Differential Propositional Calculus

[Jon Awbrey]

Cf: Differential Propositional Calculus : Overview
At: http://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-%e2%80%a2-overview/

|| The most fundamental concept in cybernetics is that of "difference",
|| either that two things are recognisably different or that one thing
|| has changed with time.
||
|| W. Ross Ashby : An Introduction to Cybernetics
|| ( http://pespmc1.vub.ac.be/books/IntroCyb.pdf )

Here's the outline of I sketch I wrote on differential propositional calculi,
which extend propositional calculi by adding 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.
I wrote this as an intuitive introduction to differential logic, which is
my best effort so far at dealing with ancient and persistent problems of
dealing with diversity and mutability in logical terms. I'll be looking
at ways to improve this draft as I serialize it to my blog.

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

Casual Introduction
===================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Casual_Introduction )

Cactus Calculus
===============
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Cactus_Calculus )

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

Formal_Development
==================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Formal_Development )

Elementary Notions
==================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Elementary_Notions )

Special Classes of Propositions
===============================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Special_Classes_of_Propositions )

Differential Extensions
=======================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Differential_Extensions )

Appendices
==========
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Appendices )

References
==========
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_References )

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 Propositional Calculus : 1
At: http://inquiryintoinquiry.com/2020/02/18/differential-propositional-calculus-%e2%80%a2-1/

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.

Casual Introduction
===================

Consider the situation represented by the venn diagram in Figure 1.

Figure 1. Local Habitations, And Names
https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-figure-1.jpg

The area of the rectangle represents a universe of discourse, X. The universe under discussion may be a population of
individuals having various additional properties or it may be a collection of locations occupied by various individuals.
The area of the "circle" represents the individuals having the property q or the locations in the corresponding region
Q. Four individuals, a, b, c, d, are singled out by name. It happens that b and c currently reside in region Q while a
and d do not.

Now consider the situation represented by the venn diagram in Figure 2.

Figure 2. Same Names, Different Habitations
https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-figure-2.jpg

Figure 2 differs from Figure 1 solely in the circumstance that the object c is outside the region Q while the object d
is inside the region Q. So far, nothing says our encountering these Figures in this order is other than purely
accidental but if we interpret this sequence of frames as a "moving picture" representation of their natural order in a
temporal process then it would be natural to suppose a and b have remained as they were with regard to the quality q
while c and d have changed their standings in that respect. In particular, c has moved from the region where q is true
to the region where q is false while d has moved from the region where q is false to the region where q is true.

Figure 3 returns to the situation in Figure 1, but this time interpolates a new quality specifically tailored to account
for the relation between Figure 1 and Figure 2.

Figure 3. Back, To The Future
https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-figure-3.jpg

This new quality, dq, is an example of a differential quality, since its absence or presence qualifies the absence or
presence of change occurring in another quality. As with any other quality, it is represented in the venn diagram by
means of a "circle" distinguishing two halves of the universe of discourse, in this case, the portions of X outside and
inside the region dQ.

Figure 1 represents a universe of discourse, X, together with a basis of discussion, {q}, for expressing propositions
about the contents of that universe. Once the quality q is given a name, say, the symbol "q", we have the basis for a
formal language specifically cut out for discussing X in terms of q. This language is more formally known as the
propositional calculus with alphabet {"q"}.

In the context marked by X and {q} there are but four different pieces of information that can be expressed in the
corresponding propositional calculus, namely, the propositions: false, not q, q, true. Referring to the sample of
points in Figure 1, the constant proposition 'false' holds of no points, the proposition 'not q' holds of a and d, the
proposition 'q' holds of b and c, and the constant proposition 'true' holds of all points in the sample.

Figure 3 preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, {q,
dq}. In corresponding fashion, the initial propositional calculus is extended by means of the enlarged alphabet, {"q",
"dq"}. Any propositional calculus over two basic propositions allows for the expression of 16 propositions all
together. Just by way of salient examples, we can pick out the most informative proposition applying to each sample
point at the initial moment of observation. Table 4 shows these initial state descriptions, using overlines to express
logical negations.

Table 4. Initial State Descriptions
https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-e280a2-initial-state-descriptions.png

Table 5 shows the rules of inference responsible for giving the differential quality dq its meaning in practice.

