#### Differential Logic

Cf: Differential Logic • Overview
https://inquiryintoinquiry.com/2020/03/20/differential-logic-overview/

LoF Group,

| The following series of posts on Differential Logic were
| shared to my other lists back in 2020 when the LoF Group
| was experiencing its bout of “listlessness”. I'll copy
| them here partly by way of general background and also
| for context in answering Lyle's last set of questions.

The previous series of posts on Differential Propositional Calculus
( https://inquiryintoinquiry.com/?s=Differential+Propositional+Calculus )
brought us to the threshold of the subject without quite stepping over,
but I wanted to lay out the necessary ingredients in the most concrete,
intuitive, and visual way possible before taking up the abstract forms.

One of my readers on Facebook told me “venn diagrams are obsolete” and
of course we all know they become unwieldy as our universes of discourse
expand beyond four or five dimensions. Indeed, one of the first lessons
I learned when I set about implementing CSP's graphs and GSB's forms on the
computer was that 2-dimensional representations of logic are a death trap
in numerous conceptual and computational ways. Still, venn diagrams do us
good service in visualizing the relationships among extensional, functional,
and intensional aspects of logic. A facility with those relationships is
critical to the computational applications and statistical generalizations
of logic commonly used in mathematical and empirical practice.

At any rate, intrepid readers will have amped up their visual imaginations
well enough at this point to pick their way through the cactus patch ahead.
The link above or the transcript below outlines my last, best introduction
to Differential Logic, which I'll be working to improve as I serialize it
to my blog.

Resource
========

Differential Logic
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
Part 1 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1 )
Part 2 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2 )
Part 3 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3 )

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey

Lyle Anderson

Jon,
Your differential logic seems to start from this statement: "What is the value of the proposition  at a distance of  and  from the cell  where you are standing?

My question to you is what is the "distance" metric that you are imposing on a proposition?  Aren't we dealing with un-ordered sets?  Un-ordered sets do not have a distance metric.  Your admonition to "Don't thing about it -- just compute" leads to a nice diagram, but it has no meaning in mathematics or the real world.

https://groups.io/g/lawsofform/attachment/317/0/JonAwbreyDifferentialVennDiagram.png

Best regards,
Lyle

Cf: Differential Logic • 1
https://inquiryintoinquiry.com/2020/03/22/differential-logic-1/

Introduction
============
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Introduction

Differential logic is the component of logic whose object is
the description of variation — for example, the aspects of change,
difference, distribution, and diversity — in universes of discourse
subject to logical description. A definition that broad naturally
incorporates any study of variation by way of mathematical models,
but differential logic is especially charged with the qualitative
aspects of variation pervading or preceding quantitative models.
To the extent a logical inquiry makes use of a formal system,
its differential component treats the principles governing the
use of a differential logical calculus, that is, a formal system
with the expressive capacity to describe change and diversity in
logical universes of discourse.

Simple examples of differential logical calculi are furnished by
differential propositional calculi. A differential propositional
calculus is a propositional calculus extended by a set of terms for
describing aspects of change and difference, for example, processes
taking place in a universe of discourse or transformations mapping
a source universe to a target universe. Such a calculus augments
ordinary propositional calculus in the same way the differential
calculus of Leibniz and Newton augments the analytic geometry
of Descartes.

References
[1] https://oeis.org/wiki/Universe_of_discourse
[2] https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
[3] https://oeis.org/wiki/Propositional_calculus

Cf: Differential Logic • 2
https://inquiryintoinquiry.com/2020/03/23/differential-logic-2/

Cactus Language for Propositional Logic
=======================================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

The development of differential logic is facilitated by having a moderately
efficient calculus in place at the level of boolean-valued functions and
elementary logical propositions. One very efficient calculus on both
conceptual and computational grounds is based on just two types of
logical connectives, both of variable k-ary scope. The syntactic
formulas of this calculus map into a family of graph-theoretic
structures called “painted and rooted cacti” which lend visual
representation to the functional structures of propositions
and smooth the path to efficient computation.

