Differential Propositional Calculus

[Jon Awbrey]

Differential Propositional Calculus • Overview
• https://inquiryintoinquiry.com/2023/11/12/differential-propositional-calculus-overview-a/

❝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

All,

Here's the outline of a 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 the ancient and
persistent problems of treating 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

cc: https://www.academia.edu/community/5NdX3L
cc: https://mathstodon.xyz/@Inquiry/111403787005796924

[Jon Awbrey]

Differential Propositional Calculus • 1
• https://inquiryintoinquiry.com/2023/11/15/differential-propositional-calculus-a-1/

All,

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/2023/11/differential-propositional-calculus-e280a2-figure-1.png

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.

Regards,

Jon

cc: https://www.academia.edu/community/l8gn9L

[Jon Awbrey]

Differential Propositional Calculus • 2
• https://inquiryintoinquiry.com/2023/11/16/differential-propositional-calculus-a-2/

All,

Casual Introduction (cont.)

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

Figure 2. Same Names, Different Habitations
• https://inquiryintoinquiry.files.wordpress.com/2023/11/differential-propositional-calculus-e280a2-figure-2.png

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.

Regards,

Jon

cc: https://www.academia.edu/community/5k1B9L

[Jon Awbrey]

Differential Propositional Calculus • 3
• https://inquiryintoinquiry.com/2023/11/17/differential-propositional-calculus-a-3/

All,

Casual Introduction (cont.)

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/2023/11/differential-propositional-calculus-e280a2-figure-3.png

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 just four distinct
pieces of information which can be expressed in the corresponding
propositional calculus, namely, the constant proposition False,
the negative proposition ¬q, the positive proposition q, and the
constant proposition True.

For example, referring to the points in Figure 1, the constant
proposition False holds of no points, the negative proposition ¬q
holds of a and d, the positive 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”}.

Regards,

Jon

cc: https://www.academia.edu/community/LZb2ZL

[Jon Awbrey]

Differential Propositional Calculus • 4
• https://inquiryintoinquiry.com/2023/11/18/differential-propositional-calculus-a-4/

All,

Casual Introduction (cont.)

Figure 3 extends the basis of description for the
space X to a set of two qualities {q, dq} and the
corresponding terms of description to an alphabet
of two symbols {“q”, “dq”}.

Figure 3. Back, To The Future
• https://inquiryintoinquiry.files.wordpress.com/2023/11/differential-propositional-calculus-e280a2-figure-3.png

Any propositional calculus over two basic propositions
allows for the expression of 16 propositions all together.
Salient among those propositions in the present setting
are the four which single out the individual sample points
at the initial moment of observation. Table 4 lists the
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

Regards,

Jon

cc: https://www.academia.edu/community/lQNG05

[Jon Awbrey]

Differential Propositional Calculus • 5
• https://inquiryintoinquiry.com/2023/11/19/differential-propositional-calculus-a-5/

All,

Casual Introduction (concl.)

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

Regards,

Jon

cc: https://www.academia.edu/community/5ANwNV

[Jon Awbrey]

Differential Propositional Calculus • 6
• https://inquiryintoinquiry.com/2023/11/20/differential-propositional-calculus-a-6/

All,

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 sequence of propositional expressions (e_1, e_2, ..., e_k) is
taken to mean exactly one of the propositions e_1, e_2, ..., e_k is false,
in other words, their minimal negation is true.

• A concatenated sequence of propositional expressions e_1 e_2 ... e_k is
taken to mean every one of the propositions e_1, e_2, ..., e_k is true,
in other words, their logical conjunction is true.

Table 6. Syntax and Semantics of a Calculus for Propositional Logic
• https://inquiryintoinquiry.files.wordpress.com/2022/10/syntax-and-semantics-of-a-calculus-for-propositional-logic-4.0.png

All other propositional connectives can be obtained through
combinations of the above 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 ε or λ 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. 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 —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Zeroth Order Logic
• https://oeis.org/wiki/Zeroth_order_logic

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

Regards,

Jon

cc: https://www.academia.edu/community/5ANGoV

[Jon Awbrey]

Differential Propositional Calculus • 7
• https://inquiryintoinquiry.com/2023/11/21/differential-propositional-calculus-a-7/

All,

Note. Please see the blog post linked above for the proper
formats of the notations used below, as they depend on numerous
typographical distinctions lost in the following transcript.

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 assume the signs are collected in the form of
a finite alphabet, ‡A‡ = {“a_1”, ..., “a_n”}. The signs
are interpreted as denoting logical features, for example,
properties of objects in the universe of discourse or simple
propositions about those objects. Corresponding to the alphabet
‡A‡ there is then a set of logical features, †A† = {a_1, ..., a_n}.

A set of logical features †A† = {a_1, ..., a_n} affords a basis
for generating an n-dimensional universe of discourse, written
A• = [†A†] = [a_1, ..., a_n]. It is useful to consider a universe
of discourse as a categorical object incorporating both the set of
points A = <a_1, ..., a_n> and the set of propositions A↑ = {f : A→B}
implicit with the ordinary picture of a venn diagram on n features.

Accordingly, the universe of discourse A• may be regarded as
an ordered pair (A, A↑) having the type (Bⁿ, (Bⁿ → B)) and this
last type designation may be abbreviated as Bⁿ +→ B, or even more
succinctly as [Bⁿ]. For convenience, the data type of a finite
set on n elements may be indicated by either one of the equivalent
notations, [n] or *n*.

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

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Functional Conception of Propositional Calculus
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#A_Functional_Conception_of_Propositional_Calculus

Regards,

Jon

cc: https://www.academia.edu/community/54Gadl

[Jon Awbrey]

Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2023/11/23/differential-propositional-calculus-a-8/

All,

Formal Development (cont.)

