the 1st. This is expressed in the above Table. A proposition would usually be called intensive if its predicate were an object of connotation; hence we have three kinds of propositions given by these two; namely, Analytic. Synthetic Intensive. Extensive. There is no such thing as an analytic extensive proposition. For an analytic proposition containing no object of denotation is merely the expression of a relation of comprehension. Of course from an analytic proposition a synthetic one may be immediately inferred. From: Man is mortal, we may infer: All men are mortals, but the predicate 'mortals' is not a mere result of the analysis of men. I have here slightly narrowed Kant's definition of the analytic judgment so as to make it not merely needless but impossible to test one by experience. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 272–274).

### Selection 20

 We come now to an objection to the division of propositions which I have just given which will require us to examine the matter somewhat more deeply. It may be said: the copula in all cases establishes an identity between two terms. Hence as in one of the propositions the object of denotation is the subject and the object of connotation the predicate, these two objects are identical and hence the division into three kinds is a distinction without a difference. In order to answer this objection we must revert to that distinction between thing, image, and form established in the lecture upon the definition of logic. A representation is anything which may be regarded as standing for something else. Matter or thing is that for which a representation might stand prescinded from all that could constitute a relation with any representation. A form is the relation between a representation and thing prescinded from both representation and thing. An image is a representation prescinded from thing and form. Derived directly from this abstractest triad was another less abstract. This is Object—Equivalent-Representation—Logos. The object is a thing corresponding to a representation regarded as actual. The equivalent representation is a representation in any language equivalent to a representation regarded as actual. A Logos is a form constituting the relation between an object and a representation regarded as actual. Every symbol may be said in three different senses to be determined by its object, its equivalent representation, and its logos. It stands for its object, it translates its equivalent representation, it realizes its logos. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 274).

### Selection 21

 Every symbol may be said in three different senses to be determined by its object, its equivalent representation, and its logos. It stands for its object, it translates its equivalent representation, it realizes its logos. As every symbol is determined in these three ways, Symbols, as such, are subject to three laws one of which is the conditio sine qua non of its standing for anything, the second of its translating anything, and the third of its realizing anything. The first law is Logic, the second Universal Rhetoric, the third Universal Grammar. But an object is a thing informed and represented. An equivalent representation is an image which is itself represented and realized, and a logos is a form, embodied in an object and representation. Hence the object of a symbol implies in itself both thing, form, and image. And hence regarded as containing one or other of these three elements it may be distinguished as material object, formal object, and representative object. Now so far as the object of a symbol contains the thing, so far the symbol stands for something and so far it denores. So far as its object embodies a form, so far the symbol has a meaning and so far it connotes. Thus we see that the denotative object and the connotative object are in fact identical; and therefore an analytic, an intensive synthetic, and an extensive proposition may all represent the same fact and yet the mode in which they are obtained and the relation of the proposition to that fact are necessarily very different. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 274–275).

### Selection 22

 But since the object contains three elements, thing, image, form, we ought to have another kind of object besides the denotative and connotative. What is this? If we suppose ourselves to know no more of man than what is contained in the definition Man is the rational animal, then we might divide man into man risible and man non-risible.               man                     ___________________|___________________           /                   \        man risible              man non-risible    And then the connotation of man would be less than that of either man risible or man non-risible. And conversely man risible and man non-risible would have a less extension than man. But we afterwards find that the class man non-risible does not exist and is impossible. Henceforward the idea of man and that of risible man are changed. The extension of risible man has become equal to that of men and the comprehension of man has become equal to that of risible man. And how has this change in the relations of the terms been effected? Before the information we knew (let us say) that there were certain risible men whom we may denote by A and there were other men who might or might not be risible whom we will denote by BB’ [— perhaps B + B’ was intended?]. We have now found that BB’ are also risible. When we said all men before we meant A + B + B’; when we say all men now we mean the same. The extension of man then has not changed. When we said risible men before we denoted A + B ?, that is to say the whole of A but none of B for certain; but now when we say risible men we denote A + B + B’. Hence the extension of risible men has increased, so as to become equal to that of men. On the other hand the intension of risible man is now as it was before, composed of risible, rational, and animal; while the comprehension of man which before contained only rational and animal, now contains risible also. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 275–276).

### Selection 23

 Thus the process of information disturbs the relations of extension and comprehension for a moment and the class which results from the equivalence of two others has a greater intension than one and a greater extension than the other. Hence, we may conveniently alter the formula for the relations of extension and comprehension; thus, instead of saying that one is the reciprocal of the other, or: comprehension × extension = constant, we may say: comprehension × extension = information. We see then that all symbols besides their denotative and connotative objects have another; their informative object. The denotative object is the total of possible things denoted. The connotative object is the total of symbols translated or implied. The informative object is the total of forms manifested and is measured by the amount of intension the term has, over and above what is necessary for limiting its extension. For example the denotative object of man is such collections of matter the word knows while it knows them i.e. while they are organized. The connotative object of man is the total form which the word expresses. The informative object of man is the total fact which it embodies; or the value of the conception which is its equivalent symbol. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 276).

### Selection 24

 Abstract words such as truth, honor, by the way, are somewhat difficult to understand. It seems to me that they are simply fictions. Every word must denote some thing; these are names for certain fictitious things which are supposed for the purpose of indicating that the object of a concrete term is meant as it would be did it contain either no information or a certain amount of information. Thus "charity is a virtue" means "What is charitable is virtuous — by the definition of charity and not by reason of what is known about it". Hence, only analytical propositions are possible of abstract terms; and on this account they are peculiarly useful in metaphysics where the question is what can we know without any information. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 276–277).

### Selection 25

 Coming back now to propositions, we should first remark that just as the framing of a term is a process of symbolization so also is the framing of a proposition. No proposition is supposed to leave its terms as it finds them. Some symbol is determined by every proposition. Hence, since symbols are determined by their objects; and there are three objects of symbols, the connotative, denotative, informative; it follows that there will be three kinds of propositions, such as alter the denotation, the information, and the connotation of their terms respectively. But when information is determined both connotation and information [— perhaps "denotation" ?] are determined; hence the three kinds will be 1st Such as determine connotation, 2nd Such as determine denotation, 3rd Such as determine both denotation and connotation. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 277).

### Selection 26

 The difference between connotation, denotation, and information supplies the basis for another division of terms and propositions; a division which is related to the one we have just considered in precisely the same way as the division of syllogism into 3 figures is related to the division into Deduction, Induction, and Hypothesis. Every symbol which has connotation and denotation has also information. For by the denotative character of a symbol, I understand application to objects implied in the symbol itself. The existence therefore of objects of a certain kind is implied in every connotative denotative symbol; and this is information. Now there are certain imperfect or false symbols produced by the combination of true symbols which have lost either their denotation or their connotation. When symbols are combined together in extension, as for example in the compound term "cats and dogs", their sum possesses denotation but no connotation or at least no connotation which determines their denotation. Hence, such terms, which I prefer to call enumerative terms, have no information, and it remains unknown whether there be any real kind corresponding to cats and dogs taken together. On the other hand, when symbols are combined together in comprehension, as for example in the compound "tailed men", the product possesses connotation but no denotation, it not being therein implied that there may be any tailed men. Such conjunctive terms have therefore no information. Thirdly, there are names purporting to be of real kinds, as men; and these are perfect symbols. Enumerative terms are not truly symbols but only signs; and Conjunctive terms are copies; but these copies and signs must be considered in symbolistic because they are composed of symbols. When an enumerative term forms the subject of a grammatical proposition, as when we say "cats and dogs have tails", there is no logical unity in the proposition at all. Logically, therefore, it is two propositions and not one. The same is the case when a conjunctive proposition forms the predicate of a sentence; for to say "hens are feathered bipeds" is simply to predicate two unconnected marks of them. When an enumerative term as such is the predicate of a proposition, that proposition cannot be a denotative one, for a denotative proposition is one which merely analyzes the denotation of its predicate, but the denotation of an enumerative term is analyzed in the term itself; hence if an enumerative term as such were the predicate of a proposition, that proposition would be equivalent in meaning to its own predicate. On the other hand, if a conjunctive term as such is the subject of a proposition, that proposition cannot be connotative, for the connotation of a conjunctive term is already analyzed in the term itself, and a connotative proposition does no more than analyze the connotation of its subject. Thus, we have: Conjunctive, Simple, Enumerative propositions so related to: Denotative, Informative, Connotative propositions that what is on the left hand of one line cannot be on the right hand of the other. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 278–279).

### Selection 27

 We are now in a condition to discuss the question of the grounds of scientific inference. The problem naturally divides itself into parts: 1st To state and prove the principles upon which the possibility in general of each kind of inference depends, 2nd To state and prove the rules for making inferences in particular cases. The first point I shall discuss in the remainder of this lecture; the second I shall scarcely be able to touch upon in these lectures. Inference in general obviously supposes symbolization; and all symbolization is inference. For every symbol as we have seen contains information. And in the last lecture we saw that all kinds of information involve inference. Inference, then, is symbolization. They are the same notions. Now we have already analyzed the notion of a symbol, and we have found that it depends upon the possibility of representations acquiring a nature, that is to say an immediate representative power. This principle is therefore the ground of inference in general. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 279–280).

### Selection 28

 But there are three distinct kinds of inference; inconvertible and different in their conception. There must, therefore, be three different principles to serve for their grounds. These three principles must also be indemonstrable; that is to say, each of them so far as it can be proved must be proved by means of that kind of inference of which it is the ground. For if the principle of either kind of inference were proved by another kind of inference, the former kind of inference would be reduced to the latter; and since the different kinds of inference are in all respects different this cannot be. You will say that it is no proof of these principles at all to support them by that which they themselves support. But I take it for granted at the outset, as I said at the beginning of my first lecture, that induction and hypothesis have their own validity. The question before us is why they are valid. The principles, therefore, of which we are in search, are not to be used to prove that the three kinds of inference are valid, but only to show how they come to be valid, and the proof of them consists in showing that they determine the validity of the three kinds of inference. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 280).

### Selection 29

 But these three principles must have this in common that they refer to symbolization for they are principles of inference which is symbolization. As grounds of the possibility of inference they must refer to the possibility of symbolization or symbolizability. And as logical principles they must relate to the reference of symbols to objects; for logic has been defined as the science of the general conditions of the relations of symbols to objects. But as three different principles they must state three different relations of symbols to objects. Now we have already found that a symbol has three different relations of objects; namely connotation, denotation, and information which are its relations to the object considered as a thing, a form, and an equivalent representation. Hence, it is obvious that these three principles must relate to the symbolizability of things, of forms, and of symbols. Our next business is to find which is which. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 280–281).

### Selection 30

 Our next business is to find which is which. For this purpose we must consider that each principle is to be proved by the kind of inference which it supports. The ground of deductive inference then must be established deductively; that is by reasoning from determinant to determinate, or in other words by reasoning from definition. But this kind of reasoning can only be applied to an object whose character depends upon its definition. Now of most objects it is the definition which depends upon the character; and so the definition must therefore itself rest on induction or hypothesis. But the principle of deduction must rest on nothing but deduction, and therefore it must relate to something whose character depends upon its definition. Now the only objects of which this is true are symbols; they indeed are created by their definition; while neither forms nor things are. Hence, the principle of deduction must relate to the symbolizability of of symbols. The principle of hypothetic inference must be established hypothetically, that is by reasoning from determinate to determinant. Now it is clear that this kind of reasoning is applicable only to that which is determined by what it determines; or that which is only subject to truth and falsehood so far as its determinate is, and is thus of itself pure 'zero'. Now this is the case with nothing whatever except the pure forms; they indeed are what they are only in so far as they determine some symbol or object. Hence the principle of hypothetic inference must relate to the symbolizability of forms. The principle of inductive inference must be established inductively, that is by reasoning from parts to whole. This kind of reasoning can apply only to those objects whose parts collectively are their whole. Now of symbols this is not true. If I write man here and dog here that does not constitute the symbol of man and dog, for symbols have to be reduced to the unity of symbolization which Kant calls the unity of apperception and unless this be indicated by some special mark they do not constitute a whole. In the same way forms have to determine the same matter before they are added; if the curtains are green and the wainscot yellow that does not make a yellow-green. But with things it is altogether different; wrench the blade and handle of a knife apart and the form of the knife has disappeared but they are the same thing — the same matter — that they were before. Hence, the principle of induction must relate to the symbolizability of things. All these principles must as principles be universal. Hence they are as follows:— All things, forms, symbols are symbolizable. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 281–282).

### Selection 31

 All these principles must as principles be universal. Hence they are as follows:— All things, forms, symbols are symbolizable. The next step is to prove each of these principles. First then, to prove deductively that all symbols are symbolizable. In every syllogism there is a term which is predicate and subject. But a predicate is a symbol of its subject. Hence, in every deduction a symbol is symbolized. Now deduction is valid independently of the matter of the judgment. Hence all symbols are symbolizable. Next; to prove inductively that all things are symbolizable. For this purpose we must take all the collocations of things we can and judge by them. Now all these collocations of things have been selected upon some principle; this principle of selection is a predicate of them and a concept. Being a concept it is a symbol. And it partakes of that peculiarity of symbols that it must have information. We have no concepts which do not denote some things as well as connoting; because all our thought begins with experience. But a symbol which has connotation and denotation contains information. Whatever symbol contains information contains more connotation than is necessary to limit its possible denotation to those things which it may denote. That is every symbol contains more than is sufficient for a principle of selection. Hence every selected collocation of things must have something more than a mere principle of selection, it must have another common quality. Now by induction this common quality may be predicated of the whole possible denotation of the concept which serves as principle of selection. And thus every collocation of things we can select is symbolized by its principle of selection. Now by induction we pass from this statement that all things we can take are symbolizable to the principle that all things are symbolzable. Q.E.D. This argument though inductive in form is of the highest possible validity, for no case can possibly arise to contradict it. Thirdly, we have to prove hypothetically that all forms are symbolizable. For this purpose we must consider that 'forms' are nothing unless they are embodied, and then they constitute the synthesis of the matter. Hence the knowledge of them cannot be directly given but must be obtained by hypothesis. Now we have to explain this fact, that all forms are to be regarded as subjects for hypothesis, by a hypothesis. For this purpose, we should reflect that whatever is symbolizable is symbolized by terms and their combinations. Now we saw at the last lecture that the process of obtaining a new term is a hypothetic inference. So that everything which is symbolizable is to be regarded as a subject for hypothesis. This accounts for the same thing being true of forms, if we make the hypothesis that all forms are symbolizable. Q.E.D. This argument though only an hypothesis could not have been stronger for the conclusion involves no matter of fact at all. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 282–283).

### Selection 32

 Thus the three grounds of inference are proved. All have been made certain. But the manner in which they have attained to certainty indicates a very different general strength of the three kinds of inference. The hypothetic argument became certain only by speaking of that which has no sense except when this principle is true. The inductive argument became certain only by taking into account all that could possibly be known. The deductive argument alone was strictly demonstrative. Thus we have in order of strength Deduction, Induction, Hypothesis. Deduction, in fact, is the only demonstration; yet no one thinks of questioning a good induction, while hypothesis is proverbially dangerous. Hypotheses non fingo, said Newton, striving to place his theory on a basis of strict induction. Yet it is hypotheses with which we must start; the baby when he lies turning his fingers before his eyes is making a hypothesis as to the connection of what he sees and what he feels. Hypotheses give us our facts. Induction extends our knowledge. Deduction makes it distinct. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 283).

### Selection 33

 In every induction we have given some remarkable fact or piece of information: S is B, where B is an object of connotation. We infer that something else: T is B. Let us suppose that T contains more information than S. Then, if T is no more extensive than S, "T is B" is a better judgment than "S is B" because it contains more information without predicating B of anything doubtful. Thus, it is better to say "All men are mortal" than "all rational animals are mortal" for the former implies the latter and contains no more possibility of error and is more distinct. But in every case of induction T is also more extensive than S. Then in case S is a true symbol and "S is B" is a single true judgment, this judgment or proposition must be the result of induction, as we saw in the last lecture that all propositions are. The question is, therefore, which is the preferable theory, "S is B" or "T is B". The greater information of T causes the latter theory to contain more truth but its greater extension renders it liable to more error. If in T the extension of S is increased more than the information is, the connotation will be diminished and 'vice versa'. Accordingly the greater the connotation of T relatively to that of S, the better is the theory proposed, "T is B". Which of the two theories to select in any case will depend upon the motives which influence us. In a desperate practical case, if one's life depends upon taking the right one, he ought to select the one whose subject has the greatest connotation. In a cool speculation where safety is the essential; the least extensive should be taken. So much for the preference between two theories. But in proceeding from fact to theory — in such a case as that about neat, swine, sheep, and deer — S is a mere enumerative term and has no connotation at all. In this case therefore T increases the connotation of S absolutely and "T is B" ought therefore to be absolutely preferred to "S is B" and be accepted assertorically; as long as there is no question between this theory and some other and as long as it is not opposed by some other induction. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 285).

Nota Bene. For the sake of readability in this transcription, I supply quotation marks around formulas and change a couple of Greek letters to Roman characters, using T for Sigma and Q for Pi.

### Selection 34

 In the case of hypothesis we have given some remarkable state of things: X is P, where X is an object of denotation; we explain this by supposing that: X is Q, and Q always contains more information than P. If Q, therefore, has no more comprehension than P, it is better to say "X is Q" than "X is P". It is clearer to say that Every man is mortal than to say that Every man is either a good mortal or a bad mortal. But in the case of hypothesis, Q always comprehends more than P. To decide then between the two; we have to consider whether Q has more denotation than P for if it has, the information of P is increased more in Q than its comprehension is and vice versa; and we must be decided which to take by our motives. This is the case of a preference between hypotheses. But in the first proceedure from facts, P is a mere conjunctive term, destitute of any denotation before this proposition. Hence in this case the information is increased absolutely, the connotation only relatively, and the hypothesis is absolutely needed and must be taken as a pis aller unless opposed by some other argument and until a better one presents itself. Polarization for instance is a series of phenomena which it is impossible to name or define without the use of a hypothesis. (Peirce 1865, "Harvard Lecture 10. Grounds of Induction", CE 1, 285–286).

Nota Bene. For the sake of readability in this transcription, I supply quotation marks around formulas and change a couple of Greek letters to Roman characters, using T for Sigma and Q for Pi.

