# Differential Logic

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### Jon Awbrey

Mar 21, 2020, 10:36:32 AM3/21/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic • Overview
At: http://inquiryintoinquiry.com/2020/03/20/differential-logic-%e2%80%a2-overview/

All,

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.

Part 1 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1 )

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

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

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 )

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

Part 2 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2 )

Propositional Forms on Two Variables (
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Propositional_Forms_on_Two_Variables )

Transforms Expanded over Differential Features (
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Transforms_Expanded_over_Differential_Features )

Transforms Expanded over Ordinary Features (
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Transforms_Expanded_over_Ordinary_Features )

Operational Representation ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Operational_Representation )

Part 3 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3 )

Development • Field Picture (
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Development_.E2.80.A2_Field_Picture )

Proposition and Tacit Extension (
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Proposition_and_Tacit_Extension )

Enlargement and Difference Maps (
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Enlargement_and_Difference_Maps )

Tangent and Remainder Maps ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Tangent_and_Remainder_Maps )

Least Action Operators ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Least_Action_Operators )

Goal-Oriented Systems ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Goal-Oriented_Systems )

Document History ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Document_History )

Regards,

Jon

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

### Jon Awbrey

Mar 22, 2020, 12:00:19 PM3/22/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic • 1
At: http://inquiryintoinquiry.com/2020/03/22/differential-logic-%e2%80%a2-1/

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.

Regards,

Jon

### Jon Awbrey

Mar 23, 2020, 3:30:24 PM3/23/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic ??? 2
At: http://inquiryintoinquiry.com/2020/03/23/differential-logic-%e2%80%a2-2/

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 (e_1, e_2,
..., e_k) and meaning exactly one of the propositions e_1, e_2, ..., e_k 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
e_1, e_2, ..., e_k as shown below.

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 e_1 e_2 ... e_k and
meaning all of the propositions e_1, e_2, ..., e_k 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.

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.

Regards,

Jon
Cactus Ej Lobe Connective.jpg
Cactus Ej Node Connective.jpg

### Jon Awbrey

Mar 24, 2020, 2:30:26 PM3/24/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic : 3
At: http://inquiryintoinquiry.com/2020/03/24/differential-logic-%e2%80%a2-3/

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/2020/03/syntax-and-semantics-of-a-calculus-for-propositional-logic-2.0.png

The simplest expression for logical truth is the empty word, typically denoted by epsilon or lambda 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).

Regards,

Jon
Syntax and Semantics of a Calculus for Propositional Logic 2.0.png

### Jon Awbrey

Mar 26, 2020, 1:30:27 PM3/26/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic : 4
At: http://inquiryintoinquiry.com/2020/03/26/differential-logic-%e2%80%a2-4/

Differential Expansions of Propositions
=======================================

Bird's 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:

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 x 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:

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:

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

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

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

Next question: What is the difference between the value of the proposition pq over there, at a distance of dp and dq,
and the value of the proposition pq where you are standing, all expressed in the form of a general formula, of course?
Here is the appropriate formulation:

Cactus Graph ((p,dp)(q,dq),pq)
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdpqdqpq.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, substituting 1 for p and 1 for q in the graph amounts to erasing the labels p and q, as
shown here:

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

And this is equivalent to the following graph:

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

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

Regards,

Jon
Cactus Graph Existential P and Q.jpg
Venn Diagram P and Q.jpg
Cactus Graph (p,dp)(q,dq).jpg
Cactus Graph (p,dp).jpg
Cactus Graph ((p,dp)(q,dq),pq).jpg
Cactus Graph (( ,dp)( ,dq), ).jpg
Cactus Graph ((dp)(dq)).jpg
Cactus Graph pq Diff ((dp)(dq)).jpg

### Jon Awbrey

Mar 29, 2020, 12:34:31 PM3/29/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic : 5
At: http://inquiryintoinquiry.com/2020/03/28/differential-logic-%e2%80%a2-5/

Note. I gave it the old college try at transcribing
the following math formulas but I recommend following

Differential Expansions of Propositions
=======================================

Worm's 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.

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

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 x B -> B or f : B^2 -> B. The concrete type takes into account the qualitative dimensions or the "units" of
the case, which can be explained as follows.

* Let P be the set of values {(p), p} = {not p, p} isomorphic to B = {0, 1}.
* Let Q be the set of values {(q), q} = {not q, q} isomorphic to B = {0, 1}.

Then interpret the usual propositions about p, q as functions of the concrete type f : P x 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 Delta, written here as D.
* The enlargement operator Epsilon, 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 x Q, its "(first order) differential extension" EX is constructed
according to the following specifications:

* EX = X x dX

where:

* X = P x Q
* dX = dP x 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 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 x Q by drawing arrows that cross 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 indicated in the following figure.

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 x 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 xor dp, q xor 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 xor dp)(q xor dq)

The corresponding cactus graph is shown below.

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

Given the proposition f(p, q) over X = P x 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 xor dp, q xor dq) xor 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 xor dp)(q xor dq) xor pq

The corresponding cactus graph is shown below.

