Operator Variables in Logical Graphs

[Jon Awbrey]

Operator Variables in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/06/operator-variables-in-logical-graphs-1/

All,

In lieu of a field study requirement for my bachelor's degree I spent
two years in various state and university libraries reading everything
I could find by and about Peirce, poring most memorably through reels
of microfilmed Peirce manuscripts Michigan State had at the time, all
in trying to track down some hint of a clue to a puzzling passage in
Peirce's “Simplest Mathematics”, most acutely coming to a head with
that bizarre line of type at CP 4.306, which the editors of Peirce's
“Collected Papers”, no doubt compromised by the typographer's reluctance
to cut new symbols, transmogrified into a script more cryptic than even
the manuscript's original hieroglyphic.

I found one key to the mystery in Peirce's use of “operator variables”,
which he and his students Christine Ladd–Franklin and O.H. Mitchell
explored in depth. I will shortly discuss that theme as it affects
logical graphs but it may be useful to give a shorter and sweeter
explanation of how the basic idea typically arises in common
logical practice.

Consider De Morgan's rules:

• ¬(A ∧ B) = ¬A ∨ ¬B

• ¬(A ∨ B) = ¬A ∧ ¬B

The common form exhibited by the two rules could be captured in a single
formula by taking “o₁” and “o₂” as variable names ranging over a family
of logical operators, then asking what substitutions for o₁ and o₂ would
satisfy the following equation.

• ¬(A o₁ B) = ¬A o₂ ¬B

We already know two solutions to this “operator equation”, namely,
(o₁, o₂) = (∧, ∨) and (o₁, o₂) = (∨, ∧). Wouldn't it be just
like Peirce to ask if there are others?

Having broached the subject of “logical operator variables”,
I will leave it for now in the same way Peirce himself did:

❝I shall not further enlarge upon this matter at this point,
although the conception mentioned opens a wide field; because
it cannot be set in its proper light without overstepping the
limits of dichotomic mathematics.❞ (Peirce, CP 4.306).

Further exploration of operator variables and operator invariants
treads on grounds traditionally known as second intentional logic
and “opens a wide field”, as Peirce says. For now, however, I will
tend to that corner of the field where our garden variety logical
graphs grow, observing the ways in which operative variations and
operative themes naturally develop on those grounds.

Regards,

Jon

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[Louis Kauffman]

Dear John,
I am writing to comment that there are some quite interesting situations that generalize the DeMorgan Duality.
One well-known one is this. Let R* denote the real number with a formal symbol @ denoting “infinity” adjoined so that
@+@ = @
@+ 0 = @
@ + x = @ when x is an ordinary real number.
1/@ = 0.
(of course you cannot do anything with @ or the system collapses. one can easily give the constraints.)
Define ~x = 1/x,
x + y = usual sum otherwise.
define x*y = xy/(x+y) = 1/((1/x) + (1/y)).
Then we have
x*y = ~(~x + ~y) so that the system (R*, ~, +, *) satisfies DeMorgan duality
and it is a Boolean algebra when restricted to {0,@}.
Note also that ~ fixes 1 and -1. This algebraic system occurs of course in electrical calculations and also
in the properties of tangles in knot theory, as you can read in the last part of my included paper “Knot Logic”.
I expect there is quite a bit more about this kind of duality in various (categorical) places.
Best,
Lou Kauffman

[Jon Awbrey]

Operator Variables in Logical Graphs • 2
• https://inquiryintoinquiry.com/2024/04/07/operator-variables-in-logical-graphs-2/

Operand Variables —

In George Spencer Brown's “Laws of Form” the relation between
the primary arithmetic and the primary algebra is founded on
the idea that a variable name appearing as an operand in an
algebraic expression indicates the contemplated absence or
presence of any expression in the arithmetic, with the
understanding that each appearance of the same variable
name indicates the same state of contemplation with
respect to the same expression of the arithmetic.

For example, consider the following expression:

Figure 1. Cactus Graph (a(a))
• https://inquiryintoinquiry.com/wp-content/uploads/2019/06/box-aa.jpg

We may regard this algebraic expression as a general expression
for an infinite set of arithmetic expressions, starting like so:

Figure 2. Cactus Graph Series (a(a))
• https://inquiryintoinquiry.com/wp-content/uploads/2019/06/box-aa-series.jpg

Now consider what that says about the following algebraic law:

Figure 3. Cactus Graph Equation (a(a)) = _
• https://inquiryintoinquiry.com/wp-content/uploads/2019/06/box-aa-.jpg

It permits us to understand the algebraic law as saying,
in effect, that every one of the arithmetic expressions
of the contemplated pattern evaluates to the very same
canonical expression as the upshot of that evaluation.
That is, as far as I know, just about as close as we
come to a conceptually and ontologically minimal way
of understanding the relation between an algebra and
its corresponding arithmetic.