### Selection 35

 The last lecture was devoted to the fundamental inquiry of the whole course, that of the grounds of inference. We first distingushed three kinds of reference which every true symbol has to its object. In the first place, every true symbol is applicable to some real thing. Hence, every symbol whether true or not asserts itself to be applicable to some real thing. This is the denotation of the symbol. All that we know of things is as denotative objects of symbols. And thus all denotation is comparative, merely. One symbol has more denotation than another or is more extensive when it asserts itself to be applicable to all the things of which the first asserts itself to be applicable and also to others. In the second place, every genuine symbol relates or purports to relate to some form embodied in its object. This is its connotation. It is, in fact, only by means of this reference to a form that a symbol acquires its applicability to the thing. The more form a symbol relates to, the greater its intension, comprehension, or connotation. Other things being equal, the greater the comprehension of a symbol the less its extension. For since its denotation is created by its connotation, the more the latter is determined, the more the former is limited. But this rule does not always hold good. For just as there are real kinds in nature, that is to say classes which differ from all others in more respects than one, so there are symbols which imply that their collected objects are real kinds and thus they connote more forms than one, either of which would be sufficient to limit their extension to the extent to which it is limited. Hence if a symbol changes in information it may change either its extension or comprehension without changing both and thus the reciprocal relation of extension and comprehension only holds good when the information is not changed. Information then may be defined as the amount of comprehension a symbol has over and above what limits its extension. A symbol not only may have information but it must have it. For every symbol must have denotation, that is, must imply the existence of some thing to which it is applicable. It may be a mere fiction; we may know it to be fiction; it may be intended to be a fiction and the very form of the word may hint that intention as in the case of abstract terms such as whiteness, nonentity, and the like. In these cases, we pretend that we hold realistic opinions for the sake of indicating that our propositions are meant to be explicatory or analytic. But the symbol itself always pretends to be a true symbol and hence implies a reference to real things. Thus, no matter how general a symbol may be, it must have some connotation limiting its denotation; it must refer to some determinate form; but it must also connote reality in order to denote at all; but all that has any determinate form has reality and thus this reality is a part of the connotation which does not limit the extension of the symbol. And so every symbol has information. To say that a symbol has information is as much as to say that it implies that it is equivalent to another symbol different in connotation. (Peirce 1865, "Harvard Lecture 11", CE 1, 286–288).

### Selection 36

 There are certain pseudo-symbols which are formed by combinations of symbols, and which must therefore be considered in logic, which lack either denotation or connotation. Thus, cats and stoves is a symbol wanting in connotation because it does not purport to relate to any definite quality. Tailed men wants denotation; for though it implies that there are men and that there are tailed things, it does not deny that these classes are mutually exclusive. All such terms are totally wanting in information. In short the formula: Connotation × Denotation = Information holds good thoroughly. (Peirce 1865, "Harvard Lecture 11", CE 1, 288).

### Selection 37

 The difference between subject and predicate was also considered in the last lecture. The subject is usually defined as the term determined by the proposition, but as the predicates of A, E, and I are also determined, this definition is inadequate. We were led to substitute for it the following:— The subject is the term determined in connotation and determining denotation; the predicate is the term determined in denotation and determining in connotation. We found that a term may be subject by virtue of being denotative or by virtue of being informative and that a term may be predicate by virtue either of being connotative or informative. But the reference of both subject and predicate cannot be informative. Thus we have three kinds of judgments: IC DC DI In the first case the subject is informative, the predicate connotative; that is to say, the connotation of the symbol which forms the subject is explicated in the predicate. Such judgments, usually called explicatory or analytic, I call connotative. In the second case the subject is denotative, the predicate connotative; that is to say, the thing which is denoted by the subject is said to embody the form connoted by the predicate. I call these judgments informative. In the third case the subject is denotative, the predicate is informative. That is, the thing which the subject denotes is offered as an example of the application of the symbol which forms the predicate. I call such judgments denotative. (Peirce 1865, "Harvard Lecture 11", CE 1, 288–289).

### Selection 38

Having thus far established:

 1st. The distinction of thing, form, and representation; together with the subsidiary one of object, logos, and image; 2nd. The distinction of sign, symbol, copy; [index, symbol, icon]; 3rd. The definition of logic as the general condition of the reference of symbols to objects; 4th. The difference between deduction, induction, and hypothesis; 5th. The fact that every mental representation is a symbol in a loose sense, and that every conception is so strictly; 6th. The fact that hypothesis gives terms or problematic propositions; inductions propositions strictly speaking — assertory propositions; and deduction apodictic propositions or syllogisms proper. That thus every elementary conception implies hypothesis and every judgment induction; 7th. The relations of denotation, connotation, and information; and 8th. The peculiarities of simple, enumerative, and conjunctive terms;

we found ourselves in a condition to solve the question of the grounds of inference by putting together these materials. (Peirce 1865, CE 1, 289).

### Selection 39

Peirce continues his remarks on the problem of the grounds of inference:

 In the first place with reference to the nature of the problem itself. It is not required to prove that deduction, induction, or hypothesis are valid. On the contrary, they are to be accepted as conditions of thought. It had been shown in previous lectures that they are so. Nor was a mode of calculating the probability of an induction or hypothesis now demanded; this being a merely subsidiary problem at best and one which may for ought we could yet see, be absurd. What we now wanted was an articulate statement and a satisfactory demonstration of those transcendental laws which give rise to the possibility of each kind of inference. Those grounds of possibility we found to be that All things, forms, symbols are symbolizable. For these laws must refer to symbolization because symbolization and inference are the same. As grounds of possibility they must refer to the possibility of symbolization. As logical laws they must consider the reference of symbols in general to objects. Now symbols in general have three relations to objects; namely so far as the latter contain things, forms, symbols. Finally as general principles they must be universal. (Peirce 1865, CE 1, 289–290).

### Selection 40

 Each ground-principle must be proved entirely by that same kind of inference which it supports. But we cannot arrive at any conclusion by mere deduction except about symbols. We cannot arrive at any conclusion by mere induction except about things. And we cannot arrive at any conclusion by mere hypothesis except about forms. Hence the ground of deduction relates to symbols; that of induction to things; that of hypothesis to forms. The three principles were proved by the several kinds of inference with certainty. The inductive proof attained certainty by considering all the instances that could be taken. And the hypothetic inference attained certainty by having only a subjective character. The influence of the three principles was shown in the case of deduction by the rule of Nota notae without which there could be no deduction. In the case of Induction by the affirmative denotative proposition which must always be the first premiss. And in the case of Hypothesis by the Universal connotative proposition which must always be the second premiss. (Peirce 1865, CE 1, 290).

### Selection 41

 Every induction, then, and every hypothesis yields a certain amount of truth. I might also show that no induction or hypothesis is completely true except such as we call cognitions a priori. For the chance against it is infinite. Hence, the question what is the 'probability' of an induction or hypothesis is senseless and the true question is how much truth does an induction contain. For the same reasons by how much truth should not be meant what proportion of inferences therefrom are true but simply of how much value are certain premisses in giving us truth by induction or hypothesis. We must distinguish therefore the truth which an inductive or hypothetic conclusion may have by accident from that which it must have from the nature of the facts explained. The former cannot properly be estimated. The latter can. For to consider first induction; if the same conclusion result inductively as the least truthful explanation possible of two different sets of facts, it is plain that a certain amount of truth it is obliged to have on account of each instance, that is on account of the extension of the subject of the fact. And each instance determines a certain amount of truth independently of the others. So that the number of different kinds of instances measures the least amount of truth the induction can have. In the same way with hypothesis the number of different properties explained measures the least possible truth of the hypothesis. (Peirce 1865, CE 1, 293–294).

### Selection 42

 In this way truth is measured upon a scale of numbers from one to infinity. And thus we cannot measure the ratio of the truth to the falsehood but only the ratio between the pregnancy of two sets of facts. Of any particular conclusion therefore we can only judge by ascertaining by further experience whether it can be improved. But the comparative usefulness of the facts upon which it proceeds may be estimated with an approach to precision. We may sum up then by the rule that the value of facts is in proportion to their number; and that from given facts the best inference when all possible retrenchment has been made, is the one which being inductive has the most comprehensive subject and which being hypothetic has the most extensive predicate. This seems to complete the logical theory of inference ... (Peirce 1865, CE 1, 294).

### Selection 43

 I fear I have wearied you in these lectures by dwelling so much upon merely logical forms. But to the pupil of Kant as to the pupil of Aristotle the Analytic of Logic is the foundation of Metaphysics. We find ourselves in all our discourse taking certain points for granted which we cannot have observed. The question therefore is what may we take for granted independent of all experience. The answer to this is metaphysics. But it is plain that we can thus take for granted only what is involved in logical forms. Hence, the necessity of studying these forms. In these lectures, one set of Logical forms has been pretty thoroughly studied; that of Hypothesis, Deduction, Induction. Another set has been partly studied, that of Denotation, Information, Connotation. Corresponding to these there are evidently certain conceptions of objects in general. To denotation corresponds the conception of an object, to information the conception of a real kind, and to connotation the conception of a logos or quality. So to Induction corresponds the conception of a Law, to Hypothesis the conception of a Case under a Law, and to Deduction the conception of a Result. There are also principles of the Judgment corresponding to these conceptions of which we have instances in the laws that all things, forms, symbols are symbolizable. All the principles that can be so derived from the forms of logic must be valid for all experience. For experience has used logic. Everything else admits of speculative doubt. (Peirce 1865, CE 1, 302).

## Anthematic Notes

### Anthematic Note 1

 Each man has his own peculiar character. It enters into all he does. It is in his consciousness and not a mere mechanical trick, and therefore it is by the principles of the last lecture a cognition; but as it enters into all his cognition, it is a cognition of things in general. It is therefore the man's philosophy, his way of regarding things; not a philosophy of the head alone — but one which pervades the whole man. This idiosyncrasy is the idea of the man, and if this idea is true he lives forever; if false, his individual soul has but a contingent existence. (Peirce 1866, CE 1, 501).

### Anthematic Note 2

 That the idiosyncrasy of a man — his peculiar character — is his peculiar philosophy, is best seen in the earliest stages of its formation before those complications have been developed which render it difficult to seize upon it. The cunning speeches of children just as they begin to talk often startle one by their philosophical nature. The drawer of Harper's Magazine has been filled for years with the sayings of "our three year old" — who seems blessed with perennial three-year-old-ness — but if all these stories are true, they are very valuable as showing the character of the childish mind in general, and particularly the philosophical tendencies of children. I shall not trouble you with the recitation of any of these funny stories — they are stale and therefore flat; but I will mention a case, which has nothing laughable in it — but which illustrates remarkably well how the peculiar differences of men are differences of philosophian method. (Peirce 1866, CE 1, 501).

### Anthematic Note 3

 A certain child who is rather backward in learning to speak, — not from dullness, but from a want of aptitude in imitating the words which it hears, — has got to use three words only; and what are these? Name, story, and matter. He says name when he wishes to know the name of a person or thing; story when he wishes to hear a narration or description; and matter — a highly abstract and philosophical term — when he wishes to be acquainted with the cause of anything. Name, story, and matter, therefore, make the foundation of this child's philosophy. What a wonderful thing that his individuality should have been shown so strongly, at that age, in selecting those three words out of all the equally common ones which he heard about him. Already he has made his list of categories, which is the principal part of any philosophy. (Peirce 1866, CE 1, 501).

### Anthematic Note 4

 Constantly, in using these words, this philosophy becomes more and more impressed upon him until, when he arrives at maturity of intellect, he may be able to show that it is a profound and legitimate classification. Tell me a man's name, his story, and his matter or character; and I know about all there is to know of him. Aristotle says there are two questions to be asked concerning anything: the oti and the dioti, the what and the why — the account of premisses and the rational account or explanation; or as this child would say the 'story' and the matter; but Aristotle has not noticed that previous to either of these questions must come the fixing of the attention upon the object — the determination of the mind to it as an object — and the demand for this determination is asking for its name. Here we have therefore in this child, a philosophy which furnishes an emendation upon the mighty Aristotle — the leader of the thought of ages, the prince of philosophers. (Peirce 1866, CE 1, 501–502).

### Anthematic Note 5

 But why should I presume to expound that soul's philosophy; could I enter fully into it he would have no private personality — he would not be the mysterious Island that every soul is to every other. No, I dare not attempt to fathom the awful depths of that child's possibilities; when he grows up, in some way and to some degree he will manifest his character, his philosophy; then we can judge as much of it as we can see, but its intrinsic worth we never can judge; it is hid forever in the bosom of its God. (Peirce 1866, CE 1, 502).

### Anthematic Note 6

 In dialectica autem vestra nullam existimavit esse nec ad melius vivendum nec ad commodius disserendum viam. Logic, on which your school lays such stress, he [Epicurus] held to be of no effect either as a guide to conduct or as an aid to thought. (Cicero, De Finibus, 1.19.63). Cicero, De Finibus Bonorum et Malorum, With an English Translation by H. Rackham, William Heinemann, London, UK, 1914, 1983.
 Who hath learnt any wit or understanding in Logique? Where are her faire promises? Nec ad melius vivendum, nec ad commodius disserendum: Neither to live better nor to dispute fitter. Montaigne, Essays, Book 3, Chapter 8. Eprint.
 Gentlemen and ladies, I announce to you this theory of immortality for the first time. It is poorly said, poorly thought; but its foundation is the rock of truth. And at least it will serve to illustrate what use might be made by mightier hands of this reviled science, logic, nec ad melius vivendum, nec ad commodius disserendum. (Peirce 1866, CE 1, page 502).

## Incidental Notes

### Incidental Note 1

I've arrived, yet again, at a problem that has occupied my attention, every now and then, since my very first readings of Peirce, and that is the question of whether and, if so, to what extent, a sign can be property of an object. The answer appears to depend on the strength of the senses in which we take the circle of thoughts like "to have", "to own", "to possess", or the substantives "possession", "property", and so on. In the weaker senses of the underlying schematism, signs can easily, all too easily be properties of objects, though one will likely hear the qualifications "accidental", "relative", "secondary", or words to that effect, quickly dispensed as a way to hedge the bet. To specify a stronger sense of eigen-valid ownership, emphatic terms like "categorical", "consensual", "genuine", "natural", "objective", "real", "universal", and a host of others may be recruited to drive home the point.

But the question behind the question is: What qualifies anything to be objective?

Here are just a few of my own thoughts on the matter.

I notice that I begin to consider calling something objective whenever there are lots and lots of different ways of looking at it, which is to say, if you think about it, that there are many different signs of it that can be sensibly related among each another, to wit, no objectivity without interoperability.

So consider this Semiotic Proof Of The Objectivity Of God: If there really were Nine Billion Names Of God, as in the Arthur Clarke story that I read as a child, then I would consider that a sufficient proof of God's objectivity. AC being British, I reckon this means 9 x 10^12 names, but I will have to check, as it's been a while since I last read the story.

• Incidental Musements:

### Incidental Note 2

Before I go on with Peirce's story of information, I want to stop for a while, at least long enough to redraw a favorite old picture of mine, that illustrates what all of this has to do with artificial and natural kinds, as they have been classically and humorously typified by the example that is commonly known as the case of the "Plucked Chicken".

The following Figure is largely self-explanatory.

o-------------------------------------------------o
|                                                 |
|                    Creature                     |
|                        o                        |
|                       / \                       |
|                      /   \                      |
|                     /     \                     |
|                    /       \                    |
|                   /         \                   |
|                  /           \                  |
|        Apterous o             o Biped           |
|                 |\           /|                 |
|                 | \         / |                 |
|                 |  \       /  |                 |
|                 |   \     /   |                 |
|                 |    \   /    |                 |
|                 |     \ /     |                 |
|                 |      o G    |                 |
|                 |     / \     |                 |
|                 |    /   \    |                 |
|                 |   /     \   |                 |
|                 |  /       \  |                 |
|                 | /         \ |                 |
|                 |/           \|                 |
|     Human Being o             o Plucked Chicken |
|                                                 |
| A   =   Apterous    =   featherless animal      |
| B   =   Bipedal     =   two-legged being        |
| C   =   Critter     =   creature, creation      |
| G   =   GLB(A, B)   =   A |^| B                 |
| H   =   Human Being                             |
| P   =   Plucked Chicken                         |
|                                                 |
o-------------------------------------------------o
Figure 1.  On Being Human


The way the joke goes, the straight man "defines" a human being H as an "apterous biped" A B, a two-legged critter without feathers, and then the wiseguy hits him over the head with a plucked chicken, and by dint of this koan, he achieves enlightenment about the marks that distinguish kindness of the artless kind from the crasser kinds of artificial kindness. Leastwise, at any rate, that's the way that I heard it.

Our focus at present is on the extra measure of constraint, in other words, the information, that comes between Pow(X), the full lattice of all possible subsets of the universe X, and Nat(X), the more constrained, determined, or informed lattice of "natural kinds" that we commonly acknowledge in our more practical outlooks on this universe of discourse.

The next two Figures present different ways of viewing the situation.

Think of the initial set-up as being cast in a lattice of arbitrary sets. Within that setting, the "greatest lower bound" (GLB) of the extensions of A and B is their set-theoretic intersection, G = GLB(A, B) = A |^| B. This G covers the desired class H but also admits the risible category P.