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

Venn Diagram Difference pq @ pq
https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-pq-difference-pq-40-pq-1.jpg

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

The venn diagram shows the analysis of the inclusive disjunction "dp or dq" into the following exclusive disjunction:

* (dp and not dq) xor (dq and not dp) xor (dp and dq)

The resultant differential proposition may be read to say "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
Venn Diagram F = P and Q.jpg
Cactus Graph F = P and Q.jpg
Venn Diagram p q dp dq.jpg
Cactus Graph Ef = (p,dp)(q,dq).jpg
Cactus Graph Df = ((p,dp)(q,dq),pq).jpg
Venn Diagram PQ Difference pq @ pq.jpg
Cactus Graph PQ Difference pq @ pq.jpg

### Jon Awbrey

Apr 4, 2020, 11:00:31 AM4/4/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic • 6
At: http://inquiryintoinquiry.com/2020/04/03/differential-logic-%e2%80%a2-6/

Differential Expansions of Propositions
=======================================

Panoptic View • Difference Maps
===============================

In the last section we computed what is variously called the "difference map", the "difference proposition", or the
"local proposition" Df_x of the proposition f(p, q) = pq at the point x where p = 1 and q = 1.

In the universe of discourse X = P × Q, the four propositions pq, p (q), (p) q, (p)(q) indicating the "cells", or the
smallest distinguished regions of the universe, are called "singular propositions". These serve as an alternative
notation for naming the points (1, 1), (1, 0), (0, 1), (0, 0), respectively.

Thus we can write D}f_x = Df|_x = Df|_(1, 1) = Df|_pq, so long as we know the frame of reference in force.

In the example f(p, q) = pq, the value of the difference proposition Df_x at each of the four points x in X may be
computed in graphical fashion as shown below.

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

Cactus Graph Difference pq @ pq = ((dp)(dq))
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-difference-pq-40-pq-dpdq.jpg

Cactus Graph Difference pq @ p(q) = (dp)dq
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dpdq.jpg

Cactus Graph Difference pq @ (p)q = dp(dq)
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dpdq-1.jpg

Cactus Graph Difference pq @ (p)(q) = dp dq
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dp-dq.jpg

The easy way to visualize the values of these graphical expressions is just to notice the following equivalents.

Cactus Graph Lobe Rule
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-lobe-rule.jpg

Cactus Graph Spike Rule
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-spike-rule.jpg

Laying out the arrows on the augmented venn diagram, one gets a picture of a differential vector field.

Venn Diagram Difference pq
https://inquiryintoinquiry.files.wordpress.com/2020/04/venn-diagram-difference-pq-1.jpg

The Figure shows the points of the extended universe EX = P × Q × dP × dQ indicated by
the difference map Df : EX → B, namely, the following six points or singular propositions.

1. p q dp dq
2. p q dp (dq)
3. p q (dp) dq
4. p (q) (dp) dq
5. (p) q dp (dq)
6. (p)(q) dp dq

The information borne by Df should be clear enough from a survey of these six points — they tell you what you have to do
from each point of X in order to change the value borne by f(p, q), that is, the move you have to make in order to reach
a point where the value of the proposition f(p, q) is different from what it is where you started.

We have been studying the action of the difference operator D on propositions of the form f : P × Q → B, as illustrated
by the example f(p, q) = pq = the conjunction of p and q. The resulting difference map Df is a "(first order)
differential proposition", that is, a proposition of the form Df : P × Q × dP × dQ → B.

The augmented venn diagram shows how the "models" or "satisfying interpretations" of Df distribute over the extended
universe of discourse EX = P × Q × dP × dQ. Abstracting from that picture, the difference map Df can be represented in
the form of a "digraph" or "directed graph", one whose points are labeled with the elements of X = P × Q and whose
arrows are labeled with the elements of dX = dP × dQ, as shown in the following Figure.

Directed Graph Difference pq
https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-difference-pq.jpg

The same 6 points of the extended universe EX = P × Q × dP × dQ given by
the difference map Df : EX → B can be described by the following formula.

Df = p q · ((dp)(dq))
+ p (q) · (dp) dq
+ (p) q · dp (dq)
+ (p)(q) · dp dq

https://en.wikipedia.org/api/rest_v1/media/math/render/svg/347100e1473cdb29b37a928a60eb0661486a1937

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to
reveal previously unsuspected aspects of the proposition’s meaning. We will encounter more and more of these
Cactus Graph Df = ((p,dp)(q,dq),pq).jpg
Cactus Graph Difference pq @ pq = ((dp)(dq)).jpg
Cactus Graph Difference pq @ p(q) = (dp)dq.jpg
Cactus Graph Difference pq @ (p)q = dp(dq).jpg
Cactus Graph Difference pq @ (p)(q) = dp dq.jpg
Cactus Graph Ej Lobe Rule.jpg
Cactus Graph Ej Spike Rule.jpg
Venn Diagram Difference pq.jpg
Directed Graph Difference pq.jpg

### Jon Awbrey

Apr 6, 2020, 10:24:18 AM4/6/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic • 7
At: http://inquiryintoinquiry.com/2020/04/05/differential-logic-%e2%80%a2-7/

Differential Expansions of Propositions
=======================================

Panoptic View • Enlargement Maps
================================

The "enlargement" or "shift" operator E exhibits a wealth of interesting and useful properties in its own right, so it
pays to examine a few of the more salient features playing out on the surface of our initial example, f(p, q) = pq.

