Dear Mauro,
Here I think Helmut is describing the transition
from forms of enclosure on a plane sheet of paper,
such as those used by Peirce and Spencer Brown, to
their topological duals in the form of rooted trees.
I gave a brief discussion of this transformation in
the following couple of places.
Logical Graphs • Introduction
https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/
Logical Graphs • Duality : Logical and Topological
https://oeis.org/wiki/Logical_Graphs#Duality:_logical_and_topological
This is the first step we need to carry out in the process
of converting planar maps to graph-theoretic data structures.
Further transformations will take us from trees to cactus graphs,
which implement a highly efficient family of logical primitives
called "minimal negation operators".
Minimal Negation Operator
https://oeis.org/wiki/Minimal_negation_operator
Regards,
Jon
On 3/21/2021 12:21 PM, Mauro Bertani wrote:
> So, if I want transform a circle in a line i have to use a function
> f:B_n -> B? This is the base of temporal logic? I'm using f:N_n -> N
>
> Regards
> Mauro
Il sab 20 mar 2021, 19:45 Jon Awbrey <[email protected]> ha scritto:
Cf: Differential Propositional Calculus • Discussion 4
http://inquiryintoinquiry.com/2021/03/20/differential-propositional-calculus-discussion-4/
<QUOTE HR:>
1. I think I like very much your Cactus Graphs. Meaning that I
am in the process of understanding them, and finding it much
better not to have to draw circles, but lines.
So, if I want transform a circle in a line i have to use a function f:B_n
-> B? This is the base of temporal logic? I'm using f:N_n -> N
Regards
Mauro
2. Less easy for me is the differential calculus.
Where is the consistency between (x,y) and (x, y, z)?
(x, y) means that x and y are not equal and (x, y, z)
means that one of them is false.
Unequality and truth/falsity for me are two concepts
so different I cannot think them together or see
a consistency between them.
3. What about (w, x, y, z)?
4. Can you give a grammar, like, what does a comma mean,
what do brackets mean, what does writing letters following
each other with an empty space but no comma mean, and so on?
5. Same with Cactus Graphs, though I think, they might be
self-explaining for me — everything is self-explaining,
depending on intellectual capacity, but mine is limited.
</QUOTE>
Dear Helmut,
Many thanks for your detailed comments and questions. They help me see
the places where more detailed explanations are needed. I added numbers
to your points above for ease of reference and possible future reference
in case I can't get to them all in one pass.
Cactus Graphs
=============
I'm glad you found the cactus graphs to your liking.
It was a critical transition for me when I passed from
trees to cacti in my graphing and programming and it
came about by recursively applying a trick of thought
I learned from Peirce himself. These days I call it a
“Meta-Peircean Move” to apply one of Peirce's heuristics
of choice or standard operating procedures to the state
resulting from previous applications. All that makes for
a longer story I made a start at telling in the following
series of posts.
Animated Logical Graphs
https://inquiryintoinquiry.com/2019/05/28/animated-logical-graphs-14/
[Links omitted for this email. See the blog post linked at the top for
the list.]
https://inquiryintoinquiry.com/2019/07/31/animated-logical-graphs-27/
Well, the clock in the hall struck time for lunch some time ago,
so I think I'll answer its call, break here, and continue later …
Regards,
Jon
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