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] qualified by the
logical features a_1, ..., a_n is a set A plus the set of all
functions from the space A to the boolean domain B = {0, 1}.
There are 2ⁿ elements in A, often pictured as the cells of
a venn diagram or the nodes of a hypercube. There are 2^(2ⁿ)
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.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Functional Conception of Propositional Calculus
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#A_Functional_Conception_of_Propositional_Calculus

Qualitative Logic and Quantitative Analogy
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Qualitative_Logic_and_Quantitative_Analogy

Formal Terms and Flexible Types
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Philosophy_of_Notation_:_Formal_Terms_and_Flexible_Types


Regards,

Jon

cc: https://www.academia.edu/community/ldJmKV

[Jon Awbrey]

Differential Propositional Calculus • 9
• https://inquiryintoinquiry.com/2023/11/24/differential-propositional-calculus-a-9/

All.

Note. As always, please see the blog post linked above
for the proper mathematical formatting.

Special Classes of Propositions —

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^(2ⁿ) propositions in [a_1, ..., a_n] are several families
numbering 2ⁿ 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 Bⁿ 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 ℓ : Bⁿ → B may be written as sums.
• https://inquiryintoinquiry.files.wordpress.com/2020/02/linear-propositions-may-be-written-as-sums.png

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

Singular Propositions s : Bⁿ → B may be written as products.
• 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, ¬a_1 ∧ ¬a_2 ∧ ¬a_3.

The basic propositions a_i : Bⁿ → 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† = {a_1, ..., a_n}. A singular proposition
with respect to the basis †A† will not remain singular if †A†
is extended by a number of new and independent features. Even if
one keeps to the original set of pairwise options {a_i}∪{¬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.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Special Classes of Propositions
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Special_Classes_of_Propositions

Basis Relativity and Type Ambiguity
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Basis_Relativity_and_Type_Ambiguity

Regards,

Jon

cc: https://www.academia.edu/community/V1nYdL

[Jon Awbrey]

Differential Propositional Calculus • 10
• http://inquiryintoinquiry.com/2023/11/25/differential-propositional-calculus-a-10/

All,

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 —

The linear propositions ℓ : Bⁿ → B 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 on Three Variables
• 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.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

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

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Special Classes of Propositions
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Special_Classes_of_Propositions

Regards,

Jon

cc: https://www.academia.edu/community/laDOpL

[Jon Awbrey]

Differential Propositional Calculus • 11
• https://inquiryintoinquiry.com/2023/11/26/differential-propositional-calculus-a-11/

All,

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 —

The positive propositions p : Bⁿ → B 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 2³ = 8 positive propositions, taking the shapes shown in Figure 9.

Figure 9. Positive Propositions on Three Variables
• 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.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

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

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Special Classes of Propositions
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Special_Classes_of_Propositions

Regards,

Jon

cc: https://www.academia.edu/community/ln1AjL

[Jon Awbrey]

Differential Propositional Calculus • 12
• https://inquiryintoinquiry.com/2023/11/27/differential-propositional-calculus-a-12/

All,

Special Classes of Propositions (concl.)

Last and literally least in extent, we examine the family of
singular propositions in a 3‑dimensional universe of discourse.

In our model of propositions as mappings from a universe of discourse X
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 —

The singular propositions s : Bⁿ → B 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 2³ = 8 singular propositions. Their venn diagrams are shown
in Figure 10.

Figure 10. Singular Propositions on Three Variables
• 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)

Regards,

Jon

cc: https://www.academia.edu/community/L6WJ8l

[Jon Awbrey]

Differential Propositional Calculus • 13
• http://inquiryintoinquiry.com/2023/11/28/differential-propositional-calculus-a-13/

All,

Note. Please see the blog post linked above for the proper formats

of the notations used below, as they depend on many typographical

distinctions lost in the following transcript.

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.

The initial alphabet ‡A‡ = {“a₁”, …, “aₙ”} is extended by a first order
differential alphabet d‡A‡ = {“da₁”, …, “daₙ”} resulting in a first order
extended alphabet E‡A‡ defined as follows.

• E‡A‡ = ‡A‡ ∪ d‡A‡ = {“a₁”, …, “aₙ”, “da₁”, …, “daₙ”}.

The initial basis †A† = {a₁, …, aₙ} is extended by a first order
differential basis d†A† = {da₁, …, daₙ} resulting in a first order
extended basis E†A† defined as follows.

• E†A† = †A† ∪ d†A† = {a₁, …, aₙ, da₁, …, daₙ}.

The initial space A = ⟨a₁, …, aₙ⟩ is extended by a first order
differential space or tangent space dA = ⟨da₁, …, daₙ⟩ at each point
of A, resulting in a first order extended space EA defined as follows.

• EA = A × dA = ⟨E†A†⟩ = ⟨†A† ∪ d†A†⟩ = ⟨a₁, …, aₙ, da₁, …, daₙ⟩.

Finally, the initial universe A• = [a₁, …, aₙ] is extended by a first order
differential universe or tangent universe dA• = [da₁, …, daₙ] at each point
of A•, resulting in a first order extended universe EA• defined as follows.

• EA• = [E†A†] = [†A† ∪ d†A†] = [a₁, …, aₙ, da₁, …, daₙ].

This gives EA• a type defined as follows.

• [Bⁿ × Dⁿ] = (Bⁿ × Dⁿ +→ B) = (Bⁿ × Dⁿ, Bⁿ × Dⁿ → B).

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
propositions f : EA → B, we arrive at the foothills of differential logic.