The first kind of connective takes the form of a parenthesized sequence
of propositional expressions, written (e₁, e₂, …, eₖ) and meaning exactly
one of the propositions e₁, e₂, …, eₖ is false, in short, their “minimal
negation” is true. An expression of this form maps into a cactus structure
called a “lobe”, in this case, “painted” with the colors e₁, e₂, …, eₖ as
shown below.

Figure 1. Lobe Connective
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-lobe-connective.jpg

The second kind of connective is a concatenated sequence of propositional expressions,
written e₁ e₂ … eₖ and meaning all of the propositions e₁, e₂, …, eₖ are true, in short,
their logical conjunction is true. An expression of this form maps into a cactus structure
called a “node”, in this case, “painted” with the colors e_1, e_2, ..., e_k as shown below.

Figure 2. Node Connective
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-node-connective.jpg

All other propositional connectives can be obtained through combinations
of these two forms. As it happens, the parenthesized form is sufficient
to define the concatenated form, making the latter formally dispensable,
but it's convenient to maintain it as a concise way of expressing more
complicated combinations of parenthesized forms. While working with
expressions solely in propositional calculus, it's easiest to use
plain parentheses for logical connectives. In contexts where
ordinary parentheses are needed for other purposes an alternate
typeface (...) may be used for the logical operators.

References
[1] https://oeis.org/wiki/Boolean-valued_function
[2] https://oeis.org/wiki/Minimal_negation_operator
[3] https://oeis.org/wiki/Logical_conjunction

Lyle Anderson

On Tue, Jun 15, 2021 at 11:36 AM, Jon Awbrey wrote:
Differential logic is the component of logic whose object is
the description of variation — for example, the aspects of change,
difference, distribution, and diversity — in universes of discourse
subject to logical description.
Jon, Logic is  "a science that deals with the principles and criteria of validity of inference and demonstration the science of the formal principles of reasoning." https://www.merriam-webster.com/dictionary/logic
There is no "variation" included within the principles and criteria of validity of inference and demonstration, therefore your "differential logic" can not be a component, i.e., subset, of logic.  You give examples of your "variation" as the "aspects" of "change", "distribution," and "diversity" in "universes of discourse" subject to "logical description".

Let's examine "aspect": 1. a particular status or phase in which something appears or may be regarded; 2. appearance to the eye or mind.

Let's examine "change":  the act, process, or result of making different in some particular:

Let's examine "variation".  When you follow the nested definitions you get that it is an abstraction on "diverse" which means "composed of distinct or unlike elements or qualities."

Let's examine "universes of discourse:"
In the formal sciences, the domain of discourse, also called the universe of discourseuniversal set, or simply universe, is the set of entities over which certain variables of interest in some formal
treatment may range.  https://en.wikipedia.org/wiki/Domain_of_discourse#Universe_of_discourse

Now, examine the definition of the Calculus of Variations:
a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbersWhereas elementary calculus is about infinitesimally small changes in the values of functions without changes in the function itself, calculus of variations is about infinitesimally small changes in the function itself, which are called variations
https://en.wikipedia.org/wiki/Calculus_of_variations

Based on this preliminary analysis of your definition, I conclude that you are attempting to describe a calculus of the variation of logical functions (a second order abstraction or calculus) using a graphical language useful only in describing a first order Boolean calculus.  As the wikipedia article suggests, there is an extremely well defined universe of techniques within the already discovered calculus of variations.  Perhaps some of these techniques are applicable to functions resulting only in true or false, but Venn diagrams and Cactus graphs are not.

Best regards,
Lyle

Lyle Anderson

Jon,
Simply repeating what you wrote before will not overcome the problem with your initial definition of "differential logic".    If that is deficient, then no amount of artful construction will repair it.
Best regards,
Lyle

Dear Lyle,

Thank you for your questions and for taking the trouble to read that far.
I started a post in response to your questions about the definition of
distance in propositional contexts but since you referred to points
further down the road in a text I hadn't yet posted to the group
I thought I should establish the context for folks who may not
have gotten that far yet.

So I'll get back to that ...