Regards,

Jon

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[Jon Awbrey]

Operator Variables in Logical Graphs • 3
• https://inquiryintoinquiry.com/2024/04/10/operator-variables-in-logical-graphs-3/

❝And if he is told that something is the way it is, then he thinks:
Well, it could probably just as easily be some other way. So the
sense of possibility might be defined outright as the capacity to
think how everything could “just as easily” be, and to attach no
more importance to what is than to what is not.❞

— Robert Musil • “The Man Without Qualities”

All,

To get a clearer view of the relation between primary arithmetic
and primary algebra consider the following extremely simple
algebraic expression.

Figure 4. Cactus Graph (a)
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-a.jpg

Here we see the variable name ‘a’ appearing as an “operand name”
in the expression ‘(a)’. In functional terms, generally speaking,
an operand name like ‘a’ might also be called an “argument name” but
it's best to avoid the potentially confusing connotations of the word
“argument” here, since it also refers in logical discussions to a more
or less specific pattern of reasoning.

In effect, the algebraic variable name indicates the contemplated
absence or presence of any arithmetic expression taking its place
in the surrounding template, which expression is proxied well enough
by its formal value, of which values we know but two. Putting it all
together, the algebraic expression ‘(a)’ varies between the following
two choices.

Figure 5. Cactus Graph Set () , (())
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-.jpg

The above selection of arithmetic expressions is what it means to
contemplate the absence or presence of the arithmetic constant ‘( )’
in the place of the operand ‘a’ in the algebraic expression ‘(a)’.
But what would it mean to contemplate the absence or presence of
the operator ‘( )’ in the algebraic expression ‘(a)’?

That is the question I'll take up next.

Regards,

Jon

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[Jon Awbrey]

Operator Variables in Logical Graphs • 4
• https://inquiryintoinquiry.com/2024/04/11/operator-variables-in-logical-graphs-4/

Re: Operator Variables in Logical Graphs • 3
• https://inquiryintoinquiry.com/2024/04/10/operator-variables-in-logical-graphs-3/

All,

Last time we contemplated the penultimately simple algebraic
expression “(a}” as a name for a set of arithmetic expressions,
specifically, (a) = {(), (())}, taking the equal sign in the
appropriate sense.

Figure 6. Cactus Graph Equation (a) = {(),(())}
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-a-1.jpg

Then we asked the corresponding question about the operator “( )”.
The above set of arithmetic expressions is what it means to contemplate

the absence or presence of the arithmetic constant “( )” in the place of
the operand “a” in the algebraic expression “(a)”. But what would it mean
to contemplate the absence or presence of the operator “( )” in the algebraic
expression “(a)”?

Evidently, a variation between the absence and the presence of the
operator “( )” in the algebraic expression “(a)” refers to a variation
between the algebraic expression “a” and the algebraic expression “(a)”,
somewhat as pictured below.

Figure 7. Cactus Graph Equation ¿a? = {a,(a)}
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-a-queaa.jpg

But how shall we signify such variations in a coherent calculus?

Regards,

Jon

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[Jon Awbrey]

Operator Variables in Logical Graphs • 5
• https://inquiryintoinquiry.com/2024/04/12/operator-variables-in-logical-graphs-5/

Re: Operator Variables in Logical Graphs • 4
• https://inquiryintoinquiry.com/2024/04/11/operator-variables-in-logical-graphs-4/

All,

We have encountered the question of how to extend our
formal calculus to take account of operator variables.

In the days when I scribbled my logical graphs on the backs
of computer punchcards, the first thing I tried was drawing big
loopy script characters, placing some inside the loops of others.
Lower case alphas, betas, gammas, deltas, and so on worked best.
Graphs like that conveyed the idea that a character-shaped boundary
drawn around an enclosed space can be viewed as absent or present
depending on whether the formal value of the character in question
is unmarked or marked. The same idea can be conveyed by attaching
characters directly to the edges of graphs.

For example, the next Figure shows how we might suggest an
algebraic expression of the form “(q)” where the absence or
presence of the operator “( )” depends on the value of the
algebraic expression “p”, the operator “( )” being absent
whenever p is unmarked and present whenever p is marked.

Figure 8. Cactus Graph (q)_p = {q,(q)}
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-q-que-pqq.jpg

It was clear from the outset that this sort of tactic
would need a lot of work to become a usable calculus,
especially when it came time to feed those punchcards
back into the computer.