Table 5. Differential Inference Rules
https://inquiryintoinquiry.files.wordpress.com/2020/02/differential-propositional-calculus-e280a2-differential-inference-rules.png

[Jon Awbrey]

Cf: Differential Propositional Calculus : 2
At: http://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-%e2%80%a2-2/

Cactus Calculus
===============

Table 6 outlines a syntax for propositional calculus based on
two types of logical connectives, both of variable k-ary scope.

* A bracketed list of propositional expressions in the form (e_1, e_2, ..., e_k)
indicates exactly one of the propositions e_1, e_2, ..., e_k is false.

* A concatenation of propositional expressions in the form e_1 e_2 ... e_k
indicates all the propositions e_1, e_2, ..., e_k are true, in other words,
their logical conjunction ( https://oeis.org/wiki/Logical_conjunction ) is true.

Table 6. Syntax and Semantics of a Calculus for Propositional Logic
https://inquiryintoinquiry.files.wordpress.com/2020/02/syntax-and-semantics-of-a-calculus-for-propositional-logic.png

All other propositional connectives can be obtained through combinations of these two forms. Strictly speaking, the
concatenation form is dispensable in light of the bracket form, but it is convenient to maintain it as an abbreviation
for more complicated bracket expressions. While working with expressions solely in propositional calculus, it is
easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes
"teletype" parentheses ( ... ) or barred parentheses (| ... |) may be used for logical operators.

The briefest expression for logical truth is the empty word, abstractly denoted epsilon or lambda in formal languages,
where it forms the identity element for concatenation. It may be given visible expression in this context by means of
the logically equivalent form (( )), 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 (
https://oeis.org/wiki/Exclusive_disjunction ). For example, we have the following paraphrases of algebraic expressions:

x + y = (x, y)

x + y + z = ((x, y), z) = (x, (y, z))

It is important to note the last expressions are not equivalent to the triple bracket (x, y, z).

More information about this syntax for propositional calculus can be found at the following locations.

Resources
=========

* Minimal Negation Operators
( https://oeis.org/wiki/Minimal_negation_operator )

* Zeroth Order Logic
( https://oeis.org/wiki/Zeroth_order_logic )

* Table A1. Propositional Forms on Two Variables
(
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Appendices#Table_A1._Propositional_Forms_on_Two_Variables

[Jon Awbrey]

Cf: Differential Propositional Calculus : 3
At: http://inquiryintoinquiry.com/2020/02/24/differential-propositional-calculus-%e2%80%a2-3/

I am working my way toward one of the places where Peirce's logic and semiotics,
Spencer Brown's Laws of Form, and Ashby's cybernetics meet, but there are a few
more courses of conceptual and notational bricks to lay down before we have the
proper foundation.

Formal Development
==================

The preceding discussion outlined the ideas leading to the differential extension
of propositional logic. The next task is to lay out the concepts and terminology
needed to describe various orders of differential propositional calculi.

Elementary Notions
==================

Logical description of a universe of discourse begins with a collection of logical signs.
For simplicity in a first approach, we may assume these logical signs are collected in
the form of a finite alphabet, \mathfrak{A} = {"a_1", ..., "a_n"}. Each of these signs
is interpreted as denoting a logical feature, for example, a property that objects of
the universe of discourse may have or a proposition about objects in the universe of
discourse. There is then corresponding to the alphabet \mathfrak{A} a set of logical
features, \mathcal{A} = {a_1, ..., a_n}.

Note. Breaking here because the rest of this post requires too much math formatting.
Please see the blog post linked above or the wiki version at the following location:

Differential Propositional Calculus : Part 2
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2

<...>

Table 7 summarizes the notations needed to describe ordinary propositional calculi in a systematic fashion.

Table 7. Propositional Calculus : Basic Notation
https://inquiryintoinquiry.files.wordpress.com/2020/02/propositional-calculus-basic-notation.png

[Jon Awbrey]

Cf: Differential Propositional Calculus : 4
At: http://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-%e2%80%a2-4/

We gradually march toward the point where
plain old propositional calculus meets up
with differential propositional calculus.

[Note: As usual, please follow the above link for a better-formatted copy.]

Special Classes of Propositions
===============================

Before moving on, let's unpack some of the assumptions, conventions, and implications involved in the array of concepts
and notations introduced above.

