Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 4

Posted on February 27, 2024 by Jon Awbrey

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 the inferential rudiments combine to form the more complex patterns of analogy and inquiry.

Anthem

Blog Dialogs

Blog Series

Blog Surveys

OEIS Wiki

Ontolog Forum

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Survey of Theme One Program • 6

Posted on February 26, 2024 by Jon Awbrey

This is a Survey of blog and wiki posts relating to the Theme One Program I worked on all through the 1980s.  The aim was to develop fundamental algorithms and data structures for integrating empirical learning with logical reasoning.  I had earlier developed separate programs for basic components of those tasks, namely, 2-level formal language learning and propositional constraint satisfaction, the latter using an extension of C.S. Peirce’s logical graphs as a syntax for propositional logic.  Thus arose the question of how well it might be possible to get “empiricist” and “rationalist” modes of operation to cooperate.  The long-term vision is the design and implementation of an Automated Research Tool able to double as a platform for Inquiry Driven Education.

Wiki Hub

Documentation

Blog Series

Blog Dialogs

Applications

References

  • Awbrey, S.M., and Awbrey, J.L. (May 1991), “An Architecture for Inquiry • Building Computer Platforms for Discovery”, Proceedings of the Eighth International Conference on Technology and Education, Toronto, Canada, pp. 874–875.  Online.
  • Awbrey, J.L., and Awbrey, S.M. (January 1991), “Exploring Research Data Interactively • Developing a Computer Architecture for Inquiry”, Poster presented at the Annual Sigma Xi Research Forum, University of Texas Medical Branch, Galveston, TX.
  • Awbrey, J.L., and Awbrey, S.M. (August 1990), “Exploring Research Data Interactively • Theme One : A Program of Inquiry”, Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training, Society for Applied Learning Technology, Washington, DC, pp. 9–15.  Online.

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Survey of Differential Logic • 7

Posted on February 25, 2024 by Jon Awbrey

This is a Survey of work in progress on Differential Logic, resources under development toward a more systematic treatment.

Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description.  A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.  To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a differential logical calculus — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Elements

Blog Series

Architectonics

Applications

Blog Dialogs

Explorations

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Sign Relations, Triadic Relations, Relation Theory • Discussion 12

Posted on February 24, 2024 by Jon Awbrey

Re: Sign Relations, Triadic Relations, Relation Theory • 1

A note from a longtime correspondent points out a search of the available texts turns up no use of the plural form “semiotics” by Peirce and just one place where he uses the plural form “Semeiotics”.  That prompts me to make the following excuse for my use or abuse of Peirce’s terms, as the case may be.

Peirce has always been one of my chief resources in the quest to understand how logic and math and science work.  There is much to be gained by getting his distinctive ideas across to active practitioners in those fields.  In doing that I find it better to tweak the words a bit, if that’s what it takes to preserve the idea, than to hallow the words at the risk of losing the idea.

As far as semiotics by any name goes, what seems to work best without too much clanging in modern ears is parsing semiotics in line with words like mathematics and cybernetics, plus we can now use the singular form as the adjective semiotic.

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Sign Relations, Triadic Relations, Relation Theory • 4

Posted on February 22, 2024 by Jon Awbrey

For ease of reference, here are two variants of Peirce’s 1902 definition of a sign, which he gives in the process of defining logic.

Selections from C.S. Peirce, “Carnegie Application” (1902)

No. 12.  On the Definition of Logic

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.  It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non‑psychological conception of logic has virtually been quite generally held, though not generally recognized.  (NEM 4, 20–21).

No. 12.  On the Definition of Logic [Earlier Draft]

Logic is formal semiotic.  A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C.  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident.  The word “formal” in the definition is also defined.  (NEM 4, 54).

Reference

  • Peirce, C.S. (1902), “Parts of Carnegie Application” (L 75), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.

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Sign Relations, Triadic Relations, Relation Theory • 3

Posted on February 20, 2024 by Jon Awbrey

The middle ground between relations in general and the sign relations informing logic, inquiry, and communication is occupied by triadic relations, also called ternary or 3‑place relations.

Triadic relations are some of the most pervasive in mathematics, over and above the importance of sign relations for logic.

For a primer on triadic relations, with examples from mathematics and semiotics, see the article linked below.

