Logical Graphs • First Impressions 4 • https://inquiryintoinquiry.com/2024/09/03/logical-graphs-first-impressions-… Duality : Logical and Topological (cont.) — Last time we took up the axiom or initial equation shown below. Logical Graph • Initial Equation I₂ • https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-arithm… We noted it could be written inline as “( ( ) ) = ” or set off in the following text display. ( ( ) ) = When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with the greatest of ease, we begin by transforming the planar graphs into their topological duals. Planar regions of the original graph become nodes or points of the dual graph and boundaries between planar regions of the original graph become edges or lines between nodes of the dual graph. For example, overlaying the corresponding dual graphs on the plane‑embedded graphs shown above, we get the following composite picture. Initial Equation I₂ • Planar Graphs Overlaid By Dual Trees • https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-i2-pla… Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the “root node” of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike‑throughs. Extracting the dual graphs from their composite matrix, we get the following equation. Initial Equation I₂ • Dual Trees • https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-i2-tre… Resources — Logical Graphs • https://oeis.org/wiki/Logical_Graphs Survey of Animated Logical Graphs • https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs… Regards, Jon cc: https://www.academia.edu/community/lJX77E cc: https://mathstodon.xyz/@Inquiry/113051725984818777

Logical Graphs • First Impressions 6 • https://inquiryintoinquiry.com/2024/09/05/logical-graphs-first-impressions-… Duality : Logical and Topological (concl.) — We have now treated in some detail various forms of the axiom or initial equation which is formulated in string form as “( ( ) ) = ”. For the sake of comparison, let's record the planar and dual forms of the axiom which is formulated in string form as “( )( ) = ( )”. First the plane‑embedded maps: Logical Graph • Initial Equation I₁ • https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-arithm… Next the plane maps and their dual trees superimposed: Initial Equation I₁ • Planar Graphs Overlaid By Dual Trees • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-i1-pla… Finally the rooted trees by themselves: Initial Equation I₁ • Dual Trees • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-i1-tre… And here are the parse trees with their traversal strings indicated: Initial Equation I₁ • Dual Trees With Parentheses • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-i1-tre… We have at this point enough material to begin thinking about the forms of analogy, iconicity, metaphor, morphism, whatever we may call them, which bear on the use of logical graphs in their various incarnations, for example, those Peirce described as “entitative graphs” and “existential graphs”. Resources — Logical Graphs • https://oeis.org/wiki/Logical_Graphs Survey of Animated Logical Graphs • https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs… Regards, Jon cc: https://www.academia.edu/community/VvdD4q cc: https://mathstodon.xyz/@Inquiry/113051725984818777

Logical Graphs • First Impressions 8 • https://inquiryintoinquiry.com/2024/09/06/logical-graphs-first-impressions-… Computational Representation (concl.) — At the next level of concreteness, a pointer-record data structure can be represented as follows. Pointer Structure 1 • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-pointe… This portrays index₀ as the address of a record which contains the following data. • datum₁, datum₂, datum₃, …, and so on. What makes it possible to represent graph-theoretic forms as dynamic data structures is the fact that an address is just another datum to be stored on a record, and so we may have a state of affairs like the following. Pointer Structure 2 • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-pointe… Returning to the abstract level, it takes three nodes to represent the three data records illustrated above: one root node connected to a couple of adjacent nodes. Items of data not pointing any further up the tree are treated as labels on the record-nodes where they reside, as shown below. Pointer Structure 3 • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-pointe… Notice that drawing the arrows is optional with rooted trees like these, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, with “up” being the direction “away from the root”. Resources — Logical Graphs • https://oeis.org/wiki/Logical_Graphs Survey of Animated Logical Graphs • https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs… cc: https://www.academia.edu/community/5ADjXW cc: https://mathstodon.xyz/@Inquiry/113051725984818777

Logical Graphs • First Impressions 10 • https://inquiryintoinquiry.com/2024/09/08/logical-graphs-first-impressions-… Primary Arithmetic as Semiotic System (cont.) — The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names which come in handy for referring to the two opposing directions of applying the axioms. Axiom I₁ • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-axiom-… Axiom I₂ • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-axiom-… Resources — Logical Graphs • https://oeis.org/wiki/Logical_Graphs Survey of Animated Logical Graphs • https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs… Regards, Jon cc: https://www.academia.edu/community/l8P809 cc: https://mathstodon.xyz/@Inquiry/113051725984818777

Logical Graphs • First Impressions 12 • https://inquiryintoinquiry.com/2024/09/10/logical-graphs-first-impressions-… Primary Algebra as Pattern Calculus — Experience teaches that complex objects are best approached in a gradual, laminar, modular fashion, one step, one layer, one piece at a time, especially when that complexity is irreducible, when all our articulations and all our representations will be cloven at joints disjoint from the structure of the object itself, with some assembly required in the synthetic integrity of the intuition. That's one good reason for spending so much time on the first half of zeroth order logic, instanced here by the primary arithmetic, a level of formal structure Peirce verged on intuiting at numerous points and times in his work on logical graphs but Spencer Brown named and brought more completely to life. Another reason for lingering a while longer in these primitive forests is that an acquaintance with “bare trees”, those unadorned with literal or numerical labels, will provide a basis for understanding what's really at issue in oft‑debated questions about the “ontological status of variables”. It is probably best to illustrate the theme in the setting of a concrete case. To do that let's look again at the previous example of reductive evaluation taking place in the primary arithmetic. Primary Arithmetic Reduction Example • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-semiot… After we've seen a few sign-transformations of roughly that shape we'll most likely notice it doesn't really matter what other branches are rooted next to the lone edge off to the side — the end result will always be the same. Eventually it will occur to us to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following. Pattern Calculus Example • https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-patter… Observations like that, made about an arithmetic of any variety and germinated by their summarizations, are the root of all algebra. Resources — Logical Graphs • https://oeis.org/wiki/Logical_Graphs Survey of Animated Logical Graphs • https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs… Regards, Jon cc: https://www.academia.edu/community/lQ9QO6 cc: https://mathstodon.xyz/@Inquiry/113051725984818777

Logical Graphs • First Impressions 14 • https://inquiryintoinquiry.com/2024/09/12/logical-graphs-first-impressions-… Formal Development — Discussion of the topic continues at Logical Graphs • Formal Development • https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development… Resources — Logical Graphs • https://oeis.org/wiki/Logical_Graphs Futures Of Logical Graphs • https://oeis.org/wiki/Futures_Of_Logical_Graphs Propositional Equation Reasoning Systems • https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems Charles Sanders Peirce • Bibliography • https://mywikibiz.com/Charles_Sanders_Peirce • https://mywikibiz.com/Charles_Sanders_Peirce_%28Bibliography%29 Logical Graphs • First Impressions 14 • https://inquiryintoinquiry.com/2024/09/12/logical-graphs-first-impressions-… Formal Development — Discussion of the topic continues at Logical Graphs • Formal Development • https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development… Resources — Logical Graphs • https://oeis.org/wiki/Logical_Graphs Futures Of Logical Graphs • https://oeis.org/wiki/Futures_Of_Logical_Graphs Propositional Equation Reasoning Systems • https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems Charles Sanders Peirce • Bibliography • https://mywikibiz.com/Charles_Sanders_Peirce • https://mywikibiz.com/Charles_Sanders_Peirce_%28Bibliography%29 Regards, Jon cc: https://www.academia.edu/community/lz94Ox cc: https://mathstodon.xyz/@Inquiry/113051725984818777