Regards,

Jon

cc: https://www.academia.edu/community/lOvrKL

[Jon Awbrey]

Differential Propositional Calculus • 14
• http://inquiryintoinquiry.com/2023/11/30/differential-propositional-calculus-a-14/

All,

Differential Extensions —

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

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

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

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Differential Extension of Propositional Calculus
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#A_Differential_Extension_of_Propositional_Calculus


Regards,

Jon

cc: https://www.academia.edu/community/LppYYL

[Jon Awbrey]

Differential Propositional Calculus • 15
• https://inquiryintoinquiry.com/2023/12/04/differential-propositional-calculus-15/

Fire over water:
The image of the condition before transition.
Thus the superior man is careful
In the differentiation of things,
So that each finds its place.

— I Ching ䷿ Hexagram 64

Differential Extension of Propositional Calculus —

This much preparation is enough to begin introducing my
subject, if I excuse myself from giving full arguments
for my definitional choices until a later stage.

To express the goal in a turn of phrase, the aim is to
develop a differential theory of qualitative equations,
one which can parallel the application of differential
geometry to dynamical systems. The idea of a tangent
vector is key to the work and a major goal is to find
the right logical analogues of tangent spaces, bundles,
and functors. The strategy is taken of looking for the
simplest versions of those constructions which can be
discovered within the realm of propositional calculus,
so long as they serve to fill out the general theme.

Reference —

Wilhelm, R., and Baynes, C.F. (trans.), The I Ching,
or Book of Changes, Foreword by C.G. Jung, Preface
by H. Wilhelm, 3rd edition, Bollingen Series XIX,
Princeton University Press, Princeton, NJ, 1967.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Differential Extension of Propositional Calculus
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#A_Differential_Extension_of_Propositional_Calculus


Regards,

Jon

cc: https://www.academia.edu/community/l7pvk5

[Jon Awbrey]

Differential Propositional Calculus • 16
• https://inquiryintoinquiry.com/2023/12/05/differential-propositional-calculus-16/

All,

Note. The text below employs ‡A‡ for Fraktur or Gothic letter A
and †A† for Calligraphic or Script letter A, but see the
blog post linked above for the proper character formats.

Differential Propositions • Qualitative Analogues of Differential Equations —

The differential extension of a universe of discourse [†A†]
is constructed by extending its initial alphabet ‡A‡ to include
a set of symbols for “differential features”, or “basic changes”
capable of occurring in [†A†]. The added symbols are taken to
denote primitive features of change, qualitative attributes of
motion, or propositions about how items in the universe of
discourse may change or move in relation to features noted
in the original alphabet.

With that in mind we define the corresponding “differential alphabet”
or “tangent alphabet” d‡A‡ = {“da₁”, …, “daₙ”}, in principle just an
arbitrary alphabet of symbols, disjoint from the initial alphabet
‡A‡ = {“a₁”, …, “aₙ”} and given the meanings just indicated.

In practice the precise interpretation of the symbols in d‡A‡ is
conceived to be changeable from point to point of the underlying
space A. Indeed, for all we know, the state space A might well
be the state space of a language interpreter, one concerned with
the idiomatic meanings of the dialect generated by ‡A‡ and d‡A‡.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Differential Propositions • Qualitative Analogues of Differential Equations
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Differential_Propositions_:_Qualitative_Analogues_of_Differential_Equations


Regards,

Jon

cc: https://www.academia.edu/community/LY90rL

[Jon Awbrey]

Differential Propositional Calculus • 17
• https://inquiryintoinquiry.com/2023/12/06/differential-propositional-calculus-17/

Differential Propositions • Tangent Spaces —

The “tangent space” to A at one of its points x, sometimes
written Tₓ(A), takes the form dA = ⟨d†A†⟩ = ⟨da₁, …, daₙ⟩.
Strictly speaking, the name “cotangent space” is probably
more correct for this construction but since we take up
spaces and their duals in pairs to form our universes of
discourse it allows our language to be pliable here.

Proceeding as we did with the base space A, the tangent space dA
at a point of A may be analyzed as the following product of distinct
and independent factors.

• dA = ∏ dAₖ = dA₁ × … × dAₙ.

Each factor dAₖ is a set consisting of two differential propositions,
dAₖ = {(daₖ), daₖ}, where (daₖ) is a proposition with the logical value
of ¬daₖ. Each component dAₖ has the type B, operating under the ordered
correspondence {(daₖ), daₖ} ≅ {0, 1}. A measure of clarity is achieved,
however, by acknowledging the differential usage with a superficially
distinct type D, whose sense may be indicated as follows.

• D = {(daₖ), daₖ} = {same, different} = {stay, change} = {stop, step}.

Viewed within a coordinate representation, spaces of type Bⁿ and Dⁿ may
appear to be identical sets of binary vectors, but taking a view at that
level of abstraction would be like ignoring the qualitative units and the
diverse dimensions that distinguish position and momentum, or the different
roles of quantity and impulse.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Differential Propositions • Tangent Spaces
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Tangent_Spaces

Regards,

Jon

cc: https://www.academia.edu/community/5Rkm2V

[Jon Awbrey]

Differential Propositional Calculus • 18
• https://inquiryintoinquiry.com/2023/12/08/differential-propositional-calculus-18/

The Extended Universe of Discourse —

The “extended basis” E†A† of a universe of discourse †A†
is formed by taking the initial basis †A† together with the
differential basis d†A†. Thus we have the following formula.

• E†A† = †A† ∪ d†A† = {a₁, …, aₙ, da₁, …, daₙ}.

This supplies enough material to construct the “differential
extension” EA of the space A, also called the “tangent bundle”
of A, in the following fashion.

