Logical Graphs • First Impressions

[Jon Awbrey]

Logical Graphs • First Impressions 1
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Introduction • Moving Pictures of Thought —

A “logical graph” is a graph-theoretic structure in one
of the systems of graphical syntax Charles Sanders Peirce
developed for logic.

In numerous papers on “qualitative logic”, “entitative graphs”,
and “existential graphs”, 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 this line of development,
a variety of formal systems have branched out from what is abstractly
the same formal base of graph-theoretic structures. This article
examines the common basis of these 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.

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 2
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

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 that 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 that C.S. Peirce used a streamer-cross symbol
where Spencer Brown used a carpenter's square marker to roughly
the same formal purpose, for instance, but the main theme of
interest at the level of pure form is indifferent to variations
of that order.

In Lieu of a Beginning —

Consider 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

[Jon Awbrey]

Logical Graphs • First Impressions 3
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

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 commonly represented by
a plane sheet of paper, can be represented in linear text
as 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 a text display:

( ( ) ) =

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 4
• https://inquiryintoinquiry.com/2023/08/24/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-arithmetic-initial-i2-framed.png

We noted it could be written inline as “( ( ) ) = ”

or set off in a 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. The planar
regions of the original graph correspond to nodes (or points) of the dual
graph, and the boundaries between planar regions in the original graph
correspond to edges (or lines) between the 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

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 5

• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Duality : Logical and Topological (cont.) —

It is easy to see the relationship between the parenthetical representations
of Peirce's logical graphs, clippedly picturing the ordered containments of
their formal contents, and the corresponding dual graphs, constituting 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

This ritual is called “traversing” the tree, and the string read off
is often 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 often called the “parse graph” of the string.
The users of this jargon tend to use it loosely, often using “parse string”
to mean the string that gets parsed into the associated graph.

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 6
• https://inquiryintoinquiry.com/2023/08/24/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-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”.

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 7
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

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 which can be 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”.

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 8
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Computational Representation (concl.) —

At the next level of concreteness, a pointer‑record data structure
can be represented as follows.

Pointer Example 1
• https://inquiryintoinquiry.files.wordpress.com/2023/09/logical-graph-pointer-structure-1.png

This portrays “index_0” as the address of a record which contains
the following data.

• datum_1, datum_2, datum_3, …, and so on.

What makes it possible to represent graph-theoretical structures
as data structures in computer memory is the fact that an address
is just another datum, and so we may have a state of affairs like
the following.

Pointer Example 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. The items of data that do not point
any further up the tree are then treated as labels on the record-nodes
where they reside, as shown below.

Pointer Example 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 same direction as “away
from the root”.

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 9
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

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 ourselves 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”.

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 10
• https://inquiryintoinquiry.com/2023/08/24/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-i1-2.0.png

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

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 11
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

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 the 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, this 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.

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 12
• https://inquiryintoinquiry.com/2023/08/24/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, represented 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 this 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.

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 13
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

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.

Regards,

Jon

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[Jon Awbrey]

Logical Graphs • First Impressions 14
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Formal Development —

Discussion of the topic continues at Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-2/

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

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