A suitably generic definition of the extended universe of discourse is afforded by the following set-up.

* Let X = X_1 × ... × X_k.
* Let dX = dX_1 × ... × dX_k.
* Then EX = X × dX
= X_1 × ... × X_k × dX_1 × ... × dX_k

For a proposition of the form f : X_1 × ... × X_k → B, the "(first order) enlargement" of f is the proposition Ef : EX →
B defined by the following equation.

* Ef(x_1, ..., x_k, dx_1, ..., dx_k)
= f(x_1 + dx_1, ..., x_k + dx_k)
= f(x_1 xor dx_1, ..., x_k xor dx_k)

The "differential variables" dx_j are boolean variables of the same type as the ordinary variables x_j. Although it is
conventional to distinguish the (first order) differential variables with the operational prefix "d" this way of
notating differential variables is entirely optional. It is their existence in particular relations to the initial
variables, not their names, which defines them as differential variables.

In the example of logical conjunction, f(p, q) = pq, the enlargement Ef is formulated as follows.

* Ef(p, q, dp, dq) = (p + dp)(q + dq) = (p xor dp)(q xor dq)

Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is
permissible to “multiply things out” in the usual manner to arrive at the following result.

* Ef(p, q, dp, dq) = p·q + p·dq + q·dp + dp·dq

To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the
above expression for Ef in the same way we did for Df. To that end, the value of Ef_x at each x in X may be computed in
graphical fashion as shown below.

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

Cactus Graph Enlargement pq @ pq = (dp)(dq)
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq.jpg

Cactus Graph Enlargement pq @ p(q) = (dp)dq
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq-1.jpg

Cactus Graph Enlargement pq @ (p)q = dp(dq)
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq-2.jpg

Cactus Graph Enlargement pq @ (p)(q) = dp dq
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dp-dq.jpg

Collating the data of this analysis yields a boolean expansion or disjunctive normal form (DNF) equivalent to the
enlarged proposition Ef.

* Ef = p q · Ef @ p q
+ p (q) · Ef @ p (q)
+ (p) q · Ef @ (p) q}
+ (p)(q) · Ef @ (p)(q)

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element ¬dp∧¬dq is
drawn as a loop at the point pq.

Directed Graph Enlargement pq
https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-enlargement-pq.jpg

We may understand the enlarged proposition Ef as telling us all the ways of reaching a model of the proposition f from
the points of the universe X.

Regards,

Jon
Cactus Graph Ef = (p,dp)(q,dq).jpg
Cactus Graph Enlargement pq @ pq = (dp)(dq).jpg
Cactus Graph Enlargement pq @ p(q) = (dp)dq.jpg
Cactus Graph Enlargement pq @ (p)q = dp(dq).jpg
Cactus Graph Enlargement pq @ (p)(q) = dp dq.jpg
Directed Graph Enlargement pq.jpg

### Jon Awbrey

Apr 8, 2020, 10:12:19 AM4/8/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic • 8
At: http://inquiryintoinquiry.com/2020/04/08/differential-logic-%e2%80%a2-8/

Propositional Forms on Two Variables
====================================

To broaden our experience with simple examples, let's examine the sixteen functions of concrete type P × Q → B and
abstract type B × B → B. The time we took contemplating logical conjunction from a variety of differential angles will
pay dividends as we study its kindred family of forms in the same lights.

Table A1 arranges the propositional forms on two variables in a convenient order, giving equivalent expressions for each
boolean function in several systems of notation.

Table A1. Propositional Forms on Two Variables
https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a1.-propositional-forms-on-two-variables.png

Regards,

Jon
Table A1. Propositional Forms on Two Variables.png

### joseph simpson

Apr 8, 2020, 4:38:17 PM4/8/20
to structura...@googlegroups.com, Cybernetic Communications, Ontolog Forum, Peirce List, SysSciWG
Jon:

I like the chart, things are starting to make a little more sense to me.

Further, I read your paper," An Architecture for Inquiry : Building Computer Platforms for Discovery" from Research Gate.

Take care, be good to yourself and have fun,

Joe

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To view this discussion on the web visit https://groups.google.com/d/msgid/structural-modeling/a5022d2b-8ca0-d75d-7947-3cffdd94de26%40att.net.

--
Joe Simpson

# All progress, therefore, depends on unreasonable people.”

George Bernard Shaw
Web Site:

### Jon Awbrey

Apr 9, 2020, 10:54:13 AM4/9/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic • Discussion 1
At: http://inquiryintoinquiry.com/2020/04/09/differential-logic-%e2%80%a2-discussion-1/

Re: Joseph Simpson

Thanks, Joe, glad you liked the table, I've got a million of 'em! I'll be setting another mess of tables directly as we
continue studying the effects of differential operators on families of propositional forms.

