Differential Logic

[Jon Awbrey]

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 )

Further Reading ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Further_Reading )

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

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache

[Jon Awbrey]

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]

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

[Jon Awbrey]

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

[Jon Awbrey]

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

[Jon Awbrey]

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
the link above for a much more readable copy.

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

[Jon Awbrey]

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
alternative readings as we go.

[Jon Awbrey]

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
https://en.wikipedia.org/api/rest_v1/media/math/render/svg/6b32eadd66a62c7cefc8c31d52f46708d68f3adc

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

[Jon Awbrey]

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

[joseph simpson]

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.

The examples in the paper help to add additional context.

Take care, be good to yourself and have fun,

Joe

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Joe Simpson

“Reasonable people adapt themselves to the world. 

Unreasonable people attempt to adapt the world to themselves. 

All progress, therefore, depends on unreasonable people.”

George Bernard Shaw

Git Hub link:

https://github.com/jjs0sbw

Research Gate link:

https://www.researchgate.net/profile/Joseph_Simpson3

YouTube Channel

https://www.youtube.com/user/jjs0sbw

Web Site:

https://systemsconcept.org/

[Jon Awbrey]

Cf: Differential Logic • Discussion 1
At: http://inquiryintoinquiry.com/2020/04/09/differential-logic-%e2%80%a2-discussion-1/

Re: Joseph Simpson
At: https://groups.google.com/d/msg/structural-modeling/xB5tRt4mcEM/IfaF8YlLBgAJ

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

[ https://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery ]

[Jon Awbrey]

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

[Jon Awbrey]

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
undergrad years.

Regards,

Jon

[Jon Awbrey]

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

[Jon Awbrey]

Cf: Differential Logic • Discussion 3
At: http://inquiryintoinquiry.com/2020/06/17/differential-logic-%e2%80%a2-discussion-3/

Re: R.J. Lipton
https://rjlipton.wordpress.com/about-me/
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]

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 )

[Jon Awbrey]

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

[Jon Awbrey]

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

[Jon Awbrey]

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)

[Jon Awbrey]

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]

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

[Jon Awbrey]

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