Suppose that we are clued into the fact that not all sets in Pow(X) are admissible, allowable, material, natural, pertinent, or relevant to the aims of the discussion in view, and that only some mysterious 'je ne sais quoi' subset of "natural kinds", Nat(X) c Pow(X), is at stake, a limitation that, whatever else it does, excludes the set P and all of that ilk from beneath GLB(A, B). Though it is difficult to say exactly how we are supposed to apply this global information, we "know" it in the sense of being able to detect its local effects, for instance, giving us the more "natural" lattice structures that are shown on the right sides of Figures 2 and 3. Relative to these "natural orders", we can observe that H = GLB(A, B), more precisely, the result of the lattice operation associated with the conjunction, GLB, or intersection of A and B gives us just the lattice element H. Thus in this more natural setting the proposed definition works okay.

o-------------------------------------------------o
|                                                 |
|          Pow       >>>--->>>       Nat          |
|                                                 |
|           C                         C           |
|           o                         o           |
|          / \                       / \          |
|         /   \                     /   \         |
|        /     \                   /     \        |
|       /       \                 /       \       |
|      /         \               /         \      |
|     /           \             /           \     |
|  A o             o B       A o             o B  |
|    |\           /|           |            /     |
|    | \         / |           |           /      |
|    |  \       /  |           |          /       |
|    |   \     /   |           |         /        |
|    |    \   /    |           |        /         |
|    |     \ /     |           |       /          |
|    |      o G    |           |      /           |
|    |     / \     |           |     /            |
|    |    /   \    |           |    /             |
|    |   /     \   |           |   /              |
|    |  /       \  |           |  /               |
|    | /         \ |           | /                |
|    |/           \|           |/                 |
|  H o             o P       H o                  |
|                                                 |
o-------------------------------------------------o
Figure 2.  Arbitrary Kinds Versus Natural Kinds


An alternative way to look at the transformation in our views as we pass from the arbitrary lattice Pow(X) to the natural lattice Nat(X) is presented in Figure 3, where the equal signs (=) suggest that the nodes for G and H are logically identified with each other. In this picture, the measure of the interval that previously existed between G and H, now shrunk to nil, affords a rough indication of the local quantity of information that went into forming the natural result.

o-------------------------------------------------o
|                                                 |
|          Pow       >>>--->>>       Nat          |
|                                                 |
|           C                         C           |
|           o                         o           |
|          / \                       / \          |
|         /   \                     /   \         |
|        /     \                   /     \        |
|       /       \                 /       \       |
|      /         \               /         \      |
|     /           \             /           \     |
|  A o             o B       A o             o B  |
|    |\           /|            \           /     |
|    | \         / |             \         /      |
|    |  \       /  |              \       /       |
|    |   \     /   |               \     /        |
|    |    \   /    |                \   /         |
|    |     \ /     |                 \ /          |
|    |      o G    |                G o           |
|    |     / \     |                  =           |
|    |    /   \    |                  =           |
|    |   /     \   |                  =           |
|    |  /       \  |                  =           |
|    | /         \ |                  =           |
|    |/           \|                  =           |
|  H o             o P              H o           |
|                                                 |
o-------------------------------------------------o
Figure 3.  Arbitrary Kinds Versus Natural Kinds


### Incidental Note 3

Seeing as how I have trouble understanding any thing at all of much complexity without a picture to guide me -- and not to victimize me! said Jon apotropaically -- whether it's that I can't get into it in the first place, or that I can't hang onto it for very long if I do, I cannot help but to keep trying to form a clearer picture of what Peirce is saying about these relationships of the kinds of signs and the aspects of the sign relation to the kinds of inference that serve a function in the "logic of science", or inquiry.

It was not my intention to keep you in suspense quite so long about the sorts of things that go into the "objective framework" (OF) of my diploid arrangement, indeed, I have discussed this on numerous prior occasions, but this time I wanted make the explanation of the plan as clear as I possibly could, and that has obliged me to do a bit of stalling before I attempted my next installment.

So it's back to the drawing board, and the half-wetted diptych, to see if I can paint a congenial picture of icons and indices. This time around I will temporarily set aside trying to fathom all of the ins and outs of Peirce's relatively intricate cases of conjunctive terms and disjunctive terms that fall short of being genuine symbols, and just try to detail my own way of seeing the forms of icons and indices in this dual frame.

How to begin? It is said that the usual vertebrate brain begins with a pleroma of neural connections that has to be trimmed as its creature grows and learns. Just by way of analogy, then, nothing more literal than that, let me draw a Figure that is meant to suggest a sign relation with all possible 3-ads of some 3-ple product space !O!x!S!x!I!.

With the powers invested in me by my poet's license and the full extent of pictographic conventionality that I may have in my command, let me draw the following picture to suggest a sign relation Q = !O!x!S!x!I!. Taken in its own right, Q has the structure of a 3-partite hypergraph, but the Figure below is intended merely to approximate selected aspects of its plenipotential structure, suggesting a complete bigraph, that is, a complete 2-partite graph, stretching between the points of !O! and the points of !S! = !I!, finished up with a complete graph on the points of a syntactic set !S! = !I!. According to long-standing conventions, these graphs can be written as K_m,n = K(!O!, !S!) and K_n = K(!S!), where m, n are the number of points in !O! and !S! = !I!, respectively.

Among those in the know, the technical gnomen for a pleromic sign relation of this empty-full variety is a "muddle".

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|              o                             s )            |
|              o     ·                 ·     s ))           |
|              o     ·     ·     ·     ·     s )))          |
|              o     ·     ·     ·     ·     s ))))         |
|              o     ·     ·     ·     ·     s )))))        |
|              o · · · · · · · · · · · · · · s ))))))       |
|              o     ·     ·     ·     ·     s )))))        |
|              o     ·     ·     ·     ·     s ))))         |
|              o     ·     ·     ·     ·     s )))          |
|              o     ·                 ·     s ))           |
|              o                             s )            |
|                                                           |
|                                                           |
| Muddled Sign Relation Q = !O!x!S!x!I!                     |
o-----------------------------------------------------------o


### Incidental Note 4

 The Critique Of Pure Reason Immanuel Kant (1724–1804) Translated by J.M.D. Meiklejohn Preface to the First Edition, 1781 Human reason, in one sphere of its cognition, is called upon to consider questions, which it cannot decline, as they are presented by its own nature, but which it cannot answer, as they transcend every faculty of the mind. It falls into this difficulty without any fault of its own. It begins with principles, which cannot be dispensed with in the field of experience, and the truth and sufficiency of which are, at the same time, insured by experience. With these principles it rises, in obedience to the laws of its own nature, to ever higher and more remote conditions. But it quickly discovers that, in this way, its labours must remain ever incomplete, because new questions never cease to present themselves; and thus it finds itself compelled to have recourse to principles which transcend the region of experience, while they are regarded by common sense without distrust. It thus falls into confusion and contradictions, from which it conjectures the presence of latent errors, which, however, it is unable to discover, because the principles it employs, transcending the limits of experience, cannot be tested by that criterion. The arena of these endless contests is called Metaphysic. http://www.philosophy.ru/library/kant/01/cr_pure_reason.html

### Incidental Note 5

No, I mean *really, really* irritating doubts ...

 The precursors of hatred arise from the infant's response to what William James (1890) called the "booming buzzing confusion" that assaults the infant's sensorium at perception's birth. The "stranger anxiety" evident as early as eight months indicates that the mental capacity to perceive differences in objects and to organize subjective psychic forces has already begun. Freud (1915) tells us that "hate, as a relation to objects, is older than love" (p. 139). Freud continues, "As an expression of the reaction of unpleasure evoked by objects, it always remains in an intimate relation with the self-preservative instincts; so that sexual and ego-instincts can readily develop an antithesis which repeats that of love and hate". Eloise Moore Agger (issue ed.), "Prologue", Special Issue on "Hatred And Its Rewards", Psychoanalytic Inquiry 20(3), 2000. Eprint.

We began, as always, 'in mudias res', in that irritatingly doubtful state of "booming buzzing confusion" that clued us in mostly to the anterior projection of William James' inciteful Psychology and we woke into a stream of consciousness staring at the appended picture of a "muddled sign relation" Q = !O!x!S!x!I!.

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|              o                             s )            |
|              o     ·                 ·     s ))           |
|              o     ·     ·     ·     ·     s )))          |
|              o     ·     ·     ·     ·     s ))))         |
|              o     ·     ·     ·     ·     s )))))        |
|              o · · · · · · · · · · · · · · s ))))))       |
|              o     ·     ·     ·     ·     s )))))        |
|              o     ·     ·     ·     ·     s ))))         |
|              o     ·     ·     ·     ·     s )))          |
|              o     ·                 ·     s ))           |
|              o                             s )            |
|                                                           |
|                                                           |
| Muddled Sign Relation Q = !O!x!S!x!I!                     |
o-----------------------------------------------------------o


There are many ways that a muddle can resolve itself, if you'll excuse the animistical sympathetic fallacy of yielding the muddle credit for its own resolution.

One may regard the process of resolution as the differential reinforcement of certain connections in preference to others, or as the emphatic differentiation of certain figures in the carpet or the tapestry that is "finding itself" being woven on the loom of this tangled skein, or brain, as the case be.

A long time before people had their minds quite set on our present notions of set theory, they used to speak of "general denotation" or "plural reference", in which a sign was related to a manifold variety of objects, whether "equivocally", in different senses, or "univocally, in the same sense, connotation, or definition of the sign (for example, the term or the word) in question, very roughly as might be suggested by the following Figure:

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|              o                                            |
|              o     ·                                      |
|              o     ·     ·                                |
|              o     ·     ·     ·                          |
|              o     ·     ·     ·     ·                    |
|              o · · · · · · · · · · · · · · s              |
|              o     ·     ·     ·     ·                    |
|              o     ·     ·     ·                          |
|              o     ·     ·                                |
|              o     ·                                      |
|              o                                            |
|                                                           |
|                                                           |
| General Denotation Or Plural Reference                    |
o-----------------------------------------------------------o


So this is one sort of pattern of highlights, reinforcement, or saliency that we often find spontaneously generating itself and emerging from the muddle like some dragonfly from a pond's muck.

The roughly dual pattern of pregnance comes soon to mind, where this would show something like the next arrangement of emphases.

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|                                            s )            |
|                                      ·     s ))           |
|                                ·     ·     s )))          |
|                          ·     ·     ·     s ))))         |
|                    ·     ·     ·     ·     s )))))        |
|              o · · · · · · · · · · · · · · s ))))))       |
|                    ·     ·     ·     ·     s )))))        |
|                    ·     ·     ·     ·     s ))))         |
|                          ·     ·     ·     s )))          |
|                                      ·     s ))           |
|                                            s )            |
|                                                           |
|                                                           |
| Referential And Semiotic Equivalence Classes              |
o-----------------------------------------------------------o


This is a genroic type of a motif that we shall find to be of extremely fruitful use again and again, where a bunch of signs ripens and falls into various and sundry "equivalence classes", either because they all denote the same object or because they all connote one another, or most happily of all, both together. These are known as "referential equivalence classes" (REC's)and "semiotic equivalence classes" (SEC's), respectively.

### Incidental Note 6

Ah Bartleby! Ah Humanity!

Let us examine a special case of "general denotation" or "plural reference", one in which we select a sample, perhaps but a single representative object, to serve as a sign of the entire collection from which it was apothematized.

Here is the Figure:

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|              o                                            |
|              o     ·                                      |
|              o     ·     ·                                |
|              o     ·     ·     ·                          |
|              o     ·     ·     ·     ·                    |
|              o = = = = = = = = = = = = = = s              |
|              o     ·     ·     ·     ·                    |
|              o     ·     ·     ·                          |
|              o     ·     ·                                |
|              o     ·                                      |
|              o                                            |
|                                                           |
|                                                           |
| General Denotation Or Plural Reference Via A Sample       |
o-----------------------------------------------------------o


The very same entity now serves in a double role, hopefully without too much duplicity, but we all know how that can be. Polled as a member of its own constituency, it functions as any other object of any other sign. Invested in the office of a typical representative, it serves its term as any term might, standing for the body politic from which it was lift.

Every setting of a general denotation or a plural reference, not just the ones of this exemplary species, is amenable to having its part in the muddle sorted out along the lines of at least two different "factorization schemes". These are easier visualized than verbalized, so here is the Figure:

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|              q = Humanity                                 |
|             /|\    ·                                      |
|            / | \         ·                                |
|           /  |  \              ·                          |
|          /   |   \                   ·                    |
|         ooooooooooo }}}}}}}}}}}}}}}}}}}}}} s =  Bartleby  |
|          ·   ·   ·                         |              |
|                 ·  ·  ·                    |              |
|                        · · ·               |              |
|                               ···          |              |
|                                      ·     |              |
|                                            i = 'Humanity' |
|                                                           |
|                                                           |
| Factorization Of A Fiber Via Objects And Via Signs        |
o-----------------------------------------------------------o


I could explain more today, but I would prefer not to.

### Incidental Note 7

Let me correct one major slip and a few minor typos in the Figure that I gave last time, and then proceed to explain what I can see rightly in this picture of two distinct ways of factoring a fiber. The more serious mislabeling is that the interpretant sign i that was newly inserted in the interpretive framework should have been set equal to the sign value "Humanity" and not the property value Humanity. The quotes are needed in order to emphasize that i is a sign or a concept, or being interpreted that way, and not being regarded as an objective attribute, intension, property, or quality, not as viewed via the interpretive frame of the scope.

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|              q = Humanity                                 |
|             /|\    ·                                      |
|            / | \         ·                                |
|           /  |  \              ·                          |
|          /   |   \                   ·                    |
|         ooooooooooo }}}}}}}}}}}}}}}}}}}}}} s =  Bartleby  |
|          ·   ·   ·                         |              |
|                 ·  ·  ·                    |              |
|                        · · ·               |              |
|                               ···          |              |
|                                      ·     |              |
|                                            i = 'Humanity' |
|                                                           |
|                                                           |
| Factorization Of A Fiber Via Objects And Via Signs        |
o-----------------------------------------------------------o


That should take care of the immediate potential for confusion.

### Incidental Note 8

I see that I am using a few bits of language about factors and fibers that I have not mentioned in quite a while, and not only that but I am using it in ways that I learned at different times from different folks, and so I am using it slightly equivocally. So let me try to straighten that out before going on with the story of Bartleby, his preferentials, and the apothematic dimensionals of his human, all too human humanity.

Let's say that we have an ordinary function f : X -> Y. If we pick out one y from the target or the codomain Y, then it either has x's from the source or the domain X that are assigned, or mapped, or sent to it by f or it doesn't. Let's say it does. Then here is the picture of what I will frequently call the "fiber (of f) at y":

o-----------------------------o-----------------------------o
|        Source Domain        |        Target Codomain      |
o-----------------------------o-----------------------------o
|                                                           |
|              x                                            |
|              x     ·                                      |
|              x     ·     ·                                |
|              x     ·     ·     ·                          |
|              x     ·     ·     ·     ·                    |
|              x     ·     ·     ·     ·     y              |
|              x     ·     ·     ·     ·                    |
|              x     ·     ·     ·                          |
|              x     ·     ·                                |
|              x     ·                                      |
|              x                                            |
|                                                           |
|                                                           |
| Functional Fiber                                          |
o-----------------------------------------------------------o


Very often the reason that one is interested in these varieties of fibers under a given function is so that one can follow them "upstream" or "backward", functionally speaking, in other words, toward the "source" of the functional value under investigation. That leads rather naturally to the other mathematical usage for the word "fiber" that I have in mind. Here are the definitions as I formulated them in my dissertation proposal:

 The "fiber" of a codomain element y in Y under a function f : X -> Y is the subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X. In other language that is often used, the fiber of y under f is called the "antecedent set", the "inverse image", the "level set", or the "pre-image" of y under f. All of these equivalent concepts are defined as follows: Fiber of y under f = f^(-1)(y) = {x in X : f(x) = y}. In the special case where f is the indicator function f_Q of the set Q c X, the fiber of %1% under indicator function f_Q is just the set Q back again: Fiber of %1% under f_Q = (f_Q)^(-1)(%1%) = {x in X : f_Q (x) = %1%} = Q. In this specifically boolean setting, as in the more generally logical context, where "truth" under any name is especially valued, it is worth devoting a specialized notation to the "fiber of truth" in a proposition, to mark the set that it indicates with a particular ease and explicitness. For this purpose, I introduce the use of "fiber bars" or "ground signs", written as "[| ... |]" around a sentence, or the sign of a proposition, and whose application is defined as follows: If f : X -> %B%, then [| f |] = f^(-1)(%1%) = {x in X : f(x) = %1%}. The definition of a fiber, in either the general or the boolean case, is a purely nominal convenience for referring to the antecedent subset, the inverse image under a function, or the pre-image of a functional value. The definition of an operator on propositions, signified by framing the signs of propositions with fiber bars or ground signs, remains a purely notational device, and yet the notion of a fiber in a logical context serves to raise an interesting point. By way of illustration, it is legitimate to rewrite the above definition in the following form: If f : X -> %B%, then [| f |] = f^(-1)(%1%) = {x in X : f(x)}. The set-builder frame "{x in X : ... }" requires a sentence to fill in the blank, as with the sentence "f(x) = %1%" that serves to fill the frame in the initial definition of a logical fiber. And what is a sentence but the expression of a proposition, in other words, the name of an indicator function? As it happens, the sign "f(x)" and the sentence "f(x) = %1%" represent the very same value to this context, for all x in X, that is, they are equal in their truth or falsity to any reasonable interpreter of signs or sentences in this context, and so either one of them can be tendered for the other, in effect, exchanged for the other, within this frame. http://suo.ieee.org/email/msg07409.html http://suo.ieee.org/email/msg07416.html

### Incidental Note 9

Last time we looked at an ordinary function f : X -> Y, and we glommed onto a single fiber of f, considered in one of two ways: (1) a set of ordered pairs F c X x Y such that <x, y> in F if and only f(x) = y, or else (2) a subset of X, horrifically asciified as f^(-1)(y) c X.

o-----------------------------o-----------------------------o
|        Source Domain        |        Target Codomain      |
o-----------------------------o-----------------------------o
|                                                           |
|              x                                            |
|              x     ·                                      |
|              x     ·     ·                                |
|              x     ·     ·     ·                          |
|              x     ·     ·     ·     ·                    |
|              x     ·     ·     ·     ·     y              |
|              x     ·     ·     ·     ·                    |
|              x     ·     ·     ·                          |
|              x     ·     ·                                |
|              x     ·                                      |
|              x                                            |
|                                                           |
|                                                           |
| Functional Fiber                                          |
o-----------------------------------------------------------o


Any function f : X -> Y, which is, after all, exactly the same as the relation f c X x Y, can be treated as an assortment of fibers of the first sort. So it is easy to grasp the elementary fact of category theory that any function whatsoever can be factored into an epic (surjective, "onto") and a monic (injective, "one to one") sequence of composed functions, as illustrated here for one fiber:

o-------------------o-------------------o-------------------o
|   Source Domain   |   Medial Domain   |   Target Domain   |
o-------------------o---------------------------------------o
|                                                           |
|         x                                                 |
|         x   ·                                             |
|         x   ·   ·                                         |
|         x   ·   ·   ·                                     |
|         x   ·   ·   ·   ·                                 |
|         x   ·   ·   ·   ·   m · · · · · · · · > y         |
|         x   ·   ·   ·   ·                                 |
|         x   ·   ·   ·                                     |
|         x   ·   ·                                         |
|         x   ·                                             |
|         x                                                 |
|                                                           |
|                                                           |
| Functional Fiber Factors                                  |
o-----------------------------------------------------------o


In sum, an arbitrary f : X -> Y can always be factored into a pair of functions of the types g : X -> M and h : M -> Y, where g is surjective, h is injective, and f = h o g, here using the "left-composition" convention according to which the composition h o g is defined by (h o g)(x) = h(g(x)).