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Sign Relations, Triadic Relations, Relation Theory • 2

Posted on February 19, 2024 by Jon Awbrey

I always have trouble deciding whether to start with the genus and drive down to the species or begin with concrete examples and accompany Sisyphus up Mt. Abstraction.

To start at the wide end of the funnel, the following article takes up relations in general, focusing on the discrete mathematical varieties we find most useful in applications, for example, as background for empirical data sets and relational data bases.

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Sign Relations, Triadic Relations, Relation Theory • 1

Posted on February 18, 2024 by Jon Awbrey

To understand how signs work in Peirce’s theory of triadic sign relations, or “semiotics”, we have to understand, in order of increasing generality, sign relations, triadic relations, and relations in general, each as conceived in Peirce’s logic of relative terms and the corresponding mathematics of relations.

Toward that understanding, here are the current versions of articles I long ago contributed to Wikipedia and Wikiversity and continue to develop at a number of other places.

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Tone Token Type • Discussion 1

Posted on February 17, 2024 by Jon Awbrey

Re: FB | Daniel Everett

DE:
People who believe that Peirce’s terms firstness, secondness, and thirdness are complicated might have overlooked the fact that they almost certainly already use two of the three terms via Peirce’s other terms type (thirdness) and token (secondness).  What is missing is only Peirce’s other term tone, which refers to firstness.

These distinctions are crucial.  Take linguistic fieldwork.  When the fieldworker first hears something or sees something but has no idea about it other than it is “strange” or unexpected, that is a tone/firstness.  When the linguist proposes the phones of a language, the list are tokens/secondnesses.  When the linguist proposes phonemes, those are types/thirdnesses.  (And underlying form would be a thirdness/type and the surface form a secondness/token.)

Daniel,

The way Peirce shades the matter of signs along the lines of a Tone‑Token‑Type spectrum is a topic of recurring discussion.  There’s a selection of Peirce quotes and a few comments from me on the following page.

Re: C.S. Peirce • Note 1

CSP:
For a “possible” Sign I have no better designation than a Tone,
though I am considering replacing this by “Mark”.

I’ve seen some readers be confused by Peirce’s sometime alternative of Mark for Tone, thinking he meant something like a scratch‑mark on paper, but he is using Mark in the sense of Character(istic), Distinctive Feature, or Quality.  I don’t know whether he had it in mind but that particular use was also common among 19th Century mathematicians in the early years of the subject known as the Representation Theory of Groups.

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Sign Relations • Semiotic Equivalence Relations 2

Posted on February 16, 2024 by Jon Awbrey

A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular.

In general, if E is an equivalence relation on a set X then every element x of X belongs to a unique equivalence class under E called the equivalence class of x under E.  Convention provides the square bracket notation for denoting such equivalence classes, in either the form [x]_E or the simpler form [x] when the subscript E is understood.  A statement that the elements x and y are equivalent under E is called an equation or an equivalence and may be expressed in any of the following ways.

\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}

Thus we have the following definitions.

\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}

In the application to sign relations it is useful to extend the square bracket notation in the following ways.  If L is a sign relation whose connotative component L_{SI} is an equivalence relation on S = I, let [s]_L be the equivalence class of s under L_{SI}.  In short, [s]_L = [s]_{L_{SI}}.  A statement that the signs x and y belong to the same equivalence class under a semiotic equivalence relation L_{SI} is called a semiotic equation (SEQ) and may be written in either of the following forms.

\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}

In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes which can be useful.  Namely, when there is known to exist a particular triple (o, s, i) in a sign relation L, it is permissible to let [o]_L be defined as [s]_L.  This lets the notation for semiotic equivalence classes harmonize more smoothly with the frequent use of similar devices for the denotations of signs and expressions.

Applying the array of equivalence notations to the sign relations for A and B will serve to illustrate their use and utility.

Connotative Components Con(L_A) and Con(L_B)

The semiotic equivalence relation for interpreter \mathrm{A} yields the following semiotic equations.

\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}

or

\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}

In this way it induces the following semiotic partition.

\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}.

The semiotic equivalence relation for interpreter \mathrm{B} yields the following semiotic equations.

\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}

or

\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}

In this way it induces the following semiotic partition.

\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}.

Semiotic Partitions for Interpreters A and B

Resources

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