Information = Comprehension × Extension

[Jon Awbrey]

Information = Comprehension × Extension • Preamble
• https://inquiryintoinquiry.com/2026/06/16/information-comprehension-x-extension-preamble-a/

Perhaps the best perspective from which to bring
the connection between the theory of signs and the
theory of inquiry into its proper focus is Peirce's
own Theory of Information, which he began setting
forth in lectures at Harvard and the Lowell Institute
in 1865 and 1866. Peirce encapsulates the elements
of his theory in the following formula.

• Information = Comprehension × Extension

In the Resources below I link to my study of Peirce's
1865–1866 Lectures on the Logic of Science, with
selections from the lectures and my commentary
on them.

Ten 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.

References —

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.

Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”,
Proceedings of the American Academy of Arts and Sciences, Vol. 7,
pp. 416–432. Archive. Online.
• https://web.archive.org/web/20200116141600/https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm
• https://www.jstor.org/stable/20179572

Resources —

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

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2025/05/04/survey-of-pragmatic-semiotic-information-9/

Regards,

Jon

cc: https://www.academia.edu/community/lnbq20
cc: https://www.researchgate.net/post/Information_Comprehension_Extension2

[Jon Awbrey]

Information = Comprehension × Extension • Selection 1
• https://inquiryintoinquiry.com/2026/06/20/information-comprehension-x-extension-selection-1-b/

I am going to begin with six selections from Peirce's
Lectures of 1866 where he addresses the roles of signs
informing sign relations and their relation to the rules
of inference governing inquiries.

⁂ ⁂ ⁂

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 those 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)

References —

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.

Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”,
Proceedings of the American Academy of Arts and Sciences, Vol. 7,
pp. 416–432. Archive. Online.
• https://web.archive.org/web/20200116141600/https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm
• https://www.jstor.org/stable/20179572

Resources —

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

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2025/05/04/survey-of-pragmatic-semiotic-information-9/

Regards,

Jon

cc: https://www.academia.edu/community/VXa9Xm
cc: https://www.researchgate.net/post/Information_Comprehension_Extension2

[Jon Awbrey]

Information = Comprehension × Extension • Selection 2
• https://inquiryintoinquiry.com/2026/07/24/information-comprehension-x-extension-selection-2-b/

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)

References —

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.

Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”,
Proceedings of the American Academy of Arts and Sciences, Vol. 7,
pp. 416–432. Archive. Online.
• https://web.archive.org/web/20200116141600/https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm
• https://www.jstor.org/stable/20179572

Resources —

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

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2025/05/04/survey-of-pragmatic-semiotic-information-9/

Regards,

Jon

cc: https://www.academia.edu/community/5N49pK
cc: https://www.researchgate.net/post/Information_Comprehension_Extension2

[Jon Awbrey]

Information = Comprehension × Extension • Selection 3
• https://inquiryintoinquiry.com/2026/07/26/information-comprehension-x-extension-selection-3-b/

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)

References —

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.

Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”,
Proceedings of the American Academy of Arts and Sciences, Vol. 7,
pp. 416–432. Archive. Online.
• https://web.archive.org/web/20200116141600/https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm
• https://www.jstor.org/stable/20179572

Resources —

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

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2025/05/04/survey-of-pragmatic-semiotic-information-9/

Regards,

Jon

cc: https://www.academia.edu/community/VW0Xm9
cc: https://www.researchgate.net/post/Information_Comprehension_Extension2

[Jon Awbrey]

Information = Comprehension × Extension • Selection 4
• https://inquiryintoinquiry.com/2026/07/27/information-comprehension-x-extension-selection-4-b/

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 that. 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)

References —

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.

Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”,
Proceedings of the American Academy of Arts and Sciences, Vol. 7,
pp. 416–432. Archive. Online.
• https://web.archive.org/web/20200116141600/https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm
• https://www.jstor.org/stable/20179572

Resources —

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

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2025/05/04/survey-of-pragmatic-semiotic-information-9/

Regards,

Jon

cc: https://www.academia.edu/community/5wAGv9
cc: https://www.researchgate.net/post/Information_Comprehension_Extension2

[Jon Awbrey]

Information = Comprehension × Extension • Selection 5
• https://inquiryintoinquiry.com/2026/07/28/information-comprehension-x-extension-selection-5-b/

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)

References —

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.

Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”,
Proceedings of the American Academy of Arts and Sciences, Vol. 7,
pp. 416–432. Archive. Online.
• https://web.archive.org/web/20200116141600/https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm
• https://www.jstor.org/stable/20179572

Resources —

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

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2025/05/04/survey-of-pragmatic-semiotic-information-9/

Regards,

Jon

cc: https://www.academia.edu/community/lOkz1D
cc: https://www.researchgate.net/post/Information_Comprehension_Extension2

[Jon Awbrey]

Information = Comprehension × Extension • Selection 6
• https://inquiryintoinquiry.com/2026/07/29/information-comprehension-x-extension-selection-6-b/

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)

References —

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.

Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”,
Proceedings of the American Academy of Arts and Sciences, Vol. 7,
pp. 416–432. Archive. Online.
• https://web.archive.org/web/20200116141600/https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm
• https://www.jstor.org/stable/20179572

Resources —

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

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2025/05/04/survey-of-pragmatic-semiotic-information-9/

Regards,

Jon

cc: https://www.academia.edu/community/V140KG
cc: https://www.researchgate.net/post/Information_Comprehension_Extension2