## Inquiry Driven Systems • Comment 6

Dear Peter,

It’s funny you should mention Tennyson’s poem in the context of an author’s view of publication as I once laid out a detailed interpretation of the poem as a metaphor on the poet’s quest to communicate.  I know I wrote a shorter, sweeter essay on that somewhere I can’t find right now but here’s one of my more turgid dilatations where I used the poem as an “epitext” — a connected series of epigraphs — for a discussion of what I called Ostensibly Recursive Texts (ORTs).

Tennyson’s poem The Lady of Shalott is akin to an ORT, but a bit more remote, since the name styled as “The Lady of Shalott”, that the author invokes over the course of the text, is not at first sight the title of a poem, but a title its character adopts and afterwards adapts as the name of a boat.  It is only on a deeper reading that this text can be related to or transformed into a proper ORT.  Operating on a general principle of interpretation, the reader is entitled to suspect the author is trying to say something about himself, his life, and his work, and that he is likely to be exploiting for this purpose the figure of his ostensible character and the vehicle of his manifest text.  If this is an aspect of the author’s intention, whether conscious or unconscious, then the reader has a right to expect several forms of analogy are key to understanding the full intention of the text.

cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science

## Animated Logical Graphs • 52

Peirce’s explorations in logic and the theory of signs opened several directions of generalization from logics of complete information (LOCI) to theories of partial information (TOPI).  Naturally we hope these avenues of approach will eventually converge on a unified base camp from which to reach greater heights of understanding, but that is still a work in progress, at least for me.

Any passage from logic as a critical, formal, or normative theory of controlled semiotic conduct to the descriptive study of signs “in the wild” involves relaxing logical norms to statistical norms.

One of the headings under which Peirce expands the scope of logic to something more general — whether keeping or losing the name of “logic” is a secondary consideration — is found in his study of Generality and Vagueness as affecting signs not fully primed for logical use.  There’s a bit about that at the following places.

As you can see, in this direction of generalization Peirce considers relaxing both the principle of contradiction and the principle of excluded middle.

### Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
• Peirce (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) • Structural Modeling (1) (2) • Systems (1) (2)

## Animated Logical Graphs • 51

Synchronicity being what it is, a long-running discussion on the Peirce List just gave me a handy bridge to a topic I’ve been meaning to take up in several other connections.  So I’m adding my comment to this series, along with links to additional resources.

Re: Peirce List (1) (2) (3) (4) (5)

Pursuing the discussion of many things:
of laws — and graphs — and reasoning —
of contradictions — and abductions —
and why the third is given not —
and whether figs have wings —

It might not be non sequitur to remember that place in Peirceland where we walk the line between classical and intuitionistic logic, namely, the boundary marked by the principle we have come to call Peirce’s Law.

Here’s links to bits of fol-de-rule, with graphs and everything —

### Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
• Peirce (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) • Structural Modeling (1) (2) • Systems (1) (2)

## Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 2

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.

## Abduction, Deduction, Induction, Analogy, Inquiry • 30

Re: “Proofs Are Not As Important As Discoveries” (PANAIAD).

Conjecture and Discovery fall under the heading of Abductive Inference (AI’s missing grape).

About which more when I pull my heading out of the Pandemic …

In the mean time, here’s a budget of links from previous discussions.

## Animated Logical Graphs • 50

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30) (45) (46) (47) (48) (49)

In the last of our six ways of looking at the Peirce duality between entitative and existential interpretations, we consider the previous Table of Logical Graphs and Venn Diagrams sorted in Orbit Order.

Logical Graphs • Entitative and Existential Venn Diagrams • Orbit Order
$\text{Logical Graph}$ $\text{Entitative Interpretation}$ $\text{Existential Interpretation}$