• EA = ⟨E†A†⟩ = ⟨†A† ∪ d†A†⟩ = ⟨a₁, …, aₙ, da₁, …, daₙ⟩

and also

• EA = A × dA = A₁ × … × Aₙ × dA₁ × … × dAₙ.

That gives EA the type Bⁿ × Dⁿ.

Finally, the “extended universe” EA• = [E†A†] is the full
collection of points and functions, or interpretations and
propositions, based on the extended set of features E†A†,
a fact summed up in the following notation.

• EA• = [E†A†] = [a₁, …, aₙ, da₁, …, daₙ].

That gives the extended universe EA• the following type.

• (Bⁿ × Dⁿ +→ B) = (Bⁿ × Dⁿ, (Bⁿ × Dⁿ → B)).

A proposition in the extended universe [E†A†] is called
a “differential proposition” and forms the logical analogue
of a system of differential equations, constraints, or relations
in ordinary calculus.

With these constructions, the differential extension EA and
the space of differential propositions (EA → B), we arrive at
the launchpad of our space explorations.

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

The adjective “differential” or “tangent” is systematically attached
to every construct based on the differential alphabet d‡A‡, taken by
itself. In like fashion, the adjective “extended” or the substantive
“bundle” is systematically attached to any construct associated with
the full complement of 2n features.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

The Extended Universe of Discourse
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Extended_Universe

Regards,

Jon

cc: https://www.academia.edu/community/ln3w35

[Jon Awbrey]

Differential Propositional Calculus • 19
• https://inquiryintoinquiry.com/2023/12/10/differential-propositional-calculus-19/

Failing to fetch me at first keep encouraged,
Missing me one place search another,
I stop some where waiting for you

— Walt Whitman • Leaves of Grass

Life on Easy Street —

The finite character of the extended universe [E†A†] makes
the task of solving differential propositions relatively
straightforward. The solution set of the differential
proposition q : EA → B is the set of models q⁻¹(1) in EA.
Finding all models of q, the extended interpretations in
EA which satisfy q, can be carried out by a finite search.

Being in possession of complete algorithms for propositional calculus
modeling, theorem checking, and theorem proving makes the analytic task
fairly simple in principle, even if the question of efficiency in the
face of arbitrary complexity remains another matter entirely.

The NP‑completeness of propositional satisfiability may weigh against
the prospects of a single efficient algorithm capable of covering the
whole space [E†A†] with equal facility but there appears to be much
room for improvement in classifying special forms and in developing
algorithms tailored to their practical processing.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1

Differential Logic • Life on Easy Street
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Easy_Street

Regards,

Jon

cc: https://www.academia.edu/community/LGZ3eV

[Jon Awbrey]

Differential Propositional Calculus • 20
• https://inquiryintoinquiry.com/2023/12/11/differential-propositional-calculus-20/

Back to the Beginning • Exemplary Universes —

❝I would have preferred to be enveloped in words,
borne way beyond all possible beginnings.❞

— Michel Foucault • The Discourse on Language

To anchor our understanding of differential logic let's look at
how the various concepts apply in the simplest possible concrete
cases, where the initial dimension is only 1 or 2. In spite of
the obvious simplicity of these cases it is possible to observe
how central difficulties of the subject begin to arise already
at this stage.

Resources —

Differential Logic and Dynamic Systems

• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Exemplary Universes
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Exemplary_Universes

Regards,

Jon

cc: https://www.academia.edu/community/ld9Pe5

[Jon Awbrey]

Differential Propositional Calculus • 21
• https://inquiryintoinquiry.com/2023/12/12/differential-propositional-calculus-21/

A One‑Dimensional Universe —

There was never any more inception than there is now,
Nor any more youth or age than there is now;
And will never be any more perfection than there is now,
Nor any more heaven or hell than there is now.

— Walt Whitman • Leaves of Grass

Let †X† = {A} be a logical basis containing one boolean variable or
logical feature A. The basis element A may be regarded as a simple
proposition or coordinate projection A : B → B. Corresponding to the
basis †X† = {A} is the alphabet ‡X‡ = {“A”} which serves whenever we
need to make explicit mention of the symbols used in our formulas and
representations.

The space X = ⟨A⟩ = {¬A, A} of points (cells, vectors, interpretations)
has cardinality 2ⁿ = 2¹ = 2 and is isomorphic to B = {0, 1}. Moreover,
X may be identified with the set of singular propositions {s : B → B}.

The space of linear propositions X* = {ℓ : B → B} = {0, A} is
algebraically dual to X and also has cardinality 2. Here, “0”
is interpreted as denoting the constant function 0 : B → B,
amounting to the linear proposition of rank 0, while A is
the linear proposition of rank 1.

Last but not least we have the positive propositions {p : B → B}
= {A, 1} of rank 1 and 0, respectively, where “1” is understood
as denoting the constant function 1 : B → B.

All told there are 2^{2ⁿ} = 2^{2¹} = 4 propositions in the
universe of discourse [†X†], collectively forming the set
X↑ = {f : X→B} = {0, ¬A, A, 1}.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • One-Dimensional Universe
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#One_Dim_U

Regards,

Jon

cc: https://www.academia.edu/community/lKDXw5

[Jon Awbrey]

Differential Propositional Calculus • 22
• https://inquiryintoinquiry.com/2023/12/14/differential-propositional-calculus-22/

A One-Dimensional Universe (cont.)

