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ICE. Note 1

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 x Extension.
  

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ICE. Note 2

| 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'.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 467.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 3

| 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.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 467-468.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 4

| Yet there are combinations of words and combinations of conceptions
| which are not strictly speaking symbols. These are of two kinds
| of which I will give you instances. We have first cases like:
|
| 'man and horse and kangaroo and whale',
|
| and secondly, cases like:
|
| 'spherical bright fragrant juicy tropical fruit'.
|
| The first of these terms has no comprehension which is adequate to the
| limitation of the extension. In fact, men, horses, kangaroos, and whales
| have no attributes in common which are not possessed by the entire class
| of mammals. For this reason, this disjunctive term, man and horse and
| kangaroo and whale, is of no use whatever. For suppose it is the subject
| of a sentence; suppose we know that men and horses and kangaroos and
| whales have some common character. Since they have no common character
| which does not belong to the whole class of mammals, it is plain that
| 'mammals' may be substituted for this term. Suppose it is the predicate
| of a sentence, and that we know that something is either a man or a horse
| or a kangaroo or a whale; then, the person who has found out this, knows
| more about this thing than that it is a mammal; he therefore knows which
| of these four it is for these four have nothing in common except what
| belongs to all other mammals. Hence in this case the particular one
| may be substituted for the disjunctive term. A disjunctive term,
| then, -- one which aggregates the extension of several symbols, --
| may always be replaced by a simple term.
|
| Hence if we find out that neat are herbivorous, swine are herbivorous,
| sheep are herbivorous, and deer are herbivorous; we may be sure that there
| is some class of animals which covers all these, all the members of which are
| herbivorous. Now a disjunctive term -- such as 'neat swine sheep and deer',
| or 'man, horse, kangaroo, and whale' -- is not a true symbol. It does not
| denote what it does in consequence of its connotation, as a symbol does;
| on the contrary, no part of its connotation goes at all to determine what
| it denotes -- it is in that respect a mere accident if it denote anything.
| Its 'sphere' is determined by the concurrence of the four members, man,
| horse, kangaroo, and whale, or neat swine sheep and deer as the case
| may be.
|
| Now those who are not accustomed to the homologies of the conceptions of
| men and words, will think it very fanciful if I say that this concurrence
| of four terms to determine the sphere of a disjunctive term resembles the
| arbitrary convention by which men agree that a certain sign shall stand
| for a certain thing. And yet how is such a convention made? The men
| all look upon or think of the thing and each gets a certain conception
| and then they agree that whatever calls up or becomes an object of that
| conception in either of them shall be denoted by the sign. In the one
| case, then, we have several different words and the disjunctive term
| denotes whatever is the object of either of them. In the other case,
| we have several different conceptions -- the conceptions of different
| men -- and the conventional sign stands for whatever is an object of
| either of them. It is plain the two cases are essentially the same,
| and that a disjunctive term is to be regarded as a conventional sign
| or index. And we find both agree in having a determinate extension
| but an inadequate comprehension.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 468-469.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 5

| 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.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 469.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 6

| 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.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 470.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 7

| 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.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 470-471.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 8

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:

t_1 = spherical
t_2 = bright
t_3 = fragrant
t_4 = juicy
t_5 = tropical
t_6 = fruit

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ICE. Note 9

Let us now consider Peirce's alternate example of a disjunctive term,
"neat, swine, sheep, deer", which he commonly borrows from classical
and scholastic discussions as a stock example of inductive reasoning.

| 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.

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ICE. Note 10

I continue with the out lay of my incidental musings
on the theme of "approximal inference rules" (AIR's).

| 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.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 467-468.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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.

Jon Awbrey

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ICE. Note 11

| 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.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 458-459.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 12

| 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'.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 459-460.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 13

| 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
|
| C.S. Peirce, 'Chronological Edition', CE 1, 460.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 14

| 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.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 461.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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ICE. Note 15

| 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.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 461.
|
| Charles Sanders Peirce,
|"The Logic of Science, or, Induction and Hypothesis",
| Lowell Institute Lectures of 1866, pages 357-504 in:
|
|'Writings of Charles S. Peirce: A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

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