For anyone wondering, "Where's the Peirce?" — he is the Atlas on whose shoulders the whole world of differential logic
turns. The elegant efficiency of Peirce's logical graphs, augmented by Spencer Brown and extended to cactus graphs,
made it feasible for the first time to take on the extra complexities of differential propositional calculus. So that
theme is a constant throughout the development of differential logic.

Hope you and yours are safe and sound,

Jon

On 4/8/2020 4:38 PM, joseph simpson wrote:
> Jon:
>
> I like the chart, things are starting to make a little more sense to me.
>
> Building Computer Platforms for Discovery" from Research Gate.

### Jon Awbrey

Apr 11, 2020, 10:13:09 AM4/11/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic • 9
At: http://inquiryintoinquiry.com/2020/04/11/differential-logic-%e2%80%a2-9/

Propositional Forms on Two Variables
====================================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Propositional_Forms_on_Two_Variables

Table A2 arranges the propositional forms on two variables according to another plan, sorting propositions with similar
shapes into seven subclasses. Thereby hangs many a tale, to be told in time.

Table A2. Propositional Forms on Two Variables
https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a2.-propositional-forms-on-two-variables-1.png

Regards,

Jon

Table A2. Propositional Forms on Two Variables.png

### Jon Awbrey

Apr 13, 2020, 9:30:29 PM4/13/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
On 4/9/2020 11:02 AM, Edwina Taborsky wrote:
>
> Yes - I think Peirce and Spencer Brown work very well together.
>
> Thanks for all your work.
>
> Edwina

Thanks, Edwina. I first encountered Peirce's Collected Papers
sometime during my freshman year in one of the quieter corners
of the Michigan State Math Library where I used to hide out to
study and shortly after a friend showed me the description of
Spencer Brown's "Laws of Form" in the 1st Whole Earth Catalog
and I sent off for it right away. I would spend the next ten
years trying to figure out what either one of them was saying.
In my view, Spencer Brown penetrated to the deepest strata of
Peirce's core ideas about logic, recognizing its operational
aspect and relational power in a way we've seldom seen since.
Not too coincidentally, those aspects and powers were a big
part of what I wrote my Senior Thesis on at the end of my

Regards,

Jon

### Jon Awbrey

Apr 26, 2020, 7:30:40 AM4/26/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic • 10
At: http://inquiryintoinquiry.com/2020/04/25/differential-logic-%e2%80%a2-10/

It’s been a while, so let’s review …

Tables A1 and A2 showed two ways of organizing the sixteen boolean functions or propositional forms on two variables, as
expressed in several notations. For ease of reference, here are fresh copies of those tables.

Table A1. Propositional Forms on Two Variables
https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a1.-propositional-forms-on-two-variables.png

Table A2. Propositional Forms on Two Variables
https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a2.-propositional-forms-on-two-variables-1.png

We took as our first example the boolean function f_8(p, q) = pq corresponding to the logical conjunction p ∧ q and
examined how the differential operators E and D act on f_8. Each differential operator takes a boolean function of two
variables f_8(p, q) and gives back a boolean function of four variables, Ef_8(p, q, dp, dq) or Df_8(p, q, dp, dq),
respectively.

In the next several posts we’ll extend our scope to the full set of boolean functions on two variables and examine how
the differential operators E and D act on that set. There being some advantage to singling out the enlargement or shift
operator E in its own right, we’ll begin by computing Ef for each function f in the above tables.

Regards,

Jon
Table A1. Propositional Forms on Two Variables.png
Table A2. Propositional Forms on Two Variables.png

### Jon Awbrey

Jun 17, 2020, 2:45:28 PM6/17/20
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
Cf: Differential Logic • Discussion 3
At: http://inquiryintoinquiry.com/2020/06/17/differential-logic-%e2%80%a2-discussion-3/

Re: R.J. Lipton
Re: P<NP
https://rjlipton.wordpress.com/2020/06/16/pnp/

Instead of boolean circuit complexity I would look at logical graph complexity, where those logical graphs are
constructed from minimal negation operators.

Physics once had a frame problem (complexity of dynamic updating) long before AI did but physics learned to reduce
complexity through the use of differential equations and group symmetries (combined in Lie groups).
One of the promising features of minimal negation operators is their relationship to differential operators. So I've
been looking into that. Here’s a link, a bit in medias res, but what I've got for now.

