Information = Comprehension × Extension • Comment

[Jon Awbrey]

Information = Comprehension × Extension • Comment 1
• https://inquiryintoinquiry.com/2026/08/02/information-comprehension-x-extension-comment-1-b/

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

Selection 1 ends with Peirce drawing the following conclusion
about the links between information, comprehension, inference,
and symbolization.

❝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)

At this point in his inventory of scientific reasoning, Peirce is
relating the nature of inference, information, and inquiry to the
character of the signs mediating the process in question, a process
he describes as “symbolization”.

In the interest of clarity let's draw from Peirce's account
a couple of quick sketches, designed to show how the examples
he gives of conjunctive terms and disjunctive terms might look
if they were cast within a lattice‑theoretic framework.

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

Looking back on Selection 5, let's first examine Peirce's example of
a conjunctive term — “spherical, bright, fragrant, juicy, tropical fruit” —
within a lattice framework. We have the following six terms.

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

Suppose z is the logical conjunction of the above 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, 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.❞ (Peirce 1866, p. 470).

In other words, if something x is said to be z then we may guess
fairly surely x is really an orange, in short, x has all the
additional features otherwise summed up quite succinctly in
the much more constrained term y, where y means “an orange”.

Figure 1 shows the implication ordering of logical terms
in the form of a “lattice diagram”.

Figure 1. Conjunctive Term z, Taken as Predicate
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-1.jpg

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” of
the conjoined terms, z = glb{t₁, t₂, t₃, t₄, t₅, t₆}, and what
we might regard as the natural conjunction or natural glb of
those terms, namely, y, “an orange”.

In sum there is an extra measure of constraint which goes into forming
the natural kinds lattice from the free lattice which logic and set theory
would otherwise impose as a default background. The local manifestations
of that global information are meted out over the structure of the natural
lattice by just such abductive gaps as the one we observe between z and y.

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/l7GWpe
cc: https://www.researchgate.net/post/Information_Comprehension_Extension_Comment

[Jon Awbrey]

Information = Comprehension × Extension • Comment 2
• https://inquiryintoinquiry.com/2026/08/07/information-comprehension-x-extension-comment-2-b/

Let's examine Peirce's second example of a disjunctive term —
“neat, swine, sheep, deer” — within the style of lattice framework
we used before.

❝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.❞ (468–469).

❝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.❞ (469).

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

In view of the analogical symmetries the disjunctive term shares
with the conjunctive case, we can run through this example in
fairly short order. We have the following four terms.

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

Suppose u is the logical disjunction of the above four terms.

u = ((s₁)(s₂)(s₃)(s₄))

Figure 2 shows the implication ordering of logical terms
in the form of a lattice diagram.

Figure 2. Disjunctive Term u, Taken as Subject
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-2.jpg

Here we have a situation which is dual to the structure of
the conjunctive example. There is a gap between the logical
disjunction u, in lattice terminology, the “least upper bound”
of the disjoined terms, u = lub{s₁, s₂, s₃, s₄}, and what we might
regard as the natural disjunction or natural lub of those terms,
namely, v, “cloven‑hoofed”.

Once again, the sheer implausibility of imagining the disjunctive term u
would ever be embedded exactly as such in a lattice of natural kinds leads
to the evident “naturalness” of the induction to the implication v ⇒ w,
namely, the rule that cloven‑hoofed animals are herbivorous.

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/V9P90w
cc: https://www.researchgate.net/post/Information_Comprehension_Extension_Comment

[Jon Awbrey]

Information = Comprehension × Extension • Comment 3
• https://inquiryintoinquiry.com/2026/08/08/information-comprehension-x-extension-comment-3-b/

Peirce identifies inference with a process he describes as
“symbolization”. Let us consider what that might imply.

❝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”.❞
(467)

Even if it were only a rough analogy between inference and
symbolization, a principle of logical continuity, what is known
in physics as a “correspondence principle”, would suggest parallels
between steps of reasoning in the neighborhood of exact inferences
and signs in the vicinity of genuine symbols. That would lead us to
expect a correspondence between degrees of inference and degrees of
symbolization extending from exact to approximate (“non‑demonstrative”)
inferences and from genuine to approximate (“degenerate”) 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.❞ (467–468)

In addition to 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 also of the kinds of signs leading up to
genuine symbols, can all be recognized as pervasive Kantian themes.