$\texttt{(} ~ \texttt{)}$

$\text{true}$
$f_{15}$
$\text{false}$
$f_{0}$

$\texttt{(} x \texttt{)(} y \texttt{)}$

$\lnot x \lor \lnot y$
$f_{7}$
$\lnot x \land \lnot y$
$f_{1}$

$\texttt{(} x \texttt{)} y$

$x \Rightarrow y$
$f_{11}$
$x \nLeftarrow y$
$f_{2}$

$x \texttt{(} y \texttt{)}$

$x \Leftarrow y$
$f_{13}$
$x \nRightarrow y$
$f_{4}$

$x y$

$x \lor y$
$f_{14}$
$x \land y$
$f_{8}$

$\texttt{(} x \texttt{)}$

$\lnot x$
$f_{3}$
$\lnot x$
$f_{3}$

$x$

$x$
$f_{12}$
$x$
$f_{12}$

$\texttt{(} x \texttt{,} y \texttt{)}$

$x = y$
$f_{9}$
$x \ne y$
$f_{6}$

$\texttt{((} x \texttt{,} y \texttt{))}$

$x \ne y$
$f_{6}$
$x = y$
$f_{9}$

$\texttt{(} y \texttt{)}$

$\lnot y$
$f_{5}$
$\lnot y$
$f_{5}$

$y$

$y$
$f_{10}$
$y$
$f_{10}$

$\texttt{(} x y \texttt{)}$

$\lnot (x \lor y)$
$f_{1}$
$\lnot (x \land y)$
$f_{7}$

$\texttt{(} x \texttt{(} y \texttt{))}$

$x \nLeftarrow y$
$f_{2}$
$x \Rightarrow y$
$f_{11}$

$\texttt{((} x \texttt{)} y \texttt{)}$

$x \nRightarrow y$
$f_{4}$
$x \Leftarrow y$
$f_{13}$

$\texttt{((} x \texttt{)(} y \texttt{))}$

$x \land y$
$f_{8}$
$x \lor y$
$f_{14}$

$\text{false}$
$f_{0}$
$\text{true}$
$f_{15}$

### Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce (1) (2) (3) (4) (5) (6) (7) (8) (9) • Structural Modeling (1) (2) • Systems (1) (2)

## Animated Logical Graphs • 49

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30) (45) (46) (47) (48)

Dualities are symmetries of order two and symmetries bear on complexity by reducing its measure in proportion to their order.  The inverse relationship between symmetry and all those dissymmetries from dispersion and diversity to entropy and uncertainty is governed in cybernetics by the Law of Requisite Variety, the medium of which exchanges C.S. Peirce invested in his formula, Information = Comprehension × Extension.

The duality between entitative and existential interpretations of logical graphs is one example of a mathematical symmetry but it’s not unusual to find symmetries within symmetries and it’s always rewarding to find them where they exist.  To that end let’s take up our Table of Venn Diagrams and Logical Graphs on Two Variables and sort the rows to bring together diagrams and graphs having similar shapes.  What defines their similarity is the action of a mathematical group whose operations transform the elements of each class among one another but intermingle no dissimilar elements.  In the jargon of transformation groups these classes are called orbits.  We find the sixteen rows partition into seven orbits, as shown below.

Venn Diagrams and Logical Graphs on Two Variables • Orbit Order
$\text{Boolean Function}$ $\text{Entitative Graph}$ $\text{Existential Graph}$

$f_{0}$ $\text{false}$ $\text{false}$

$f_{1}$ $\lnot (x \lor y)$ $\lnot x \land \lnot y$

$f_{2}$ $\lnot x \land y$ $\lnot x \land y$

$f_{4}$ $x \land \lnot y$ $x \land \lnot y$

$f_{8}$ $x \land y$ $x \land y$

$f_{3}$ $\lnot x$ $\lnot x$

$f_{12}$ $x$ $x$

$f_{6}$ $x \ne y$ $x \ne y$

$f_{9}$ $x = y$ $x = y$

$f_{5}$ $\lnot y$ $\lnot y$

$f_{10}$ $y$ $y$

$f_{7}$ $\lnot x \lor \lnot y$ $\lnot (x \land y)$

$f_{11}$ $x \Rightarrow y$ $x \Rightarrow y$

$f_{13}$ $x \Leftarrow y$ $x \Leftarrow y$

$f_{14}$ $x \lor y$ $x \lor y$

$f_{15}$ $\text{true}$ $\text{true}$

### Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce (1) (2) (3) (4) (5) (6) (7) (8) (9) • Structural Modeling (1) (2) • Systems (1) (2)

## Animated Logical Graphs • 48

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30) (45) (46) (47)

A more graphic picture of Peirce duality is given by the next Table, which shows how logical graphs map to venn diagrams under entitative and existential interpretations.  Column 1 shows the logical graphs for the sixteen boolean functions on two variables.  Column 2 shows the venn diagrams associated with the entitative interpretation and Column 3 shows the venn diagrams associated with the existential interpretation.