To finish off the topic of factoring functions for now, I will give the commutative diagram and the additional bit of explanation that I gave once before, to wit:

We began with the trusim from category theory, at least, the sorts of "concrete categories" of sets and functions that will be most salient in the minds of most everybody: That an arbitrary arrow factors into a couple of pieces, an epic on which a monic ensues, 'Iliad' and 'Odyssey', if you will, and if you catch my drift, and whether you will or not, 'tis true.

|                   f
|               arbitrary
|         X o-------------->o Y
|            \             ^
|             \           /
|       g      \         /    h
|   surjective  \       /  injective
|     "epic"     \     /    "monic"
|                 \   /
|                  v /
|                   o
|                   M


Now, there's a catch here -- there's always a catch, the way I see it -- leastwise, once we begin to think so systematically as to be working inside any sort of category at all, instead of merely picking up on this or that isolated instance of an arbalistrary functional arrow, then this ostensibly trivial truism becomes contingent on the list of a "suitable transitional object" (STO), like M in our example, and of the "requisite intermedi-arrows" (RIA's), like g and h, explicitly listed within the formal category in question. Otherwise, "you just cannot get there from here" is the only thing that answers to your desire for mediation.

### Incidental Note 10

We have been contemplating a few of the more facile relationships among functions, their fibers, and their factorizations. Just to be as clear about all this as we possibly can, let us look at one very simple but perfectly generic picture of the global situation.

The next Figure illustrates a function f : X -> Y with this data:

X = {x_1, x_2, x_3, x_4, x_5}
Y = {y_1, y_2, y_3, y_4, y_5, y_6}
f = {(x_1, y_2), (x_2, y_2), (x_3, y_2), (x_4, y_5), (x_5, y_5)}
o-----------------------------o-----------------------------o
|        Source Domain        |        Target Codomain      |
o-----------------------------o-----------------------------o
|                                                           |
|          x_1 o--------------------------·  o y_1          |
|                                          \                |
|                                           \               |
|          x_2 o-----------------------------o y_2          |
|                                           /               |
|                                          /                |
|          x_3 o--------------------------·  o y_3          |
|                                                           |
|                                                           |
|          x_4 o--------------------------·  o y_4          |
|                                          \                |
|                                           \               |
|                                            o y_5          |
|                                           /               |
|                                          /                |
|          x_5 o--------------------------·  o y_6          |
|                                                           |
|                                                           |
| Functional Fibers                                         |
o-----------------------------------------------------------o


Just by way of introducing a few bits of useful terminology, I take the liberty of expressing the following observations:

 Dom(f) = Domain(f) = X Cod(f) = Codomain(f) = Y Ran(f) = Range(f) = {y_2, y_5} Cor(f) = Corange(f) = X

Naturally, Dom(f) = Cor(f) for any relation f that happens to be a function, but I am introducing these terms as employed in a more general relational context.

The fibers of f are either one of these constructions:

1. Relational Fibers:
f & y_2 = {(x_1, y_2), (x_2, y_2), (x_3, y_2)}
f & y_5 = {(x_4, y_5), (x_5, y_5)}
2. Partitional Fibers:
f · y_2 = {x_1, x_2, x_3}
f · y_5 = {x_4, x_5}

There are, of course, many different systems of notation and terminology for these things.

### Incidental Note 11

From the cosmic urn of all possible functions we draw the unique random function f : X -> Y.

o-----------------------------o-----------------------------o
|        Source Domain        |        Target Codomain      |
o-----------------------------o-----------------------------o
|                                                           |
|          x_1 o--------------------------·  o y_1          |
|                                          \                |
|                                           \               |
|          x_2 o-----------------------------o y_2          |
|                                           /               |
|                                          /                |
|          x_3 o--------------------------·  o y_3          |
|                                                           |
|                                                           |
|          x_4 o--------------------------·  o y_4          |
|                                          \                |
|                                           \               |
|                                            o y_5          |
|                                           /               |
|                                          /                |
|          x_5 o--------------------------·  o y_6          |
|                                                           |
|                                                           |
| Functional Fibers                                         |
o-----------------------------------------------------------o


Now we form the canonical decomposition or factorization of f into a surjective function followed by an injective function.

o-------------------o-------------------o-------------------o
|   Source Domain   |   Middle Domain   |   Target Domain   |
o-------------------o---------------------------------------o
|                                                           |
|          x_1 o-----------·                 o y_1          |
|                           \                               |
|                            \ m_1                          |
|          x_2 o--------------o------------->o y_2          |
|                            /                              |
|                           /                               |
|          x_3 o-----------·                 o y_3          |
|                                                           |
|                                                           |
|          x_4 o-----------·                 o y_4          |
|                           \                               |
|                            \ m_2                          |
|                             o------------->o y_5          |
|                            /                              |
|                           /                               |
|          x_5 o-----------·                 o y_6          |
|                                                           |
|                                                           |
| Functional Factors                                        |
o-----------------------------------------------------------o


Here we have the following data:

f = h o g
f : X -> Y, arbitrary
g : X -> M, surjective
h : M -> Y, injective
X = {x_1, x_2, x_3, x_4, x_5}
M = {m_1, m_2}
Y = {y_1, y_2, y_3, y_4, y_5, y_6}
f = {(x_1, y_2), (x_2, y_2), (x_3, y_2), (x_4, y_5), (x_5, y_5)}
g = {(x_1, m_1), (x_2, m_1), (x_3, m_1), (x_4, m_2), (x_5, m_2)}
h = {(m_1, y_2), (m_2, y_5)}

I think that you can see from this sufficient bit that it lies in the very nature of a function for it factorizing to permit.

### Incidental Note 12

I think that we have fairly well convinced ourselves — at least, I am reasonably sure that some of us have — that every function can be factored into an "onto" followed by a "one-to-one" mapping, as shown here:

o-------------------o-------------------o-------------------o
|   Source Domain   |   Middle Domain   |   Target Domain   |
o-------------------o---------------------------------------o
|                                                           |
|          x_1 o-----------·                 o y_1          |
|                           \                               |
|                            \ m_1                          |
|          x_2 o--------------o------------->o y_2          |
|                            /                              |
|                           /                               |
|          x_3 o-----------·                 o y_3          |
|                                                           |
|                                                           |
|          x_4 o-----------·                 o y_4          |
|                           \                               |
|                            \ m_2                          |
|                             o------------->o y_5          |
|                            /                              |
|                           /                               |
|          x_5 o-----------·                 o y_6          |
|                                                           |
|                                                           |
| Factured Fiber Trails                                     |
o-----------------------------------------------------------o


So patent is the pending of Damocles' Razor on our modern incre-mentalities that we would scarcely dare to think of it this way without a little bit of prodding, but it is possible to treat this functional fractionation process as a case of transmuting a 2-adic relation f c X x Y into a 3-adic relation L c X x M x Y.

In our present example we have the data:

f c X x Y
X = {x_1, x_2, x_3, x_4, x_5}
Y = {y_1, y_2, y_3, y_4, y_5, y_6}
f = {(x_1, y_2), (x_2, y_2), (x_3, y_2), (x_4, y_5), (x_5, y_5)}

and

L c X x M x Y
X = {x_1, x_2, x_3, x_4, x_5}
M = {m_1, m_2}
Y = {y_1, y_2, y_3, y_4, y_5, y_6}
L = {(x_1, m_1, y_2), (x_2, m_1, y_2), (x_3, m_1, y_2), (x_4, m_2, y_5), (<x_5, m_2, y_5)}

### Incidental Note 13

 Comprehension. The sum of characteristics which connote a class notion symbolized by a general term. Also, the features common to a number of instances or objects. Thus, the connotation (qv) or intension (qv) of a concept. (Otto F. Kraushaar, in D.D. Runes (ed.) Dictionary of Philosophy, 1962).

I had the impression that Peirce's usage was conforming to some standard tradition, Scholastic or Kantian or both, but have not taken the time to trace it -- I imagine that somebody hereabouts would probably have it on the tip of their tongue, anyways. My affliction with OSSD (obsessive syntactic symmetry disorder) made it difficult for me to switch from the equal temperaments of extension/intension, but I have the sense that peirce is making a technical distinction between "an intension", as being any one property of an object of a concept, and "the comprehension", as being the collection or the conjunction of "all" of the properties of the object that are relevant to some context of discussion. But that is just my unchecked sense of what he's saying, and I'm still just guessing. So I went with Kraushaar, as he seems to cover the Kantian line. And there's always a chance that Runes is derivative of Peirce, via the century dictionary and other sources like that. Detective work needed here.

## Reflective Note

What Peirce achieved in this line of inquiry was to develop a theory of information out of purely logical considerations. The majority of the flashes of insight that he propagates in 1865-1866 will not be seen again until the first shots of our own information and computing revolutions. These are nothing less than intimations of the "capacity limitations" of signals, symbols, and their users, due in part to the physical nature of actual sign tokens, and in part to the fact that we fallible and mortal finite information creatures are forever out of existential necessity forced to learn and to think under conditions of imperfect information, and thus are we ever bound to "reason under uncertainty", with all of the usual afflictions of biased opinion, partial knowledge, and bounded rationality. Strangely enough, it is the very improvements in the speed and capacity of our computing and information media since the early days of these revolutions that has brought on a reactionary tendency to forget the basic principles on which the whole republic of information is founded. And thus it happens that Peirce's way of viewing information is not just "enlightened for his time", as people say, but enlightened also in comparison to ours.

## Commentary Notes

### Commentary Note 1

Peirce's incipient theory of information, that he appears to have developed by sheer force of logical insight from his early understanding of signs and scientific inquiry, is not an easy subject to grasp in its developing state. An attempt to follow his reasoning step by step might well begin with this:

 Let us now return to the information. The information of a term is the measure of its superfluous comprehension.

Today we would say that information has to do with constraint, law, redundancy. I think that Peirce is talking about more or less the same thing under the theme of superfluous comprehension, where the comprehension of a term or expression is the collection of properties, also known as intensions, that it implies about the things to which it applies.

### Commentary Note 2

 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.

When we set about comprehending the comprehension of a sign, say, a term or expression, we run into a very troublesome issue as to how many intensions (predicates, properties, qualities) an object of that sign has. For how do we quantify the number of qualities a thing has? Without some more or less artificial strait imposed on the collection of qualities, the number appears without limit.

Let's pass this by, as Peirce does, for now, and imagine that we have fixed on some way of speaking sensibly about the comprehension of a sign in a particular set of signs, the collection of which we may use as a language or a medium.

Then we can begin to talk about the amount of redundancy, the superfluidity of comprehension, if you will, as Peirce does, that belongs to a given sign, and thus to its object.

### Commentary Note 3

 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.

In a sense of primal innocence, logical laws bind the vacuum state of any medium that is capable of bearing, delivering, nurturing, and preserving signal meanings. In other words, when we use symbols, not simple signs, in a channel, language, or medium that is constrained by logical laws, these laws do more than strain, they also exact the generation of symbols upon symbols to fill the requisite logical forms, and so there will always be lots more ways than one to say any given thing you might choose to say.

### Commentary Note 4

 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.

In order to speak of the meandering channel, the abdundancy of language, the superfluidity of media, the play in the wheel of symbolism, then, it is necessary to classify the different kinds of signs, the varied ways that signs up to and including symbols, namely, those that are interpretive by dint of their very essence, can be interpreted as being referential to their objects.

On running through this familiar yet ever strange refrain for another time, I see that I have scarcely begun to trace the sinews of the linkages among the three types of signs, "the differences there are in the manner in which different representations stand for their objects", the matter of extension and comprehension, and the whole life-cycle of inquiry that engages me most.

### Commentary Note 5

Signs, inquiry, and information. Let's focus on that for a while.

To put Peirce's examples more in line with the order of his three categories, let us renumber them in the following way:

 1. The conjunctive term "spherical bright fragrant juicy tropical fruit". 2.1. The disjunctive term "man or horse or kangaroo or whale". 2.2. The disjunctive term "neat or swine or sheep or deer".

Peirce suggests an analogy or a parallelism between the corresponding elements of the following triples:

 1 Conjunctive Term : Iconical Sign : Abductive Case 2 Disjunctive Term : Indicial Sign : Inductive Rule

Here is an overview of the two patterns of reasoning, along with the first steps of an analysis in sign-theoretic terms:

1. Conjunctive term "spherical bright fragrant juicy tropical fruit".

o-----------------------------o-----------------------------o
|   Objective Framework   |  Interpretive Framework  |
o-----------------------------o-----------------------------o
|                              |
|                t_1 t_2 ... t_5 t_6  |
|                 o  o    o  o   |
|                  *  *    *  *    |
|                   * *  * *     |
|                    * *  * *      |
|                     ****       |
|                     z o        |
|                      |\       |
|                      | \ Rule    |
|                      | \ y=>z    |
|                      |  \      |
|                   Fact |  \     |
|                   x=>z |   o y    |
|                      |  /     |
|                      |  /      |
|                      | / Case    |
|                      | / x=>y    |
|                      |/       |
|                     x o        |
|                              |
o-----------------------------------------------------------o
| Conjunctive Predicate z, Abduction to the Case x => y   |
o-----------------------------------------------------------o
|                              |
| !S! = !I! = {t_1, t_2, t_3, t_4, t_5, t_6, x, y, z}  |
|                              |
| t_1 = "spherical"                    |
| t_2 = "bright"                     |
| t_3 = "fragrant"                    |
| t_4 = "juicy"                      |
| t_5 = "tropical"                    |
| t_6 = "fruit"                      |
|                              |
| x  = "subject"                     |
| y  = "orange"                     |
| z  = "spherical bright fragrant juicy tropical fruit" |
|                              |
o-----------------------------------------------------------o

 A similar line of thought may be gone through in reference to hypothesis. In this case we must start with the consideration of the term: "spherical, bright, fragrant, juicy, tropical fruit". Such a term, formed by the sum of the comprehensions of several terms, is called a conjunctive term. A conjunctive term has no extension adequate to its comprehension. Thus the only spherical bright fragrant juicy tropical fruit we know is the orange and that has many other characters besides these. Hence, such a term is of no use whatever. If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once. On the other hand, if the conjunctive term is subject and we know that every spherical bright fragrant juicy tropical fruit necessarily has certain properties, it must be that we know more than that and can simplify the subject. Thus a conjunctive term may always be replaced by a simple one. So if we find that light is capable of producing certain phenomena which could only be enumerated by a long conjunction of terms, we may be sure that this compound predicate may be replaced by a simple one. And if only one simple one is known in which the conjunctive term is contained, this must be provisionally adopted. (Peirce, CE 1, 470).

2. Disjunctive term "neat or swine or sheep or deer".

o-----------------------------o-----------------------------o
|   Objective Framework   |  Interpretive Framework  |
o-----------------------------o-----------------------------o
|                              |
|                     w o        |
|                      |\       |
|                      | \ Rule    |
|                      | \ v=>w    |
|                      |  \      |
|                   Fact |  \     |
|                   u=>w |   o v    |
|                      |  /     |
|                      |  /      |
|                      | / Case    |
|                      | / u=>v    |
|                      |/       |
|                     u o        |
|                     ****       |
|                    * *  * *      |
|                   * *  * *     |
|                  *  *    *  *    |
|                 o  o    o  o   |
|                s_1 s_2    s_3 s_4  |
|                              |
o-----------------------------------------------------------o
| Disjunctive Subject u, Induction to the Rule v => w    |
o-----------------------------------------------------------o
|                              |
| !S! = !I! = {s_1, s_2, s_3, s_4, u, v, w}       |
|                              |
| s_1 = "neat"                      |
| s_2 = "swine"                      |
| s_3 = "sheep"                      |
| s_4 = "deer"                      |
|                              |
| u  = "neat or swine or sheep or deer"         |
| v  = "cloven-hoofed"                  |
| w  = "herbivorous"                   |
|                              |
o-----------------------------------------------------------o

 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. Peirce, CE 1, 468-469).

### Commentary Note 6

Before we return to Peirce's description of a near duality between icons and indices, involving a reciprocal symmetry between intensions and extensions of concepts that becomes perturbed to the breaking and yet the growing point by the receipt of a fresh bit of information, I think that it may help to recall a few pieces of technical terminology that Peirce introduced into this discussion.

Passage 1

 It is important to distinguish between the two functions of a word: 1st to denote something — to stand for something, and 2nd to mean something — or as Mr. Mill phrases it — to connote something. What it denotes is called its Sphere. What it connotes is called its Content. Thus the sphere of the word man is for me every man I know; and for each of you it is every man you know. The content of man is all that we know of all men, as being two-legged, having souls, having language, &c., &c. It is plain that both the sphere and the content admit of more and less. … Now the sphere considered as a quantity is called the Extension; and the content considered as quantity is called the Comprehension. Extension and Comprehension are also termed Breadth and Depth. So that a wider term is one which has a greater extension; a narrower one is one which has a less extension. A higher term is one which has a less Comprehension and a lower one has more. The narrower term is said to be contained under the wider one; and the higher term to be contained in the lower one. We have then: o-----------------------------o-----------------------------o |               |               | | What is 'denoted'     | What is 'connoted'     | |               |               | | Sphere           | Content          | |               |               | | Extension         | Comprehension       | |               |               | |      ( wider      |     ( lower       | | Breadth <         | Depth <          | |      ( narrower    |     ( higher      | |               |               | | What is contained 'under' | What is contained 'in'   | |               |               | o-----------------------------o-----------------------------o  The principle of explicatory or deductive reasoning then is that: Every part of a word's Content belongs to every part of its Sphere, or: Whatever is contained in a word belongs to whatever is contained under it. Now this maxim would not be true if the Extension and Comprehension were directly proportional to one another; this is to say if the Greater the one the greater the other. For in that case, though the whole Content would belong to the whole Sphere; yet only a particular part of it would belong to a part of that Sphere and not every part to every part. On the other hand if the Comprehension and Extension were not in some way proportional to one another, that is if terms of different spheres could have the same content or terms of the same content different spheres; then there would be no such fact as a content's belonging to a sphere and hence again the maxim would fail. For the maxim to be true, then, it is absolutely necessary that the comprehension and extension should be inversely proportional to one another. That is that the greater the sphere, the less the content. Now this evidently true. If we take the term man and increase its comprehension by the addition of black, we have black man and this has less extension than man. So if we take black man and add non-black man to its sphere, we have man again, and so have decreased the comprehension. So that whenever the extension is increased the comprehension is diminished and vice versa. (Peirce 1866, "Lowell Lecture 7", CE 1, 459–460).