The “first order differential extension” of †X† is
E†X† = {x₁, dx₁} = {A, dA}. If the feature A is
interpreted as applying to some object or state
then the feature dA may be taken as an attribute
of the same object or state which tells it is
changing “significantly” with respect to the
property A, as if it bore an “escape velocity”
with respect to the state A. In practice,
differential features acquire their meaning
through a class of “temporal inference rules”.

For example, relative to a frame of observation to be left
implicit for now, if A and dA are true at a given moment,
it would be reasonable to assume (A) = ¬A will be true in
the next moment of observation. Taken all together we have
the fourfold scheme of inference shown below.

Differential Inference Rules
• https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-inference-rules-a.png

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • One-Dimensional Universe
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#One_Dim_U

Regards,

Jon

cc: https://www.academia.edu/community/LpJeY5

[Jon Awbrey]

Differential Propositional Calculus • 23
• https://inquiryintoinquiry.com/2023/12/15/differential-propositional-calculus-23/

A One-Dimensional Universe (concl.)

❝The clock indicates the moment . . . . but what does
eternity indicate?❞

— Walt Whitman • Leaves of Grass

It might be thought an independent time variable needs to be
brought in at this point but it is an insight of fundamental
importance to recognize the idea of process is logically prior
to the notion of time. A time variable is a reference to a “clock”
— a canonical, conventional process accepted or established as
a standard of measurement but in essence no different than any
other process.

This raises the question of how different subsystems in a more
global process can be brought into comparison and what it means
for one process to serve the function of a local standard for
others. But inquiries of that order serve but to wrap up puzzles
in further riddles and are obviously too involved to be handled at
our current level of approximation.

Observe how the secular inference rules, used by themselves,
involve a loss of information, since nothing in them tells
whether the momenta {¬dA, dA} are changed or unchanged in
the next moment. To know that one would have to determine
d²A, and so on, pursuing an infinite regress. In order to
rest with a finitely determinate system it is necessary to
make an infinite assumption, for example, that dⁿA = 0 for
all n greater than some fixed value M. Another way to escape
the regress is through the provision of a dynamic law, in
typical form making higher order differentials dependent
on lower degrees and estates.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • One-Dimensional Universe
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#One_Dim_U

Regards,

Jon

cc: https://www.academia.edu/community/lzJejl

[Jon Awbrey]

Differential Propositional Calculus • 24
• https://inquiryintoinquiry.com/2023/12/16/differential-propositional-calculus-24/

Example 1. A Square Rigging —

| Urge and urge and urge,
| Always the procreant urge of the world.

| — Walt Whitman • Leaves of Grass

Returning to the universe of discourse based on
a single feature A, suppose we are given the
initial condition

• A = dA

and the second order differential law

• d²A = (A)

where we use the notation (A) for ¬A.

Since the equation A = dA is logically equivalent to
the disjunction A dA ∨ (A)(dA) there are two possible
trajectories, as shown in the following Table.

Table. A Pair of Commodious Trajectories
• https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-commodious-trajectories.png

In either case the state A (dA)(d²A) is a stable attractor
or terminal condition for both starting points.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • A Square Rigging
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Example_1

Regards,

Jon

cc: https://www.academia.edu/community/LgJXvV

[Jon Awbrey]

Differential Propositional Calculus • 25
• https://inquiryintoinquiry.com/2023/12/17/differential-propositional-calculus-25/

Example 1. A Square Rigging (cont.)

Because the initial space X = ⟨A⟩ is one‑dimensional we can easily
fit the second order extension E²X = ⟨A, dA, d²A⟩ within the compass
of a single venn diagram, charting the pair of converging trajectories
as shown in the following Figure.

Example 1. The Anchor
• https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-anchor.png

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • A Square Rigging
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Example_1

Regards,

Jon

cc: https://www.academia.edu/community/LZvM3L

[Jon Awbrey]

Differential Propositional Calculus • 26
• https://inquiryintoinquiry.com/2023/12/18/differential-propositional-calculus-26/

Example 1. A Square Rigging (cont.)

If we eliminate from view the regions of E²X ruled out by the dynamic law
d²A = (A) then what remains is the quotient structure shown in the following
Figure. The picture makes it easy to see how the dynamically allowable portion
of the universe is partitioned between the respective holdings of A and d²A.
As it happens, the fact might have been expressed “right off the bat” by an
equivalent formulation of the differential law, one which uses the exclusive
disjunction to state the law as (A, d²A).

Example 1. The Tiller
• https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-tiller.png

What we have achieved in this example is to give a differential description
of a simple dynamic process. We did this by embedding a directed graph,
representing the state transitions of a finite automaton, in the share
of a boolean lattice or n‑cube cut out by nullifying all the regions
the dynamics outlaws.

With growth in the dimensions of our contemplated universes it becomes
essential, both for human comprehension and for computer implementation,
that dynamic structures of interest be represented not actually, by
acquaintance, but virtually, by description. In our present study we
are using the language of propositional calculus to express the relevant
descriptions, and to grasp the structures embodied in subsets of n‑cubes
without being forced to actualize all their points.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • A Square Rigging
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Example_1

Regards,

Jon

cc: https://www.academia.edu/community/lKgQJl

[Jon Awbrey]

Differential Propositional Calculus • 27
• https://inquiryintoinquiry.com/2023/12/19/differential-propositional-calculus-27/

Commentary On Small Models —

One reason for engaging in our present order of extremely reduced
but explicitly controlled case study is to throw light on the general
study of languages, formal and natural, in their full array of syntactic,
semantic, and pragmatic aspects. Propositional calculus is one of the
last points of departure where it is possible to see that trio of aspects
interacting in a non‑trivial way without being immediately and totally
overwhelmed by the complexity they generate. Often that complexity leads
investigators of formal and natural languages to adopt the strategy of
focusing on a single aspect and abandoning all hope of understanding the
whole, whether it’s the still living natural language or the dynamics of
inquiry crystallized in formal logic.