A universe of discourse A+ = [a_1, ..., a_n] based on the logical features a_1, ..., a_n is a set A plus the set of all
possible functions from the space A to the boolean domain B = {0, 1}. There are 2^n elements in A, often pictured as
the cells of a venn diagram or the nodes of a hypercube. There are 2^(2^n) possible functions from A to B, accordingly
pictured as all the ways of painting the cells of a venn diagram or the nodes of a hypercube with a palette of two colors.

A logical proposition about the elements of A is either true or false of each element in A, while a function f:A->B
evaluates to 1 or 0 on each element of A. The analogy between logical propositions and boolean-valued functions is
close enough to adopt the latter as models of the former and simply refer to the functions f:A->B as propositions about
the elements of A.

The full set of propositions f : A -> B contains a number of smaller classes deserving of special attention.

A "basic proposition" in the universe of discourse [a_1, ..., a_n] is one of the propositions in the set {a_1, ...,
a_n}. There are of course exactly n of these. Depending on the context, basic propositions may also be called
coordinate propositions or simple propositions.

Among the 2^(2^n) propositions in [a_1, ..., a_n] are several families numbering 2^n propositions each which take on
special forms with respect to the basis {a_1, ..., a_n}. Three of these families are especially prominent in the
present context, the "linear", the "positive", and the "singular" propositions. Each family is naturally parameterized
by the coordinate n-tuples in B^n and falls into n+1 ranks, with a binomial coefficient (n choose k) giving the number
of propositions having rank or weight k in their class.

Linear Propositions
https://inquiryintoinquiry.files.wordpress.com/2020/02/linear-propositions-may-be-written-as-sums.png

Positive Propositions
https://inquiryintoinquiry.files.wordpress.com/2020/02/positive-propositions-may-be-written-as-products.png

Singular Propositions
https://inquiryintoinquiry.files.wordpress.com/2020/02/singular-propositions-may-be-written-as-products.png

In each case the rank k ranges from 0 to n and counts the number of positive appearances of the coordinate propositions
a_1, ..., a_n in the resulting expression. For example, when n = 3 the linear proposition of rank 0 is 0, the positive
proposition of rank 0 is 1, and the singular proposition of rank 0 is (a_1)(a_2)(a_3), that is, not a_1 and not a_2 and
not a_3.

The basic propositions a_i : B^n -> B are both linear and positive. So these two kinds of propositions, the linear and
the positive, may be viewed as two different ways of generalizing the class of basic propositions.

Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical
basis {a_1, ..., a_n}. A singular proposition with respect to the basis {a_1, ..., a_n} will not remain singular if the
basis is extended by a number of new and independent features. Even if one keeps to the original set of pairwise
options {a_i} + {not a_i} to pick out a new basis, the sets of linear propositions and positive propositions are both
determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional
choice of a cell as origin.

[Jon Awbrey]

Cf: Differential Propositional Calculus : 5
At: http://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-%e2%80%a2-5/

Special Classes of Propositions (cont.)
=======================================

Let's pause at this point and get a better sense of how our special classes of propositions are structured and how they
relate to propositions in general. We can do this by recruiting our visual imaginations and drawing up a sufficient
budget of venn diagrams for each family of propositions. The case for 3 variables is exemplary enough for a start.

Linear Propositions
===================

Linear Propositions May Be Written As Sums
https://inquiryintoinquiry.files.wordpress.com/2020/02/linear-propositions-may-be-written-as-sums.png

One thing to keep in mind about these sums is that the values in B = {0, 1} are added "modulo 2", that is, in such a way
that 1 + 1 = 0.

In a universe of discourse based on three boolean variables, p, q, r, the linear propositions take the shapes shown in
Figure 8.

Figure 8. Linear Propositions : B^3 -> B
https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagrams-e280a2-p-q-r-e280a2-linear-propositions.jpg

At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following
three forms:

* (p, (q, r))

* ((p, q), r)

* p + q + r

Next are the three linear propositions of rank 2, which may be expressed by the following three forms, respectively:

* (p, r)

* (q, r)

* (p, q)

Next are the three linear propositions of rank 1, which
are none other than the three basic propositions, p, q, r.

At the bottom is the linear proposition of rank 0, the everywhere false proposition or the constant 0 function, which
may be expressed by the form ( ) or by a simple 0.

[Jon Awbrey]

Cf: Differential Propositional Calculus : 6
At: http://inquiryintoinquiry.com/2020/03/02/differential-propositional-calculus-%e2%80%a2-6/

Special Classes of Propositions (cont.)
=======================================

Next we take up the family of positive propositions and follow the same plan as before, tracing the rule of their
formation in the case of a 3-dimensional universe of discourse.