Regards,

Jon

On 6/15/2021 5:05 PM, Lyle Anderson wrote:
Jon,
Simply repeating what you wrote before will not overcome the problem with your initial definition of "differential logic".    If that is deficient, then no amount of artful construction will repair it.
Best regards,
Lyle

Cf: Differential Logic • 3
https://inquiryintoinquiry.com/2020/03/24/differential-logic-3/

Cactus Language for Propositional Logic
=======================================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

Table 1 shows the cactus graphs, the corresponding cactus expressions,
their logical meanings under the so-called “existential interpretation”,
and their translations into conventional notations for a sample of
basic propositional forms.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic
https://inquiryintoinquiry.files.wordpress.com/2021/03/syntax-and-semantics-of-a-calculus-for-propositional-logic-3.0.png

The simplest expression for logical truth is the empty word,
typically denoted by ε or λ in formal languages, where it is
the identity element for concatenation. To make it visible
in context, it may be denoted by the equivalent expression
“(())”, or, especially if operating in an algebraic context,
by a simple “1”. Also when working in an algebraic mode, the
plus sign “+” may be used for exclusive disjunction. Thus we
have the following translations of algebraic expressions into
cactus expressions.

• a + b = (a, b)

• a + b + c = (a, (b, c)) = ((a, b), c)

It is important to note the last expressions
are not equivalent to the 3-place form (a, b, c).

Lyle Anderson

On Wed, Jun 16, 2021 at 08:10 AM, Jon Awbrey wrote:
The simplest expression for logical truth is the empty word
Jon,
Is there a similar expression for differential logic?
Is there a "delta" true?  Is there a differential false?
When you say "differential" do you really mean "indeterminate?
What is the relationship between your "differential logic" and GSB's "re-entry?"
Just trying to help.
May the Form be with you!
Best regards,
Lyle

Cf: Differential Logic • Discussion 4
http://inquiryintoinquiry.com/2021/06/16/differential-logic-discussion-4/

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2021-06/thrd4.html#00078
::: Mauro Bertani
https://list.iupui.edu/sympa/arc/peirce-l/2021-06/msg00109.html

<QUOTE MB:>
About Lobe Connective and Node Connective and their consequences,
I have a question:

You say that genus and species are evaluated by the proposition (a, (b),(c)).

The following proposition would no longer be appropriate: a (b, c).

And another question about differential calculus:

When we talk about A and dA we talk about A and (A)
or is it more similar to A and B?
</QUOTE>

Dear Mauro,

The proposition (a, (b),(c)) describes a genus a divided into species b and c.

The proposition a (b, c) says a is always true while just one of b or c is true.

The first proposition leaves space between the whole universe and the genus a
while the second proposition identifies the genus a with the whole universe.

The differential proposition dA is one we use to describe a change of state
(or a state of change) from A to (A) or the reverse.

Resources
=========

• Logic Syllabus ( https://oeis.org/wiki/Logic_Syllabus )
• Logical Graphs ( https://oeis.org/wiki/Logical_Graphs )
• Minimal Negation Operators ( https://oeis.org/wiki/Minimal_negation_operator )

Lyle Anderson

On Wed, Jun 16, 2021 at 01:32 PM, Jon Awbrey wrote:
The differential proposition dA is one we use to describe a change of state
(or a state of change) from A to (A) or the reverse.
Jon,
Does this mean that if A is the proposition "The sky is blue.", then dA would be the statement "The shy is not blue."?  Don't you already have a notation for this in A and (A)?  From where does "state" and "change of state" come in relation to a proposition?
Best regards,
Lyle

Cf: Differential Logic • Discussion 5
http://inquiryintoinquiry.com/2021/06/17/differential-logic-discussion-5/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/differential_logic/83557540
::: Lyle Anderson ( https://groups.io/g/lawsofform/message/330 )

<QUOTE JA:>
The differential proposition dA is one we use to describe
a change of state (or a state of change) from A to (A) or
the reverse.
</QUOTE>

<QUOTE LA:>
Does this mean that if A is the proposition “The sky is blue”,
then dA would be the statement “The sky is not blue”? Don't
you already have a notation for this in A and (A) ? From
where does “state” and “change of state” come in relation
to a proposition?
</QUOTE>

Dear Lyle,

The differential variable dA : X → B = {0, 1} is a derivative variable,
a qualitative analogue of a velocity vector in the quantitative realm.

