Information = Comprehension × Extension

[Jon Awbrey]

Information = Comprehension × Extension • Preamble
• https://inquiryintoinquiry.com/2024/10/04/information-comprehension-x-extension-preamble/

All,

Eight summers ago I hit on what struck me as a new insight into one
of the most recalcitrant problems in Peirce's semiotics and logic of
science, namely, the relation between “the manner in which different
representations stand for their objects” and the way in which different
inferences transform states of information. I roughed out a sketch of
my epiphany in a series of blog posts then set it aside for the cool of
later reflection. Now looks to be a choice moment for taking another look.

A first pass through the variations of representation and reasoning detects the
axes of iconic, indexical, and symbolic manners of representation on the one hand
and the axes of abductive, inductive, and deductive modes of inference on the other.
Early and often Peirce suggests a natural correspondence between the main modes of
inference and the main manners of representation but his early arguments differ from
his later accounts in ways deserving close examination, partly for the extra points in
his line of reasoning and partly for his explanation of indices as signs constituted by
convening the variant conceptions of sundry interpreters.

Resources —

Inquiry Blog • Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

OEIS Wiki • Information = Comprehension × Extension
• https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

C.S. Peirce • Upon Logical Comprehension and Extension
• https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm

Regards,

Jon

cc: https://www.academia.edu/community/LGqOKr
cc: https://mathstodon.xyz/@Inquiry/113249701127551380

[Jon Awbrey]

Information = Comprehension × Extension • Selection 1
• https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-extension-selection-1-a/

All,

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

❝Let us now return to the information. The information of a term is the
measure of its superfluous comprehension. That is to say that the proper
office of the comprehension is to determine the extension of the term.

❝For instance, you and I are men because we possess those attributes —
having two legs, being rational, &c. — which make up the comprehension
of “man”. Every addition to the comprehension of a term lessens its
extension up to a certain point, after that further additions increase
the information instead.

❝Thus, let us commence with the term “colour”; add to the comprehension
of this term, that of “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, p. 467)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”,
Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce :
A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project,
Indiana University Press, Bloomington, IN, 1982.

Resources —

Inquiry Blog • Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

OEIS Wiki • Information = Comprehension × Extension
• https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

C.S. Peirce • Upon Logical Comprehension and Extension
• https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm

Regards,

Jon

cc: https://www.academia.edu/community/Lm0YXe
cc: https://mathstodon.xyz/@Inquiry/113249701127551380

[Jon Awbrey]

Information = Comprehension × Extension • Selection 2
• https://inquiryintoinquiry.com/2024/10/06/information-comprehension-x-extension-selection-2-a/

Re: Information = Comprehension × Extension • Selection 1
• https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-extension-selection-1-a/

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

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

❝For this purpose, I must call your attention to the differences there are
in the manner in which different representations stand for their objects.

❝In the first place there are likenesses or copies — such as “statues”, “pictures”,
“emblems”, “hieroglyphics”, and the like. Such representations stand for their
objects only so far as they have an actual resemblance to them — that is agree
with them in some characters. The peculiarity of such representations is that
they do not determine their objects — they stand for anything more or less;
for they stand for whatever they resemble and they resemble everything
more or less.

❝The second kind of representations are such as are set up by a convention of men
or a decree of God. Such are “tallies”, “proper names”, &c. The peculiarity of
these “conventional signs” is that they represent no character of their objects.

❝Likenesses denote nothing in particular; “conventional signs” connote nothing
in particular.

❝The third and last kind of representations are “symbols” or general representations.
They connote attributes and so connote them as to determine what they denote. To this
class belong all “words” and all “conceptions”. Most combinations of words are also
symbols. A proposition, an argument, even a whole book may be, and should be,
a single symbol.❞

(Peirce 1866, pp. 467–468)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”,
Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce :
A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project,
Indiana University Press, Bloomington, IN, 1982.

Resources —

Inquiry Blog • Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

OEIS Wiki • Information = Comprehension × Extension
• https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension

C.S. Peirce • Upon Logical Comprehension and Extension
• https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm

Regards,

Jon

cc: https://www.academia.edu/community/V1EKBy
cc: https://mathstodon.xyz/@Inquiry/113249701127551380

[Jon Awbrey]

Information = Comprehension × Extension • Selection 3
• https://inquiryintoinquiry.com/2024/10/07/information-comprehension-x-extension-selection-3-a/

All,

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

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

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

❝man and horse and kangaroo and whale,

❝and secondly, cases like:

❝spherical bright fragrant juicy tropical fruit.

❝The first of these terms has no comprehension which is adequate to the limitation
of the extension. In fact, men, horses, kangaroos, and whales have no attributes
in common which are not possessed by the entire class of mammals. For this reason,
this disjunctive term, man and horse and kangaroo and whale, is of no use whatever.

