Animated Logical Graphs • 63

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (57)(58)(59)(60)(61)(62)

We’ve been using the duality between entitative and existential interpretations of logical graphs to get a handle on the mathematical forms pervading logical laws.  A few posts ago we took up the tools of groups and symmetries and transformations to study the duality and we looked to the space of 2-variable boolean functions as a basic training grounds.  On those grounds the translation between interpretations presents as a group $G$ of order two acting on a set $X$ of sixteen logical graphs denoting boolean functions.

Last time we arrived at a Table showing how the group $G$ partitions the set $X$ into ten orbits of logical graphs.  Here again is that Table.

$\text{Peirce Duality as Group Symmetry} \stackrel{_\bullet}{} \text{Orbit Order}$

I invited the reader to investigate the relationship between the group order $|G| = 2,$ the number of orbits $10,$ and the total number of fixed points $16 + 4 = 20.$  In the present case the product of the group order (2) and the number of orbits (10) is equal to the sum of the fixed points (20) — Is that just a fluke?  If not, why so?  And does it reflect a general rule?

We can make a beginning toward answering those questions by inspecting the incidence relation of fixed points and orbits in the Table above.  Each singleton orbit accumulates two hits, one from the group identity and one from the other group operation.  But each doubleton orbit also accumulates two hits, since the group identity fixes both of its two points.  Thus all the orbits are double-counted by counting the incidence of fixed points and orbits.  In sum, dividing the total number of fixed points by the order of the group brings us back to the exact number of orbits.

Resources

cc: Cybernetics (1) (2)Laws of FormFB | Logical Graphs • Ontolog Forum (1) (2)
• Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) • Structural Modeling (1) (2)
• Systems Science (1) (2)

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