Regards,

Jon

cc: https://www.academia.edu/community/VjQ1Kp
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[Jon Awbrey]

Operator Variables in Logical Graphs • 6
• https://inquiryintoinquiry.com/2024/04/14/operator-variables-in-logical-graphs-6/

All,

Another tactic I tried by way of porting operator variables into
Peirce's logical graphs and Spencer Brown's logical forms was to
hollow out a leg of the latter's crosses, gnomons, or markers,
whatever you wish to call them, as shown below.

Figure 9. Transitional Form (q)_p = {q,(q)}
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-q-qua-p.jpg

The initial idea I had in mind was the same as before, that the
operator over q would be counted as absent when p evaluates to
a space and present when p evaluates to a cross.

However, much in the same way operators with a shade of negativity tend
to be more generative than the purely positive brand, it turned out more
useful to reverse the initial polarity of operation, letting the operator
over q be counted as absent when p evaluates to a cross and present when
p evaluates to a space.

So that is the convention I'll adopt from this point on.

Regards,

Jon

cc: https://www.academia.edu/community/lQrgXE
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[Jon Awbrey]

Operator Variables in Logical Graphs • 7
• https://inquiryintoinquiry.com/2024/04/15/operator-variables-in-logical-graphs-7/

Re: Operator Variables in Logical Graphs • 6
• https://inquiryintoinquiry.com/2024/04/14/operator-variables-in-logical-graphs-6/

All,

A funny thing just happened. Let's see if we can tell where.

We started with the algebraic expression “(a)”, where the operand “a”
suggests the contemplated absence or presence of an arbitrary arithmetic
expression. Next we contemplated the absence or presence of the operator
“( )” in “(a)” to be determined by the value of a newly introduced variable,
say “b”, which is placed in a new slot of a newly extended operator form, as
suggested by the following Figure.

Figure 10. Control Form (a)_b
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-a-quo-b.jpg

What happened here is this. Our contemplation of a constant operator
as being potentially variable gave rise to the contemplation of a newly
introduced but otherwise quite ordinary operand variable, albeit in a
newly-fashioned formula. In its interpretation for logic the newly
formed operation may be viewed as an extension of ordinary negation,
one in which the negation of the first variable is “controlled” by
the value of the second variable.

We may regard this development as marking a form of “controlled reflection”,
or a form of “reflective control”. From here on out we'll use the inline
syntax “(a , b)” to indicate the corresponding operation on two variables,
whose formal operation table is given below.

Table 11. Formal Operation Table (a,b)
• https://inquiryintoinquiry.com/wp-content/uploads/2021/03/formal-operation-table-a-b.png

• The Entitative Interpretation (En),
for which Space = False and Cross = True,
calls this operation “logical equality”.

• The Existential Interpretation (Ex),
for which Space = True and Cross = False,
calls this operation “logical difference”.

Regards,

Jon

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[Jon Awbrey]

Operator Variables in Logical Graphs • 8
• https://inquiryintoinquiry.com/2024/04/16/operator-variables-in-logical-graphs-8/

Re: Operator Variables in Logical Graphs • 7
• https://inquiryintoinquiry.com/2024/04/15/operator-variables-in-logical-graphs-7/

A trick of discovery I learned by observing Peirce's working methods,
more than anything he wrote outright, might be put in the following words.

• Take what is constant, Treat it as variable, See if anything remains the same.

The step of controlled reflection we took with the previous post
can be repeated at will, as suggested by the following series of forms.

Figure 12. Reflective Series (a) to (a, b, c, d)
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/reflective-series-a-to-abcd.jpg

Written inline, we have the series “(a)”, “(a , b)”, “(a , b , c)”,
“(a , b , c , d)”, and so on, whose general form is “(x₁ , x₂ , … , xₖ)”.
With this move we have passed beyond the graph-theoretical form of
rooted trees to what graph theorists know as “rooted cacti”.

I will discuss this “cactus language” and its logical interpretations next.

Regards,

Jon

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[Jon Awbrey]

Operator Variables in Logical Graphs • 9
• https://inquiryintoinquiry.com/2024/04/17/operator-variables-in-logical-graphs-9/

All,

The following Table will suffice to show how the “streamer‑cross”
forms C.S. Peirce used in his essay on “Qualitative Logic” and
Spencer Brown used in his “Laws of Form”, as they are extended
through successive steps of controlled reflection, translate into
syntactic strings and rooted cactus graphs.

Table 13. Syntactic Correspondences
• https://inquiryintoinquiry.com/wp-content/uploads/2021/04/syntactic-correspondences-2.0.png

Regards,

Jon

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[Jon Awbrey]

Operator Variables in Logical Graphs • 10
• https://inquiryintoinquiry.com/2024/04/18/operator-variables-in-logical-graphs-10/

Re: Operator Variables in Logical Graphs • 9
• https://inquiryintoinquiry.com/2024/04/17/operator-variables-in-logical-graphs-9/

All,

Let's examine the Formal Operation Table for the third in
our series of reflective forms to see if we can elicit the
general pattern.

