Minimal Negation Operators

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Jon Awbrey

Sep 24, 2021, 3:25:52 PM9/24/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Minimal Negation Operators • 1


To accommodate moderate levels of complexity in the application of
logical graphs to practical problems our Organon needs a class of
organules called “minimal negation operators”. I outlined the
history of their early development from Peirce's alpha graphs
for propositional calculus in a previous series of posts.
The next order of business is to sketch their properties
in a systematic fashion and to illustrate their uses.
As it turns out, taking MNOs as primitive operators
enables extremely efficient expressions for many
natural constructs and affords a bridge between
boolean domains of two values and domains with
finite numbers of values, for example, finite
sets of individuals.

Brief Introduction

A “minimal negation operator” (ν) is a logical connective
which says “just one false” of its logical arguments.
The first four cases are described below.

0. If the list of arguments is empty, as expressed in the form ν(),
then it cannot be true that exactly one of the arguments is false,
so ν() = false.

1. If p is the only argument then ν(p) says p is false,
so ν(p) expresses the logical negation of the proposition p.
Written in several different notations, we have the following
equivalent expressions.

ν(p) = not(p) = ¬p = ~p = p′

2. If p and q are the only two arguments then ν(p, q) says
exactly one of p, q is false, so ν(p, q) says the same
thing as p ≠ q. Expressing ν(p, q) in terms of ands (∙),
ors (∨), and nots (¬) gives the following form.

ν(p, q) = p′∙q ∨ p∙q′

It is permissible to omit the dot (∙) in contexts
where it is understood, giving the following form.

ν(p, q) = p′q ∨ pq′

The venn diagram for ν(p, q) is shown in Figure 1.

Figure 1. ν(p, q)

3. The venn diagram for ν(p, q, r) is shown in Figure 2.

Figure 2. ν(p, q, r)

The center cell is the region where all three arguments
p, q, r hold true, so ν(p, q, r) holds true in just the
three neighboring cells. In other words:

ν(p, q, r) = p′qr ∨ pq′r ∨ pqr′


• Minimal Negation Operators


Venn Diagram (P,Q).jpg
Venn Diagram (P,Q,R).jpg

Jon Awbrey

Sep 25, 2021, 6:28:32 PM9/25/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Minimal Negation Operators • 2

The brief description of minimal negation operators given in the
previous post is enough to convey the rule of their construction.
For future reference, a more formal definition is given below.

Initial Definition

The “minimal negation operator” ν (Greek nu) is a multigrade operator
(νₖ | k∈N), where each νₖ (for k in the set of non-negative integers N)
is a k-ary boolean function defined by the rule that νₖ(x₁, …, xₖ) = 1
if and only if exactly one of the arguments xₘ is 0.

In contexts where the initial letter ν is understood, the
minimal negation operators may be indicated by argument lists
in parentheses. In what follows a distinctive typeface will be
used for logical expressions based on minimal negation operators,
for example, (x , y , z ) = ν(x, y, z).

The first four members of this family of operators are shown below.
The third and fourth columns give paraphrases in two other notations,
where tildes and primes, respectively, indicate logical negation.

Figure 1. Minimal Negation Operators ν₀, ν₁, ν₂, ν₃


Logic Syllabus

Boolean Function

Multigrade Operator

Minimal Negation Operator

Survey of Animated Logical Graphs


Minimal Negation Operators 0, 1, 2, 3.png

Jon Awbrey

Sep 26, 2021, 4:56:29 PM9/26/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Minimal Negation Operators • 3


It will take a few more rounds of stage-setting before we are
able to entertain concrete examples of applications but the
following may indicate the direction of generalization
embodied in minimal negation operators.

To begin, let’s observe two ways of generalizing the logical operation
commonly known as exclusive disjunction (XOR) or symmetric difference (Δ).

Let B = the boolean domain {0, 1}.

Exclusive disjunction is a boolean function Δ : B × B → B
isomorphic to the algebraic field addition + : B × B → B,
also known as addition mod 2. Adding the language of
minimal negation operators to the mix we have the
following equivalent expressions.

