Logical Graphs • First Impressions

[Jon Awbrey]

Logical Graphs • First Impressions 1
• https://inquiryintoinquiry.com/2024/08/30/logical-graphs-first-impressions-1/

Moving Pictures of Thought —

A logical graph is a graph‑theoretic structure in one of the systems
of graphical syntax Charles S. Peirce developed for logic.

Introduction —

In numerous papers on qualitative logic, entitative graphs, and
existential graphs, C.S. Peirce developed several versions of
a graphical formalism, or a graph‑theoretic formal language,
designed to be interpreted for logic.

In the century since Peirce initiated their line of development, a variety
of formal systems have branched out from what is abstractly the same formal
base of graph‑theoretic structures. The posts to follow explore the common
basis of those formal systems from a bird's eye view, focusing on the aspects
of form shared by the entire family of algebras, calculi, or languages, however
they happen to be viewed in a given application.

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/V1Eowd
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 2
• https://inquiryintoinquiry.com/2024/08/31/logical-graphs-first-impressions-2/

Abstract Point of View —

The bird's eye view in question is more formally known as the perspective
of formal equivalence, from which remove one overlooks many distinctions
which appear momentous in more concrete settings. Expressions inscribed
in different formalisms whose syntactic structures are algebraically or
topologically isomorphic are not recognized as being different from each
other in any significant sense.

An eye to historical detail will note in passing that C.S. Peirce used
a “streamer‑cross” symbol where Spencer Brown used a “carpenter's square”
marker to roughly the same formal purpose, to give just one example, but
the main theme of interest at the level of pure form is indifferent to
variations of that order.

In Lieu of a Beginning —

We may start by contemplating the following two formal equations.

Figure 1
• https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-figure-1-framed.png

Figure 2
• https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-figure-2-framed.png

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/lOZkBp
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 3
• https://inquiryintoinquiry.com/2024/09/01/logical-graphs-first-impressions-3/

Duality : Logical and Topological —

In using logical graphs there are two types of duality to consider,
logical duality and topological duality.

Graphs of the order Peirce considered, those embedded in a continuous manifold
like that represented by a plane sheet of paper, can be represented in linear
text by what are called “traversal strings” and parsed into “pointer structures”
in computer memory.

A blank sheet of paper can be represented in linear text as a blank space, but
that way of doing it tends to be confusing unless the logical expression under
consideration is set off in a separate display.

For example, consider the axiom or initial equation shown below.

Logical Graph • Arithmetic Initial I₂
• https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-arithmetic-initial-i2-framed.png

This can be written inline as “( ( ) ) = ” or set off in the following
text display.

( ( ) ) =

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/VX2aAP
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 4
• https://inquiryintoinquiry.com/2024/09/03/logical-graphs-first-impressions-4/

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-arithmetic-initial-i2-framed.png

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-plane-tree-framed.png

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-tree-framed.png

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/lJX77E
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 5
• https://inquiryintoinquiry.com/2024/09/04/logical-graphs-first-impressions-5/

Duality : Logical and Topological (cont.) —

It is easy to see the relation between the parenthetical expressions of
Peirce's logical graphs, showing their contents in order of containment,
and the corresponding dual graphs, forming a species of rooted trees
to be described in greater detail below.

In the case of our last example, a moment's contemplation of the following
picture will lead us to see how we can get the corresponding parenthesis
string by starting at the root of the tree, climbing up the left side of
the tree until we reach the top, then climbing back down the right side
of the tree until we return to the root, all the while reading off the
symbols, in this case either “(” or “)”, we happen to encounter in our
travels.

Initial Equation I₂ • Dual Tree With Parentheses
• https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-i2-tree-parens-framed.png

The above ritual is called “traversing” the tree, and the string read off
is called the “traversal string” of the tree. The reverse ritual, which
passes from the string to the tree, is called “parsing” the string, and
the tree constructed is called the “parse graph” of the string. The users
of that jargon tend to use it loosely, often using “parse string” to mean
the string whose parsing creates the associated graph.

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/LY24AK
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 6
• https://inquiryintoinquiry.com/2024/09/05/logical-graphs-first-impressions-6/

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-arithmetic-initial-i1-framed.png

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-plane-tree-framed.png

Finally the rooted trees by themselves:

Initial Equation I₁ • Dual Trees

• https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-i1-tree-framed.png

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-tree-parens-framed.png

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/VvdD4q
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 7
• https://inquiryintoinquiry.com/2024/09/06/logical-graphs-first-impressions-7/

Computational Representation —

The parse graphs we've been looking at so far bring us one step closer
to the pointer graphs it takes to make the above types of maps and trees
live in computer memory but they are still a couple of steps too abstract
to detail the concrete species of dynamic data structures we need.

The time has come to flesh out the skeletons we have drawn up to this point.

