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#deduction

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Continued thread

Interpreter and Interpretant • Selection 10.2
inquiryintoinquiry.com/2025/02
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In logical terms, the analogy of experience proceeds by inducing a Rule about the validity of a current knowledge base and then by deducing a Fact, the applicability of that knowledge base to a current experience.

Step 1 is Inductive, abstracting a Rule from a Case and a Fact.

• Case : E_past ⇒ E_poss, Chosen events fairly sample Collective events.
• Fact : E_past ⇒ K_pres, Chosen events support the Knowledge regime.
────────────────────────────────────────────────
• Rule : E_poss ⇒ K_pres, Collective events support the Knowledge regime.

Step 2 is Deductive, admitting a Case to a Rule and arriving at a Fact.

• Case : E_pres ⇒ E_poss, Current events fairly sample Collective events.
• Rule : E_poss ⇒ K_pres, Collective events support the Knowledge regime.
────────────────────────────────────────────────
• Fact : E_pres ⇒ K_pres, Current events support the Knowledge regime.

References —

Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
pdcnet.org/inquiryct/content/i
academia.edu/57812482/Interpre

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY.
gutenberg.org/files/37423/3742

Resources —

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
inquiryintoinquiry.com/2024/02

Survey of Semiotics, Semiosis, Sign Relations
inquiryintoinquiry.com/2024/01



Analogy of Experience
Inquiry Into Inquiry · Interpreter and Interpretant • Selection 10
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Interpreter and Interpretant • Selection 10.1
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Transfer —

Returning to the scene of Dewey's “Sign of Rain” example, let's continue examining how the transfer of knowledge through the analogy of experience works in that case.

By way of a recap, we began by considering a fragment K_pres of the reasoner's knowledge base which is logically equivalent to a conjunction of two rules.

• K_pres ⇔ (B ⇒ A) ∧ (B ⇒ D).

K_pres may be thought of as a piece of knowledge or item of information allowing for the possibility of certain conditions, expressed in the form of a logical constraint on the present universe of discourse.

Next we found it convenient to express all logical statements in terms of their models, that is, in terms of the primitive circumstances or elements of experience over which they hold true.

• Let E_past be the chosen set of experiences, or the circumstances in mind under “past experience”.

• Let E_poss be the collective set of experiences, or the prospective total of possible circumstances.

• Let E_pres be the current experience, or the circumstances immediately present to the reasoner.

If we think of the knowledge base K_pres as referring to the “regime of experience” over which it is valid, then the sets of models involved in the analogy may be ordered according to the relationships of set inclusion or logical implication existing among them.

Figure 4 shows the subsumption relations involved in the analogy of experience.

Figure 4. Analogy of Experience
inquiryintoinquiry.files.wordp



Continued thread

Interpreter and Interpretant • Selection 9.2
inquiryintoinquiry.com/2025/01

Transfer —

In logical terms, the analogy of experience proceeds by inducing a Rule about the validity of a current knowledge base and then by deducing a Fact, the applicability of that knowledge base to a current experience.

Step 1 is Inductive, abstracting a Rule from a Case and a Fact.

• Case : E_past ⇒ E_poss, Chosen events fairly sample Collective events.
• Fact : E_past ⇒ K_pres, Chosen events support the Knowledge regime.
────────────────────────────────────────────────
• Rule : E_poss ⇒ K_pres, Collective events support the Knowledge regime.

Step 2 is Deductive, admitting a Case to a Rule and arriving at a Fact.

• Case : E_pres ⇒ E_poss, Current events fairly sample Collective events.
• Rule : E_poss ⇒ K_pres, Collective events support the Knowledge regime.
────────────────────────────────────────────────
• Fact : E_pres ⇒ K_pres, Current events support the Knowledge regime.

References —

Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
pdcnet.org/inquiryct/content/i
academia.edu/57812482/Interpre

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY.
gutenberg.org/files/37423/3742

Resources —

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
inquiryintoinquiry.com/2024/02

Survey of Semiotics, Semiosis, Sign Relations
inquiryintoinquiry.com/2024/01



Inquiry Into Inquiry · Interpreter and Interpretant • Selection 9
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Interpreter and Interpretant • Selection 9.1
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Transfer —

Let's examine how the transfer of knowledge through the analogy of experience works in the case of Dewey's “Sign of Rain” example.