Passage 2

 The highest terms are therefore broadest and the lowest terms the narrowest. We can take a term so broad that it contains all other spheres under it. Then it will have no content whatever. There is but one such term — with its synonyms — it is Being. We can also take a term so low that it contains all other content within it. Then it will have no sphere whatever. There is but one such term — it is Nothing. o------------------------o------------------------o |            |            | | Being        | Nothing       | |            |            | | All breadth     | All depth      | |            |            | | No depth       | No breadth      | |            |            | o------------------------o------------------------o  We can conceive of terms so narrow that they are next to nothing, that is have an absolutely individual sphere. Such terms would be innumerable in number. We can also conceive of terms so high that they are next to being, that is have an entirely simple content. Such terms would also be innumerable. o------------------------o------------------------o |            |            | | Simple terms     | Individual terms   | |            |            | o------------------------o------------------------o  But such terms though conceivable in one sense — that is intelligible in their conditions — are yet impossible. You never can narrow down to an individual. Do you say Daniel Webster is an individual? He is so in common parlance, but in logical strictness he is not. We think of certain images in our memory — a platform and a noble form uttering convincing and patriotic words — a statue — certain printed matter — and we say that which that speaker and the man whom that statue was taken for and the writer of this speech — that which these are in common is Daniel Webster. Thus, even the proper name of a man is a general term or the name of a class, for it names a class of sensations and thoughts. The true individual term the absolutely singular this & that cannot be reached. Whatever has comprehension must be general. In like manner, it is impossible to find any simple term. This is obvious from this consideration. If there is any simple term, simple terms are innumerable for in that case all attributes which are not simple are made up of simple attributes. Now none of these attributes can be affirmed or denied universally of whatever has any one. For let A be one simple term and B be another. Now suppose we can say All A is B; then B is contained in A. If, therefore, A contains anything but B it is a compound term, but A is different from B, and is simple; hence it cannot be that All A is B. Suppose No A is B, then not-B is contained in A; if therefore A contains anything besides not-B it is not a simple term; but if it is the same as not-B, it is not a simple term but is a term relative to B. Now it is a simple term and therefore Some A is B. Hence if we take any two simple terms and call one A and the other B we have: Some A is B and Some A is not B or in other words the universe will contain every possible kind of thing afforded by the permutation of simple qualities. Now the universe does not contain all these things; it contains no well-known green horse. Hence the consequence of supposing a simple term to exist is an error of fact. There are several other ways of showing this besides the one that I have adopted. They all concur to show that whatever has extension must be composite. (Peirce 1866, "Lowell Lecture 7", CE 1, 460–461).

Passage 3

 The moment, then, that we pass from nothing and the vacuity of being to any content or sphere, we come at once to a composite content and sphere. In fact, extension and comprehension — like space and time — are quantities which are not composed of ultimate elements; but every part however small is divisible. The consequence of this fact is that when we wish to enumerate the sphere of a term — a process termed division — or when we wish to run over the content of a term — a process called definition — since we cannot take the elements of our enumeration singly but must take them in groups, there is danger that we shall take some element twice over, or that we shall omit some. Hence the extension and comprehension which we know will be somewhat indeterminate. But we must distinguish two kinds of these quantities. If we were to subtilize we might make other distinctions but I shall be content with two. They are the extension and comprehension relatively to our actual knowledge, and what these would be were our knowledge perfect. Logicians have hitherto left the doctrine of extension and comprehension in a very imperfect state owing to the blinding influence of a psychological treatment of the matter. They have, therefore, not made this distinction and have reduced the comprehension of a term to what it would be if we had no knowledge of fact at all. I mention this because if you should come across the matter I am now discussing in any book, you would find the matter left in quite a different state. (Peirce 1866, "Lowell Lecture 7", CE 1, 462).

### Commentary Note 7

I find one more patch of material from Peirce's early lectures that we need to cover the subject of indices. I include a piece of the context, even if it overlaps a bit with fragments that still live in recent memory.

Passage 4

 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, "Lowell Lecture 7", CE 1, 468–469).

### Commentary Note 8

I'm going to make yet another try at following the links that Peirce makes among conventions, disjunctive terms, indexical signs, and inductive rules. For this purpose, I'll break the text up into smaller pieces, and pick out just those parts of it that have to do with the indexical aspect of things.

Before I can get to this, though, I will need to deal with the uncertainty that I am experiencing over the question as to whether a connotation is just another notation, and thus belongs to the interpretive framework, that is, the SI-plane, or whether it is an objective property, a quality of objects of terms. I have decided to finesse the issue by forcing my own brand of interpretation on the next text, where the trouble starts:

Passage 1

 It is important to distinguish between the two functions of a word: 1st to denote something — to stand for something, and 2nd to mean something — or as Mr. Mill phrases it — to connote something. What it denotes is called its Sphere. What it connotes is called its Content. Thus the sphere of the word man is for me every man I know; and for each of you it is every man you know. The content of man is all that we know of all men, as being two-legged, having souls, having language, &c., &c. It is plain that both the sphere and the content admit of more and less. … Now the sphere considered as a quantity is called the Extension; and the content considered as quantity is called the Comprehension. Extension and Comprehension are also termed Breadth and Depth. So that a wider term is one which has a greater extension; a narrower one is one which has a less extension. A higher term is one which has a less Comprehension and a lower one has more. The narrower term is said to be contained under the wider one; and the higher term to be contained in the lower one. We have then: o-----------------------------o-----------------------------o |               |               | | What is 'denoted'     | What is 'connoted'     | |               |               | | Sphere           | Content          | |               |               | | Extension         | Comprehension       | |               |               | |      ( wider      |     ( lower       | | Breadth <         | Depth <          | |      ( narrower    |     ( higher      | |               |               | | What is contained 'under' | What is contained 'in'   | |               |               | o-----------------------------o-----------------------------o  The principle of explicatory or deductive reasoning then is that: Every part of a word's Content belongs to every part of its Sphere, or: Whatever is contained in a word belongs to whatever is contained under it. Now this maxim would not be true if the Extension and Comprehension were directly proportional to one another; this is to say if the Greater the one the greater the other. For in that case, though the whole Content would belong to the whole Sphere; yet only a particular part of it would belong to a part of that Sphere and not every part to every part. On the other hand if the Comprehension and Extension were not in some way proportional to one another, that is if terms of different spheres could have the same content or terms of the same content different spheres; then there would be no such fact as a content's belonging to a sphere and hence again the maxim would fail. For the maxim to be true, then, it is absolutely necessary that the comprehension and extension should be inversely proportional to one another. That is that the greater the sphere, the less the content. Now this evidently true. If we take the term man and increase its comprehension by the addition of black, we have black man and this has less extension than man. So if we take black man and add non-black man to its sphere, we have man again, and so have decreased the comprehension. So that whenever the extension is increased the comprehension is diminished and vice versa. (Peirce 1866, "Lowell Lecture 7", CE 1, 459–460).

I am going to treat Peirce's use of the quantity consideration as a significant operator that transforms its argument from the syntactic domain SI to the objective domain O.

 Now the sphere considered as a quantity is called the Extension; and the content considered as quantity is called the Comprehension.

Taking this point of view, then, I will consider the Extensions of terms and the Comprehensions of terms, to be quantities, in effect, objective formal elements that are subject to being compared with one another within their respective domains. In particular, I will view them as elements of partially ordered sets. On my reading of Peirce's text, the word content is still ambiguous from context of use to context of use, but I will simply let that be as it may, hoping that it will suffice to fix the meaning of the more technical term comprehension.

This is still experimental — I'll just have to see how it works out over time.

### Commentary Note 9

2. Conventions, Disjunctive Terms, Indexical Signs, Inductive Rules

2.1. "man and horse and kangaroo and whale" (intensional conjunction).

Nota Bene. In this particular choice of phrasing, Peirce is using the intensional "and", meaning that the compound term has the intensions that are shared by all of the component terms, in this way producing a term that bears the greatest common intension of the terms that are connected in it. This is formalized as the greatest lower bound in a lattice of intensions, dual to the union of sets or least upper bound in a lattice of extensions.

It is perhaps more common today to use the extensional "or" in order to express the roughly equivalent compound concept:

2.1. "men or horses or kangaroos or whales" (extensional disjunction).

 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" ... [This term] 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. C.S. Peirce, 'Chronological Edition', CE 1, 468.

Let us first assemble a minimal syntactic domain S that is sufficient to begin discussing this example:

S = {"m", "h", "k", "w", "S", "M", "P"}

Here, I have introduced the abbreviations:

"m" = "man"
"h" = "horse"
"k" = "kangaroo"
"w" = "whale"
"S" = "man or horse or kangaroo or whale"
"M" = "Mammal"
"P" = "Predicate shared by man, horse, kangaroo, whale"

Let's attempt to keep tabs on things by using angle brackets for the comprehension of a term, and square brackets for the extension of a term.

For brevity, let x = ["x"], in general.

Here is an initial picture of the situation, so far as I can see it:

o-----------------------------o-----------------------------o
|   Objective Framework   |  Interpretive Framework  |
o-----------------------------o-----------------------------o
|                              |
|       P <------------o------------ "P"       |
|       = \              |\       |
|      = \             |\      |
|      =   \             | \      |
|     =   \            | \     |
|     =     \            |  \     |
|    P     M <------o--------------|--- "M"    |
|     \     =            |  /     |
|     \   =            | /     |
|      \   =             | /      |
|      \ =             |/      |
|       \ =              |/       |
|       S <------------o------------ "S"       |
|      ** **            ** **      |
|     ** **          ** **     |
|    * *   * *        * *   * *    |
|   * *   * *      * *   * *   |
|  o  o     o  o    o  o     o  o  |
|  m  h     k  w    "m" "h"   "k" "w"  |
|                              |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P"    |
o-----------------------------------------------------------o
|                              |
| !S! = !I! = {"m", "h", "k", "w", "S", "M", "P"}    |
|                              |
| "m" = "man"                       |
| "h" = "horse"                      |
| "k" = "kangaroo"                    |
| "w" = "whale"                      |
|                              |
| "S" = "man or horse or kangaroo or whale"        |
| "M" = "Mammal"                     |
| "P" = "Predicate shared by man, horse, kangaroo, whale" |
|                              |
o-----------------------------------------------------------o


In effect, relative to the lattice of natural (non-phony) kinds, any property P, predicated of S, can be "lifted" to a mark of M.

### Commentary Note 10

2. Conventions, Disjunctive Terms, Indexical Signs, Inductive Rules (cont.)

 We come next to consider inductions. In inferences of this kind we proceed as if upon the principle that as is a sample of a class so is the whole class. The word class in this connection means nothing more than what is denoted by one term, — or in other words the sphere of a term. Whatever characters belong to the whole sphere of a term constitute the content of that term. Hence the principle of induction is that whatever can be predicated of a specimen of the sphere of a term is part of the content of that term. And what is a specimen? It is something taken from a class or the sphere of a term, at random — that is, not upon any further principle, not selected from a part of that sphere; in other words it is something taken from the sphere of a term and not taken as belonging to a narrower sphere. Hence the principle of induction is that whatever can be predicated of something taken as belonging to the sphere of a term is part of the content of that term. But this principle is not axiomatic by any means. Why then do we adopt it? To explain this, we must remember that the process of induction is a process of adding to our knowledge; it differs therein from deduction — which merely explicates what we know — and is on this very account called scientific inference. Now deduction rests as we have seen upon the inverse proportionality of the extension and comprehension of every term; and this principle makes it impossible apparently to proceed in the direction of ascent to universals. But a little reflection will show that when our knowledge receives an addition this principle does not hold. … The reason why this takes place is worthy of notice. Every addition to the information which is incased in a term, results in making some term equivalent to that term. … Thus, every addition to our information about a term is an addition to the number of equivalents which that term has. Now, in whatever way a term gets to have a new equivalent, whether by an increase in our knowledge, or by a change in the things it denotes, this always results in an increase either of extension or comprehension without a corresponding decrease in the other quantity. (Peirce 1866, "Lowell Lecture 7", CE 1, 462–464).

2.1. "man and horse and kangaroo and whale" (aggregarious animals).

It seems to me now that my previous explanation of the use of "and" in this example was far too complicated and contrived. So let's just say that the conjunction "and" is being used in its aggregational sense.

I will also try an alternate style of picture for the lifting property, by means of which, relative to the lattice of natural (non-ad-hoc) kinds, a property P, naturally predicated of S, can be elevated to apply to M.

o-----------------------------o-----------------------------o
|   Objective Framework   |  Interpretive Framework  |
o-----------------------------o-----------------------------o
|                              |
|       P <------------o------------ "P"       |
|       |\              |\       |
|       |\             |\      |
|       | \             | \      |
|       | \            | \     |
|       |  \            |  \     |
|       |  M <------o--------------|--- "M"    |
|       |  =            |  /     |
|       | =            | /     |
|       | =             | /      |
|       |=             |/      |
|       |=              |/       |
|       S <------------o------------ "S"       |
|      ** **            ** **      |
|     ** **          ** **     |
|    * *   * *        * *   * *    |
|   * *   * *      * *   * *   |
|  o  o     o  o    o  o     o  o  |
|  m  h     k  w    "m" "h"   "k" "w"  |
|                              |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P"    |
o-----------------------------------------------------------o
|                              |
| !S! = !I! = {"m", "h", "k", "w", "S", "M", "P"}    |
|                              |
| "m" = "man"                       |
| "h" = "horse"                      |
| "k" = "kangaroo"                    |
| "w" = "whale"                      |
|                              |
| "S" = "man or horse or kangaroo or whale"        |
| "M" = "Mammal"                     |
| "P" = "Predicate shared by man, horse, kangaroo, whale" |
|                              |
o-----------------------------------------------------------o


I believe that we can now begin to see the linkage to inductive rules. When a sample S is fairly or randomly drawn from the membership M of some population and when every member of S is observed to have the property P, then it is naturally rational to expect that every member of M will also have the property P. This is the principle behind all of our more usual statistical generalizations, giving us the leverage that it takes to lift predicates from samples to a membership sampled.

Now, the aggregate that is designated by "man, horse, kangaroo, whale", even if it's not exactly a random sample from the class of mammals, is drawn by design from sufficiently many and sufficiently diverse strata within the class of mammals to be regarded as a quasi-random selection. Thus, it affords us with a sufficient basis for likely generalizations.

### Commentary Note 11

At this point it will help to jump ahead a bit in time, and to take in the more systematic account of the same material from Peirce's "New List of Categories" (1867).

 I shall now show how the three conceptions of reference to a ground, reference to an object, and reference to an interpretant are the fundamental ones of at least one universal science, that of logic. (Peirce 1867, CP 1.559).

We will have occasion to consider this paragraph in detail later, but for the present purpose let's hurry on down to the end of it.

 In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. In deductive argument, the conclusion is represented by the premisses as by a general sign under which it is contained. In hypotheses, something like the conclusion is proved, that is, the premisses form a likeness of the conclusion. Take, for example, the following argument: [Abduction of a Case] M is, for instance, P1, P2, P3, and P4; S is P1, P2, P3, and P4: Therefore, S is M. Here the first premiss amounts to this, that "P1, P2, P3, and P4" is a likeness of M, and thus the premisses are or represent a likeness of the conclusion. That it is different with induction another example will show: [Induction of a Rule] S1, S2, S3, and S4 are taken as samples of the collection M; S1, S2, S3, and S4 are P: Therefore, All M is P. Hence the first premiss amounts to saying that "S1, S2, S3, and S4" is an index of M. Hence the premisses are an index of the conclusion. (Peirce 1867, CP 1.559).

1. Abductive Inference and Iconic Signs

Peirce's analysis of the patterns of abductive argument can be understood according to the following paraphrase:

• Abduction of a Case:
Fact: SP1, SP2, SP3, SP4
Rule: MP1, MP2, MP3, MP4
-------------------------------------------------
Case: SM
If X ⇒ each of A, B, C, D, …,
then we have the following equivalents:
1. X ⇒ the greatest lower bound (glb) of A, B, C, D, …
2. X ⇒ the logical conjunction ABCD ∧ …
3. XQ = ABCD ∧ …

More succinctly, letting Q = P1P2P3P4, the argument is summarized by the following scheme:

• Abduction of a Case:
Fact: SQ
Rule: MQ
--------------
Case: SM

In this piece of Abduction, it is the glb or the conjunction of the ostensible predicates that is the operative predicate of the argument, that is, it is the predicate that is common to both the Fact and the Rule of the inference.

Finally, the reason why one can say that Q is an iconic sign of the object M is that Q can be taken to denote M by virtue of the qualities that they share, namely, P1, P2, P3, P4.

Notice that the iconic denotation is symmetric, at least in principle, that is, icons are icons of each other as objects, at least potentially, whether or not a particular interpretive agent is making use of their full iconicity during a particular phase of semeiosis.

The abductive situation is diagrammed in Figure 11.1.

o-------------------------------------------------o
|                         |
|      P_1  P_2     P_3  P_4      |
|      o  o     o  o      |
|       *  *     *  *       |
|        * *   * *        |
|         * *   * *         |
|          * * * *          |
|           ** **           |
|           Q o            |
|            |\            |
|            |\           |
|            | \           |
|            | \          |
|            |  \          |
|            |  o M        |
|            |  /          |
|            |            |
|            | /           |
|            |            |
|            |/            |
|           S o            |
|                         |
o-------------------------------------------------o
| Figure 1. Abduction of the Case S => M     |
o-------------------------------------------------o


In a diagram like this, even if one does not bother to show all of the implicational or the subject-predicate relationships by means of explicit lines, then one may still assume the transitive closure of the relations that are actually shown, along with any that are noted in the text that accompanies it.

2. Inductive Inference and Indexic Signs

Peirce's analysis of the patterns of inductive argument can be understood according to the following paraphrase:

• Induction of a Rule:
Case: S1M, S2M, S3M, S4M
Fact: S1P, S2P, S3P, S4P
-------------------------------------------------
Rule: MP
If X <= each of A, B, C, D, …,
then we have the following equivalents:
1. X <= the least upper bound (lub) of A, B, C, D, …
2. X <= the logical disjunction ABCD ∨ …
3. X <= L = ABCD ∨ …

More succinctly, letting L = P1P2P3P4, the argument is summarized by the following scheme:

• Induction of a Rule:
Case: LM
Fact: LP
--------------
Rule: MP

In this bit of Induction, it is the lub or the disjunction of the ostensible subjects that is the operative subject of the argument, to wit, the subject that is common to both the Case and the Fact of the inference.

