Information Equals Comprehension Times Extension

Another angle from which to approach the incidence of signs and inquiry is by way of Peirce's "theory of information" — yes, that's just what he called it, 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:

Information = Comprehension × 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.

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 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` * ` `*` ` ` ` `*` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` * `*` ` `*` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` * * ` * * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ** ** ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` 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{t_j : 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{s_j : 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 v ⇒ w, 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).

Selection 18

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.

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. (Peirce 1866, "Lowell Lecture 7", CE 1, 466–467).

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).</p>

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:
- http://math.cofc.edu/faculty/kasman/MATHFICT/mf82.html
- http://www.lsi.usp.br/~rbianchi/clarke/ACC.Biography.html

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 i

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

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

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;