Information = Comprehension × Extension • Comment
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Jon Awbrey
2:56 PM (1 hour ago) 2:56 PM
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
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
• 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
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