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