Positive Propositions
=====================

Positive Propositions May Be Written As Products
https://inquiryintoinquiry.files.wordpress.com/2020/02/positive-propositions-may-be-written-as-products.png

In a universe of discourse based on three boolean variables, p, q, r, there are 2^3 = 8 positive propositions. Their
venn diagrams are shown in Figure 9.

Figure 9. Positive Propositions : B^3 -> B
https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-positive-propositions.jpg

At the top is the venn diagram for the positive proposition of rank 3, corresponding to the boolean product or logical
conjunction pqr.

Next are the venn diagrams for the three positive propositions of rank 2, corresponding to the three boolean products,
pr, qr, pq, respectively.

Next are the three positive propositions of rank 1,

which are none other than the three basic propositions, p, q, r.

At the bottom is the positive proposition of rank 0, the everywhere true proposition or the constant 1 function, which
may be expressed by the form (( )) or by a simple 1.

[Jon Awbrey]

Cf: Differential Propositional Calculus : 7
At: http://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-%e2%80%a2-7/

Special Classes of Propositions (concl.)
========================================

Last and in a way least, we examine the family of singular propositions in a 3-dimensional universe of discourse.

In our model of propositions as mappings of a universe of discourse to a set of two values, in other words, indicator
functions of the form f : X -> B, singular propositions are those singling out the minimal distinct regions of the
universe, represented by single cells of the corresponding venn diagram.

Singular Propositions
=====================

Singular Propositions May Be Written As Products
https://inquiryintoinquiry.files.wordpress.com/2020/02/singular-propositions-may-be-written-as-products.png

In a universe of discourse based on three boolean variables, p, q, r, there are 2^3 = 8 singular propositions. Their
venn diagrams are shown in Figure 10.

Figure 10. Singular Propositions : B^3 -> B
https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagrams-e280a2-p-q-r-e280a2-singular-propositions.jpg

At the top is the venn diagram for the singular proposition of rank 3, corresponding to the boolean product pqr and
identical with the positive proposition of rank 3.

Next are the venn diagrams for the three singular propositions of rank 2, which may be expressed by the following three
forms, respectively:

* pr (q),

* qr (p),

* pq (r).

Next are the three singular propositions of rank 1, which may be expressed by the following three forms, respectively:

* q (p)(r),

* p (q)(r),

* r (p)(q).

At the bottom is the singular proposition of rank 0, which may be expressed by the following form:

* (p)(q)(r).

[Jon Awbrey]

Cf: Differential Propositional Calculus : 8
At: http://inquiryintoinquiry.com/2020/03/07/differential-propositional-calculus-%e2%80%a2-8/

Differential Extensions
=======================

An initial universe of discourse, A+, supplies the groundwork for any number of further extensions, beginning with the
"first order differential extension", EA+. The construction of EA+ can be described in the following stages:

Note. Breaking here because the rest of this
post requires too much math formatting.
Please see the blog post linked above or

the wiki copy at the following location:

Differential Propositional Calculus : Part 2 : Differential_Extensions
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Differential_Extensions

<...>

A proposition in a differential extension of a universe of discourse is called a "differential proposition" and forms
the analogue of a system of differential equations in ordinary calculus. With these constructions, the "first order
extended universe" EA+ and the "first order differential proposition" f : EA->B, we have arrived, in concept at least,
at the foothills of "differential logic".

Table 11 summarizes the notations needed to describe the first order differential extensions of propositional calculi in
a systematic manner.

Table 11. Differential Extension : Basic Notation
https://inquiryintoinquiry.files.wordpress.com/2020/03/differential-extension-basic-notation.png

[Jon Awbrey]

Cf: Differential Propositional Calculus : Discussion 1
At: http://inquiryintoinquiry.com/2020/03/12/differential-propositional-calculus-%e2%80%a2-discussion-1/

|| The most fundamental concept in cybernetics is that of "difference",
|| either that two things are recognisably different or that one thing
|| has changed with time.
||
|| W. Ross Ashby : An Introduction to Cybernetics
|| ( http://pespmc1.vub.ac.be/books/IntroCyb.pdf )

Re: Cybernetics Communications
https://groups.google.com/d/topic/cybcom/8QIMPmPhvjU/overview
Re: Klaus Krippendorff
https://groups.google.com/d/msg/cybcom/8QIMPmPhvjU/cES14-ANAwAJ

KK: To me, differences are the result of drawing distinctions.
They don't exist unless you actively draw them. So, the
act of drawing distinctions is more fundamental than the
differences thereby created.