From the perspective I find most useful here, a language is a
syntactic system designed or evolved in part to express a set
of descriptions. When the explicit symbols of a language have
extensions in its object world which are actually infinite, or
when the implicit categories and generative devices of a linguistic
theory have extensions in its subject matter which are potentially
infinite, then the finite characters of terms, statements, arguments,
grammars, logics, and rhetorics force a surplus intension to color all
its symbols and functions, across the spectrum from object language to
metalinguistic reflection.

In the aphorism of W. von Humboldt often cited by Chomsky, language
requires “the infinite use of finite means”. That is necessarily
true when the extensions are infinite, when the referential symbols
and grammatical categories of a language possess infinite sets of
models and instances. But it also voices a practical truth when the
extensions, though finite at every stage, tend to grow at exponential
rates.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Commentary On Small Models
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#COSM

Regards,

Jon

cc: https://www.academia.edu/community/l8m8z5

[Jon Awbrey]

Differential Propositional Calculus • 28
• https://inquiryintoinquiry.com/2023/12/20/differential-propositional-calculus-28/

Commentary On Small Models —

The consequence of dealing with “practically infinite extensions” becomes crucial in building neural network systems
capable of learning and adapting, since the adaptive competence of any intelligent system is limited to the objects and
domains it is able to represent. If we seek to design systems which operate intelligently with the full deck of
propositions dealt by intact universes of discourse then we must supply those systems with succinct representations and
efficient transformations in that domain.

Beyond the ability to learn and adapt, which taken at the ebb so often devolves into bare conformity and confirmation
bias, the ability to inquire and reason makes even more demands on propositional representation. The project of
constructing inquiry driven systems forces us to contemplate the level of generality embodied in logical propositions —
we can see this because the progress of inquiry is driven by evident discrepancies among expectations, intentions, and
observations, and each of those components of systematic knowledge takes on the fully generic character of an empirical
summary or an axiomatic theory.

A compression scheme by any other name is a symbolic representation — and this is what the differential extension of
propositional calculus is intended to supply. But why is this particular program of mental calisthenics worth carrying
out in general?

The provision of a uniform logical framework for describing time‑evolving systems makes the task of understanding
complex systems easier than it would otherwise be when we try to tackle each new system de novo, “from scratch” as we
say. Having a uniform medium ready to hand helps both in looking for invariant representations of individual cases and
also in finding points of comparison among diverse structures otherwise appearing to be isolated systems. All this goes
to facilitate the search for compact knowledge, to apply what is learned from individual cases to the general realm.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Commentary On Small Models
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#COSM

Regards,

Jon

cc: https://www.academia.edu/community/Lb7Ygl

[Jon Awbrey]

Differential Propositional Calculus • 29
• https://inquiryintoinquiry.com/2023/12/21/differential-propositional-calculus-29/

Back to the Feature —

❝I guess it must be the flag of my disposition, out of hopeful
green stuff woven.❞

— Walt Whitman • Leaves of Grass

Let's assume the sense intended for differential features is
well enough established in the intuition for now that we may
continue outlining the structure of the differential extension
[E†X†] = [A, dA]. Over the extended alphabet E†X† = {x₁, dx₁}
= {A, dA} of cardinality 2ⁿ = 2 we generate the set of points
EX of cardinality 2²ⁿ = 4 which bears the following chain of
equivalent descriptions.

• EX = ⟨A, dA⟩
= {(A), A} × {(dA), dA}
= {(A)(dA), (A)dA, A(dA), A∙dA}.

The space EX may be given the nominal type B×D, at root isomorphic
to B×B = B². An element of EX may be regarded as a “disposition at
a point” or a “situated direction”, in effect, a singular mode of
change occurring at a single point in the universe of discourse.
In practice the modality of those changes may be interpreted in
various ways, for example, as expectations, intentions, or
observations with respect to the behavior of a system.

To complete the construction of the extended universe of discourse
EX• = [A, dA] the basic dispositions in EX need to be extended to
the full set of differential propositions EX↑ = {g : EX → B}, each
of type B×D → B. There are 2^{2²ⁿ} = 16 propositions in EX↑, as
detailed in the following Table.

Table. Differential Propositions
• https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-differential-propositions.png

Aside from changing the names of variables and shuffling the order of rows,
the Table follows the format previously used for boolean functions of two
variables. The rows are grouped to reflect natural similarity classes
holding among the propositions. In a future discussion the classes will
be given additional explanation and motivation as the orbits of a certain
transformation group acting on the set of 16 propositions. Notice that
four of the propositions, in their logical expressions, resemble those
given in the table for X↑. Thus the first set of propositions {fₖ} is
automatically embedded in the present set {gₘ} and the corresponding
inclusions are indicated at the far left margin of the Table.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Back to the Feature
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Feature

Regards,

Jon

cc: https://www.academia.edu/community/5kzK0l

[Jon Awbrey]

Differential Propositional Calculus • 30
• https://inquiryintoinquiry.com/2023/12/22/differential-propositional-calculus-30/

Tacit Extensions —

❝I would really like to have slipped imperceptibly into this lecture,
as into all the others I shall be delivering, perhaps over the years ahead.❞

— Michel Foucault • The Discourse on Language

Re: Table of Differential Propositions
• https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-differential-propositions.png

In viewing the previous Table of Differential Propositions
it is important to notice the subtle distinction in type
between a function fₖ : X → B and its inclusion as a function
gₘ : EX → B, even though they share the same logical expression.