XOR(p, q) = Δ(p, q) = p + q = ν(p, q) = (p, q)

Minimal Negation ν(p, q) as Parity Indicator

Generalizing the function p + q of two variables to more
variables extends the sequence of functions in the fashion
p + q + r, p + q + r + s, p + q + r + s + t, and so on.
These are known as “parity sums”, returning a value of 0
when there are an even number of 1’s in the sum and returning
a value of 1 when there are an odd number of 1’s in the sum.

Minimal Negation ν(p, q) as Border Indicator

The equivalent expressions (p, q) = ν(p, q) = p + q = p Δ q = p XOR q
may be read with a different connotation, indicating the venn diagram
cells adjacent to the conjunction p ∧ q. Generalizing the function
(p, q) of two variables to more variables extends the sequence of
functions in the fashion (p, q, r), (p, q, r, s), (p, q, r, s, t),
and so on. That sequence of operators differs from the sequence of
parity sums once it passes the 2-variable case.

The triple sum may be written in terms of 2-place minimal negations as follows.

p + q + r = ((p, q), r) = (p, (q, r))

It is important to recognize the triple sum expressions and the
3-place minimal negation (p, q, r) have very different meanings.



Jon Awbrey

Sep 27, 2021, 2:45:25 PM9/27/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Minimal Negation Operators • 4


I'm including a more detailed definition of minimal negation operators
in terms of conventional logical operations largely because readers of
particular tastes have asked for it in the past. But it can easily be
skipped until one has a felt need for it. Skimmed lightly, though, it
can serve to illustrate a major theme in logic and mathematics, namely,
the Relativity of Complexity or the Relativity of Primitivity to the
basis we have chosen for constructing our conceptual superstructures.

⁂ ⁂ ⁂

Defining minimal negation operators over a more conventional basis
is next in order of exposition, if not necessarily in order of every
reader’s reading. For what it’s worth and against the day when it may
be needed, here is a definition of minimal negations in terms of ∧, ∨,
and ¬.

Formal Definition

To express the general form of νₙ in terms of familiar operations,
it helps to introduce an intermediary concept.

Definition. Let the function ¬ₘ : Bⁿ → B be defined for each
integer m in the interval [1, n] by the following equation.

• ¬ₘ(x₁, …, xₘ, …, xₙ) = x₁ ∧ … ∧ xₘ₋₁ ∧ ¬xₘ ∧ xₘ₊₁ ∧ … ∧ xₙ.

Then νₙ : Bⁿ → B is defined by the following equation.

• νₙ(x₁, …, xₙ) = ¬₁(x₁, …, xₙ) ∨ … ∨ ¬ₘ(x₁, …, xₙ) ∨ … ∨ ¬ₙ(x₁, …, xₙ).

We may take the boolean product x₁ ∙ … ∙ xₙ or the logical conjunction
x₁ ∧ … ∧ xₙ to indicate the point x = (x₁, …, xₙ) in the space Bⁿ, in
which case the minimal negation νₙ(x₁, …, xₙ) indicates the set of points in
Bⁿ which differ from x in exactly one coordinate. This makes νₙ(x₁, …, xₙ)
a discrete functional analogue of a point-omitted neighborhood in ordinary
real analysis, more precisely, a point-omitted distance-one neighborhood.
Viewed in that light the minimal negation operator can be recognized as
a differential construction, an observation opening a very wide field.

The remainder of this discussion proceeds on the algebraic convention
making the plus sign (+) and the summation symbol (∑) both refer to
addition mod 2. Unless otherwise noted, the boolean domain B = {0, 1}
is interpreted for logic in such a way that 0 = false and 1 = true.
This has the following consequences.

• The operation x + y is a function equivalent to the exclusive disjunction of
x and y, while its fiber of 1 is the relation of inequality between x and y.

• The operation ∑ₘ xₘ = x₁ + … + xₙ maps the bit sequence (x₁, …, xₙ)
to its parity.

The following properties of the minimal negation operators
νₙ : Bⁿ → B may be noted.

• The function ν₂(x, y) is the same as that associated with
the operation x + y and the relation x ≠ y.