I often return to that line from Ashby. This time I thought it made an apt segue from the scene of propositional
calculus, where universes of discourse are ruled by collections of distinctive features, to the differential extension
of propositional calculus, which enables us to describe trajectories within and transformations between our logical
universes.

So I agree with Klaus Krippendorff about "which came first", the distinctions drawn or the states distinguished in space
or time. The primitive character of distinctions is especially salient in this setting since our formalism for
propositional calculus is built on the forms of distinction pioneered by C.S. Peirce and augmented by George Spencer Brown.

Resource
========

* Differential Propositional Calculus
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
* Part 1
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1
* Part 2
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2

Regards,

Jon

[Jon Awbrey]

Cf: Differential Propositional Calculus • Discussion 2
At: http://inquiryintoinquiry.com/2020/03/18/differential-propositional-calculus-%e2%80%a2-discussion-2/

All,

The times are rife with distraction, so let's pause and retrace how we got to this place.

Our last reading in "Cybernetics" brought us in sight of a convergence or complementarity between the triadic relations
in Peirce's semiotics and those in Ashby's regulator games. There's a lot more to explore in that direction and I plan
to get back to it soon.

The two threads intertwined here, Cybernetics and Differential Logic, both spun off a thread on Pragmatic Truth, asking
what theories of truth are compatible with Peircean disciplines of pragmatic thinking. That's a topic with a tangled
history but the latest local tangle is documented in the following posts and excerpts.

Pragmatic Theory Of Truth • 13
https://inquiryintoinquiry.com/2019/11/03/pragmatic-theory-of-truth-%e2%80%a2-13/

Pragmatic inquiry into a candidate concept of truth would begin by applying the pragmatic maxim to clarify the concept
as far as possible and a pragmatic definition of truth, should any result, would find its place within Peirce's theory
of logic as formal semiotics, in other words, stated in terms of a formal theory of triadic sign relations.

Pragmatic Theory Of Truth • 14
https://inquiryintoinquiry.com/2019/11/06/pragmatic-theory-of-truth-%e2%80%a2-14/

There are many conceptions of truth — linguistic, model-theoretic, proof-theoretic — for the moment I'm focused on
cybernetics, systems, and experimental sciences and this is where the pragmatic conception of truth fits what we
naturally do in those sciences remarkably well.

The main thing in those activities is the relationship among symbol systems, the world, and our actions, whether in
thought, among ourselves, or between ourselves and the world. So the notion of truth we want here is predicated on
three dimensions: the patch of the world we are dealing with in a given application, the systems of signs we are using
to describe that domain, and the transformations of signs we find of good service in bearing information about that
piece of the world.

Pragmatic Theory Of Truth • 18
https://inquiryintoinquiry.com/2019/11/14/pragmatic-theory-of-truth-%e2%80%a2-18/

We do not live in axiom systems. We do not live encased in languages, formal or natural. There is no reason to think
we will ever have exact and exhaustive theories of what's out there, and the truth, as we know, is “out there”. Peirce
understood there are more truths in mathematics than are dreamt of in logic and Gödel's realism should have put the last
nail in the coffin of logicism, but some ways of thinking just never get a clue.

That brings us to the question —

* What are formalisms and all their embodiments in brains and computers good for?

For that I'll turn to cybernetics …

Survey of Cybernetics
https://inquiryintoinquiry.com/2020/03/18/survey-of-cybernetics-%e2%80%a2-1/

The Survey linked above recaps the reading of Ashby's "Cybernetics" up to the present date.

Meanwhile, the inquiry into Pragmatic Truth branched off at another point when a question from Stephen Paul King
demanded an answer in terms of Differential Logic. That point of departure is documented in the following post.

Differential Logic • Comment 4
https://inquiryintoinquiry.com/2020/01/04/differential-logic-%e2%80%a2-comment-4/

This updates the state of the threads linking pragmatic truth, cybernetics, and differential logic. Disentangling them
to any large extent has always been difficult if not impossible, at least for me.

Regards,

Jon

[joseph simpson]

Jon:

Thanks for continuing to provide this excellent commentary.