Let's say x ϵ R is a real value giving the membrane potential
in a particular segment of a nerve cell's axon and A : R → B
is a categorical variable predicating whether the site is in
the activated state, A(x) = 1, or not, A(x) = 0. We observe
the site at discrete intervals, a few milliseconds apart, and
obtain the following data.

• At time t₁ the site is in a resting state, A(x) = 0.
• At time t₂ the site is in an active state, A(x) = 1.
• At time t₃ the site is in a resting state, A(x) = 0.

On current information we have no way of predicting the state at
time t₂ from the state at time t₁ but we know action potentials
are inherently transient so we can fairly well guess the state
of change at time t₂ is dA = 1, in other words, about to be
changing from A to (A). The site's qualitative “position”
and “velocity” at time t₂ can now be described by means
of the compound proposition A dA.

Resources
=========

Logic Syllabus
https://oeis.org/wiki/Logic_Syllabus

Logical Graphs
https://oeis.org/wiki/Logical_Graphs

Minimal Negation Operators
https://oeis.org/wiki/Minimal_negation_operator

Differential Logic
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3

Lyle Anderson

On Thu, Jun 17, 2021 at 09:15 AM, Jon Awbrey wrote:
The differential variable dA : X → B = {0, 1} is a derivative variable,
a qualitative analogue of a velocity vector in the quantitative realm.
Jon,
I got a "D" in English Lit in college. The defining incident that gave me a clue of the outcome of the course was when I had given a flowery and sunny description of the meaning of a poem, and the professor had shouted at me, "NO! Anderson, it's about a rape!" I wanted to lead with that story because I wanted you to know I have been where you are and lived to tell about it. I also wanted to let you know that I am not shouting at you. With that said to set the context, let me begin my comment on your last post.

Now I am beginning to understand where you have gone wrong.  You are trying to cram second-order entities into first-order constructs.  Your Cactus Graphs are first order constructs.  Your example is an oscillating function, which is a second-order construct that has two axes, value and time, but you are treating it at the same level of abstraction as a simple proposition.  You obfuscate your confusion in your own mind by using undefined terms such as "derivative variable."

Since this is a "Laws of Form" forum, perhaps it would be well to look at and discuss what George Spencer-Brown has to say about "differential logic".  It is all in Chapter 11, Equations of the second degree.  Lets pick it up when GSB starts to talk about Frequency and Velocity, since that seems to fit into you example of neural signaling.

Then, GSB introduces the concept of a Function, that he means in the usual sense that mathematicians and computer scientists use the term.

Doesn't Figure 1 look like what you are describing with you neural example?  You are describing an oscillator function.

This leads to Real and Imaginary value, and the Memory function, which ties nicely into your example of a cell in a neural network.

Face it Jon, you are in good company.  Alfred North Whitehead and Bertrand Russell spent all that time writing Principia Mathematica, and Kurt Friedrich Gödel showed how it was incomplete.  At least Russell lived to see and appreciate the correction by George Spencer-Brown with Laws of Form.

I stand ready to discuss any aspect of this that you wish to discuss.

May the Form be with you!
Best regards,
Lyle
Best regards,
Lyle

Cf: Differential Logic • 4
https://inquiryintoinquiry.com/2020/03/26/differential-logic-4/

Differential Expansions of Propositions
=======================================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Differential_Expansions_of_Propositions

Bird’s Eye View
===============
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Bird.27s_Eye_View

An efficient calculus for the realm of logic represented by boolean functions
and elementary propositions makes it feasible to compute the finite differences
and the differentials of those functions and propositions.

For example, consider a proposition of the form “p and q”
graphed as two letters attached to a root node, as shown below.

Figure 1. Cactus Graph Existential p and q
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-existential-p-and-q.jpg

Written as a string, this is just the concatenation “p q”.

The proposition pq may be taken as a boolean function f(p, q)
having the abstract type f : B × B → B, where B = {0, 1} is
read in such a way that 0 means false and 1 means true.

Imagine yourself standing in a fixed cell of the corresponding
venn diagram, say, the cell where the proposition pq is true,
as shown in the following Figure.

