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

Operator Variables in Logical Graphs • 1.2
• https://inquiryintoinquiry.com/2024/04/06/operator-variables-in-logical-graphs-1/

Consider De Morgan's rules:

• ¬(A ∧ B) = ¬A ∨ ¬B

• ¬(A ∨ B) = ¬A ∧ ¬B

The common form exhibited by the two rules could be captured in a single formula by taking “o₁” and “o₂” as variable names ranging over a family of logical operators, then asking what substitutions for o₁ and o₂ would satisfy the following equation.

• ¬(A o₁ B) = ¬A o₂ ¬B

We already know two solutions to this “operator equation”, namely, (o₁, o₂) = (∧, ∨) and (o₁, o₂) = (∨, ∧). Wouldn't it be just like Peirce to ask if there are others?

Having broached the subject of “logical operator variables”, I will leave it for now in the same way Peirce himself did:

❝I shall not further enlarge upon this matter at this point, although the conception mentioned opens a wide field; because it cannot be set in its proper light without overstepping the limits of dichotomic mathematics.❞ (Peirce, CP 4.306).

Further exploration of operator variables and operator invariants treads on grounds traditionally known as second intentional logic and “opens a wide field”, as Peirce says. For now, however, I will tend to that corner of the field where our garden variety logical graphs grow, observing the ways in which operative variations and operative themes naturally develop on those grounds.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #LogicalOperatorVariables

@Bene @ehud

C.S. #Peirce

In the Way of Inquiry • Initial Unpleasantness
• https://inquiryintoinquiry.com/2023/01/08/in-the-way-of-inquiry-initial-unpleasantness-a/

Clouds and thunder:
The image of Difficulty at the Beginning.
Thus the superior man
Brings order out of confusion.

— I Ching ䷂ Hexagram 3

Inquiry begins in doubt, a debit of certainty and a drought of information, never a pleasant condition to acknowledge, and one of the primary obstacles to inquiry may be reckoned as owing to the onus one naturally feels on owning up to that debt. Human nature far prefers to revel in the positive features of whatever scientific knowledge it already possesses and the mind defers as long as possible the revolt it feels arising on facing the uncertainties that still persist, the “nots” and “not yets” it cannot as yet and ought not deny.

Reference —

The I Ching, or Book of Changes, R. Wilhelm and C.F. Baynes (trans.), Foreword by C.G. Jung, Bollingen Series 19, Princeton University Press, Princeton, NJ. 1st edition 1950, 2nd edition 1961, 3rd edition 1967.

Overview —
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Obstacles —
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_5#Obstacles

#Peirce #Inquiry #InquiryIntoInquiry #InquiryDrivenSystems
#Semiotics #SignRelations #Semiositis #ObstaclesToInquiry

In the Way of Inquiry • Obstacles
• https://inquiryintoinquiry.com/2023/01/07/in-the-way-of-inquiry-obstacles-a/

❝Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you already incline to think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy:

❝Do not block the way of inquiry.❞

C.S. Peirce, Collected Papers, CP 1.135–136.
From an unpaginated ms. “F.R.L.”, c. 1899.

Often the biggest obstacle to learning more is the need to feel you already know. And yet there are some things you know, at least, compared to other things, and it makes sense to use what you already know well enough to learn what you need to know better. The question is, how do you know which is which? What test can tell what is known so well it can be trusted in learning what is not?

One way to test a supposed knowledge is to try and formulate it in such a way it can be taught to other people. A related test, harder in some ways but easier in others, is to try and formalize knowledge so completely that even a computer can go through the motions supposed to be definitive of its practice.

Both ways of testing a supposition of knowledge depend on putting knowledge in forms which can be communicated or transported from one medium or system of interpretation to another. Knowledge already in a concrete form takes no more than a simple reformation or transformation, otherwise it takes a more radical metamorphosis, from a wholly disorganized condition to the first inklings of a portable or sharable form.

#Peirce #Inquiry #InquiryIntoInquiry #InquiryDrivenSystems
#Semiotics #SignRelations #Semiositis #ObstaclesToInquiry

@NicoleCRust @knutson_brain

I'm guessing different people must draw the distinction differently. Coming from a #Peirce #Whitehead #Pragmatism + #ProcessThinking tradition, I personally view computations as processes.

For example, people of that persuasion will tend to give a process like #Inquiry equal billing with the corresponding product we all know and love as #Knowledge and maybe even consider it more fundamental.