Logical Graphs • Interpretive Duality

Jon Awbrey

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Oct 26, 2023, 1:15:33 PM10/26/23

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Logical Graphs • Interpretive Duality 1
• https://inquiryintoinquiry.com/2023/10/26/logical-graphs-interpretive-duality-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.files.wordpress.com/2020/11/boolean-functions-on-two-variables.png

Regards,

Jon

cc: https://www.academia.edu/community/5k4z9V

Boolean Functions on Two Variables.png

Jon Awbrey

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Oct 29, 2023, 12:50:10 PM10/29/23

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Logical Graphs • Interpretive Duality 2
• https://inquiryintoinquiry.com/2023/10/29/logical-graphs-interpretive-duality-2/

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 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
• https://inquiryintoinquiry.files.wordpress.com/2020/11/venn-diagrams-and-logical-graphs-on-two-variables.png

Regards,

Jon

cc: https://www.academia.edu/community/lP9QM5

Venn Diagrams and Logical Graphs on Two Variables.png

Jon Awbrey

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Nov 1, 2023, 5:40:28 PM11/1/23

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Logical Graphs • Interpretive Duality 3
• https://inquiryintoinquiry.com/2023/11/01/logical-graphs-interpretive-duality-3/

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

Re: Logical Graphs • Interpretive Duality
• https://inquiryintoinquiry.com/2023/10/26/logical-graphs-interpretive-duality-1/
• https://inquiryintoinquiry.com/2023/10/29/logical-graphs-interpretive-duality-2/

All,

To see how our choice of interpretation bears on cases beyond
the bare minimum 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.files.wordpress.com/2023/10/peirces-law-e280a2-dual-graphs.png

Regards,

Jon

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

Peirce's Law • Dual Graphs.png

Jon Awbrey

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Nov 9, 2023, 1:04:47 PM11/9/23

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Logical Graphs • Interpretive Duality 4
• https://inquiryintoinquiry.com/2023/11/08/logical-graphs-interpretive-duality-4/

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

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.

Peirce's Law • Parallel Proofs.png

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