• Differential Logic • Cactus Language
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

Resources
=========

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

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

• Survey of Animated Logical Graphs
https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

• Survey of Differential Logic
https://inquiryintoinquiry.com/2020/02/08/survey-of-differential-logic-%e2%80%a2-2/

• Survey of Theme One Program
https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/

Regards,

Jon

### Jon Awbrey

Jun 15, 2021, 10:54:36 AM6/15/21
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG, Laws of Form
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

### Jon Awbrey

Jun 15, 2021, 2:36:18 PM6/15/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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
 https://oeis.org/wiki/Universe_of_discourse
 https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
 https://oeis.org/wiki/Propositional_calculus

### Jon Awbrey

Jun 15, 2021, 4:56:24 PM6/15/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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
 https://oeis.org/wiki/Boolean-valued_function
 https://oeis.org/wiki/Minimal_negation_operator
 https://oeis.org/wiki/Logical_conjunction
Cactus Ej Lobe Connective.jpg
Cactus Ej Node Connective.jpg

### Jon Awbrey

Jun 16, 2021, 11:10:47 AM6/16/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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)
Syntax and Semantics of a Calculus for Propositional Logic 3.0.png

### Jon Awbrey

Jun 16, 2021, 4:32:15 PM6/16/21
to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG, Laws of Form
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 )

### Jon Awbrey

Jun 17, 2021, 12:15:25 PM6/17/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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
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

### Jon Awbrey

Jun 17, 2021, 6:45:44 PM6/17/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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

### Jon Awbrey

Jun 18, 2021, 1:49:07 PM6/18/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
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
Venn Diagram F = P and Q.jpg
Cactus Graph F = P and Q.jpg
Venn Diagram p q dp dq.jpg

### Jon Awbrey

Jun 19, 2021, 1:02:33 PM6/19/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
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
Venn Diagram p q dp dq.jpg
Cactus Graph Ef = (p,dp)(q,dq).jpg
Cactus Graph Df = ((p,dp)(q,dq),pq).jpg
Venn Diagram Difference pq @ pq.jpg
Cactus Graph Difference pq @ pq.jpg

### Jon Awbrey

Jun 19, 2021, 3:12:36 PM6/19/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
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/
::: 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)

### Jon Awbrey

Jun 20, 2021, 12:00:26 PM6/20/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • Discussion 7
https://inquiryintoinquiry.com/2021/06/20/differential-logic-discussion-7/

Re: Differential Logic • 1
https://inquiryintoinquiry.com/2020/03/22/differential-logic-1/
Re: FB | Pattern Languages for Systemic Transformation
::: Steve Kramer

<QUOTE SK:>
Can differential logic be described using category theory?
To what other logical or mathematical modalities does
differential logic relate? Give an example.
Partial credit will be given.
</QUOTE>

Dear Steve,

The ultimate category-theoretic generalization of the
functional derivative in calculus and the tangent vector
in differential geometry is called a “tangent functor”.
Finding the proper logical analogue of a tangent functor
is the main business of my essay on Differential Logic
and Dynamic Systems.

Differential Logic and Dynamic Systems
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_4
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_5
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Appendices
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_References

But that opus is a lot to take in all at once so over the years
I've tried writing a number of easier pieces to work up to it more
gradually. The first intuitive inklings of the subject are provided
by the following overture.

Differential Propositional Calculus
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Appendices
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_References

The following series provides a more systematic treatment of substantial issues.
Regards,

Jon

### Jon Awbrey

Jun 26, 2021, 4:16:14 PM6/26/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • 6
https://inquiryintoinquiry.com/2020/04/03/differential-logic-6/

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

Panoptic View • Difference Maps
===============================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Panoptic_View_.E2.80.A2_Difference_Maps

In the previous section we computed what is variously described
as the “difference map”, the “difference proposition”, or the
“local proposition” Dfₓ of the proposition f(p, q) = pq at
the point x where p = 1 and q = 1.

In the universe of discourse X = P × Q, the four propositions
pq, p(q), (p)q, (p)(q) indicating the “cells”, or the smallest
distinguished regions of the universe, are called “singular
propositions”. These serve as an alternative notation for
naming the points (1, 1), (1, 0), (0, 1), (0, 0), respectively.

Thus we can write Dfₓ = Df|_x = Df|_(1, 1) = Df|_pq,
so long as we know the frame of reference in force.

In the example f(p, q) = pq, the value of the difference proposition Dfₓ
at each of the four points x in X may be computed in graphical fashion
as shown below.

Figure 1.0 Cactus Graph Df = ((p, dp)(q, dq), pq)
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-df-pdpqdqpq-1.jpg

Figure 1.1 Cactus Graph Difference pq @ pq = ((dp)(dq))
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-difference-pq-40-pq-dpdq.jpg

Figure 1.2 Cactus Graph Difference pq @ p(q) = (dp)dq
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dpdq.jpg

Figure 1.3 Cactus Graph Difference pq @ (p)q = dp(dq)
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dpdq-1.jpg

Figure 1.4 Cactus Graph Difference pq @ (p)(q) = dp dq
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dp-dq.jpg

The easy way to visualize the values of these graphical expressions
is just to notice the following equivalents.

Figure 2. Cactus Graph Lobe Rule
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-lobe-rule.jpg

Figure 3. Cactus Graph Spike Rule
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-spike-rule.jpg

Laying out the arrows on the augmented venn diagram,
one gets a picture of a differential vector field.

Figure 4. Venn Diagram Difference pq
https://inquiryintoinquiry.files.wordpress.com/2020/04/venn-diagram-difference-pq-1.jpg

The Figure shows the points of the extended universe
EX = P × Q × dP × dQ indicated by the difference map
Df : EX → B, namely, the following six points or
singular propositions.