Table 14. Formal Operation Table (a,b,c) Variant 1
• https://inquiryintoinquiry.com/wp-content/uploads/2021/04/formal-operation-table-a-b-c-e280a2-variant-1.png

Alternatively, if we think in terms of the corresponding
cactus graphs, writing “o” for an unmarked node and “|”
for a terminal edge, we get the following Table.

Table 15. Formal Operation Table (a,b,c) Variant 2
• https://inquiryintoinquiry.com/wp-content/uploads/2021/04/formal-operation-table-a-b-c-e280a2-variant-2.png

Evidently, the rule is that “(a , b , c)” denotes the value
denoted by “o” if and only if exactly one of the variables
a, b, c has the value denoted by “|”, otherwise “(a , b , c)”
denotes the value denoted by “|”. Examining the whole series
of reflective forms shows this to be the general rule.

• In the Entitative Interpretation (En), where o = false and | = true,
“(x₁ , … , xₙ)” translates as “not just one of the xₖ is true”.

• In the Existential Interpretation (Ex), where o = true and | = false,
“(x₁ , … , xₙ)” translates as “just one of the xₖ is not true”.

Regards,

Jon

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[Jon Awbrey]

Operator Variables in Logical Graphs • 11
• https://inquiryintoinquiry.com/2024/04/20/operator-variables-in-logical-graphs-11/

Re: Futures Of Logical Graphs • Themes and Variations
• https://oeis.org/wiki/Futures_Of_Logical_Graphs
• https://oeis.org/wiki/Futures_Of_Logical_Graphs#Themes_and_variations

All,

This post and the next wrap up the Themes and Variations section
of my speculation on Futures of Logical Graphs. I made an effort
to “show my work”, reviewing the steps I took to arrive at the
present perspective on logical graphs, whistling past the least
productive of the blind alleys, cul‑de‑sacs, detours, and forking
paths I explored along the way. It can be useful to tell the story
that way, partly because others may find things I missed down those
roads, but it does call for a recap of the main ideas I would like
readers to take away.

Partly through my reflection on Peirce's use of operator variables
I was led to what I called a “reflective extension of logical graphs”,
amounting to a graphical formal language called the “cactus language”
or “cactus syntax” after its principal graph‑theoretic data structure.

The abstract syntax of cactus graphs can be interpreted for logical use in
a couple of ways, both of which arise from generalizing the negation operator
“( )” in a particular direction, treating “( )” as the controlled, moderated,
or reflective negation operator of order 1 and adding another operator for
each integer greater than 1. The resulting family of operators is symbolized
by bracketed argument lists of the forms “( )”, “( , )”, “( , , )”, and so on,
where the number of places is the “order” of the reflective negation operator
in question.

Two rules suffice for evaluating cactus graphs.

• The rule for evaluating a k‑node operator, corresponding to
an expression of the form “x₁ x₂ … xₖ₋₁ xₖ”, is as follows.

Figure 16. Node Evaluation Rule
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-xj-node-evaluation-rule.jpg

• The rule for evaluating a k‑lobe operator, corresponding to
an expression of the form “(x₁ , x₂ , … , xₖ₋₁ , xₖ)”, is as follows.

Figure 17. Lobe Evaluation Rule
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/box-xj-lobe-evaluation-rule.jpg

Regards,

Jon

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[Jon Awbrey]

Operator Variables in Logical Graphs • 12
• https://inquiryintoinquiry.com/2024/04/21/operator-variables-in-logical-graphs-12/

Re: Operator Variables in Logical Graphs • 11
• https://inquiryintoinquiry.com/2024/04/20/operator-variables-in-logical-graphs-11/

All,

The rules given in the previous post for evaluating cactus graphs
were given in purely formal terms, that is, by referring to the
mathematical forms of cacti without mentioning their potential
for logical meaning.

As it turns out, two ways of mapping cactus graphs to logical
meanings are commonly found in practice. These two mappings
of mathematical structure to logical meaning are formally dual
to each other and known as the “Entitative” and “Existential”
interpretations respectively.

The following Table compares the entitative and existential
interpretations of the primary cactus structures, from which
the rest of their semantics can be derived.

Table 18. Logical Interpretations of Cactus Structures
• https://inquiryintoinquiry.com/wp-content/uploads/2019/07/logical-interpretations-of-cactus-structures-en-ex.jpg

Regards,

Jon

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