#informationequalscomprehensiontimesextension — Public Fediverse posts

Information = Comprehension × Extension • Preamble 1
• 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.

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 2.3
• https://inquiryintoinquiry.com/2024/10/06/information-comprehension-x-extension-selection-2-a/

❝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

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 2.2
• https://inquiryintoinquiry.com/2024/10/06/information-comprehension-x-extension-selection-2-a/

❝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.❞

(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.

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 2.1
• 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.❞

(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.

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 1.2
• https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-extension-selection-1-a/

❝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.

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Information = Comprehension × Extension • Selection 1.1
• https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-extension-selection-1-a/

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.❞

(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

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

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

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

#Peirce #Logic #Inference #Inquiry #Abduction #Induction #Deduction #LogicOfScience
#Information #Comprehension #Extension #InformationEqualsComprehensionTimesExtension
#Semiotics #SignRelations #Icon #Index #Symbol #PragmaticSemioticInformation

Pragmatic Semiotic Information • Comment 3
• https://inquiryintoinquiry.com/2024/03/22/pragmatic-semiotic-information-comment-3/

Memories are coming back to me more through the association of ideas than ordered by time or place. I can sense, almost touch a tangle of thoughts interlaced with each other — the “information first” approach to ontology, the “arrows only”, element‑free angle on category theory, Peirce's relativity of generals and individuals dispatching nominalism once and for all — but there is at core a hard knot of ideas so tightly wound it makes it difficult to articulate the links or see the untying if there is one to make.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • Comment 2
• https://inquiryintoinquiry.com/2024/03/20/pragmatic-semiotic-information-comment-2/

I was at the time working as a “scanner” in the High Energy Physics Lab at Michigan State, sitting in a darkened room measuring tracks of particle interactions projected on a lighted scanning table from reels and reels of bubble chamber photographs gathered at CERN in a massive mad dash accelerator experiment some years before. For my part it was a menial job, 4pm to midnight every worklong day, but even a minion can imagine himself sharing in a hunt for the Ω⁻ particle, or whatever the Grail or Questying Beastie was at the time.

Meanwhile, in another part of the grove, I was spending my daylight hours checking off the final boxes for my Bachelor's degree, the main thing being to get my paper on Peirce, “Complications of the Simplest Mathematics”, approved as a substitute for a field study requirement. That had taken me two years' work in MSU's media library, poring through the microfilm reels of Peirce's Nachlass in search of enlightenment about a single puzzling paragraph I tripped over in his Collected Papers.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • Comment 1
• https://inquiryintoinquiry.com/2024/03/18/pragmatic-semiotic-information-comment-1/

I remember it was back in ’76 when I began to notice a subtle shift of focus in the computer science journals I was reading, from discussing “X” to discussing “Information About X”, a transformation I noted mentally as \( X \to \mathrm{Info}(X) \) whenever I ran across it. I suppose that small arc of revolution had been building for years but it struck me as crossing a threshold to a more explicit, self‑conscious stage about that time.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • 2.1
• https://inquiryintoinquiry.com/2024/03/04/pragmatic-semiotic-information-2/

What is information that a sign may bear it?

Three more questions arise at this juncture.

• How is a sign empowered to contain information?
• What is the practical context of communication?
• Why do we care about these bits of information?

A very rough answer to these questions might begin as follows.

Human beings are initially concerned solely with their own lives but then a world obtrudes on their subjective existence and so they find themselves forced to take an interest in the objective realities of its nature.

In pragmatic terms our initial aim, concern, interest, object, or “pragma” is expressed by the verbal infinitive “to live”, but the infinitive is soon reified into the derivative substantial forms of “nature”, “reality”, “the world”, and so on. Against that backdrop we find ourselves cast as the protagonists on a “scene of uncertainty”.

The situation may be pictured as a juncture from which a manifold of options fan out before us. It may be an issue of “truth”, “duty”, or “hope”, the last codifying a special type of uncertainty as to “what regulative principle has any chance of success”, but the chief uncertainty is that we are called on to make a choice and all too often we have very little clue which of the options is most fit to pick.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • 1.2
• http://inquiryintoinquiry.com/2024/03/03/pragmatic-semiotic-information-1/

A question of what's true is a “descriptive question” and there exist what are called “descriptive sciences” devoted to answering descriptive questions about any domain of phenomena one might care to name.

A question of what's to do, in other words, what must be done by way of achieving a given aim, is a “normative question” and there exist what are called “normative sciences” devoted to answering normative questions about any domain of problems one might care to address.

Since information plays its role on a stage set by uncertainty, a big part of saying what information is will necessarily involve saying what uncertainty is. There is little chance the vagaries of a word like “uncertainty”, given the nuances of its ordinary, poetic, and technical uses, can be corralled by a single pen, but there do exist established models and formal theories which manage to address definable aspects of uncertainty and these do have enough uses to make them worth looking into.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Pragmatic Semiotic Information • 1.1
• http://inquiryintoinquiry.com/2024/03/03/pragmatic-semiotic-information-1/

Information • What's it good for?