Figure 2. Venn Diagram p and q
https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-and-q.jpg

Now ask yourself: What is the value of the proposition pq
at a distance of dp and dq from the cell pq where you are
standing?

Don't think about it — just compute:

Figure 3. Cactus Graph (p, dp)(q, dq)
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdpqdq-1.jpg

The cactus formula (p, dp)(q, dq) and its corresponding graph arise
by replacing p with p + dp and q with q + dq in the boolean product
or logical conjunction pq and writing the result in the two dialects
of cactus syntax. This follows because the boolean sum p + dp is
equivalent to the logical operation of exclusive disjunction, which
parses to a cactus graph of the following form.

Figure 4. Cactus Graph (p, dp)
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdp-1.jpg

Next question: What is the difference between the value of
the proposition pq over there, at a distance of dp and dq from
where you are standing, and the value of the proposition pq where
you are, all expressed in the form of a general formula, of course?
The answer takes the following form.

Figure 5. Cactus Graph ((p, dp)(q, dq), pq)
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdpqdqpq-1.jpg

There is one thing I ought to mention at this point: Computed over B,
plus and minus are identical operations. This will make the relation
between the differential and the integral parts of the appropriate
calculus slightly stranger than usual, but we will get into that later.

Last question, for now: What is the value of this expression from your
current standpoint, that is, evaluated at the point where pq is true?
Well, replacing p with 1 and q with 1 in the cactus graph amounts to
erasing the labels p and q, as shown below.

Figure 6. Cactus Graph (( , dp)( , dq), )
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-dp-dq-1-1.jpg

And this is equivalent to the following graph.

Figure 7. Cactus Graph ((dp)(dq))
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-dpdq-1.jpg

We have just met with the fact
that the differential of the AND
is the OR of the differentials.

• p and q ---Diff---> dp or dq

Figure 8. Cactus Graph pq Diff ((dp)(dq))
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pq-diff-dpdq-1.jpg

It will be necessary to develop a more refined analysis of
that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan's rule,
it is no accident, as differentiation and negation turn out to be
closely related operations. Indeed, one can find discussions of
logical difference calculus in the Boole–De Morgan correspondence
and Peirce also made use of differential operators in a logical
context, but the exploration of these ideas has been hampered by
a number of factors, not the least of which has been the lack of
a syntax adequate to handle the complexity of expressions evolving
in the process.

Note. Due to the large number of Figures I won't attach them here,
but see the blog post linked at top of the page for the Figures and
also for the proper math formatting.

Regards,

Jon

Lyle Anderson

On Thu, Jun 17, 2021 at 03:45 PM, Jon Awbrey wrote:
Don't think about it — just compute:
Jon,
Please try thinking.  What you wrote does not compute.  What GSB wrote does compute.  Let's try The Laws of Form.
Best regards,
Lyle

Cf: Differential Logic • 5
https://inquiryintoinquiry.com/2020/03/28/differential-logic-5/

Differential Expansions of Propositions
=======================================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Differential_Expansions_of_Propositions

Worm's Eye View
===============
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Worm.27s_Eye_View

Let's run through the initial example again, keeping an eye
on the meanings of the formulas which develop along the way.
We begin with a proposition or a boolean function f(p, q) = pq
whose venn diagram and cactus graph are shown below.

Figure 1. Venn Diagram f = pq
https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-f-p-and-q.jpg

Figure 2. Cactus Graph f = pq
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-f-p-and-q.jpg

A function like this has an abstract type and a concrete type.
The abstract type is what we invoke when we write things like
f : B × B → B or f : B² → B. The concrete type takes into
account the qualitative dimensions or “units” of the case,
which can be explained as follows.

• Let P be the set of values {(p), p} = {not p, p} ≅ B.
• Let Q be the set of values {(q), q} = {not q, q} ≅ B.

Then interpret the usual propositions about p, q
as functions of the concrete type f : P × Q → B.

We are going to consider various “operators” on these functions.
An operator F is a function which takes one function f into
another function Ff.

The first couple of operators we need to consider are logical
analogues of two which play a founding role in the classical
finite difference calculus, namely:

• The “difference operator” Δ, written here as D.
• The “enlargement operator”, written here as E.

