Interpretive Duality in Logical Graphs

[Jon Awbrey]

Interpretive Duality in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-graphs-1/

All,

The duality between Entitative and Existential interpretations
of logical graphs is a good example of a mathematical symmetry,
in this case a symmetry of order two. Symmetries of this and
higher orders give us conceptual handles on excess complexity
in the manifold of sensuous impressions, making it well worth
the effort to seek them out and grasp them where we find them.

Both Peirce and Spencer Brown understood the significance of
the mathematical unity underlying the dual interpretation of
logical graphs. Peirce began with the Entitative option and
later switched to the Existential choice while Spencer Brown
exercised the Entitative option in his “Laws of Form”.

In that vein, here's a Rosetta Stone to give us a grounding in
the relationship between boolean functions and our two readings
of logical graphs.

Boolean Functions on Two Variables
• https://inquiryintoinquiry.com/wp-content/uploads/2020/11/boolean-functions-on-two-variables.png

Regards,

Jon

cc: https://www.academia.edu/community/5REb1n

[Jon Awbrey]

Interpretive Duality in Logical Graphs • 2
• https://inquiryintoinquiry.com/2024/04/23/interpretive-duality-in-logical-graphs-2/

Re: Interpretive Duality in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-graphs-1/

All,

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 the previous post 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
• https://inquiryintoinquiry.com/wp-content/uploads/2020/11/venn-diagrams-and-logical-graphs-on-two-variables.png

Regards,

Jon

cc: https://www.academia.edu/community/VvOwq8

[Jon Awbrey]

Interpretive Duality in Logical Graphs • 3
• https://inquiryintoinquiry.com/2024/04/24/interpretive-duality-in-logical-graphs-3/

Re: Interpretive Duality in Logical Graphs

• https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-graphs-1/
• https://inquiryintoinquiry.com/2024/04/23/interpretive-duality-in-logical-graphs-2/

All,

For a sense of how the choice of interpretation bears on cases
beyond the bare minimum complexity let us start with the familiar
example of Peirce's law, commonly expressed in the following form.

• ((p ⇒ q) ⇒ p) ⇒ p

The following two formal equations show how Peirce's law may
be expressed in terms of logical graphs, operating under the
entitative and existential interpretations, respectively.

Peirce's Law • Dual Graphs
• https://inquiryintoinquiry.com/wp-content/uploads/2023/11/peirces-law-e280a2-dual-graphs-2.0.png

Resources —

Peirce's Law
• https://inquiryintoinquiry.com/2023/10/18/peirces-law-a/

Regards,

Jon

cc: https://www.academia.edu/community/lPxGaY

[Jon Awbrey]

Interpretive Duality in Logical Graphs • 4
• https://inquiryintoinquiry.com/2024/04/25/interpretive-duality-in-logical-graphs-4/

Re: Interpretive Duality in Logical Graphs
• https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-graphs-1/
• https://inquiryintoinquiry.com/2024/04/23/interpretive-duality-in-logical-graphs-2/

• https://inquiryintoinquiry.com/2024/04/24/interpretive-duality-in-logical-graphs-3/

All,

Last time we took up Peirce's law, ((p ⇒ q) ⇒ p) ⇒ p, and
saw how it might be expressed in two different ways, under

the entitative and existential interpretations, respectively.

The next thing to do is see how our choice of interpretation
bears on the patterns of proof we might find. A sense of the
possibilities may be gotten by displaying the two styles of
proof in parallel columns, as shown below.

Peirce's Law • Parallel Proofs
• https://inquiryintoinquiry.com/wp-content/uploads/2023/11/peirces-law-e280a2-parallel-proofs.png

Resources —

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon

cc: https://www.academia.edu/community/VjQX0k

[Jon Awbrey]

Interpretive Duality in Logical Graphs • 5
• https://inquiryintoinquiry.com/2024/04/26/interpretive-duality-in-logical-graphs-5/

Re: Interpretive Duality in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-graphs-1/

All,

Another way of looking at interpretive duality in logical graphs
is given by the following Table, showing how logical graphs denote
the sixteen boolean functions on two variables under entitative and
existential interpretations, respectively.

Logical Graphs • Entitative and Existential Interpretations
•
https://inquiryintoinquiry.com/wp-content/uploads/2020/11/logical-graphs-e280a2-entitative-and-existential-interpretations.png

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon

cc: https://www.academia.edu/community/5k8BwN

[Jon Awbrey]

Interpretive Duality in Logical Graphs • 6
• https://inquiryintoinquiry.com/2024/04/29/interpretive-duality-in-logical-graphs-6/

Re: Interpretive Duality in Logical Graphs • 2
• https://inquiryintoinquiry.com/2024/04/23/interpretive-duality-in-logical-graphs-2/

All,

A more graphic picture of interpretive duality is given by the
next Table, showing how logical graphs map to venn diagrams
under entitative and existential interpretations.

• Column 1 shows the logical graphs for the 16 boolean functions on 2 variables.

• Column 2 shows the venn diagrams associated with the entitative interpretation.

• Column 3 shows the venn diagrams associated with the existential interpretation.

Logical Graphs • Entitative and Existential Venn Diagrams
•
https://inquiryintoinquiry.com/wp-content/uploads/2020/11/logical-graphs-e280a2-entitative-and-existential-venn-diagrams.png

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon

cc: https://www.academia.edu/community/laaB63

[Jon Awbrey]

Interpretive Duality in Logical Graphs • 7
• https://inquiryintoinquiry.com/2024/04/30/interpretive-duality-in-logical-graphs-7/

Re: Interpretive Duality in Logical Graphs • 2
• https://inquiryintoinquiry.com/2024/04/23/interpretive-duality-in-logical-graphs-2/

All,

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 the usual dissymmetries from dispersion and diversity to
entropy and uncertainty is governed in cybernetics by the Law
of Requisite Variety, the medium of which exchange 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 those 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
•
https://inquiryintoinquiry.com/wp-content/uploads/2020/12/venn-diagrams-and-logical-graphs-on-two-variables-e280a2-orbit-order.png

Resources —

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

Regards,

Jon

cc: https://www.academia.edu/community/LEZn7Y

[Jon Awbrey]

Interpretive Duality in Logical Graphs • 8
• https://inquiryintoinquiry.com/2024/05/01/interpretive-duality-in-logical-graphs-8/

Re: Interpretive Duality in Logical Graphs • 6
• https://inquiryintoinquiry.com/2024/04/29/interpretive-duality-in-logical-graphs-6/

All,

The last of our six ways of looking at interpretive duality
is arrived at by taking the previous Table of Logical Graphs
and Venn Diagrams and sorting it in Orbit Order.

Logical Graphs • Entitative and Existential Venn Diagrams • Orbit Order
•
https://inquiryintoinquiry.files.wordpress.com/2020/12/logical-graphs-e280a2-entitative-and-existential-venn-diagrams-e280a2-orbit-order.png

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon

cc: https://www.academia.edu/community/V0EaGp