Naturally, we want to maintain the logical equivalence of expressions
representing the same proposition while appreciating the full diversity
of a proposition's functional and typical representatives. Both perspectives,
and all the levels of abstraction extending through them, have their reasons,
as will develop in time.

Because this special circumstance points to a broader theme,
it's a good idea to discuss it more generally. Whenever there
arises a situation like that above, where one basis †X† is a
subset of another basis †Y†, we say any proposition f : X → B
has a “tacit extension” to a proposition εf : Y → B and we say
the space (X → B) has an “automatic embedding” within the space
(Y → B).

The “tacit extension operator” ε is defined in such a way that
εf puts the same constraint on the variables of †X† within †Y†
as the proposition f initially put on †X†, while it puts no
constraint on the variables of †Y† beyond †X†, in effect,
conjoining the two constraints.

Indexing the variables as †X† = {x₁, …, xₙ} and
†Y† = {x₁, …, xₙ, …, xₙ₊ₖ} the tacit extension from
†X† to †Y† may be expressed by the following equation.

• εf(x₁, …, xₙ, …, xₙ₊ₖ) = f(x₁, …, xₙ).

On formal occasions, such as the present context of definition,
the tacit extension from †X† to †Y† is explicitly symbolized by
the operator ε : (X→B)→(Y→B), where the bases †X† and †Y† are set
in context, but it's normally understood the “ε” may be silent.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Tacit Extensions
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Tacit_Extensions

Regards,

Jon

cc: https://www.academia.edu/community/LpNd75

[Jon Awbrey]

Differential Propositional Calculus • 31
• https://inquiryintoinquiry.com/2023/12/23/differential-propositional-calculus-31/

Tacit Extensions —

Re: Table of Differential Propositions
• https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-differential-propositions.png

Returning to the Table of Differential Propositions, let's
examine how the general concept of a tacit extension applies
to the differential extension of a one‑dimensional universe
of discourse, where †X† = {A} and †Y† = E†X† = {A, dA}.

Each proposition fₖ : X → B has a canonical expression eₖ
in the set {0, (A), A, 1}. The tacit extension εfₖ : EX → B
may then be expressed as a logical conjunction of two factors,
fₖ = eₖ∙τ, where τ is a logical tautology using all the variables
of †Y† − †X†. The following Table shows how the tacit extensions
εfₖ of the propositions fₖ may be expressed in terms of the extended
basis {A, dA}.

Table. Tacit Extension of [A] to [A, dA]
• https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-tacit-extensions.png

In its bearing on the singular propositions over a universe of
discourse X the above analysis has an interesting interpretation.
The tacit extension takes us from thinking about a particular state,
like A or (A), to considering the collection of outcomes, the outgoing
changes or singular dispositions, springing or stemming from that state.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Tacit Extensions
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Tacit_Extensions

Regards,

Jon

cc: https://www.academia.edu/community/laNmp5

[Jon Awbrey]

Differential Propositional Calculus • 32
• https://inquiryintoinquiry.com/2023/12/26/differential-propositional-calculus-32/

Example 2. Drives and Their Vicissitudes —

❝I open my scuttle at night and see the far‑sprinkled systems,
And all I see, multiplied as high as I can cipher, edge but
the rim of the farther systems.❞

— Walt Whitman • Leaves of Grass

Before we leave the one‑feature case let's look at a more
substantial example, one which illustrates a general class
of curves through the extended feature spaces and provides
an opportunity to discuss important themes concerning their
structure and dynamics.

As before let †X† = {x₁} = {A}. The discussion to follow considers
a class of trajectories having the property that dⁱA = 0 for all i
greater than a fixed value n and indulges in the use of a picturesque
vocabulary to describe salient classes of those curves.

Given the above finite order condition, there is a highest order
non‑zero difference dⁿA exhibited at each point of any trajectory
one may consider. With respect to any point of the corresponding
curve let us call that highest order differential feature dⁿA the
“drive” at that point. Curves of constant drive dⁿA are then
referred to as “n‑th‑gear curves”.

Note. The fact that a difference calculus can be developed for
boolean functions is well known and was probably familiar to Boole,
who was an expert in difference equations before he turned to logic.
And of course there is the strange but true story of how the Turin
machines of the 1840s prefigured the Turing machines of the 1940s.
At the very outset of general purpose mechanized computing we find
the motive power driving the Analytical Engine of Babbage, the
kernel of an idea behind all of his wheels, was exactly his notion
that difference operations, suitably trained, can serve as universal
joints for any conceivable computation.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Drives and Their Vicissitudes
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Drives

Regards,

Jon

cc: https://www.academia.edu/community/lQYdql

[Jon Awbrey]

Differential Propositional Calculus • 33
• https://inquiryintoinquiry.com/2023/12/27/differential-propositional-calculus-33/

Example 2. Drives and Their Vicissitudes (cont.)

Expressed in terms of “drives” and “gears” our next Example
may be described as the family of 4‑th‑gear curves in the
fourth extension E⁴X = ⟨A, dA, d²A, d³A, d⁴A⟩. Those are the
trajectories generated subject to the dynamic law d⁴A = 1,
where it's understood all higher differences are equal to 0.

Since d⁴A and all higher differences dⁿA are fixed, the state
vectors vary only with respect to their projections as points
of E³X = ⟨A, dA, d²A, d³A⟩. Thus there is just enough space in
a planar venn diagram to plot all the orbits and to show how they
partition the points of E³X. It turns out there are exactly two
possible orbits, of eight points each, as shown in the following
Figure.