Finally, the reason why one can say that L is an indexical sign of the object M is that L can be taken to denote M by virtue of the instances that they share, namely, S1, S2, S3, S4.

Notice that the indexical denotation is symmetric, at least in principle, that is, indices are indices of each other as objects, at least potentially, whether or not a particular interpretive agent is making use of their full indiciality during a particular phase of semeiosis.

The inductive situation is diagrammed in Figure 11.2.

o-------------------------------------------------o
|                         |
|           P o            |
|            |\            |
|            |            |
|            | \           |
|            |            |
|            |  \          |
|            |  o M        |
|            |  /          |
|            | /          |
|            | /           |
|            |/           |
|            |/            |
|           L o            |
|           ** **           |
|          * * * *          |
|         * *   * *         |
|        * *   * *        |
|       *  *     *  *       |
|      o  o     o  o      |
|      S_1  S_2     S_3  S_4      |
|                         |
o-------------------------------------------------o
| Figure 2. Induction of the Rule M => P     |
o-------------------------------------------------o


### Commentary Note 12

Let's redraw the New List pictures of Abduction and Induction in a way that is a little less cluttered, availing ourselves of the fact that logical implications or lattice subsumptions obey a transitive law to leave unmarked what is thereby understood.

o-------------------------------------------------o
|                         |
|         P_1  ...  P_k         |
|         o  o  o         |
|          \  |  /          |
|          \ | /          |
|           \ | /           |
|           \|/           |
|            \|/            |
|           Q o            |
|            |\            |
|            |\           |
|            | \           |
|            | \          |
|            |  \          |
|            |  o M        |
|            |  ^          |
|            | /          |
|            | /           |
|            |/           |
|            |/            |
|           S o            |
|                         |
o-------------------------------------------------o
| Icon Q of Object M, Abduction of Case "S is M" |
o-------------------------------------------------o

o-------------------------------------------------o
|                         |
|           P o            |
|            |^            |
|            |\           |
|            | \           |
|            | \          |
|            |  \          |
|            |  o M        |
|            |  /          |
|            | /          |
|            | /           |
|            |/           |
|            |/            |
|           L o            |
|            /|\            |
|           /|\           |
|           / | \           |
|          / | \          |
|          /  |  \          |
|         o  o  o         |
|         S_1  ...  S_k         |
|                         |
o-------------------------------------------------o
| Index L of Object M, Induction of Rule "M is P" |
o-------------------------------------------------o


The main problem that I have with these pictures in their present form is that they do not sufficiently underscore the distinction in roles between signs and objects, and thus we may find it a bit jarring that the middle term of a syllogistic figure is described as an object of iconic and indexic signs.

### Commentary Note 13

In the process of rationalizing Peirce's account of induction to myself I find that I have now lost sight of the indexical sign relationships, so let me go back to the drawing board one more time to see if I can get the indexical and the inductive aspects of the situation back into the very same picture. Here is how we left off last time:

o-----------------------------o-----------------------------o
|   Objective Framework   |  Interpretive Framework  |
o-----------------------------o-----------------------------o
|                              |
|       P <------------------------- "P"       |
|       |\              |\       |
|       |\             |\      |
|       | \             | \      |
|       | \            | \     |
|       |  \            |  \     |
|       |  M <---------------------|--- "M"    |
|       |  =            |  /     |
|       | =            | /     |
|       | =             | /      |
|       |=             |/      |
|       |=              |/       |
|       S <------------------------- "S"       |
|      ** **            ** **      |
|     ** **          ** **     |
|    * *   * *        * *   * *    |
|   * *   * *      * *   * *   |
|  o  o     o  o    o  o     o  o  |
|  m  h     k  w    "m" "h"   "k" "w"  |
|  S_1 S_2   S_3 S_4   "S_1" "S_2"  "S_3" "S_4" |
|                              |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P"    |
o-----------------------------------------------------------o
|                              |
| !S! = !I! = {"m", "h", "k", "w", "S", "M", "P"}    |
|                              |
| "m" = "man"                       |
| "h" = "horse"                      |
| "k" = "kangaroo"                    |
| "w" = "whale"                      |
|                              |
| "S" = "man or horse or kangaroo or whale"        |
| "M" = "Mammal"                     |
| "P" = "Predicate shared by man, horse, kangaroo, whale" |
|                              |
o-----------------------------------------------------------o


In this double-entry account we are more careful to distinguish the signs that belong to the interpretive framework (IF) from the objects that belong to the objective framework (OF). One benefit of this scheme is that it immediately resolves many of the conceptual puzzles that arise from confusing the roles of objects and the roles of signs in the relevant sign relation.

For example, we observe the distinction between the objects S, M, P and the signs "S", "M", "P". The objects may be regarded as extensive classes or as intensive properties, as the context demands. The signs may be regarded as sentences or as terms, in accord with the application and the ends in view.

It is as if we collected a stratified sample S of the disjoint type "man, horse, kangaroo, whale" from the class M of mammals, and observed the property P to hold true of each of them. Now we know that this could be a statistical fluke, in other words, that S is just an arbitrary subset of the relevant universe of discourse, and that the very next M you pick from outside of S might not have the property P. But that is not very likely if the sample was fairly or randomly drawn. So the objective domain is not a lattice like the power set of the universe but something more constrained, of a kind that makes induction and learning possible, a lattice of natural kinds, you might say. In the natural kinds lattice, then, the lub of S is close to M.

Now that I have this much of the picture assembled in one frame, it occurs to me that I might be confusing myself about what are the sign relations of actual interest in this situation. After all, samples and signs are closely related, as evidenced by the etymological connection between them that goes back at least as far as Hippocrates.

I need not mention any further the more obvious sign relations that we use just to talk about the objects in the example, for the signs and the objects in these relations of denotation are organized according to their roles in the diptych of objective and interpretive frames. But there are, outside the expressly designated designations, the ways that samples of species tend to be taken as signs of their genera, and these sign relations are discovered internal to the previously marked object domain.

Let us look to Peirce's New List of the next year for guidance:

 In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. … [Induction of a Rule, where the premisses are an index of the conclusion.] S1, S2, S3, and S4 are taken as samples of the collection M; S1, S2, S3, and S4 are P: Therefore, All M is P. Hence the first premiss amounts to saying that "S1, S2, S3, and S4" is an index of M. Hence the premisses are an index of the conclusion. (Peirce 1867, CP 1.559).

There we see an abstract example with the same logical structure and almost precisely the same labeling. It is a premiss of this argument that "S1, S2, S3, S4" is an index of M. But we are left wondering if he means the objective class M or the sign "M". If we take the quotation marks of "S1, S2, S3, S4" as giving the disjunctive term equal to "S", then we have the next picture:

o-----------------------------o-----------------------------o
|   Objective Framework   |  Interpretive Framework  |
o-----------------------------o-----------------------------o
|                              |
|       P <------------------------- "P"       |
|       |\              |\       |
|       |\             |\      |
|       | \             | \      |
|       | \            | \     |
|       |  \            |  \     |
|       |  M <---------------------|--- "M"    |
|       |  = $          |  / %    | |       | =            | /     | |       | =             | /      | |       |=  $          |/  %   |
|       |=              |/       |
|       S <------------------------- "S"       |
|      ** **   $       $ ** *%   %   |
|     * * * *        $ * * * %     | |    * *   * *     $  * *   * %    |
|   * *   * * $ $   * *   * % %  |
|  o  o     o  o    o  o     o  o  |
|  m  h     k  w   "m" "h"   "k" "w" |
|  S_1 S_2   S_3 S_4   "S_1" "S_2"   "S_3" "S_4" |
|                              |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P"    |
o-----------------------------------------------------------o


So we have two readings of what Peirce is saying:

1. The interpretation where "S" is an index of M by virtue of "S" being a property of each Sj, literally a generic sign of each of them, and by virtue of each Sj being an instance of M. The "S" to S4 to M linkage is painted  $. 2. The interpretation where "S" is an index of "M" by virtue of "S" being a property of each "Sj", literally an implicit sign of each of them, and by dint of each "Sj" being an instance of "M". The "S" to "S4" to "M" link is drawn as % % %. On third thought, there is still the possibility of a sense in which S is literally an index of M, that is, we might regard a fair sample from S as nothing less than a representative sample from M. ### Commentary Note 14 With the clarity afforded by a reflective interval, my third thought, the relatively ultimate, more perfect interpretant of the intervening struggle toward that final, hopefully neither dying nor raging light, begins to look ever more like the fitting icon of my first impression. I was trying to understand the things that Peirce said and wrote about the conventional, disjunctive, indexical, inductive complex of notions in the period 1865–1867. And I was focused for the moment on this bit:  In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. … [Induction of a Rule, where the premisses are an index of the conclusion.] S1, S2, S3, and S4 are taken as samples of the collection M; S1, S2, S3, and S4 are P: Therefore, All M is P. Hence the first premiss amounts to saying that "S1, S2, S3, and S4" is an index of M. Hence the premisses are an index of the conclusion. (Peirce 1867, CP 1.559). I've gotten as far as sketching this picture of the possible readings: o-----------------------------o-----------------------------o |   Objective Framework   |  Interpretive Framework  | o-----------------------------o-----------------------------o |                              | |       P <------------------------- "P"       | |       |\              |\       | |       |\             |\      | |       | \             | \      | |       | \            | \     | |       |  \            |  \     | |       |  M <---------------------|--- "M"    | |       |  = *           |  / %    | |       | =            | /     | |       | =             | /      | |       |=  *          |/  %   | |       |=              |/       | |       S <------------------------- "S"       | |      ** **   *        $ ** *%   %   |
|     * * * *        $ * * * %     | |    * *   * *     $  * *   * %    |
|   * *   * * * $   * *   * % %  | |  o  o     o  o    o  o     o  o  | |  m  h     k  w   "m" "h"   "k" "w" | |  S_1 S_2   S_3 S_4   "S_1" "S_2"   "S_3" "S_4" | |                              | o-----------------------------------------------------------o | Disjunctive Subject "S" and Inductive Rule "M => P"    | o-----------------------------------------------------------o  In order of increasing objectivity, here are three alternatives: 1. The interpretation where "S" is an index of "M" by virtue of "S" being a property of each "Sj", literally an implicit sign of each of them, and by dint of each "Sj" being an instance of "M". The "S" to "S4" to "M" link is drawn [% % % %]. 2. The interpretation where "S" is an index of M by virtue of "S" being a property of each Sj, literally a generic sign of each of them, and by virtue of each Sj being an instance of M. The "S" to S4 to M link is a 2-tone [$ \$ * *].
3. The interpretation where S is an index of M by virtue of S being a property of each Sj, literally a supersample of each of them, and by virtue of each Sj being an instance of M. The S to S4 to M link is shown as [* * * *].

Perhaps it is the nature of the sign situation that all three interpretations will persevere and keep some measure of merit. At the moment I am leaning toward the third interpretation as it manifests the possibility of a higher grade of objectivity.

### Commentary Note 15

I am going to stick with the Index-Induction side of the problem until I feel like I understand what's going on with this linkage between the faces of the sign relation and the phases of inquiry.

The New List (1867) account of the relationship between the kinds of signs and the kinds of arguments says this:

 In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object.

In general, if one takes the components of an Argument to be its Conclusion, its Premisses taken collectively, and its Interpretant, then they can be seen to take up the following sign relational duties:

<Conclusion, Premisses, Interpretant> = <Object, Sign, Interpretant>

This generality may be broken down according to the role of the premisses:

 The premisses may afford a likeness, index, or symbol of the conclusion.

In the case of the inductive argument, we have the following role assigments:

<Conclusion, Premisses, Interpretant> = <Object, Index, Interpretant>

Marked out in greater detail, we have the following role assignments:

Premisses (Index):

 S1, S2, S3, and S4 are taken as samples of the collection M. S1, S2, S3, and S4 are P.

Conclusion (Object):

 All M is P.

Remark:

 Hence the first premiss amounts to saying that "S1, S2, S3, and S4" is an index of M. Hence the premisses are an index of the conclusion.

One of the questions that I have at this point is whether Peirce is speaking loosely or strictly when he refers to the conclusion and the premisses of the argument in question. Strictly speaking, the conclusion has the form M ⇒ P and the premisses have the forms Sj ⇒ M and Sj ⇒ P. But taken more loosely, as often happens in contexts where the antecedent of a conditional statement is already assumed to hold true, people will sometimes refer to the consequent of a conditional conclusion as the conclusion and the consequents of conditional premisses as the premisses. In the present case, such a practice would lead to speaking of the predicate M as one of the premisses and the predicate P as the conclusion. So let us keep that interpretive option in mind as we go.

## Commentary Work Notes

### Commentary Work Note 1

Here is my current picture of the situation, so far as it goes:

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|              P <------------@------------ "P"             |
|              |\                            |\             |
|              | \                           | \            |
|              |  \                          |  \           |
|              |   \                         |   \          |
|              |    \                        |    \         |
|              |     M <------@--------------|--- "M"       |
|              |    = .                      |    / #       |
|              |   =                         |   /          |
|              |  =                          |  /           |
|              | =     .                     | /     #      |
|              |=                            |/             |
|              S <------------@------------ "S"             |
|            .. ..      .                * .. .#      #     |
|          . .   . .                 *   . .   . #          |
|        .  .     .  .           *     .  .     .  #        |
|      .   .       .   . .   *       .   .       .   # #    |
|    o    o         o    o         o    o         o    o    |
|    m    h         k    w       "m"   "h"       "k"   "w"  |
|   S_1  S_2       S_3  S_4     "S_1" "S_2"     "S_3" "S_4" |
|                                                           |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P"       |
o-----------------------------------------------------------o


I got as far as sketching a few readings of the penultimate sentence:

 Hence the first premiss amounts to saying that "S1, S2, S3, and S4" is an index of M.

Uncertain as my comprehension remains at this point, I will have to leave it in suspension for the time being. But let me make an initial pass at the final sentence, so as not to leave an utterly incomplete impression of the whole excerpt.

 Hence the premisses are an index of the conclusion.

The first premiss is this:

 S1, S2, S3, and S4 are taken as samples of the collection M.

We gather that it says that "S1, S2, S3, S4" is an index of M.

Taking this very literally, I would guess that it holds by way of the paths from "S1, S2, S3, S4" to  Sj to M.

The second premiss is this:

 S1, S2, S3, and S4 are P.

Together these premisses form an index of the conclusion, namely:

 All M is P.

And all of this is said to be so because:

 In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object.

And that is a bit that I will need to try to think about a bit before I even try to draw a picture of it.

Let me advance a few words in prospect of how I plan to address the problem of objectivity. Although I've been using the bipartite scheme of objective and interpretive frameworks, that is only a matter of convenient organization, and embodies nothing like a claim to the invariant status of either objects or signs, as we have seen plenty of examples already of just how shifty those roles can be.

As I suggested in my last note, one sort of evidence, the amassing of which tends to make me assign a matter to the objective side of my experience, is the possibility of viewing it from many diverse angles, of being able to describe it from manifold points of view, and being able to relate those angles and views in a sensible way.

In the mathematical perspective known as "category theory", the question of objectivity is handled by way of what are derivatively enough nomenclated as "universal properties".

Working within any given category, the things that are potentially worth caring about, called "objects" or "spaces", along with the corresponding metamorphoses among them, called "arrows" or "morphisms", can be treated as having an objective status to the extent that there are many distinct "views" of them, called "functors", that relate to each other in natural and especially nice ways called "natural transformations". That's it in a nutshell, very roughly, but I have forced a few details in prospect of the ways that I will have to change the setting a little for the sake of better accommodating semiotics in a suitably re-modelled category theory.

### Commentary Work Note 2

Here's the "New List" text about the relations between the types of signs and the types of inference, that is, the morphological and temporal constituents of inquiry:

 In an argument, the premisses form a representation of the conclusion, because they indicate the interpretant of the argument, or representation representing it to represent its object. The premisses may afford a likeness, index, or symbol of the conclusion. [Deduction of a Fact] In deductive argument, the conclusion is represented by the premisses as by a general sign under which it is contained. [Abduction of a Case] In hypotheses, something like the conclusion is proved, that is, the premisses form a likeness of the conclusion. Take, for example, the following argument:-- M is, for instance, P1, P2, P3, and P4; S is P1, P2, P3, and P4: ∴ S is M. Here the first premiss amounts to this, that "P1, P2, P3, and P4" is a likeness of M, and thus the premisses are or represent a likeness of the conclusion. [Induction of a Rule] That it is different with induction another example will show. S1, S2, S3, and S4 are taken as samples of the collection M; S1, S2, S3, and S4 are P: ∴ All M is P. Hence the first premiss amounts to saying that "S1, S2, S3, and S4" is an index of M. Hence the premisses are an index of the conclusion. C.S. Peirce, "New List" CP 1.559 and CE 2, p. 58. Charles Sanders Peirce, "On a New List of Categories" (1867), Collected Papers CP 1.545-567. Chronological Edition CE 2, 49-59. http://www.peirce.org/writings/p32.html http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm
o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|   P_1  P_2       P_3  P_4     "P_1" "P_2"     "P_3" "P_4" |
|    o    o         o    o         o    o         o    o    |
|     ..   .       .   ...   *      ..   .       .   #.#    |
|      . .  .     .  . .         *   . .  .     .  # .      |
|          . .   . .                 *   . .   . #          |
|       .    .. ..    . .             .  * .. .#    . #     |
|              P <------------@------------ "P"             |
|        .     ^=    .                 .     ^^    .        |
|              | =     .                     | \     #      |
|         .    |  = .                   .    |  \ .         |
|              |   =                         |   \          |
|          .   |   .= .                  .   |   .\ #       |
|              |     M <------@--------------|--- "M"       |
|           .  |  . ^                     .  |  . ^         |
|              |   /                         |   /          |
|            . | ./                        . | ./           |
|              | /                           | /            |
|             .|/                           .|/             |
|              S <------------@------------ "S"             |
|                                                           |
o-----------------------------------------------------------o
| Conjunctive Predicate "P" and Abductive Case "S => M"     |
o-----------------------------------------------------------o

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|              P <------------@------------ "P"             |
|             ^^^                           ^^^             |
|              | \                           | \            |
|            . | .\                        . | .\           |
|              |   \                         |   \          |
|           .  |  . \                     .  |  . \         |
|              |     M <------@--------------|--- "M"       |
|          .   |   .= .                  .   |   .^ #       |
|              |   =                         |   /          |
|         .    |  = .                   .    |  / .         |
|              | =     .                     | /     #      |
|        .     |=    .                 .     |/    .        |
|              S <------------@------------ "S"             |
|       .    .. ..    . .             .  * .. .#    . #     |
|          . .   . .                 *   . .   . #          |
|      . .  .     .  . .         *   . .  .     .  # .      |
|     ..   .       .   ...   *      ..   .       .   #.#    |
|    o    o         o    o         o    o         o    o    |
|   S_1  S_2       S_3  S_4     "S_1" "S_2"     "S_3" "S_4" |
|                                                           |
o-----------------------------------------------------------o
| Disjunctive Subject "S" and Inductive Rule "M => P"       |
o-----------------------------------------------------------o


### Commentary Work Note 3

Let me pause just a moment to knock out a couple of quick sketches of how I see the scene that Peirce is depicting here.