❝For suppose it is the subject of a sentence; suppose we know that men and horses
and kangaroos and whales have some common character. Since they have no common
character which does not belong to the whole class of mammals, it is plain that
mammals may be substituted for this term.

❝Suppose it is the predicate of a sentence, and that we know that something
is either a man or a horse or a kangaroo or a whale; then, the person who
has found out this, knows more about this thing than that it is a mammal;
he therefore knows which of these four it is for these four have nothing
in common except what belongs to all other mammals. Hence in this case
the particular one may be substituted for the disjunctive term.

❝A disjunctive term, then, — one which aggregates the extension
of several symbols, — may always be replaced by a simple term.

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

❝Now those who are not accustomed to the homologies of the conceptions of
men and words, will think it very fanciful if I say that this concurrence of
four terms to determine the sphere of a disjunctive term resembles the arbitrary
convention by which men agree that a certain sign shall stand for a certain thing.
And yet how is such a convention made? The men all look upon or think of the thing
and each gets a certain conception and then they agree that whatever calls up or
becomes an object of that conception in either of them shall be denoted by the sign.

❝In the one case, then, we have several different words and the disjunctive term
denotes whatever is the object of either of them. In the other case, we have
several different conceptions — the conceptions of different men — and the
conventional sign stands for whatever is an object of either of them.

❝It is plain the two cases are essentially the same, and that a disjunctive term
is to be regarded as a conventional sign or index. And we find both agree in
having a determinate extension but an inadequate comprehension.❞

(Peirce 1866, pp. 468–469)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”,
Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce :
A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project,
Indiana University Press, Bloomington, IN, 1982.

Regards,

Jon

cc: https://www.academia.edu/community/Lx81Ga
cc: https://mathstodon.xyz/@Inquiry/113249701127551380

[Jon Awbrey]

Information = Comprehension × Extension • Selection 4
• https://inquiryintoinquiry.com/2024/10/08/information-comprehension-x-extension-selection-4-a/

Re: Information = Comprehension × Extension • Selection 3
• https://inquiryintoinquiry.com/2024/10/07/information-comprehension-x-extension-selection-3-a/

All,

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

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

❝Accordingly, if we are engaged in symbolizing and we come to such a proposition as
“Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive
term may be replaced by a true symbol. But suppose we know of no symbol for neat,
swine, sheep, and deer except cloven-hoofed animals. There is but one objection
to substituting this for the disjunctive term; it is that we should, then, say
more than we have observed. In short, it has a superfluous information.

❝But we have already seen that this is an objection which must always stand in the
way of taking symbols. If therefore we are to use symbols at all we must use them
notwithstanding that. Now all thinking is a process of symbolization, for the
conceptions of the understanding are symbols in the strict sense.

❝Unless, therefore, we are to give up thinking altogether we must admit the validity
of induction. But even to doubt is to think. So we cannot give up thinking and the
validity of induction must be admitted.❞

(Peirce 1866, p. 469)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”,
Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce :
A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project,
Indiana University Press, Bloomington, IN, 1982.

Regards,

Jon

cc: https://www.academia.edu/community/VjvJbJ
cc: https://mathstodon.xyz/@Inquiry/113249701127551380

[Jon Awbrey]

Information = Comprehension × Extension • Selection 5
• https://inquiryintoinquiry.com/2024/10/09/information-comprehension-x-extension-selection-5-a/

All,

Peirce now turns to his example of a conjunctive term, which
he uses to show the connection between iconic reference and
abductive inference.

❝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, p. 470)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”,
Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce :
A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project,
Indiana University Press, Bloomington, IN, 1982.

Regards,

Jon

cc:https://www.academia.edu/community/5kdJBG
cc: https://mathstodon.xyz/@Inquiry/113249701127551380

[Jon Awbrey]

Information = Comprehension × Extension • Selection 6
• https://inquiryintoinquiry.com/2024/10/10/information-comprehension-x-extension-selection-6-a/

Re: Information = Comprehension × Extension • Selection 1
• https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-extension-selection-1-a/

All,

Selection 1 opens with Peirce proposing, “The information of a term is the
measure of its superfluous comprehension”, and it closes with his offering
the following promise.

❝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.❞

Summing up his account to this point, Peirce appears confident he's kept his promise.
Promising on our own account to give it another pass, we'll let him have the last word —
for now.

❝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, pp. 470–471)

Reference —

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”,
Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce :
A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project,
Indiana University Press, Bloomington, IN, 1982.

Regards,

Jon

cc: https://www.academia.edu/community/L6RmB8
cc: https://mathstodon.xyz/@Inquiry/113249701127551380