For concreteness, consider a fragment K_pres of the reasoner's knowledge base which is logically equivalent to a conjunction of two rules.

• K_pres ⇔ (B ⇒ A) ∧ (B ⇒ D).

K_pres may be thought of as a piece of knowledge or item of information allowing for the possibility of certain conditions, expressed in the form of a logical constraint on the present universe of discourse.

It is convenient to have the option of expressing all logical statements in terms of their models, that is, in terms of the primitive circumstances or elements of experience over which they hold true.

• Let E_past be the chosen set of experiences, or the circumstances in mind under “past experience”.

• Let E_poss be the collective set of experiences, or the prospective total of possible circumstances.

• Let E_pres be the current experience, or the circumstances immediately present to the reasoner.

If we think of the knowledge base K_pres as referring to the “regime of experience” over which it is valid, then the sets of models involved in the analogy may be ordered according to the relationships of set inclusion or logical implication existing among them.



Inquiry Into Inquiry · Interpreter and Interpretant • Selection 9
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Interpreter and Interpretant • Selection 8
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Transfer —

What exactly gives the acquisition of a knowledge base its distinctively inductive character? It is evidently the “analogy of experience” involved in applying what we've learned in the past to what confronts us in the present.

Whenever we find ourselves approaching a problem with the thought, “If past experience is any guide …” we can be sure the analogy of experience has come into play. We are seeking to find analogies between past experience as a totality and present experience as a point of application.

From a statistical point of view what we mean is this — “If past experience is a fair sample of possible experience then knowledge gained from past experience may usefully apply to present experience”. It is that mechanism which allows a knowledge base to be carried across gulfs of experience which remain indifferent to the effective contents of its rules.

Next we'll examine how the transfer of knowledge through the analogy of experience works out in the case of Dewey's “Sign of Rain” example.

References —

Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
pdcnet.org/inquiryct/content/i
academia.edu/57812482/Interpre

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY.
gutenberg.org/files/37423/3742



Inquiry Into Inquiry · Interpreter and Interpretant • Selection 8
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Interpreter and Interpretant • Selection 7
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Learning —

Rules in a knowledge base, as far as their effective content goes, can be obtained by any mode of inference. For example, consider a proposition of the following form.

• B ⇒ A, Just Before it rains, the Air is cool.

Such a proposition is usually induced from a consideration of many past events. The inductive inference may be observed to fit the following pattern.

• Case : C ⇒ B, In Certain events, it is just Before it rains.
• Fact : C ⇒ A, In Certain events, the Air is cool.
────────────────────────────────────
• Rule : B ⇒ A, Just Before it rains, the Air is cool.

However, the same proposition could also be abduced as an explanation of a singular occurrence or deduced as a conclusion of a prior theory.

References —

Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
pdcnet.org/inquiryct/content/i
academia.edu/57812482/Interpre

Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY.
gutenberg.org/files/37423/3742

Resources —

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
inquiryintoinquiry.com/2024/02

Survey of Semiotics, Semiosis, Sign Relations
inquiryintoinquiry.com/2024/01




Inquiry Into Inquiry · Interpreter and Interpretant • Selection 7
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Interpreter and Interpretant • Selection 6
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Inquiry and Induction —

To understand the bearing of inductive reasoning on the closing phases of inquiry there are a couple of observations we should make.

• Smaller inquiries are typically woven into larger inquiries, whether the whole pattern of inquiry is carried on by a single agent or by a complex community.

• There are several ways particular instances of inquiry are related to ongoing inquiries at larger scales. Three modes of interaction between component inquiries and compound inquiries may be described under the headings of Learning, Transfer, and Testing of Rules.




Inquiry Into Inquiry · Interpreter and Interpretant • Selection 6
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Interpreter and Interpretant • Selection 5
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Inquiry and Inference —

If we follow Dewey's “Sign of Rain” story far enough to consider the import of thought for action, we realize the subsequent conduct of the interpreter, progressing up through the natural conclusion of the episode — the quickening steps, the seeking of shelter in time to escape the rain — all those acts amount to a series of further interpretants for the initially recognized signs of rain and the first impressions of the actual case. Just as critical reflection develops the positive and negative signs which gather about an idea, pragmatic interpretation explores the consequential and contrasting actions which give effective and testable meaning to a person's belief in it.