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 Propositional Calculus • Overview
At: http://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-%e2%80%a2-overview/

|| The most fundamental concept in cybernetics is that of "difference",
|| either that two things are recognisably different or that one thing
|| has changed with time.
||
|| W. Ross Ashby : An Introduction to Cybernetics
|| ( http://pespmc1.vub.ac.be/books/IntroCyb.pdf )

Here's the outline of I sketch I wrote on differential propositional calculi,

which extend propositional calculi by adding 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.

I wrote this as an intuitive introduction to differential logic, which is
my best effort so far at dealing with ancient and persistent problems of
dealing with diversity and mutability in logical terms. I'll be looking
at ways to improve this draft as I serialize it to my blog.

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

Casual Introduction
===================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Casual_Introduction )

Cactus Calculus
===============
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Cactus_Calculus )

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

Formal_Development
==================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Formal_Development )

Elementary Notions
==================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Elementary_Notions )

Special Classes of Propositions
===============================

( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Special_Classes_of_Propositions )

Differential Extensions
=======================
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Differential_Extensions )

Appendices
==========
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Appendices )

References
==========
( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_References )

[Jon Awbrey]

Cf: Differential Propositional Calculus • Discussion 3
https://inquiryintoinquiry.com/2021/03/15/differential-propositional-calculus-discussion-3/

| That mathematics, in common with other art forms, can lead us
| beyond ordinary existence, and can show us something of the
| structure in which all creation hangs together, is no new idea.
| But mathematical texts generally begin the story somewhere in the
| middle, leaving the reader to pick up the thread as best he can.
| Here the story is traced from the beginning.
|
| G. Spencer Brown • Laws of Form

Re: Laws of Form
https://groups.io/g/lawsofform/topic/differential_propositional/81172986
::: Lyle Anderson
(1) https://groups.io/g/lawsofform/message/162
(2) https://groups.io/g/lawsofform/message/163

Dear Lyle,

Charles S. Peirce, with his x-ray vision, revealed for the first time
in graphic detail the mathematical forms structuring our logical organon.
Spencer Brown broadened that perspective in two directions, tracing more
clearly than Peirce’s bare foreshadowings the infrastructure of primary
arithmetic and hypothesizing the existence of imaginary logical values
in a larger algebraic superstructure.

Spencer Brown explored the algebraic extension of the boolean domain B
to a superset equipped with logical imaginaries, operating on analogy
with the algebraic extension of the real line R to the complex plane C.
Seeing as how complex variables are frequently used to model time domains
in physics and engineering, that will continue to be a likely and natural
direction of exploration.

My own work, however, led me in a different direction.
There are many different ways of fruitfully extending
a given domain. Aside from the above class of algebraic
extensions there is a class of differential extensions and,
when that proverbial road diverged, I took the differential one.

Who knows? maybe some where in that undergrowth the roads converge again …

Resources
=========

Differential Logic • Introduction
https://inquiryintoinquiry.com/2008/07/29/differential-logic/

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

[Jon Awbrey]

Cf: Differential Propositional Calculus • Discussion 4
http://inquiryintoinquiry.com/2021/03/20/differential-propositional-calculus-discussion-4/

Re: Differential Propositional Calculus
https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/
::: Discussion 3
https://inquiryintoinquiry.com/2021/03/15/differential-propositional-calculus-discussion-3/

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/thrd1.html#00020
::: Helmut Raulien
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/msg00095.html

<QUOTE HR:>

1. I think I like very much your Cactus Graphs. Meaning that I
am in the process of understanding them, and finding it much
better not to have to draw circles, but lines.

2. Less easy for me is the differential calculus.
Where is the consistency between (x,y) and (x, y, z)?
(x, y) means that x and y are not equal and (x, y, z)
means that one of them is false.
Unequality and truth/falsity for me are two concepts
so different I cannot think them together or see
a consistency between them.

3. What about (w, x, y, z)?

4. Can you give a grammar, like, what does a comma mean,
what do brackets mean, what does writing letters following
each other with an empty space but no comma mean, and so on?

5. Same with Cactus Graphs, though I think, they might be
self-explaining for me — everything is self-explaining,
depending on intellectual capacity, but mine is limited.

</QUOTE>

Dear Helmut,

Many thanks for your detailed comments and questions. They help me see
the places where more detailed explanations are needed. I added numbers
to your points above for ease of reference and possible future reference
in case I can't get to them all in one pass.

