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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

https://inquiryintoinquiry.com/2017/08/27/minimal-negation-operators-1/

All,

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)

https://inquiryintoinquiry.files.wordpress.com/2017/08/venn-diagram-pq.jpg

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

Figure 2. ν(p, q, r)

https://inquiryintoinquiry.files.wordpress.com/2017/08/venn-diagram-pqr.jpg

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′

Resource

========

• Minimal Negation Operators

https://oeis.org/wiki/Minimal_negation_operator

Regards,

Jon

https://inquiryintoinquiry.com/2017/08/27/minimal-negation-operators-1/

All,

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)

https://inquiryintoinquiry.files.wordpress.com/2017/08/venn-diagram-pq.jpg

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

Figure 2. ν(p, q, r)

https://inquiryintoinquiry.files.wordpress.com/2017/08/venn-diagram-pqr.jpg

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′

Resource

========

• Minimal Negation Operators

https://oeis.org/wiki/Minimal_negation_operator

Regards,

Jon

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

https://inquiryintoinquiry.com/2017/08/30/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

https://inquiryintoinquiry.files.wordpress.com/2021/09/minimal-negation-operators-0-1-2-3.png

Resources

=========

Logic Syllabus

https://oeis.org/wiki/Logic_Syllabus

Boolean Function

https://oeis.org/wiki/Boolean_function

Multigrade Operator

https://oeis.org/wiki/Multigrade_operator

Minimal Negation Operator

https://oeis.org/wiki/Minimal_negation_operator

Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/

Regards,

Jon

https://inquiryintoinquiry.com/2017/08/30/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

https://inquiryintoinquiry.files.wordpress.com/2021/09/minimal-negation-operators-0-1-2-3.png

Resources

=========

Logic Syllabus

https://oeis.org/wiki/Logic_Syllabus

Boolean Function

https://oeis.org/wiki/Boolean_function

Multigrade Operator

https://oeis.org/wiki/Multigrade_operator

Minimal Negation Operator

https://oeis.org/wiki/Minimal_negation_operator

Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/

Regards,

Jon

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

https://inquiryintoinquiry.com/2017/08/30/minimal-negation-operators-3/

All,

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.

Regards,

Jon

https://inquiryintoinquiry.com/2017/08/30/minimal-negation-operators-3/

All,

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.

Regards,

Jon

Sep 27, 2021, 2:45:25 PM9/27/21

Cf: Minimal Negation Operators • 4

https://inquiryintoinquiry.com/2017/09/01/minimal-negation-operators-4/

All,

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

disjoint.

Regards,

Jon

https://inquiryintoinquiry.com/2017/09/01/minimal-negation-operators-4/

All,

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

disjoint.

Regards,

Jon

Sep 30, 2021, 5:32:21 PM9/30/21

Cf: Minimal Negation Operators • 5

https://inquiryintoinquiry.com/2021/09/30/minimal-negation-operators-5/

Re: Peirce List

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/thrd10.html#00287

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/thrd11.html#00322

::: Imran Makani

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/msg00342.html

<QUOTE IM:>

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

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 …

Regards,

Jon

https://inquiryintoinquiry.com/2021/09/30/minimal-negation-operators-5/

Re: Peirce List

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/thrd10.html#00287

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/thrd11.html#00322

::: Imran Makani

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/msg00342.html

<QUOTE IM:>

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 …

Regards,

Jon

Oct 2, 2021, 8:40:16 AM10/2/21

Cf: Minimal Negation Operators • Discussion 2

https://inquiryintoinquiry.com/2021/10/02/minimal-negation-operators-discussion-2/

Re: Minimal Negation Operators

https://inquiryintoinquiry.com/2017/09/01/minimal-negation-operators-4/

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/msg00353.html 22

<QUOTE JC:>

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.

</QUOTE>

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.

Regards,

Jon

https://inquiryintoinquiry.com/2021/10/02/minimal-negation-operators-discussion-2/

Re: Minimal Negation Operators

https://inquiryintoinquiry.com/2017/09/01/minimal-negation-operators-4/

Re: Peirce List

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/thrd10.html#00287

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/thrd11.html#00322

:: Jerry Chandler
https://list.iupui.edu/sympa/arc/peirce-l/2021-09/thrd10.html#00287

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/thrd11.html#00322

https://list.iupui.edu/sympa/arc/peirce-l/2021-09/msg00353.html 22

<QUOTE JC:>

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.

</QUOTE>

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.

Regards,

Jon

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