• In contrast, ν₃(x, y, z) is not identical to x + y + z.

• More generally, the function νₙ(x₁, …, xₙ) for k > 2
is not identical to the boolean sum ∑ₘ xₘ = x₁ + … + xₙ.

• The inclusive disjunctions indicated for the νₙ of more than
one argument may be replaced with exclusive disjunctions without
affecting the meaning since the terms in disjunction are already



Jon Awbrey

Sep 30, 2021, 5:32:21 PM9/30/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Minimal Negation Operators • 5

Re: Peirce List
::: Imran Makani

In his first post on this thread Jon clearly says that
[minimal negation operators] were developed from Peirce’s
alpha graphs for propositional calculus and that he has
even outlined the history of this early development in
a previous series of posts.

Dear Imran,

Welcome to the List and heartfelt thanks for your appreciation of my
contribution to it. I'm just a person who goes to sleep every night
and wakes up every morning with issues in Peirce's work taking pride
of place in his mind. It has been that way — no doubt with less
persistence at first, there were other demands and diversions then —
since I happened on Peirce's work my first year in college and right
up until the present time when my inquiries into the consequences of
his work literally pervade my dreams and days.

If you'll excuse my anecdotage, it took me nine years to complete my
Bachelor of Arts — demands and diversions were abundant — matriculating
first in Math and Physics, taking a break in Communication Arts where
I tilted with Aristotle, at long last mustering out in a cross-cultural
cultivating radical-liberal arts college with a concentration I created
myself in “Mathematical And Philosophical Method”. The cornerstone of
that first year and the capstone of my senior thesis, “Complications of
the Simplest Mathematics”, compass the dark night and the dawn's light
of my Peirce Decade One.

Well, I've run out of time for now …
I'll continue this memoir tomorrow …



Jon Awbrey

Oct 2, 2021, 8:40:16 AM10/2/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Minimal Negation Operators • Discussion 2

Re: Minimal Negation Operators
:: Jerry Chandler 22

As a chemist, CSP often inscended hyle terminology into his logical corpse as he sought to extend the 15–17th century
historical usages of the meaning of the concept of a “term”.

One particularity of chemical synthesis is the absence of the “negative” operators on the chemical elements. Each
element is a logical constant in the language of chemistry and hence can not be negated. Yet, in the notation for
chemistry it is necessary to assert and signify the absence of a chemical unit in a logical product. This could be
referred to as a minimal negation in a logically consistent semantics of a chemical syntax.

I have no information, either positive or negative, of the meaning Jon intends to infer logically with his usage of this
non-standard semantics. However, this semantics is obviously useful in attempting to give a logical semantics for the
well‑established semiosis of hyle.

Dear Jerry,

I've been spending a lot of time lately thinking about how I first got into all the things I've gotten into over the
years. The thing that surprised me the most was how much of my life I've been immersed in raw data despite my best
efforts to rise above it in flights of theory and just plain fancy. The honors chemistry course I took my first year in
college was pretty advanced — we “hit the ground running” as my Dad used to say from his paratroop days — moving from
covalent bonding theory the first term to molecular orbital theory the second.

It was there I first encountered the triple interaction of theory, experiment, and electronic computation. Aside from
the routine programs we ran to analyze our data, drawing least squares lines through experimental scatterplots and all
that, I began my first attempts to compute with symbolic forms, trying to get Fortran to place the electron dots around
and between chemical symbols in various molecular combinations. Mostly I learned to dislike Fortan — wrong tool for the
job, I guess — and it would be years before I woke to Lisp.

At any rate, let me beg off on chemical logic or logical chemistry. My experiences in that borderland are more a tale
of fits and starts than anything conclusive and reconstructing the details would take a search through the darker
corners of my basement archives.

The matter of “non-standard semantics”, however, is a timely and topical subject to address, one it would dispel a mass
of obscurities about the link between logic and semiotics to clarify as much as we can.

To begin, we may pose the question as follows.

• In what way does a propositional calculus
based on minimal negation operators
deviate from standard semantics?

I will take that up next time, perhaps under a different heading.


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