1. p q dp dq
2. p q dp (dq)
3. p q (dp) dq
4. p (q)(dp) dq
5. (p) q dp (dq)
6. (p)(q) dp dq

The information borne by Df should be clear enough from a survey
of these six points — they tell you what you have to do from each
point of X in order to change the value borne by f(p, q), that is,
the move you have to make in order to reach a point where the value
of the proposition f(p, q) is different from what it is where you
started.

We have been studying the action of the difference operator D
on propositions of the form f : P × Q → B, as illustrated by the
example f(p, q) = pq which is known in logic as the conjunction
of p and q. The resulting difference map Df is a (first order)
differential proposition, that is, a proposition of the form
Df : P × Q × dP × dQ → B.

The augmented venn diagram shows how the models or satisfying
interpretations of Df distribute over the extended universe of
discourse EX = P × Q × dP × dQ. Abstracting from that picture,
the difference map Df can be represented in the form of a digraph
or directed graph, one whose points are labeled with the elements
of X = P × Q and whose arrows are labeled with the elements of
dX = dP × dQ, as shown in the following Figure.

Figure 5. Directed Graph Difference pq
https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-difference-pq.jpg

f = p ∙ q

Df = p ∙ q ∙ ((dp)(dq))
+ p ∙ (q) ∙ (dp) dq
+ (p) ∙ q ∙ dp (dq)
+ (p) ∙ (q) ∙ dp dq
Cactus Graph Df = ((p,dp)(q,dq),pq).jpg
Cactus Graph Difference pq @ pq = ((dp)(dq)).jpg
Cactus Graph Difference pq @ p(q) = (dp)dq.jpg
Cactus Graph Difference pq @ (p)q = dp(dq).jpg
Cactus Graph Difference pq @ (p)(q) = dp dq.jpg
Cactus Graph Ej Lobe Rule.jpg
Cactus Graph Ej Spike Rule.jpg
Venn Diagram Difference pq.jpg
Directed Graph Difference pq.jpg

### Jon Awbrey

Jul 3, 2021, 9:30:26 AM7/3/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • 7
https://inquiryintoinquiry.com/2020/04/05/differential-logic-7/

| 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.

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

Panoptic View • Enlargement Maps
================================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Panoptic_View_.E2.80.A2_Enlargement_Maps

The “enlargement” or “shift” operator E exhibits a wealth of
interesting and useful properties in its own right, so it pays
to examine a few of the more salient features playing out on the
surface of our initial example, f{p, q} = pq.

A suitably generic definition of the extended universe of discourse
is afforded by the following set-up.

• Let X = X₁ × … × Xₖ

• Let dX = dX₁ × … × dXₖ

• Then EX = X × dX

= X₁ × … × Xₖ × dX₁ × … × dXₖ

For a proposition of the form f : X₁ × … × Xₖ → B,
the “(first order) enlargement” of f is the proposition
Ef : EX → B defined by the following equation.

• Ef{x₁, …, xₖ, dx₁, …, dxₖ}

= f{x₁ + dx₁, …, xₖ + dxₖ}

= f{(x₁, dx₁), …, (xₖ, dxₖ)}

The differential variables dxₕ are boolean variables of the same type as
the ordinary variables xₕ. Although it is conventional to distinguish the
(first order) differential variables with the operational prefix “d” this way
of notating differential variables is entirely optional. It is their existence
in particular relations to the initial variables, not their names, which defines
them as differential variables.

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

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

Given that this expression uses nothing more than the boolean ring operations
of addition and multiplication, it is permissible to “multiply things out” in
the usual manner to arrive at the following result.

• Ef{p, q, dp, dq} = p∙q + p∙dq + q∙dp + dp∙dq

To understand what the enlarged or shifted proposition means in logical terms,
it serves to go back and analyze the above expression for Ef in the same way
we did for Df. To that end, the value of Efₓ at each x in X may be computed
in graphical fashion as shown below.

Figure 1.0 Cactus Graph Ef = (p,dp)(q,dq)
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-ef-pdpqdq-1.jpg

Figure 1.1 Cactus Graph Enlargement pq @ pq = (dp)(dq)
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq.jpg

Figure 1.2 Cactus Graph Enlargement pq @ p(q) = (dp)dq
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq-1.jpg

Figure 1.3 Cactus Graph Enlargement pq @ (p)q = dp(dq)
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq-2.jpg

Figure 1.4 Cactus Graph Enlargement pq @ (p)(q) = dp dq
https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dp-dq.jpg

Collating the data of this analysis yields a boolean expansion or
“disjunctive normal form” (DNF) equivalent to the enlarged proposition Ef.

• Ef = pq∙Ef_pq + p(q)∙Ef_p(q) + (p)q∙Ef_(p)q + (p)(q)∙Ef_(p)(q)

Here is a summary of the result, illustrated by means of a digraph picture,
where the “no change” element (dp)(dq) is drawn as a loop at the point p∙q.