These days, E is more often called the “shift operator”.

In order to describe the universe in which these operators operate,
it is necessary to enlarge the original universe of discourse.
Starting from the initial space X = P × Q, its “(first order)
differential extension” EX is constructed according to the
following specifications.

• EX = X × dX

where:

• X = P × Q

• dX = dP × dQ

• dP = {(dp), dp}
• dQ = {(dq), dq}

The interpretations of these new symbols can be diverse, but
the easiest option for now is just to say dp means “change p”
and dq means “change q”.

Drawing a venn diagram for the differential extension EX = X × dX
requires four logical dimensions, P, Q, dP, dQ, but it is possible
to project a suggestion of what the differential features dp and dq
are about on the 2-dimensional base space X = P × Q by drawing arrows
crossing the boundaries of the basic circles in the venn diagram for X,
reading an arrow as dp if it crosses the boundary between p and (p) in
either direction and reading an arrow as dq if it crosses the boundary
between q and (q) in either direction, as shown in the following Figure.

Figure 3. Venn Diagram p q dp dq
https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-q-dp-dq.jpg

Note: Given the length of this section I'm going to break here
and continue in the next message, but see the blog post linked
above for the a better copy of the Figures and Math Formatting.

Regards,

Jon

Lyle Anderson

On Fri, Jun 18, 2021 at 10:49 AM, Jon Awbrey wrote:
The interpretations of these new symbols can be diverse, but
the easiest option for now is just to say dp means “change p”
and dq means “change q”.
Jon,
Cutting and pasting from your previous post does not constitute thinking.  And statements such as the one above have no place in a document that even remotely asserts that it is about mathematics or logic.

Other nonsense statements in this section include: "A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like f : B × B → B or f : B² → B. The concrete type takes into account the qualitative dimensions or “units” of the case, which can be explained as follows."

The computer scientists have claimed the definition of "abstract type" and "concrete type", so this statement would be very perplexing to anyone who has learned an Object Oriented Programming language. An abstract type is a named set of characteristics, methods, and initial data that can not be instantiated into a program element, but is used to construct other abstract or concrete types. Concrete types can be instantiated into a program element. Instantiated means that there is a program statement such as "CREATE x as TYPE q" that adds the element x to the executing program with all the attributes of the type "q."

I repeat my initial overarching question: Where is this going that is new?

May the Form be with you.
Best regards,
Lyle

Cf: Differential Logic • 5
https://inquiryintoinquiry.com/2020/03/28/differential-logic-5/

Differential Expansions of Propositions (cont.)
===============================================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Differential_Expansions_of_Propositions

Worm's Eye View (cont.)
=======================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Worm.27s_Eye_View

| Note. In the plaintext transcript below, I will use braces { }
| for function arguments f{x, y} and reserve parentheses ( ) for
| cactus formulas (p) = ¬p, (p, q) = p + q = p XOR q, and so on.
| Please see the blog post linked above for proper math formats.

| Picking up from where we left off last time, we are in the
| process of developing a propositional calculus analogue of
| the classical finite difference calculus, illustrated here
| in the case of the logical conjunction f{p, q} = pq = p∧q.

Drawing a venn diagram for the differential extension EX = X × dX
requires four logical dimensions, P, Q, dP, dQ, but it is possible
to project a suggestion of what the differential features dp and dq
are about on the 2-dimensional base space X = P × Q by drawing arrows
crossing the boundaries of the basic circles in the venn diagram for X,
reading an arrow as dp if it crosses the boundary between p and (p) in
either direction and reading an arrow as dq if it crosses the boundary
between q and (q) in either direction, as shown in the following Figure.

Figure 3. Venn Diagram p q dp dq
https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-q-dp-dq.jpg

Propositions are formed on differential variables, or any combination of
ordinary logical variables and differential logical variables, in the
same ways propositions are formed on ordinary logical variables alone.
For example, the proposition (dp (dq)) says the same thing as dp ⇒ dq,
in other words, there is no change in p without a change in q.

Given the proposition f{p, q} over the space X = P × Q, the
“(first order) enlargement” of f is the proposition Ef over
the differential extension EX defined by the following formula.