Logical Graphs • Entitative and Existential Venn Diagrams
$\text{Logical Graph}$ $\text{Entitative Interpretation}$ $\text{Existential Interpretation}$

$\texttt{(} ~ \texttt{)}$

$\text{true}$
$f_{15}$
$\text{false}$
$f_{0}$

$\texttt{(} x \texttt{)(} y \texttt{)}$

$\lnot x \lor \lnot y$
$f_{7}$
$\lnot x \land \lnot y$
$f_{1}$

$\texttt{(} x \texttt{)} y$

$x \Rightarrow y$
$f_{11}$
$x \nLeftarrow y$
$f_{2}$

$\texttt{(} x \texttt{)}$

$\lnot x$
$f_{3}$
$\lnot x$
$f_{3}$

$x \texttt{(} y \texttt{)}$

$x \Leftarrow y$
$f_{13}$
$x \nRightarrow y$
$f_{4}$

$\texttt{(} y \texttt{)}$

$\lnot y$
$f_{5}$
$\lnot y$
$f_{5}$

$\texttt{(} x \texttt{,} y \texttt{)}$

$x = y$
$f_{9}$
$x \ne y$
$f_{6}$

$\texttt{(} x y \texttt{)}$

$\lnot (x \lor y)$
$f_{1}$
$\lnot (x \land y)$
$f_{7}$

$x y$

$x \lor y$
$f_{14}$
$x \land y$
$f_{8}$

$\texttt{((} x \texttt{,} y \texttt{))}$

$x \ne y$
$f_{6}$
$x = y$
$f_{9}$

$y$

$y$
$f_{10}$
$y$
$f_{10}$

$\texttt{(} x \texttt{(} y \texttt{))}$

$x \nLeftarrow y$
$f_{2}$
$x \Rightarrow y$
$f_{11}$

$x$

$x$
$f_{12}$
$x$
$f_{12}$

$\texttt{((} x \texttt{)} y \texttt{)}$

$x \nRightarrow y$
$f_{4}$
$x \Leftarrow y$
$f_{13}$

$\texttt{((} x \texttt{)(} y \texttt{))}$

$x \land y$
$f_{8}$
$x \lor y$
$f_{14}$

$\text{false}$
$f_{0}$
$\text{true}$
$f_{15}$

### Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) • Structural Modeling (1) (2) • Systems (1) (2)

## Animated Logical Graphs • 47

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30) (45) (46)

A logical concept represented by a boolean variable has its extension, the cases it covers in a designated universe of discourse, and its comprehension (or intension), the properties it implies in a designated hierarchy of predicates.  The formulas and graphs tabulated in previous posts are well-adapted to articulate the syntactic and intensional aspects of propositional logic.  But their very tailoring to those tasks tends to slight the extensional and therefore empirical applications of logic.  Venn diagrams, despite their unwieldiness as the number of logical dimensions increases, are indispensable in providing the visual intuition with a solid grounding in the extensions of logical concepts.  All that makes it worthwhile to reset our table of boolean functions on two variables to include the corresponding venn diagrams.

Venn Diagrams and Logical Graphs on Two Variables
$\text{Boolean Function}$ $\text{Entitative Graph}$ $\text{Existential Graph}$

$f_{0}$ $\text{false}$ $\text{false}$

$f_{1}$ $\lnot (x \lor y)$ $\lnot x \land \lnot y$

$f_{2}$ $\lnot x \land y$ $\lnot x \land y$

$f_{3}$ $\lnot x$ $\lnot x$

$f_{4}$ $x \land \lnot y$ $x \land \lnot y$

$f_{5}$ $\lnot y$ $\lnot y$

$f_{6}$ $x \ne y$ $x \ne y$

$f_{7}$ $\lnot x \lor \lnot y$ $\lnot (x \land y)$

$f_{8}$ $x \land y$ $x \land y$

$f_{9}$ $x = y$ $x = y$

$f_{10}$ $y$ $y$

$f_{11}$ $x \Rightarrow y$ $x \Rightarrow y$

$f_{12}$ $x$ $x$

$f_{13}$ $x \Leftarrow y$ $x \Leftarrow y$

$f_{14}$ $x \lor y$ $x \lor y$

$f_{15}$ $\text{true}$ $\text{true}$

### Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) • Structural Modeling (1) (2) • Systems (1) (2)

## Animated Logical Graphs • 46

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30) (45)

Another way of looking at Peirce duality is given by the following Table, which shows how logical graphs denote boolean functions under entitative and existential interpretations.  Column 1 shows the logical graphs for the sixteen boolean functions on two variables.  Column 2 shows the boolean functions denoted under the entitative interpretation and Column 3 shows the boolean functions denoted under the existential interpretation.

$\text{Logical Graphs} \stackrel{_\bullet}{} \text{Entitative and Existential Interpretations}$

### Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) • Structural Modeling (1) (2) • Systems (1) (2)