Figure 2. Directed Graph Enlargement pq
https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-enlargement-pq.jpg

• f = p ∙ q

• Ef = p ∙ q ∙ (dp)(dq)
+ p ∙(q) ∙ (dp) dq
+ (p)∙ q ∙ dp (dq)
+ (p)∙(q) ∙ dp dq

Cactus Graph Ef = (p,dp)(q,dq).jpg
Cactus Graph Enlargement pq @ pq = (dp)(dq).jpg
Cactus Graph Enlargement pq @ p(q) = (dp)dq.jpg
Cactus Graph Enlargement pq @ (p)q = dp(dq).jpg
Cactus Graph Enlargement pq @ (p)(q) = dp dq.jpg
Directed Graph Enlargement pq.jpg

### Jon Awbrey

Jul 6, 2021, 8:12:19 AM7/6/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • 8
https://inquiryintoinquiry.com/2020/04/08/differential-logic-8/

Propositional Forms on Two Variables
====================================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Propositional_Forms_on_Two_Variables

To broaden our experience with simple examples, let's examine
the sixteen functions of concrete type P × Q → B and abstract
type B × B → B. Our inquiry into the differential aspects of
logical conjunction will pay dividends as we study the actions
of E and D on this family of forms.

Table A1 arranges the propositional forms on two variables
in a convenient order, giving equivalent expressions for
each boolean function in several systems of notation.

Regards,

Jon

Table A1. Propositional Forms on Two Variables.png

### Jon Awbrey

Jul 6, 2021, 1:08:42 PM7/6/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • 9
https://inquiryintoinquiry.com/2020/04/11/differential-logic-9/

Propositional Forms on Two Variables
====================================
https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Propositional_Forms_on_Two_Variables

Table A2 arranges the propositional forms on two variables according to
another plan, sorting propositions with similar shapes into seven subclasses.
Thereby hangs many a tale, to be told in time.

Table A2. Propositional Forms on Two Variables
https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a2.-propositional-forms-on-two-variables-1.png

Regards,

Jon

Table A2. Propositional Forms on Two Variables.png

### Jon Awbrey

Jul 7, 2021, 8:00:24 AM7/7/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • Comment 6
https://inquiryintoinquiry.com/2021/07/07/differential-logic-comment-6/

Cf: Category Theory
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/differential.20logic
::: Jon Awbrey
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/differential.20logic/near/245114541

All,

I opened a topic in the “logic” stream of “category theory.zulipchat”
to discuss differential logic in a category theoretic environment and
began by linking a few basic resources.

The topic on logical graphs (
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/logical.20graphs ) introduced a style
of graph-theoretic syntax for propositional logic stemming from the work of Charles S. Peirce and G. Spencer Brown and
touched on a generalization of Peirce's and Spencer Browns's tree-like forms to what graph theorists know as “cactus
graphs” or “cacti”.

Somewhat serendipitously, as it turns out, this cactus syntax is just the thing we need to develop differential
propositional calculus ( https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/ ),
which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments
the analytic geometry of Descartes.

Resources
=========

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

• Survey of Differential Logic
( https://inquiryintoinquiry.com/2021/05/15/survey-of-differential-logic-3/ )

Regards,

Jon

### Jon Awbrey

Jul 8, 2021, 10:40:27 AM7/8/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • 10
https://inquiryintoinquiry.com/2020/04/25/differential-logic-10/

It’s been a while, so let’s review …

Tables A1 and A2 showed two ways of organizing the sixteen boolean functions
or propositional forms on two variables, as expressed in several notations.
For ease of reference, here are fresh copies of those Tables.

Table A1. Propositional Forms on Two Variables
https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a1.-propositional-forms-on-two-variables.png
We took as our first example the boolean function f₈{p, q} = pq corresponding
to the logical conjunction p ∧ q and examined how the differential operators
E and D act on f₈. Each differential operator takes a boolean function of
two variables f₈{p, q} and gives back a boolean function of four variables,
Ef₈{p, q, dp, dq} or Df₈{p, q, dp, dq}, respectively.
Table A1. Propositional Forms on Two Variables.png
Table A2. Propositional Forms on Two Variables.png

### Jon Awbrey

Jul 11, 2021, 12:54:52 PM7/11/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • Discussion 8
https://inquiryintoinquiry.com/2021/07/11/differential-logic-discussion-8/
::: Lyle Anderson ( https://groups.io/g/lawsofform/message/406 )

<QUOTE LA:>
Before we go on to other Boolean functions, can we nail down the relationship between p and dp and q and dq?
</QUOTE>

Dear Lyle,

I wandered into this differential wonderland by following my nose through a budget
of old readings on the calculus of finite differences. It was a long time ago in
a math library not too far away in space but no longer extant in time. None other
than Boole wrote a treatise on the subject and corresponded with De Morgan about it.
I recall picking up the “E” for “enlargement operator” somewhere in that mix. It was
a genuine epiphany. All of which brings me round to think the most felicitous entry
point may be the one I first happened on, as I recounted in the “Chapter on Linear Topics”
( https://inquiryintoinquiry.files.wordpress.com/2021/05/theme-one-guide-e280a2-linear-topics.pdf )
I linked at the end of the following post.