Cactus Graphs
=============

I'm glad you found the cactus graphs to your liking.
It was a critical transition for me when I passed from
trees to cacti in my graphing and programming and it
came about by recursively applying a trick of thought
I learned from Peirce himself. These days I call it a
“Meta-Peircean Move” to apply one of Peirce's heuristics
of choice or standard operating procedures to the state
resulting from previous applications. All that makes for
a longer story I made a start at telling in the following
series of posts.

Animated Logical Graphs
https://inquiryintoinquiry.com/2019/05/28/animated-logical-graphs-14/

[Links omitted for this email. See the blog post linked at the top for the list.]

https://inquiryintoinquiry.com/2019/07/31/animated-logical-graphs-27/

Well, the clock in the hall struck time for lunch some time ago,
so I think I'll answer its call, break here, and continue later …

Regards,

Jon

[Jon Awbrey]

Cf: Differential Propositional Calculus • Discussion 5
http://inquiryintoinquiry.com/2021/03/21/differential-propositional-calculus-discussion-5/

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/thrd1.html#00020
::: Helmut Raulien
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/msg00095.html

Re: Ontolog Forum
https://groups.google.com/g/ontolog-forum/c/T8DAe6shKFw
::: Mauro Bertani
https://groups.google.com/g/ontolog-forum/c/T8DAe6shKFw/m/zKS50rJAAwAJ

<QUOTE HR:>

1. I think I like very much your Cactus Graphs. Meaning that I
am in the process of understanding them, and finding it much
better not to have to draw circles, but lines.

2. Less easy for me is the differential calculus.
Where is the consistency between (x,y) and (x, y, z)?
(x, y) means that x and y are not equal and (x, y, z)
means that one of them is false.
Unequality and truth/falsity for me are two concepts
so different I cannot think them together or see
a consistency between them.

3. What about (w, x, y, z)?

</QUOTE>

<QUOTE MB:>

So, if I want to transform a circle into a line
I have to use a function f : Bⁿ → B ? This is the
base of temporal logic? I’m using f : Nⁿ → N.

</QUOTE>

Dear Mauro,

Here I think Helmut is describing the transition from forms
of enclosure on a plane sheet of paper, such as those used
by Peirce and Spencer Brown, to their topological duals in
the form of rooted trees. There is more detail about this
transformation at the following sites.

Logical Graphs • Introduction
https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/

Logical Graphs • Duality : Logical and Topological
https://oeis.org/wiki/Logical_Graphs#Duality:_logical_and_topological

This is the first step in the process of converting planar maps
to graph-theoretic data structures. Further transformations take
us from trees to the more general class of cactus graphs, which
implement a highly efficient family of logical primitives called
“minimal negation operators”. These are described in the following
article.

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

Regards,

Jon

[Jon Awbrey]

Cf: Differential Propositional Calculus • Discussion 6
http://inquiryintoinquiry.com/2021/03/22/differential-propositional-calculus-discussion-6/

Re: Differential Propositional Calculus • Discussions
https://inquiryintoinquiry.com/2021/03/15/differential-propositional-calculus-discussion-3/
https://inquiryintoinquiry.com/2021/03/20/differential-propositional-calculus-discussion-4/
https://inquiryintoinquiry.com/2021/03/21/differential-propositional-calculus-discussion-5/

<QUOTE HR:>

1. I think I like very much your Cactus Graphs. Meaning that I
am in the process of understanding them, and finding it much
better not to have to draw circles, but lines.

2. Less easy for me is the differential calculus.
Where is the consistency between (x,y) and (x, y, z)?
(x, y) means that x and y are not equal and (x, y, z)
means that one of them is false.
Unequality and truth/falsity for me are two concepts
so different I cannot think them together or see
a consistency between them.

3. What about (w, x, y, z)?

</QUOTE>

Dear Helmut,

Table 1 shows the cactus graphs, the corresponding cactus expressions in
“traversal string” or plain text form, their logical meanings under the
“existential interpretation”, and their translations into conventional
notations for a number of common propositional forms. I’ll change
variables to {x, a, b, c} instead of {w, x, y, z} at this point
simply because I already had a Table like that on hand.

As far as the consistency between (a, b) and (a, b, c) goes,
that’s easy enough to see — if exactly one of two boolean
variables is false then the two must have different values.

Out of time for today, so I’ll get to the rest of your questions next time.

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

Regards,

Jon