In order to draw out those themes and see how Peirce was led to
develop their leading ideas, let us bring together our previous
Figures, abstracting from their concrete details, and see if we
can figure out what is going on.

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

Figure 3. Conjunctive Predicate z, Abduction of Case x ⇒ y
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-3.jpg

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

Figure 4. Disjunctive Subject u, Induction of Rule v ⇒ w
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-4.jpg

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/5MJpxN
cc: https://www.researchgate.net/post/Information_Comprehension_Extension_Comment

[Jon Awbrey]

Information = Comprehension × Extension • Comment 4
• https://inquiryintoinquiry.com/2026/08/09/information-comprehension-x-extension-comment-4-b/

Re: Information = Comprehension × Extension • Comment 3
• https://inquiryintoinquiry.com/2026/08/08/information-comprehension-x-extension-comment-3-b/

Reflecting further on Comment 3, many things still puzzle me about
Peirce's account at this point. The question marks I added to the
Figures of that post indicate the node labels I have remaining doubts
about. For example, in Figure 3, is z really an icon of object y?
Again, in Figure 4, is u really an index of object v? There is nothing
for it but returning to Peirce's text and trying once more to follow his
reasoning.

Let's go back to Peirce's example of abductive inference and try to get a
clearer picture of why he connects it with conjunctive terms and iconic signs.

Figure 1 shows the implication ordering of logical terms in the form
of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-1.jpg

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

Figure 3. Conjunctive Predicate z, Abduction of Case x ⇒ y
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-3.jpg

The relationship between conjunctive terms and iconic signs may be
understood along the following lines. If there is anything with all
the properties described by the conjunctive term “spherical bright
fragrant juicy tropical fruit” then sign users may use that thing as
an icon of an orange, precisely because it shares those properties
with an orange. But the only natural examples of things with all
those properties are oranges themselves, so the only thing qualified
to serve as a natural icon of an orange by virtue of those very
properties is that orange itself or another orange.

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/5wA12M
cc: https://www.researchgate.net/post/Information_Comprehension_Extension_Comment

[Jon Awbrey]

Information = Comprehension × Extension • Comment 5
• https://inquiryintoinquiry.com/2026/08/10/information-comprehension-x-extension-comment-5-b/

Let's stay with Peirce's example of abductive inference
a little longer and try to clear up the more troublesome
confusions tending to arise.

Figure 1 shows the implication ordering of logical terms
in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-1.jpg

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

Figure 3. Conjunctive Predicate z, Abduction of Case x ⇒ y
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-3.jpg

One thing needs to be stressed at this point. It is important
to recognize the conjunctive term itself — namely, the syntactic
string “spherical bright fragrant juicy tropical fruit” — is not
an icon but a symbol.‡ It has its place in a formal system of
symbols, for example, a propositional calculus, where it would
normally be interpreted as a logical conjunction of six elementary
propositions, denoting anything in the universe of discourse with
all six of the corresponding properties.

The symbol “spherical bright fragrant juicy tropical fruit” denotes
objects which may be taken as icons of oranges by virtue of their
bearing those six properties in common with oranges. But there are
no objects denoted by the symbol which aren't already oranges themselves.
Thus we observe a natural reduction in the denotation of the symbol,
consisting in the absence of cases outside of oranges which have all
the properties indicated.

The above analysis provides another way to understand the abductive
inference from the Fact x ⇒ z and the Rule y ⇒ z to the Case x ⇒ y.
The lack of any cases which are z and not y is expressed by the implication
z ⇒ y. Taking that in conjunction with the Rule y ⇒ z gives the logical
equivalence y = z. But that reduces the Case x ⇒ y to the Fact x ⇒ z and
so the Case is justified.

Viewed in the light of the above analysis, Peirce's example of abductive
reasoning exhibits an especially strong form of inference, almost deductive
in character. Do all abductive arguments take that form, or may there be
weaker styles of abductive reasoning which enjoy their own levels of
plausibility? That must remain an open question at this point.

Remark —

‡ Readers will notice I have slipped at this point from using “symbol”
in the precise technical sense Peirce introduced at the beginning of
this discussion to the more ordinary sense all of us, Peirce included,
tend to use on other occasions. Should it become a big problem we can
always find a way to mark the distinction but so far it seems context
has usually sufficed to resolve any likely confusion.