• Differential Logic and Dynamic Systems • Discussion 5
https://inquiryintoinquiry.com/2021/06/13/differential-logic-and-dynamic-systems-discussion-5/

Maybe it will help to go through that ...

Regards,

Jon
Theme One Guide • Linear Topics.pdf

### Jon Awbrey

Jul 12, 2021, 3:02:24 PM7/12/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • Discussion 9
https://inquiryintoinquiry.com/2021/07/12/differential-logic-discussion-9/
:: Lyle Anderson ( https://groups.io/g/lawsofform/message/424 )

<QUOTE LA:>
All I am asking is what is your definition of “dp” in relation to “p‌”.
So far I have dp is what one has to do to get from p to (p) or from
(p) to p‌. Is that all there is to it? If that is the case, then
what you are really dealing with is some flavor of Lattice Theory.
</QUOTE>

Dear Lyle,

What we've been doing all along here is taking the things we'd normally
do in a garden variety “calculus of many variables” course with spaces like:

R, R^j, R^j → R, R^j → R^k, ...

and functoring that whole business over to B, in other words,
cranking the analogies as far as we can push them to spaces like:

B, B^j, B^j → B, B^j → B^k, ...

A few analogies are bound to break in transit through the Real-Bool barrier,
once familiar constructions morph into new-fangled configurations, and other
distinctions collapse or “condense” as Spencer Brown called it. Still enough
structure gets preserved overall to reckon the result a kindred subject.

To be continued ...

Jon

### Jon Awbrey

Jul 13, 2021, 6:18:21 PM7/13/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • Discussion 10
https://inquiryintoinquiry.com/2021/07/13/differential-logic-discussion-10/
Dear Lyle,

Let's say we're observing a system at discrete intervals of time and
testing whether its state satisfies or falsifies a given predicate or
proposition p at each moment. Then p and dp are two state variables
describing the time evolution of the system. In logical conception p
and dp are independent variables, even if empirical discovery finds
them bound by law.

What gives the differential variable dp its meaning in relation to
the ordinary variable p is not the conventional notation used here
but a class of “temporal inference rules”, in the present example,
the fourfold scheme of inference shown below.

Temporal Inference Rules
https://inquiryintoinquiry.files.wordpress.com/2021/07/temporal-inference-rules-p.png

Regards,

Jon
Temporal Inference Rules P.png

### Jon Awbrey

Jul 16, 2021, 7:16:02 PM7/16/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • Discussion 11
https://inquiryintoinquiry.com/2021/07/16/differential-logic-discussion-11/

Re: Differential Logic • Discussion 9
https://inquiryintoinquiry.com/2021/07/12/differential-logic-discussion-9/

All,

Let's look more closely at the “functor” from R to B
and the connection it makes between real and boolean
hierarchies of types. There's a detailed discussion
of this analogy in the article and section linked below.

• Differential Logic and Dynamic Systems
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview

•• The Analogy Between Real and Boolean Types
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#The_Analogy_Between_Real_and_Boolean_Types

Assorted types of mathematical objects seen often enough in practice to
earn themselves common names, along with their more common isomorphisms,
are shown in Table below.

Table 3. Analogy Between Real and Boolean Types
https://inquiryintoinquiry.files.wordpress.com/2021/07/analogy-between-real-and-boolean-types.png

Regards,

Jon
Analogy Between Real and Boolean Types.png

### Jon Awbrey

Jul 21, 2021, 9:48:43 AM7/21/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG
Cf: Differential Logic • Discussion 12
https://inquiryintoinquiry.com/2021/07/21/differential-logic-discussion-12/

Re: Category Theory
https://categorytheory.zulipchat.com/#narrow/stream/232162-learning.3A-show.20and.20tell/topic/Systems.20dynamics
::: John Baez
(1)
https://categorytheory.zulipchat.com/#narrow/stream/232162-learning.3A-show.20and.20tell/topic/Systems.20dynamics/near/196949955

(2)
https://categorytheory.zulipchat.com/#narrow/stream/232162-learning.3A-show.20and.20tell/topic/Systems.20dynamics/near/196950139

<QUOTE JB:>
One thing I'm interested in is functorially relating purely qualitative models to quantitative ones, or mixed
quantitative-qualitative models where you have some numerical information of the sort you describe, but not all of it.
That's a situation we often find ourselves in: having a mixture of quantitative and qualitative information about
what's going on in a complicated system.

When I say “functorially”, I mean for starters: there should be a functor from “quantitative models” of system dynamics
to “qualitative models”.
</QUOTE>

Dear John,

This is something I've been working on. In a turn of phrase I once concocted,
it's like passing from the qualitative theory of differential equations to the
differential theory of qualitative equations.

Cf. Differential Logic
https://categorytheory.zulipchat.com/#narrow/stream/233104-theory.3A-logic/topic/differential.20logic

Regards,
Jon