Interpreter and Interpretant • Selection 4
inquiryintoinquiry.com/2025/01

Interpretation and Inquiry —

To illustrate the role of sign relations in inquiry we begin with Dewey's elegant and simple example of reflective thinking in everyday life.

❝A man is walking on a warm day. The sky was clear the last time he observed it; but presently he notes, while occupied primarily with other things, that the air is cooler. It occurs to him that it is probably going to rain; looking up, he sees a dark cloud between him and the sun, and he then quickens his steps. What, if anything, in such a situation can be called thought? Neither the act of walking nor the noting of the cold is a thought. Walking is one direction of activity; looking and noting are other modes of activity. The likelihood that it will rain is, however, something suggested. The pedestrian feels the cold; he thinks of clouds and a coming shower.❞ (John Dewey, How We Think, 6–7).




Why do we say that induction is stronger than deduction?

It depends, because induction is stronger than deduction in several ways.

Formal strength lies in part in what sort of problem the method can work on. Induction can work on stuff that deduction can't even begin to work on.

Strength of output lies in how well the outcomes tolerate contradiction. An inductive conclusion can withstand quite a bit of contradiction. Deductive outcomes cannot withstand any contradiction at all.

These two are related, though.
An inductive outcome can be formed from input that contains contradictions. And when the outcome encounters a contradiction, this is simply added to the Body of Evidence, with all the other contradictions that in the end didn't manage to prevent conclusion.

Of course, given enough such evidence, a better outcome or set of outcomes can be produced.

But with deduction, if you hit even a single contradiction, your entire effort was wasted - the conclusion is now worthless.

This strength of induction was predicted already by Hume, who described what we now call inductive reasoning as:

  • infallible in its operations
  • discovers itself at the first appearance of life and thought
  • independent of all the laboured deductions of the understanding

@philosophy #deduction #induction

In his work Anticipatory Systems, #RobertRosen carefully introduces readers with knowledge of Cybernetics to the foundations of mathematics so that the formal limits of methodological approaches like #modelling, reasoning and #deduction in their domain of #science becomes obvious. Later he would do the same for a more general audience in Life Itself. The reasoning is very similar, clear, and formally sound. Still scientists will sheepishly adhere to believe in #reductionism.

Continued thread

Information = Comprehension × Extension • Selection 2.3
inquiryintoinquiry.com/2024/10

❝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
inquiryintoinquiry.com/2024/03

OEIS Wiki • Information = Comprehension × Extension
oeis.org/wiki/Information_%3D_

C.S. Peirce • Upon Logical Comprehension and Extension
peirce.sitehost.iu.edu/writing



Inquiry Into Inquiry · Information = Comprehension × Extension • Selection 2Over 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 …
Continued thread

Information = Comprehension × Extension • Selection 2.2
inquiryintoinquiry.com/2024/10

❝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.



Inquiry Into Inquiry · Information = Comprehension × Extension • Selection 2Over 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 …
Continued thread

Information = Comprehension × Extension • Selection 2.1
inquiryintoinquiry.com/2024/10

Re: Information = Comprehension × Extension • Selection 1
inquiryintoinquiry.com/2024/10

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.



Inquiry Into Inquiry · Information = Comprehension × Extension • Selection 2Over 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 …
Continued thread

Information = Comprehension × Extension • Selection 1.2
inquiryintoinquiry.com/2024/10

❝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.



Inquiry Into Inquiry · Information = Comprehension × Extension • Selection 1Our 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 a…
Continued thread

Information = Comprehension × Extension • Selection 1.1
inquiryintoinquiry.com/2024/10

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
inquiryintoinquiry.com/2024/03

OEIS Wiki • Information = Comprehension × Extension
oeis.org/wiki/Information_%3D_

C.S. Peirce • Upon Logical Comprehension and Extension
peirce.sitehost.iu.edu/writing



Inquiry Into Inquiry · Information = Comprehension × Extension • Selection 1Our 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 a…