The good of information is its use in reducing our uncertainty about an issue which comes before us. But uncertainty comes in many flavors and so the information which serves to reduce uncertainty can be applied in several ways. The situations of uncertainty human agents commonly find themselves facing have been investigated under many headings, literally for ages, and the categories subtle thinkers arrived at long before the dawn of modern information theory still have their uses in setting the stage of an introduction.

Picking an example of a subtle thinker almost at random, the philosopher‑scientist Immanuel Kant surveyed the questions of human existence within the span of the following three axes.

• What's true?
• What's to do?
• What's to hope?

The third question is a bit too subtle for the present frame of discussion but the first and second are easily recognizable as staking out the two main axes of information theory, namely, the dual dimensions of “information” and “control”. Roughly the same space of concerns is elsewhere spanned by the dual axes of competence and performance, specification and optimization, or just plain knowledge and skill.

#Peirce #Logic #Inquiry #Information #Uncertainty #ScientificMethod
#Semiotics #Semiosis #RelationTheory #SignRelations #TriadicRelations
#InformationEqualsComprehensionTimesExtension #PragmaticSemioticInformation

Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 4
• http://inquiryintoinquiry.com/2024/02/27/survey-of-abduction-deduction-induction-analogy-inquiry-4/

This is a Survey of blog and wiki posts on three elementary forms of inference, as recognized by a logical tradition extending from Aristotle through Charles S. Peirce. Particular attention is paid to the way these inferential rudiments combine to form the more complex patterns of analogy and inquiry.

Please follow the above link for the full set of resources.
Articles and blog series on the core ideas are linked below.

Project Reports —

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

Prospects for Inquiry Driven Systems
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems

Introduction to Inquiry Driven Systems
• https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems

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

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Blog Surveys —

Survey of Inquiry Driven Systems
• https://inquiryintoinquiry.com/2023/04/23/survey-of-inquiry-driven-systems-5/

Survey of Pragmatic Semiotic Information
• https://inquiryintoinquiry.com/2023/07/23/survey-of-pragmatic-semiotic-information-7/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

#Peirce #Logic #Abduction #Deduction #Induction #Analogy #Inquiry
#FunctionalLogic #DifferentialLogic #DynamicsOfInquiry #Semiotics
#InformationEqualsComprehensionTimesExtension #InquiryDrivenSystem

Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 3
• https://inquiryintoinquiry.com/2023/04/02/survey-of-abduction-deduction-induction-analogy-inquiry-3/

This is a Survey of blog and wiki posts on three elementary forms of inference, as recognized by a logical tradition extending from Aristotle through C.S. Peirce. Particular attention is paid to the way these inferential rudiments combine to form the more complex patterns of analogy and inquiry.

Blog Dialogs —

• Abduction, Deduction, Induction, Analogy, Inquiry
1. https://inquiryintoinquiry.com/2012/08/16/abduction-deduction-induction-analogy-inquiry-1/
2. https://inquiryintoinquiry.com/2012/11/21/abduction-deduction-induction-analogy-inquiry-2/
3. https://inquiryintoinquiry.com/2016/02/17/abduction-deduction-induction-analogy-inquiry-3/
•••
29. https://inquiryintoinquiry.com/2020/03/15/abduction-deduction-induction-analogy-inquiry-29/
30. https://inquiryintoinquiry.com/2020/12/15/abduction-deduction-induction-analogy-inquiry-30/
31. https://inquiryintoinquiry.com/2022/10/14/abduction-deduction-induction-analogy-inquiry-31/

Blog Series —

Functional Logic • Inquiry and Analogy
• https://inquiryintoinquiry.com/2023/06/20/functional-logic-inquiry-and-analogy-preliminaries-2/
1. https://inquiryintoinquiry.com/2023/06/21/functional-logic-inquiry-and-analogy-1-2/
2. https://inquiryintoinquiry.com/2023/06/22/functional-logic-inquiry-and-analogy-2-2/
3. https://inquiryintoinquiry.com/2023/06/23/functional-logic-inquiry-and-analogy-3-2/
•••
19. https://inquiryintoinquiry.com/2023/07/17/functional-logic-inquiry-and-analogy-19-2/
20. https://inquiryintoinquiry.com/2023/07/18/functional-logic-inquiry-and-analogy-20-2/
21. https://inquiryintoinquiry.com/2023/07/19/functional-logic-inquiry-and-analogy-21-2/
Discussion
• https://inquiryintoinquiry.com/2023/07/06/functional-logic-inquiry-and-analogy-discussion-1/

OEIS Wiki —

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

Prospects for Inquiry Driven Systems
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems

Introduction to Inquiry Driven Systems
• https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems

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

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

#Peirce #Logic #Abduction #Deduction #Induction #Analogy #Inquiry
#FunctionalLogic #DifferentialLogic #DynamicsOfInquiry #Semiotics
#InformationEqualsComprehensionTimesExtension #InquiryDrivenSystem