Here is the rougher draft of the two, a diptych impaneled of an object fold and a sign fold, an interpretant being, after all, just another passing moment in the life cycle of a sign, and so there is an object o, with its intensions p through q, collectively constellating the comprehension of any sign sk, above the instances, instantiations, or instants i through j of the object o, aggregatively constituting the extension of any sign sk that is said to denote o or its plurality below.

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|          p  ...  q                    · s_1               |
|           \     /               ·     ·                   |
|            ^   ^          ·     ·     ·                   |
|             \ /     ·     ·     ·     ·                   |
|              o< · · · · · · · · · · · · s_k               |
|             / \     ·     ·     ·     ·                   |
|            ^   ^          ·     ·     ·                   |
|           /     \               ·     ·                   |
|          i  ...  j                    · s_n               |
|                                                           |
o-----------------------------------------------------------o


I should mention, in no uncertain terms, that Peirce's present account does not yet count this abstract hypostasis or hypostatic object o in any explicit way, though I think it is inherent in the very form of his thinking. So let us keep an eye out, as we proceed with the story, for when, if ever, this particular character first treads on the scene.

For future reference, I will go ahead and post here this advance notice of what may well be the next stage in the developmental differentiation of this, my embyronic autopoetics, but let's just see what we shall see.

o-----------------------------------------------------------o
|                Higher Order Framework (HOF)               |
o-------------------o-------------------o-------------------o
|     Objective      Operands, Operators       Organon      |
o-------------------o-------------------o-------------------o
|                                                           |
|                       implications         incitations    |
|                     higher intensions                     |
|                        (·········)              · s_1     |
|                          \     /            ·   ·         |
|                           ^   ^         ·   ·   ·         |
|                            \ /      ·   ·   ·   ·         |
|         o <·············· pomps < · · · · · · · · s_k     |
|                            / \      ·   ·   ·   ·         |
|                           ^   ^         ·   ·   ·         |
|                          /     \            ·   ·         |
|                        (·········)              · s_n     |
|                      implementations                      |
|                       institutions         inditations    |
|                                                           |
o-----------------------------------------------------------o


### Commentary Work Note 4

Lest we get totally lost in the devilish details,
I think that it might be a good idea to remember
what I observed once before about the main theme
of this whole part of Peirce's discussion:

At this point in his discussion, Peirce is relating the nature of
inference, inquiry, and information to the character of the signs
that are invoked in support of the overall process in question,
a process that he is presently describing as "symbolization".

The links among the types of signs -- icons, indices, symbols --
the aspects of semiosis -- connotation, denotation, information --
and the types of inference -- abduction, induction, deduction --
are parts of a bigger picture that we should try to keep in view.

Outline of Peirce's Examples:

1.    Conjunctive term "spherical bright fragrant juicy tropical fruit".
2.1.  Disjunctive term "man or horse or kangaroo or whale".
2.2.  Disjunctive term "neat or swine or sheep or deer".

| 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'.
|
| (CSP, CE 1, 468-469).

Returning to our present examples, the unmentioned elements of the
universe of discourse X are the potential objects of !O!, and the
sign domain !S! will contain all of the terms that we want to put
on the sign relational Tables.  What are the interpretant signs?
Well, in public discursions we cannot literally cut and paste our
mental signs into texts, so we are stuck with using the same sorts
of signs that we ordinarily assign to the sign domain !S! as proxies
for these concepts, or "mental symbols", and so this surrogation will
constitute the customary practice in formal, public discussions.  But
the nice thing about the "formal" point of view, in other words, the
outlook that looks out for the forms of things above all, is that we
can be relatively sure that there is an approximate isomorph of the
one-foot-in-the-psyche sort of sign relation that can found among
the wholly unsubsidiary public domains.  So nothing much is lost,
formally speaking, if we forget about the possible distincture
in essence between a sign in the mind and a sign on the wall.

Given that we are now working with sign relations of the form
L c OxSxI = OxSxS, that is to say, where S = I is the common
syntactic domain, we can picture the situation as follows:

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                             |                             |
|                             |   "man"                     |
|                             |   "horse"                   |
|                             |   "kangaroo"                |
|                             |   "whale"                   |
|                             |   "mammal"                  |
|                             |                             |
|                             |   "spherical"               |
|                             |   "bright"                  |
|                             |   "fragrant"                |
|                             |   "juicy"                   |
|                             |   "tropical"                |
|                             |   "fruit"                   |
|                             |   "orange"                  |
|                             |                             |
o-----------------------------o-----------------------------o

That is to say, in the beginning there is nothing but words,
and the objective side of the universe is all form and void.

Example 2.1.  "man or horse or kangaroo or whale"

| 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
| 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.  (CSP, CE 1, 468-469).

Suppose that one "attribute in common" or "common character"
that comes to mind for men, horses, kangaroos, and whales
is the descriptor "sucks when young" (SWY).  Then we have
the following beginnings of a sign relational set-up:

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|                                          w                |
|                                          o                |
|                                          * *     Rule     |
|                                          *   *   v=>w     |
|                                          *     *          |
|                                          *       *        |
|                                     Fact *         *      |
|                                     u=>w *           o v  |
|                                          *         *      |
|                                          *       *        |
|                                          *     * Case     |
|                                          *   *   u=>v     |
|                                          * *              |
|                                          o u              |
|                                        .. ..              |
|                                      . .   . .            |
|                                    .  .     .  .          |
|                                  .   .       .   .        |
|                                o    o         o    o      |
|                               s_1  s_2       s_3  s_4     |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
| Disjunctive Subject u, Induction to the Rule v => w       |
o-----------------------------------------------------------o

S  =  I  =  {s_1, s_2, s_3, s_4, u, v, w}

s_1  =  "man"
s_2  =  "horse"
s_3  =  "kangaroo"
s_4  =  "whale"

u    =  "man or horse or kangaroo or whale"
v    =  "sucks when young"
w    =  "mammal"

Still nothing happening on the objective side of the world.
Our first inkling of a connection to the "world of matter"
comes with the notions of denotation and Peirce's "sphere".

| It is important to distinguish between the two functions of a word:
| 1st to denote something -- to stand for something, and 2nd to mean
| something -- or as Mr. Mill phrases it -- to 'connote' something.
|
| What it denotes is called its 'Sphere'.
| What it connotes is called its 'Content'.
| Thus the 'sphere' of the word 'man' is for
| me every man I know;  and for each of you it
| is every man you know.  The 'content' of 'man'
| is all that we know of all men, as being two-
| legged, having souls, having language, &c., &c.
| It is plain that both the 'sphere' and the
| 'content' admit of more and less.  ...
|
| Now the sphere considered as a quantity is called the Extension;
| and the content considered as quantity is called the Comprehension.
| Extension and Comprehension are also termed Breadth and Depth.  So that
| a wider term is one which has a greater extension;  a narrower one is
| one which has a less extension.  A higher term is one which has a
| less Comprehension and a lower one has more.
|
| The narrower term is said to be contained under the wider one;
| and the higher term to be contained in the lower one.
|
| We have then:
|
| o-----------------------------o-----------------------------o
| |                             |                             |
| |  What is 'denoted'          |  What is 'connoted'         |
| |                             |                             |
| |  Sphere                     |  Content                    |
| |                             |                             |
| |  Extension                  |  Comprehension              |
| |                             |                             |
| |           ( wider           |         ( lower             |
| |  Breadth  <                 |  Depth  <                   |
| |           ( narrower        |         ( higher            |
| |                             |                             |
| |  What is contained 'under'  |  What is contained 'in'     |
| |                             |                             |
| o-----------------------------o-----------------------------o
|
| The principle of explicatory or deductive reasoning then is that:
|
| Every part of a word's Content belongs to
| every part of its Sphere,
|
| or:
|
| Whatever is contained 'in' a word belongs to
| whatever is contained under it.
|
| CSP, CE 1, pages 459-460.


### Commentary Work Note 5

Obeying some hobgoblin of consistency, I present Peirce's second example
of a {disjunctive term, indexical sign, inductive rule} configuration on
the same pattern as the first, and then I want to look more closely at a
tricky feature of his account of comprehension, connotation, and content.

Example 2.2.  "neat or swine or sheep or deer"

| 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.  (CSP, CE 1, 468-469).

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|                                          w                |
|                                          o                |
|                                          * *     Rule     |
|                                          *   *   v=>w     |
|                                          *     *          |
|                                          *       *        |
|                                     Fact *         *      |
|                                     u=>w *           o v  |
|                                          *         *      |
|                                          *       *        |
|                                          *     * Case     |
|                                          *   *   u=>v     |
|                                          * *              |
|                                          o u              |
|                                        .. ..              |
|                                      . .   . .            |
|                                    .  .     .  .          |
|                                  .   .       .   .        |
|                                o    o         o    o      |
|                               s_1  s_2       s_3  s_4     |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
| Disjunctive Subject u, Induction to the Rule v => w       |
o-----------------------------------------------------------o

Syntactic domain, !S!  =  {s_1, s_2, s_3, s_4, u, v, w}.

s_1  =  "neat"
s_2  =  "swine"
s_3  =  "sheep"
s_4  =  "deer"

u    =  "neat or swine or sheep or deer"
v    =  "cloven-hoofed"
w    =  "herbivorous"

In this situation there is a gap between the logical disjunction u,
or, expressed in lattice terminology, the "least upper bound" (lub)
of the disjoined terms, u = lub(s_j), and what we might well call
their "natural disjunction" v = "cloven-hoofed".

The sheer implausibility of imagining that the disjunctive term u
would ever be embedded exactly per se in a lattice of natural kinds,
leads to the evident "naturalness" of the induction to v => w, namely,
the rule that cloven-hoofed animals are herbivorous.

On to the tricky feature of which I spoke.
Let us focus the explanation of "content"
that was given in the following statement:

| It is important to distinguish between the two
| functions of a word:  1st to denote something --
| to stand for something, and 2nd to mean something --
| or as Mr. Mill phrases it -- to 'connote' something.
|
| What it denotes is called its 'Sphere'.
| What it connotes is called its 'Content'.
|
| Thus the 'sphere' of the word 'man' is for me every man
| I know;  and for each of you it is every man you know.
|
| The 'content' of 'man' is all that we know of all men, as
| being two-legged, having souls, having language, &c., &c.

The question is:  What sort of thing is a connotation?
Is it a sign?  That is to say, is it yet another term?
Or is it something like an abstract attribute, namely,
a character, an intension, a property, or a quality?
And while we're asking, does it really even matter?


### Commentary Work Note 6

| It is important to distinguish between the two
| functions of a word:  1st to denote something --
| to stand for something, and 2nd to mean something --
| or as Mr. Mill phrases it -- to 'connote' something.
|
| What it denotes is called its 'Sphere'.
| What it connotes is called its 'Content'.
|
| Thus the 'sphere' of the word 'man' is for me every man
| I know;  and for each of you it is every man you know.
|
| The 'content' of 'man' is all that we know of all men, as
| being two-legged, having souls, having language, &c., &c.
|
| CSP, CE 1, page 459.

The question is:  What sort of thing is a connotation?
Is it a sign?  That is to say, is it yet another term?
Or is it something like an abstract attribute, namely,
a character, an intension, a property, or a quality?
And while we're asking, does it really even matter?

"No" is one answer worth considering.
But then:  Why does it not matter?
What reason might be given that
would excuse the indifference?

This is a question that has exercised me since
my earliest studies of Peirce.  I can remember
discussing it with my philosophy mentor at the
time and I distinctly recall having arrived at
some conclusion or other, but, alas, I haven't
the foggiest notion what exactly my revelation
amounted to.  Perhaps that is all for the best,
as the vagrancy of memory is frequently better
than the vapidity of one's banalytic anamnesia.

These days, I usually try to finesse the trick
under the trumped up rubric of a factorization.
So let me excavate my last attempts to explain
this business and see if I can improve on them.

I am gathering some wooly old links here just for my future reference.
I would not chase these if I were you.  Since I am not you, I have to,
and when I have carded them all out I will get back to you, but later.

Factorization Issues

01.  http://suo.ieee.org/email/msg02332.html
02.  http://suo.ieee.org/email/msg02334.html
03.  http://suo.ieee.org/email/msg02338.html
04.  http://suo.ieee.org/email/msg02340.html
05.  http://suo.ieee.org/email/msg02345.html
06.  http://suo.ieee.org/email/msg02349.html
07.  http://suo.ieee.org/email/msg02355.html
08.  http://suo.ieee.org/email/msg02396.html
09.  http://suo.ieee.org/email/msg02400.html
10.  http://suo.ieee.org/email/msg02430.html
11.  http://suo.ieee.org/email/msg02448.html
12.  http://suo.ieee.org/email/msg04334.html
13.  http://suo.ieee.org/email/msg04416.html
14.  http://suo.ieee.org/email/msg07143.html
15.  http://suo.ieee.org/email/msg07166.html
16.  http://suo.ieee.org/email/msg07182.html
17.  http://suo.ieee.org/email/msg07185.html
18.  http://suo.ieee.org/email/msg07186.html

http://suo.ieee.org/ontology/msg00007.html
http://suo.ieee.org/ontology/msg00025.html
http://suo.ieee.org/ontology/msg00032.html
http://suo.ieee.org/ontology/msg01926.html
http://suo.ieee.org/ontology/msg02008.html
http://suo.ieee.org/ontology/msg03285.html


### Commentary Work Note 7

My head is spinning, and I need to try and get a grip on the matter
of how we are supposed to understand words like attribute, character,
intension, feature, mark, predicate, property, quality, and so on, not
to say that all of them even belong to the same series of near synonyms,
and when we have achieved a measure of clarity about these general issues
of not quite common enough usage, then maybe we can begin to address the
question of what Peirce intends for us to understand by way of the more
specialized terms of art, to wit, comprehension, connotation, content.

| It is important to distinguish between the two
| functions of a word:  1st to denote something --
| to stand for something, and 2nd to mean something --
| or as Mr. Mill phrases it -- to 'connote' something.
|
| What it denotes is called its 'Sphere'.
| What it connotes is called its 'Content'.
|
| Thus the 'sphere' of the word 'man' is for me every man
| I know;  and for each of you it is every man you know.
|
| The 'content' of 'man' is all that we know of all men, as
| being two-legged, having souls, having language, &c., &c.
|
| CSP, CE 1, page 459.

The question is:  What sort of thing is a connotation?
Is it a sign?  That is to say, is it yet another term?
Or is it something like an abstract attribute, namely,
a character, an intension, a property, or a quality?
And while we're asking, does it really even matter?

In this context, is the word's function to mean something,
to connote something, and all that we know by means of it,
is it meant to mean that it mean a thing quiet apart from
signs, or is it meant to mean just more of the same order
of signs, ever and again yet more signs, if yet new kinds?

Rather than just make up whatever comes into my head,
let me go back to some passages that I cited earlier
and consider them more carefully in the light of all
that has transpired in the mean time.  Also, for the
sake of equipping this discussion with a comparative
anchor in contemporary usage, let me first cite this:

| Predicate.
|
| The four traditional kinds of categorical propositions are:
| All S is P, No S is P, Some S is P, Some S is not P.  In each
| of these the concept denoted by S is the 'subject' and that
| denoted by P is the 'predicate'.
|
| Hilbert and Ackermann use the word 'predicate' for
| a propositional function of one or more variables;
| Carnap uses it for the corresponding syntactical entity,
| the name or designation of such a propositional function
| (i.e., of a property or relation).
|
| Alonzo Church, in Runes, page 248.
|
| Dagobert Runes (ed.), 'Dictionary of Philosophy',
| Littlefield, Adams, & Company, Totowa, NJ, 1972.

Well, that was a helpful verdict.  Evidently a predicate
is proclaimed to be anything from a syntactic signifier
to a propositional function (property or relation) to
a concept denoted by a term in a categorical premiss.
The Latin "predicamentum" is supposed to translate
the Greek "kategoria", a manner of speaking about
a subject, initially as a "put-down in public".

I tend to think of a predicate as a sign or utterance,
but maybe it is better to avoid the word in favor of
object-sign pairs like "feature" and "feature name".


### Commentary Work Note 8

Passage 1

| We come next to consider inductions.  In inferences of this kind
| we proceed as if upon the principle that as is a sample of a class
| so is the whole class.  The word 'class' in this connection means
| nothing more than what is denoted by one term, -- or in other words
| the sphere of a term.  Whatever characters belong to the whole sphere
| of a term constitute the content of that term.  Hence the principle of
| induction is that whatever can be predicated of a specimen of the sphere
| of a term is part of the content of that term.  And what is a specimen?
| It is something taken from a class or the sphere of a term, at random --
| that is, not upon any further principle, not selected from a part of
| that sphere;  in other words it is something taken from the sphere
| of a term and not taken as belonging to a narrower sphere.  Hence
| the principle of induction is that whatever can be predicated of
| something taken as belonging to the sphere of a term is part of
| the content of that term.  But this principle is not axiomatic
| by any means.  Why then do we adopt it?
|
| CSP, CE 1, pages 462-463.