Nodes in a graph represent “records” in computer memory. A record
is a collection of data conceived to reside at a specific “address”.
The address of a record is analogous to a demonstrative pronoun,
a word like “this” or “that”, on which account programmers call
it a “pointer” and semioticians recognize it as a type of sign
called an “index”.

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/Lpz9AW
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 8
• https://inquiryintoinquiry.com/2024/09/06/logical-graphs-first-impressions-8/

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-pointer-structure-1.png

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-pointer-structure-2.png

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-pointer-structure-3.png

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-7/

cc: https://www.academia.edu/community/5ADjXW
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 9
• https://inquiryintoinquiry.com/2024/09/08/logical-graphs-first-impressions-9/

Quick Tour of the Neighborhood —

This much preparation allows us to take up the founding axioms or
initial equations which determine the entire system of logical graphs.

Primary Arithmetic as Semiotic System —

Though it may not seem too exciting, logically speaking, there are many reasons
to make oneself at home with the system of forms represented indifferently,
topologically speaking, by rooted trees, balanced strings of parentheses,
and finite sets of non‑intersecting simple closed curves in the plane.

• For one thing it gives us a non‑trivial example of a sign domain
on which to cut our semiotic teeth, non‑trivial in the sense that
it contains a countable infinity of signs.

• In addition it allows us to study a simple form of computation
recognizable as a species of “semiosis” or sign‑transforming process.

This space of forms, along with the pair of axioms which divide it
into two formal equivalence classes, is what Spencer Brown called
the “primary arithmetic”.

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/5kdq9m
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 10
• https://inquiryintoinquiry.com/2024/09/08/logical-graphs-first-impressions-10/

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-i1-2.0.png

Axiom I₂
• https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-axiom-i2-2.0.png

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/l8P809
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 11
• https://inquiryintoinquiry.com/2024/09/09/logical-graphs-first-impressions-11/

Primary Arithmetic as Semiotic System (concl.) —

Let S be the set of rooted trees and let S₀ be the 2‑element subset
consisting of a rooted node and a rooted edge. Simple intuition,
or a simple inductive proof, will assure us that any rooted tree
can be reduced by means of the axioms of the primary arithmetic
to either a root node or a rooted edge.

For example, consider the reduction which proceeds as follows.

Primary Arithmetic Reduction Example
• https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-semiotic-system.png

Regarded as a semiotic process, this amounts to a sequence of
signs, every one after the first serving as an “interpretant”
of its predecessor, ending in a final sign which may be taken
as the canonical sign for their common object, in the upshot
being the result of the computation process.

Simple as it is, the sequence exhibits the main features of
any computation, namely, a semiotic process proceeding from
an obscure sign to a clear sign of the same object, being
in its aim and effect an action on behalf of clarification.

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/lQ9vxp
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 12
• https://inquiryintoinquiry.com/2024/09/10/logical-graphs-first-impressions-12/

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-semiotic-system.png

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-pattern-calculus.png

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/lQ9QO6
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 13
• https://inquiryintoinquiry.com/2024/09/11/logical-graphs-first-impressions-13/

Primary Algebra as Pattern Calculus (concl.) —

Speaking of algebra, and having just encountered one example
of an algebraic law, we might as well introduce the axioms
of the “primary algebra”, once again deriving their substance
and their name from the works of Charles Sanders Peirce and
George Spencer Brown, respectively.

Axiom J₁
• https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-axiom-j1-2.0.png

Axiom J₂
• https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-axiom-j2-2.0.png

The choice of axioms for any formal system is to some degree
a matter of aesthetics, as it is commonly the case that many
different selections of formal rules will serve as axioms to
derive all the rest as theorems.

As it happens, the example of an algebraic law we noticed first,
“a ( ) = ( )”, as simple as it appears, proves to be provable
as a theorem on the grounds of the foregoing axioms.

We might also notice at this point a subtle difference between the primary
arithmetic and the primary algebra with respect to the grounds of justification
we have naturally if tacitly adopted for their respective sets of axioms.

The arithmetic axioms were introduced by fiat, in a quasi‑apriori fashion,
though it is of course only long prior experience with the practical uses of
comparably developed generations of formal systems that would actually induce
us to such a quasi‑primal move. The algebraic axioms, in contrast, can be seen
to derive both their motive and their justification from the observation and
summarization of patterns which are visible in the arithmetic spectrum.

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-7/

Regards,

Jon

cc: https://www.academia.edu/community/LY2YZD
cc: https://mathstodon.xyz/@Inquiry/113051725984818777

[Jon Awbrey]

Logical Graphs • First Impressions 14
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-first-impressions-14/

Formal Development —

Discussion of the topic continues at Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-b/

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-14/

Formal Development —

Discussion of the topic continues at Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-b/

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