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/lz3G6P
cc: https://www.researchgate.net/post/Information_Comprehension_Extension_Comment

[Jon Awbrey]

Information = Comprehension × Extension • Comment 6
• https://inquiryintoinquiry.com/2026/08/16/information-comprehension-x-extension-comment-6-b/

Returning to Peirce's example of inductive inference in Comment 2,
let's try to get a clearer picture of why he connects it with
disjunctive terms and indicial signs.

At this point in time I can't say I'm entirely satisfied with
my understanding of the relationship between disjunctive terms,
indicial signs, and inductive inferences as presented by Peirce
in his early accounts. What follows is just one of the simplest
and least question‑begging attempts at rational reconstruction
I've been able to devise.

Figure 2 shows the implication ordering of logical terms

in the form of a lattice diagram.

Figure 2. Disjunctive Term u, Taken as Subject
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-2.jpg

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

Figure 4. Disjunctive Subject u, Induction of Rule v ⇒ w
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-4.jpg

If there is any distinguishing feature shared by all the instances
under the disjunctive description “neat, swine, sheep, deer” then
sign users may take that feature as a predictor of being herbivorous,
precisely because all the things under the disjunctive description
are herbivorous. But everything under the disjunctive description
is cloven‑hoofed, so the cases under the disjunctive description
serve to indicate, support, or witness the utility of the induction
from cloven‑hoofed to herbivorous.

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/VDDWqj
cc: https://www.researchgate.net/post/Information_Comprehension_Extension_Comment

[Jon Awbrey]

Information = Comprehension × Extension • Comment 7
• https://inquiryintoinquiry.com/2026/08/20/information-comprehension-x-extension-comment-7-b/

Let's stay with Peirce's example of inductive inference

a little longer and try to clear up the more troublesome
confusions tending to arise.

Figure 2 shows the implication ordering of logical terms
in the form of a lattice diagram.

Figure 2. Disjunctive Term u, Taken as Subject
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-2.jpg

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

Figure 4. Disjunctive Subject u, Induction of Rule v ⇒ w
• https://inquiryintoinquiry.com/wp-content/uploads/2016/10/ice-figure-4.jpg

One final point needs to be stressed. It is important to recognize
the disjunctive term itself — the syntactic formula “neat, swine, sheep,
deer” or any logically equivalent formula — is not an index but a symbol.‡
It has the character of an artificial symbol which is constructed to fill
a place in a formal system of symbols, for example, a propositional calculus.
In that setting it would normally be interpreted as a logical disjunction of
four elementary propositions, denoting anything in the universe of discourse
which has any of the four corresponding properties.

The artificial symbol “neat, swine, sheep, deer” denotes objects which serve
as indices of the genus herbivore by virtue of their belonging to one of the
four named species of herbivore. But there is in addition a natural symbol
which serves to unify the manifold of given species, namely, the concept of
a cloven‑hoofed animal.

As a symbol or general representation, the concept of a cloven‑hoofed
animal connotes an attribute and connotes it in such a way as to
determine what it denotes. Thus we observe a natural expansion
in the connotation of the symbol, amounting to what Peirce calls
the “superfluous comprehension”, the information added by an
“ampliative” or synthetic inference.

In sum we have sufficient information to motivate an inductive inference,
from the Fact u ⇒ w and the Case u ⇒ v to the Rule v ⇒ w.

Remark —

‡ Here, once again, I have departed from using “symbol” in the

precise technical sense Peirce introduced at the beginning of

this discussion, reverting to the more ordinary sense all of

us, Peirce included, tend to use on other occasions.

Perhaps the best way to smooth the wrinkle in usage is to
mark a distinction among symbols, singling out the natural,
normal, canonical, or simple symbols within the more general
run of artificial, compound, or complex types. Taking that
tack has the beneficial side‑effect of aligning the work of
abduction and induction, at least, in a pre‑established
universe of discourse, with the mainstream of work in
computation which takes us from dubious terms to clear
signs for the objects of our interest.

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/V03gKe
cc: https://www.researchgate.net/post/Information_Comprehension_Extension_Comment