Example 2. Fourth Gear Orbits
• https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-fourth-gear-orbits.gif

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Drives and Their Vicissitudes
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Drives

Regards,

Jon

cc: https://www.academia.edu/community/VXAx1L

[Jon Awbrey]

Differential Propositional Calculus • 34
• https://inquiryintoinquiry.com/2023/12/28/differential-propositional-calculus-34/

Example 2. Drives and Their Vicissitudes (cont.)

With a little thought it is possible to devise a canonical
indexing scheme for the states in differential logical systems.
A scheme of that order allows for comparing changes of state
in universes of discourse that weigh in on different scales
of observation.

To that purpose, let us index the states q in E^m X with the dyadic
rationals (or the binary fractions) in the half‑open interval [0, 2).
Formally and canonically, a state q_r is indexed by a fraction r = s/t
whose denominator is the power of two t = 2^m and whose numerator is
a binary numeral formed from the coefficients of state in a manner
to be described next.

The “differential coefficients” of a state q are just the values d^k (q)
for k = 0 to m, where d⁰A is defined as being identical to A. To form
the binary index d₀.d₁ … d_m of the state q the coefficient d^k A(q) is
read off as the binary digit d_k associated with the place value 2^{-k}.
Expressed in algebraic formulas, the rational index r of the state q is
given by the following equivalent formulations.

Differential Coefficients • State Coordinates
•
https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-differential-coefficients-e280a2-state-coordinates.png

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Drives and Their Vicissitudes
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Drives

Regards,

Jon

cc: https://www.academia.edu/community/5kY1NL

[Jon Awbrey]

Differential Propositional Calculus • 35
• https://inquiryintoinquiry.com/2023/12/29/differential-propositional-calculus-35/

Example 2. Drives and Their Vicissitudes (concl.)

Applied to the example of 4‑th‑gear curves, the indexing scheme
results in the data of the next two Tables, showing one period
for each orbit.

Fourth Gear Orbits 1 and 2
•
https://inquiryintoinquiry.files.wordpress.com/2023/12/differential-logic-e280a2-fourth-gear-orbits-e280a2-tablestitles.png

The states in each orbit are listed as ordered pairs (p_i, q_j),
where p_i may be read as a temporal parameter indicating the
present time of the state and where j is the decimal equivalent
of the binary numeral s.

Grasped more intuitively, the Tables show each state q_s
with a subscript s equal to the numerator of its rational
index, taking for granted the constant denominator of 2⁴=16.
In that way the temporal succession of states can be reckoned
by a “parallel round‑up rule”. Namely, if (dₖ, dₖ₊₁) is any
pair of adjacent digits in the state index r then the value
of dₖ in the next state is dₖ′=dₖ+dₖ₊₁.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2

Differential Logic • Drives and Their Vicissitudes
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Drives

Regards,

Jon

cc: https://www.academia.edu/community/Vj44Wl

[Jon Awbrey]

Differential Propositional Calculus • 36
• https://inquiryintoinquiry.com/2023/12/31/differential-propositional-calculus-36/

Transformations of Discourse —

❝It is understandable that an engineer should be completely absorbed
in his speciality, instead of pouring himself out into the freedom and
vastness of the world of thought, even though his machines are being
sent off to the ends of the earth; for he no more needs to be capable
of applying to his own personal soul what is daring and new in the soul
of his subject than a machine is in fact capable of applying to itself
the differential calculus on which it is based. The same thing cannot,
however, be said about mathematics; for here we have the new method
of thought, pure intellect, the very well‑spring of the times, the
“fons et origo” of an unfathomable transformation.❞

— Robert Musil • The Man Without Qualities

Here we take up the general study of “logical transformations”,
or maps relating one universe of discourse to another. In many ways,
and especially as applied to the subject of intelligent dynamic systems,
the argument will develop the antithesis of the statement just quoted.
Along the way, if incidental to my ends, I hope the present essay can
pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed
at its head.

The goal is to answer a single question: “What is a propositional
tangent functor?” In other words, the aim is to develop a clear
conception of what manner of thing would pass in the logical realm
for a genuine analogue of the tangent functor, an object conceived
to generalize as far as possible in the abstract terms of category
theory the ordinary notions of functional differentiation and the
all too familiar operations of taking derivatives.

As a first step we examine the types of transformations we already
know as “extensions” and “projections” and we use their special cases
to illustrate several styles of logical and visual representation which
figure in the sequel.

Resources —

Differential Logic and Dynamic Systems

• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3

Differential Logic • Transformations of Discourse
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3#Transformations_of_Discourse

Regards,

Jon

cc: https://www.academia.edu/community/LZMazV

[Jon Awbrey]

Differential Propositional Calculus • 37
• https://inquiryintoinquiry.com/2024/01/01/differential-propositional-calculus-37/

Foreshadowing Transformations • Extensions and Projections of Discourse —

❝And, despite the care which she took to look behind her at
every moment, she failed to see a shadow which followed her
like her own shadow, which stopped when she stopped, which
started again when she did, and which made no more noise
than a well‑conducted shadow should.❞

— Gaston Leroux • The Phantom of the Opera

Many times in our discussion we have occasion to place one universe of
discourse in the context of a larger universe of discourse. An embedding
of the type [†X†] → [†Y†] is implied any time we make use of one basis †X†
which happens to be included in another basis †Y†.

When discussing differential relations we usually have in mind the extended
alphabet ‡Y‡ has a special construction or a specific lexical relation with
respect to the initial alphabet ‡X‡, one which is marked by characteristic
types of accents, indices, or inflected forms.

Resources —

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3

Differential Logic • Foreshadowing Transformations
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3#Foreshadowing_Transformations

Regards,

Jon

cc: https://www.academia.edu/community/V9WrEl