Passage 2

| To explain this, we must remember that the process of induction is a
| process of adding to our knowledge;  it differs therein from deduction --
| which merely explicates what we know -- and is on this very account called
| scientific inference.  Now deduction rests as we have seen upon the inverse
| proportionality of the extension and comprehension of every term;  and this
| principle makes it impossible apparently to proceed in the direction of
| ascent to universals.  But a little reflection will show that when our
|
| Thus suppose a blind man to be told that no red things are
| blue.  He has previously known only that red is a color;
| and that certain things 'A', 'B', and 'C' are red.
|
|    The comprehension of red then has been for him   'color'.
|    Its extension has been                           'A', 'B', 'C'.
|
| But when he learns that no red thing is blue, 'non-blue'
| is added to the comprehension of red, without the least
| diminution of its extension.
|
|    Its comprehension becomes   'non-blue color'.
|    Its extension remains       'A', 'B', 'C'.
|
| Suppose afterwards he learns that a fourth thing 'D' is red.
| Then, the comprehension of 'red' remains unchanged, 'non-blue color';
| while its extension becomes 'A', 'B', 'C', and 'D'.  Thus, the rule
| that the greater the extension of a term the less its comprehension
| and 'vice versa', holds good only so long as our knowledge is not
| added to;  but as soon as our knowledge is increased, either the
| comprehension or extension of that term which the new information
| concerns is increased without a corresponding decrease of the other
| quantity.
|
| The reason why this takes place is worthy of notice.  Every addition to
| the information which is incased in a term, results in making some term
| equivalent to that term.  Thus when the blind man learns that 'red' is
| not-blue, 'red not-blue' becomes for him equivalent to 'red'.  Before
| that, he might have thought that 'red not-blue' was a little more
| restricted term than 'red', and therefore it was so to him, but
| the new information makes it the exact equivalent of red.
| In the same way, when he learns that 'D' is red, the
| term 'D-like red' becomes equivalent to 'red'.
|
| to the number of equivalents which that term has.  Now, in whatever
| way a term gets to have a new equivalent, whether by an increase in
| our knowledge, or by a change in the things it denotes, this always
| results in an increase either of extension or comprehension without
| a corresponding decrease in the other quantity.
|
| For example we have here a number of circles
| dotted and undotted, crossed and uncrossed:
|
| (·X·)  (···)  (·X·)  (···)  ( X )  (   )  ( X )  (   )
|
| Here it is evident that the greater the extension the
| less the comprehension:
|
| o-------------------o-------------------o
| |                   |                   |
| | dotted            | 4 circles         |
| |                   |                   |
| o-------------------o-------------------o
| |                   |                   |
| | dotted & crossed  | 2 circles         |
| |                   |                   |
| o-------------------o-------------------o
|
| Now suppose we make these two terms 'dotted circle'
| and 'crossed and dotted circle' equivalent.  This we can
| do by crossing our uncrossed dotted circles.  In that way,
| we increase the comprehension of 'dotted circle' and at the
| same time increase the extension of 'crossed and dotted circle'
| since we now make it denote 'all dotted circles'.
|
| CSP, CE 1, pages 463-464.

Passage 3

| Thus every increase in the number of equivalents of any term increases either
| its extension or comprehension and 'conversely'.  It may be said that there
| are no equivalent terms in logic, since the only difference between such
| terms would be merely external and grammatical, while in logic terms
| which have the same meaning are identical.  I fully admit that.
| Indeed, the process of getting an equivalent for a term is
| an identification of two terms previously diverse.  It is,
| in fact, the process of nutrition of terms by which they
| get all their life and vigor and by which they put forth
| an energy almost creative -- since it has the effect of
| reducing the chaos of ignorance to the cosmos of science.
| Each of these equivalents is the explication of what there is
| wrapt up in the primary -- they are the surrogates, the interpreters
| of the original term.  They are new bodies, animated by that same soul.
| I call them the 'interpretants' of the term.  And the quantity of these
| 'interpretants', I term the 'information' or 'implication' of the term.
|
| CSP, CE 1, pages 464-465.

Passage 4

| We must therefore modify the law of
| the inverse proportionality of
| extension and comprehension
|
| Extension x Comprehension = Constant
|
| which crudely expresses the fact
| that the greater the extension the
| less the comprehension, we must write
|
| Extension x Comprehension = Information
|
| which means that when the information
| is increased there is an increase of
| either extension or comprehension
| without any diminution of the
| other of these quantities.
|
| Now, ladies and gentlemen, as it is true that
| every increase of our knowledge is an increase
| in the information of a term -- that is, is an
| addition to the number of terms equivalent to
| that term -- so it is also true that the first
| step in the knowledge of a thing, the first
| framing of a term, is also the origin of the
| information of that term because it gives the
| first term equivalent to that term.  I here
| announce the great and fundamental secret
| of the logic of science.  There is no term,
| properly so called, which is entirely destitute
| of information, of equivalent terms.  The moment
| an expression acquires sufficient comprehension
| to determine its extension, it already has more
| than enough to do so.
|
| CSP, CE 1, page 465.

Passage 5

| We are all, then, sufficiently familiar with the fact
| that many words have much implication;  but I think we
| need to reflect upon the circumstance that every word
| implies some proposition or, what is the same thing,
| every word, concept, symbol has an equivalent term --
| or one which has become identified with it, --
| in short, has an 'interpretant'.
|
| Consider, what a word or symbol is;  it is a sort of representation.
| Now a representation is something which stands for something.  I will
| not undertake to analyze, this evening, this conception of 'standing
| for' something -- but, it is sufficiently plain that it involves the
| standing 'to' something 'for' something.  A thing cannot stand for
| something without standing 'to' something 'for' that something.
| Now, what is this that a word stands 'to'?  Is it a person?  We
| usually say that the word 'homme' stands to a Frenchman for 'man'.
| It would be a little more precise to say that it stands 'to' the
| Frenchman's mind -- to his memory.  It is still more accurate
| to say that it addresses a particular remembrance or image in
| that memory.  And what 'image', what remembrance?  Plainly,
| the one which is the mental equivalent of the word 'homme' --
| in short, its interpretant.  Whatever a word addresses then
| or 'stands to', is its interpretant or identified symbol.
| Conversely, every interpretant is addressed by the word;
| for were it not so, did it not as it were overhear what
| the words says, how could it interpret what it says.
|
| There are doubtless some who cannot understand this metaphorical argument.
| I wish to show that the relation of a word to that which it addresses is
| the same as its relation to its equivalent or identified terms.  For that
| purpose, I first show that whatever a word addresses is an equivalent term, --
| its mental equivalent.  I next show that, since the intelligent reception
| of a term is the being addressed by that term, and since the explication
| of a term's implication is the intelligent reception of that term, that
| the interpretant or equivalent of a term which as we have already seen
| explicates the implication of a term is addressed by the term.
|
| The interpretant of a term, then, and that which it stands to are identical.
| Hence, since it is of the very essence of a symbol that it should stand 'to'
| something, every symbol -- every word and every 'conception' -- must have an
| interpretant -- or what is the same thing, must have information or implication.
|
|
| CSP, CE 1, pages 466-467.


### Commentary Work Note 9

I am still a few hypotheses shy of an explanation of all that
our Mister Tuesday Afternoon was saying just now, or a little
while ago, about icons and indices, and their symmetries, but
I am under the perhaps too facile impression that I have long
understood the gist of it, by dint of the particular examples
that arise in my application to systems theory, many of which
seem to fit the pattern of what Peirce seems to be describing.

And so, here for comparison is the picture of an iconic sign:

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|                   q  o                                    |
|                      ··                                   |
|                      · ·                                  |
|                      ·  ·                                 |
|                      ·   ·                                |
|                      ·    ·                               |
|                      ·     ·                              |
|                      ·      ·                             |
|                      ·       ·                            |
|                      ·        v                           |
|                      ·         o  u                       |
|                      ·        /                           |
|                      v       /                            |
|                   x  o------@                             |
|                              \                            |
|                               \                           |
|                                o  v                       |
|                                                           |
o-----------------------------------------------------------o
| Sign u is an Icon of Object x by Virtue of Property q     |
o-----------------------------------------------------------o

And here is the putatively dual figure of an indexical sign:

o-----------------------------o-----------------------------o
|     Objective Framework     |   Interpretive Framework    |
o-----------------------------o-----------------------------o
|                                                           |
|                                o  u                       |
|                               /                           |
|                              /                            |
|                   x  o------@                             |
|                      ^       \                            |
|                      ·        \                           |
|                      ·         o  v                       |
|                      ·        ^                           |
|                      ·       ·                            |
|                      ·      ·                             |
|                      ·     ·                              |
|                      ·    ·                               |
|                      ·   ·                                |
|                      ·  ·                                 |
|                      · ·                                  |
|                      ··                                   |
|                   t  o                                    |
|                                                           |
o-----------------------------------------------------------o
| Sign v is an Index of Object x by Virtue of Instance t    |
o-----------------------------------------------------------o

The reason that the indexical style of picture appears almost immediately
recognizable in a systems-theoretic context is because the state space of
a complex dynamic system is a setting in which "objects have instances".
In effect, an object is an abstract unity that comprises a collection of
components and connects a sequence of state points in an orbit over time.

Now, an "abstract unity" is a funny sort of thing -- it is partly a whole
and wholly a part, in other words, a whole in its own right that is merely
a face of some more complete and full-bodied whole.  What this means in the
present setting is that we can view the whole system and the temporal state
of the whole system as objects, the latter being an "instant" of the former.

For example, two people in a dialogue may be viewed as a "dyadic system",
and each person's experience of the interaction is a facet of the whole.
One's experience is an index of the other's experience, and vice versa,
by virtue of their actual connection in the instantaneous state of the
whole system.  And if one of them points to a common object, to which
the other independently or by dint of that pointer attends, then each
of their experiences, in that moment, becomes an index of that object.


### Commentary Work Note 10

Let us persist in investigating the circumstance that the space of symbols
is not combinatorially closed, in other words, that there are combinations
of symbols that are not strictly speaking symbols.  Seeing as how symbols
are marked by their "use" -- in the use of which word Peirce is alluding
to Kant and the big idea that the use of an argument, a book, a concept,
a sentence, a work of art, in a word, a symbol, is to reduce a manifold
of sensuous impressions to a unity -- another way to express the facts
of this dis-closure is to say that some combinations of useful usages
are in and of themselves utterly useless.  But you knew that already.

ICE.  Induction Example

ICE 19.  http://stderr.org/pipermail/inquiry/2003-March/000214.html
ICE 19.  http://suo.ieee.org/ontology/msg03780.html

| To explain this, we must remember that the process of induction is a
| process of adding to our knowledge;  it differs therein from deduction --
| which merely explicates what we know -- and is on this very account called
| scientific inference.  Now deduction rests as we have seen upon the inverse
| proportionality of the extension and comprehension of every term;  and this
| principle makes it impossible apparently to proceed in the direction of
| ascent to universals.  But a little reflection will show that when our
|
| Thus suppose a blind man to be told that no red things are
| blue.  He has previously known only that red is a color;
| and that certain things 'A', 'B', and 'C' are red.
|
|    The comprehension of red then has been for him   'color'.
|    Its extension has been                           'A', 'B', 'C'.
|
| But when he learns that no red thing is blue, 'non-blue'
| is added to the comprehension of red, without the least
| diminution of its extension.
|
|    Its comprehension becomes   'non-blue color'.
|    Its extension remains       'A', 'B', 'C'.
|
| Suppose afterwards he learns that a fourth thing 'D' is red.
| Then, the comprehension of 'red' remains unchanged, 'non-blue color';
| while its extension becomes 'A', 'B', 'C', and 'D'.  Thus, the rule
| that the greater the extension of a term the less its comprehension
| and 'vice versa', holds good only so long as our knowledge is not
| added to;  but as soon as our knowledge is increased, either the
| comprehension or extension of that term which the new information
| concerns is increased without a corresponding decrease of the other
| quantity.
|
| The reason why this takes place is worthy of notice.  Every addition to
| the information which is incased in a term, results in making some term
| equivalent to that term.  Thus when the blind man learns that 'red' is
| not-blue, 'red not-blue' becomes for him equivalent to 'red'.  Before
| that, he might have thought that 'red not-blue' was a little more
| restricted term than 'red', and therefore it was so to him, but
| the new information makes it the exact equivalent of red.
| In the same way, when he learns that 'D' is red, the
| term 'D-like red' becomes equivalent to 'red'.
|
| to the number of equivalents which that term has.  Now, in whatever
| way a term gets to have a new equivalent, whether by an increase in
| our knowledge, or by a change in the things it denotes, this always
| results in an increase either of extension or comprehension without
| a corresponding decrease in the other quantity.
|
| For example we have here a number of circles
| dotted and undotted, crossed and uncrossed:
|
| (·X·)  (···)  (·X·)  (···)  ( X )  (   )  ( X )  (   )
|
| Here it is evident that the greater the extension the
| less the comprehension:
|
| o-------------------o-------------------o
| |                   |                   |
| | dotted            | 4 circles         |
| |                   |                   |
| o-------------------o-------------------o
| |                   |                   |
| | dotted & crossed  | 2 circles         |
| |                   |                   |
| o-------------------o-------------------o
|
| Now suppose we make these two terms 'dotted circle'
| and 'crossed and dotted circle' equivalent.  This we can
| do by crossing our uncrossed dotted circles.  In that way,
| we increase the comprehension of 'dotted circle' and at the
| same time increase the extension of 'crossed and dotted circle'
| since we now make it denote 'all dotted circles'.
|
| CSP, CE 1, pages 463-464.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce:  A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
`

## References

• Abbreviations for frequently cited works:
CE n, m = Writings of Charles S. Peirce: A Chronological Edition, vol. n, page m.
CP n.m = Collected Papers of Charles Sanders Peirce, vol. n, paragraph m.
CTN n, m = Contributions to 'The Nation' , vol. n, page m.
EP n, m = The Essential Peirce: Selected Philosophical Writings, vol. n, page m.
NEM n, m = The New Elements of Mathematics by Charles S. Peirce, vol. n, page m.
SIL m = Studies in Logic by Members of the Johns Hopkins University, page m.
SS m = Semiotic and Significs … Charles S. Peirce and Lady Welby, page m.
SW m = Charles S. Peirce, Selected Writings, page m.
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP (volume).(paragraph).
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–. Cited as CE (volume), (page).
• Peirce, C.S, The New Elements of Mathematics by Charles S. Peirce, 4 volumes in 5, Carolyn Eisele (ed.), Mouton Publishers, The Hague, Netherlands, 1976. Humanities Press, Atlantic Highlands, NJ, 1976. Cited as NEM (volume), (page).
• Peirce, C.S., The Essential Peirce : Selected Philosophical Writings, Volume 1 (1867–1893), Nathan Houser and Christian Kloesel (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1992. Cited as EP 1, (page).
• Peirce, C.S., The Essential Peirce : Selected Philosophical Writings, Volume 2 (1893–1913), Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1998. Cited as EP 2, (page).
• Peirce, C.S. (1865), "On the Logic of Science", Harvard University Lectures, CE 1, 161–302.
• Peirce, C.S. (1866), "The Logic of Science, or, Induction and Hypothesis", Lowell Institute Lectures, CE 1, 357–504.
• Peirce, C.S. (1867), "On a New List of Categories", Proceedings of the American Academy of Arts and Sciences 7 (1868), 287–298. Presented, 14 May 1867. Reprinted (CP 1.545–559), (CE 2, 49–59), (EP 1, 1–10).
• Peirce, C.S. (1877), "The Fixation of Belief", Popular Science Monthly 12, 1–15, 1877. Reprinted, CP 5.358–387. Eprint.
• Peirce, C.S. (1878), "How to Make Our Ideas Clear", Popular Science Monthly 12, 286–302, 1878. Reprinted, CP 5.388–410. Eprint.
• Peirce, C.S. (1899), "F.R.L." [First Rule of Logic], unpaginated manuscript, c. 1899. Reprinted, CP 1.135–140. Eprint.
• Peirce, C.S. (1902), "Application of C.S. Peirce to the Executive Committee of the Carnegie Institution" (1902 July 15). Published, "Parts of Carnegie Application" (L75), pp. 13–73 in The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy, Carolyn Eisele (ed.), Mouton Publishers, The Hague, Netherlands, 1976. Eprint, Joseph Ransdell (ed.).

• Awbrey, Jon, and Awbrey, Susan (1995), "Interpretation as Action : The Risk of Inquiry", Inquiry : Critical Thinking Across the Disciplines 15, 40–52. Eprint.
• De Tienne, André (2006), "Peirce's Logic of Information", Seminario del Grupo de Estudios Peirceanos, Universidad de Navarra, 28 Sep 2006. Eprint.

## Document history

### Ontology List (Feb 2002 – Feb 2004)

Ontology List : Combined Links (Feb 2002 – Jun 2002)

Ontology List : Selected Links (Feb 2002 – Jun 2002)

Ontology List : Selected Links (Feb 2002 – Apr 2003)

Ontology List : Anthematic (Apr 2003)

Ontology List : Discussion (Jan 2004 – Feb 2004)

### Inquiry List (Mar 2003 – Apr 2003)

Inquiry List : C.S. Peirce, Harvard and Lowell Lectures (1865–1866)

Inquiry List : C.S. Peirce, Harvard Lectures (1865)

Inquiry List : C.S. Peirce, Lowell Lectures (1866)

Inquiry List : Reflection (Mar 2003)

1. http://stderr.org/pipermail/inquiry/2003-March/000233.html

Inquiry List : Commentary (Mar 2003 – Apr 2003)

Inquiry List : Anthematic (Apr 2003)

### Inquiry List : (Jan 2004 – Feb 2004)

Inquiry List : Discussion (Jan 2004 – Feb 2004)

### Inquiry List (Nov 2004)

Inquiry List : C.S. Peirce, Harvard and Lowell Lectures (1865–1866)

Inquiry List : C.S. Peirce, Harvard Lectures (1865)

Inquiry List : C.S. Peirce, Lowell Lectures (1866)

Inquiry List : Anthematic (Nov 2004)

Inquiry List : Commentary (Nov 2004)

### NKS Forum (Nov 2004)

NKS Forum : C.S. Peirce, Harvard and Lowell Lectures (1865–1866)

NKS Forum : Commentary (Nov 2004)

## Annotation work area

New List (1867)

Commentary Notes

• Passage 1. CE 1, 459–460.
• Passage 2. CE 1, 460–461.
• Passage 3. CE 1, 462.
• Passage 4. CE 1, 468–469.
• Passage 1. CE 1, 462–463. Induction
• Passage 2. CE 1, 463–464. Induction
• Passage 3. CE 1, 464–465.
• Passage 4. CE 1, 465.
• Passage 5. CE 1, 466–467.