• Ef{p, q, dp, dq} = f{p + dp, q + dq} = f{(p, dp), (q, dq)}

In the example f{p, q} = pq, the enlargement Ef is computed as follows.

• Ef{p, q, dp, dq} = {p + dp}{q + dq} = (p, dp)(q, dq)

Figure 4. Cactus Graph Ef = (p,dp)(q,dq)
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-ef-pdpqdq-1.jpg

Given the proposition f{p, q} over the space X = P × Q, the
“(first order) difference” of f is the proposition Df over EX
defined by the formula Df = Ef - f, or, written out in full:

• Df{p, q, dp, dq} = f{p + dp, q + dq} - f{p, q} = ( f{ (p, dp), (q, dq) }, f{p, q} )

In the example f{p, q} = pq, the difference Df is computed as follows.

• Df{p, q, dp, dq} = {p + dp}{q + dq} - pq = ((p, dp)(q, dq), pq)

Figure 5. Cactus Graph Df = ((p,dp)(q,dq),pq)
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-df-pdpqdqpq-1.jpg

At the end of the previous section we evaluated this first order difference of
conjunction Df at a single location in the universe of discourse, namely, at the
point picked out by the singular proposition pq, in terms of coordinates, at the
place where p = 1 and q = 1. This evaluation is written in the form Df|_{pq} or
Df|_{(1, 1)}, and we arrived at the locally applicable law which may be stated
and illustrated as follows.

• f{p, q} = pq = p AND q ⇒ Df|_{pq} = ((dp)(dq)) = dp OR dq

Figure 6. Venn Diagram Difference pq @ pq
https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-difference-pq-40-pq.jpg

Figure 7. Cactus Graph Difference pq @ pq
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-difference-pq-40-pq.jpg

The venn diagram shows the analysis of the
inclusive disjunction ((dp)(dq)) into the
following exclusive disjunction.

• dp (dq) + (dp) dq + dp dq

The differential proposition ((dp)(dq)) may be read as saying
“change p or change q or both”. And this can be recognized as
just what you need to do if you happen to find yourself in the
center cell and require a complete and detailed description of
ways to escape it.

Regards,

Jon

Lyle Anderson

On Sat, Jun 19, 2021 at 10:02 AM, Jon Awbrey wrote:
The differential proposition ((dp)(dq)) may be read as saying
“change p or change q or both”. And this can be recognized as
just what you need to do if you happen to find yourself in the
center cell and require a complete and detailed description of
ways to escape it.
Jon,
Is this what is new: "you happen to find yourself in the center cell [of a Venn diagram] and require a complete and detailed description of ways to escape it?"
Best regards,
Lyle

Cf: Differential Logic • Discussion 6
https://inquiryintoinquiry.com/2021/06/19/differential-logic-discussion-6/

Re: Differential Logic • 5
https://inquiryintoinquiry.com/2020/03/28/differential-logic-5/
Re: Laws of Form
https://groups.io/g/lawsofform/topic/differential_logic/83557540
::: Lyle Anderson ( https://groups.io/g/lawsofform/message/338 )

<QUOTE JA:>
The differential proposition ((dp)(dq)) may be read as saying
“change p or change q or both”. And this can be recognized as
just what you need to do if you happen to find yourself in the
center cell and require a complete and detailed description of
ways to escape it.
</QUOTE>

<QUOTE LA:>
Is this what is new: “you happen to find yourself in the center cell
[of a Venn diagram] and require a complete and detailed description of
ways to escape it”?
</QUOTE>

Dear Lyle,

What's improved, if not entirely new, is the development of appropriate
logical analogues of differential calculus and differential geometry.
There has been work on applying the calculus of finite differences to
propositions, but the traditional styles of syntax are so weighed down
by conceptual clutter that the resulting formal systems hardly get off
the ground before they become too unwieldy to stand.

That is where the formal elegance and practical efficiency of C.S. Peirce's
logical graphs and Spencer Brown's graphical forms come to save the day.
That, I think, is new. Or at least it was when I began work on it.

Regards,

Jon (the Prisoner of Vennda, No More)

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