Cf: Animated Logical Graphs • 66
http://inquiryintoinquiry.com/2021/03/09/animated-logical-graphs-66/

Re: Richard J. Lipton • The Art Of Math
https://rjlipton.wordpress.com/2020/11/12/the-art-of-math/
Re: Animated Logical Graphs
https://inquiryintoinquiry.com/2021/02/26/animated-logical-graphs-61/
https://inquiryintoinquiry.com/2021/02/28/animated-logical-graphs-62/
https://inquiryintoinquiry.com/2021/03/02/animated-logical-graphs-63/

All,

Once we bring the dual interpretations of logical graphs to the
same Table and relate their parleys to the same objects, it is
clear we are dealing with a triadic sign relation of the sort
taken up in C.S. Peirce’s “semiotics” or theory of signs.

A “sign relation” L ⊆ O × S × I, as a set L embedded in a
cartesian product O × S × I, tells how the “signs” in S
and the “interpretant signs” in I correlate with the
“objects” or objective situations in O.

There are many ways of using sign relations to model
various types of sign-theoretic situations and processes.
The following cases are often seen.

• Some sign relations model co‑referring signs or transitions
between signs within a single language or symbol system.
In that event L ⊆ O × S × I has S = I.

• Other sign relations model translations between different
languages or different interpretations of the same language,
in other words, different ways of referring the same set of
signs to a shared object domain.

The next Table extracts the sign relation L ⊆ O × S × I involved
in switching between existential and entitative interpretations
of logical graphs.

Table. Peirce Duality as Sign Relation (also attached)
https://inquiryintoinquiry.files.wordpress.com/2021/03/peirce-duality-as-sign-relation.png

• Column 1 shows the object domain O as the
set of 16 boolean functions on 2 variables.

• Column 2 shows the sign domain S as a representative set
of logical graphs denoting the objects in O according to
the existential interpretation.

• Column 3 shows the interpretant domain I as the same set
of logical graphs denoting the objects in O according to
the entitative interpretation.

Resources
=========

C.S. Peirce • Logic and Signs
https://inquiryintoinquiry.com/2012/06/01/c-s-peirce-on-the-definition-of-logic/

C.S. Peirce • Logic as Semiotic
https://inquiryintoinquiry.com/2012/06/04/c-s-peirce-logic-as-semiotic/

Sign Relation
https://oeis.org/wiki/Sign_relation

Triadic Relation
https://oeis.org/wiki/Triadic_relation

Regards,

Jon