Differential Propositional Calculus

[Jon Awbrey]

Differential Propositional Calculus • Overview
• https://inquiryintoinquiry.com/2024/11/27/differential-propositional-calculus-overview-b/

❝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

Differential logic is the component of logic whose object is
the description of variation — the aspects of change, difference,
distribution, and diversity — in universes of discourse subject
to logical description. To the extent a logical inquiry makes
use of a formal system, its differential component treats the
use of a differential logical calculus — a formal system with
the expressive capacity to describe change and diversity in
logical universes of discourse.

In accord with the strategy of approaching logical systems in stages,
first gaining a foothold in propositional logic and advancing on those
grounds, we may set our first stepping stones toward differential logic
in “differential propositional calculi” — propositional calculi extended by
sets 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.

What follows is the outline of a sketch on differential propositional
calculus intended as an intuitive introduction to the larger subject of
differential logic, which amounts in turn to my best effort so far at
dealing with the ancient and persistent problems of treating diversity
and mutability in logical terms.

Note. I'll give just the links to the main topic heads below.
Please follow the link at the top of the page for the full outline.

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/lepagz
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 1
• https://inquiryintoinquiry.com/2024/11/29/differential-propositional-calculus-1-b/

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 the 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 with the property q or the locations in
the corresponding region Q. Four individuals, a, b, c, d, are
singled out by name. As it happens, b and c currently reside
in region Q while a and d do not.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/LxkRqZ
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 2
• https://inquiryintoinquiry.com/2024/11/30/differential-propositional-calculus-2-b/

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.

Nothing says our encountering the Figures in the above order is other
than purely accidental but if we interpret the 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.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/LpkQpK
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 3
• https://inquiryintoinquiry.com/2024/12/01/differential-propositional-calculus-3-b/

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

The new quality, dq, is called a “differential quality”
by virtue of the fact its absence or presence qualifies
the absence or presence of change occurring in another
quality. As with any 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. That 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”}.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/lypQnm
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 4
• https://inquiryintoinquiry.com/2024/12/02/differential-propositional-calculus-4-b/

Casual Introduction (cont.)

In Figure 3 we saw how the basis of description for the universe
of discourse X could be extended to a set of two qualities {q, dq}
while the corresponding terms of description could be extended to
an alphabet of two symbols {“q”, “dq”}.

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

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/lOQW9E
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 5
• https://inquiryintoinquiry.com/2024/12/03/differential-propositional-calculus-5-b/

Casual Introduction (concl.)

Table 5 exhibits the rules of inference responsible for
giving the differential proposition 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

If the feature q is interpreted as applying to an object in
the universe of discourse X then the differential feature dq
may be taken as an attribute of the same object which tells it
is changing “significantly” with respect to the property q —
as if the object bore an “escape velocity” with respect to
the condition q.

For example, relative to a frame of observation to be made
more explicit later on, if q and dq are true at a given moment,
it would be reasonable to assume ¬q will be true in the next
moment of observation. Taken all together we have the fourfold
scheme of inference shown above.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/VBqdBD
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 6
• https://inquiryintoinquiry.com/2024/12/04/differential-propositional-calculus-6-b/

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 (e₁, e₂, …, eₖ)
is taken to mean exactly one of the propositions e₁, e₂, …, eₖ is false,
in other words, their “minimal negation” is true.

• A concatenated sequence of propositional expressions e₁ e₂ … eₖ
is taken to mean every one of the propositions e₁, e₂, …, eₖ 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 may be obtained through combinations of
the above two forms. As it happens, 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 bracket forms.
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, denoted ε or λ
in formal languages, where it forms the identity element for concatenation.
It may be given visible expression in textual settings 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).

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/VvWmyj
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 7
• https://inquiryintoinquiry.com/2024/12/06/differential-propositional-calculus-7-b/

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‡ = {“a₁”, …, “aₙ”}.
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† = {a₁, …, aₙ}.

A set of logical features †A† = {a₁, …, aₙ} affords a basis for generating
an n‑dimensional universe of discourse, written A° = [†A†] = [a₁, …, aₙ].
It is useful to consider a universe of discourse as a categorical object
incorporating both the set of points A = <a₁, …, aₙ> 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↑) bearing the type (Bⁿ, (Bⁿ → B)), which 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 basic 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/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/LxkZB2
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 8
• https://inquiryintoinquiry.com/2024/12/07/differential-propositional-calculus-8-b/

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° = [a₁, …, aₙ] qualified by the logical
features a₁, …, aₙ 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/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/LxkZen
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 9
• https://inquiryintoinquiry.com/2024/12/08/differential-propositional-calculus-9-b/

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 [a₁, …, aₙ]
is one of the propositions in the set {a₁, …, aₙ}. There are
of course exactly n of these. Depending on context, basic
propositions may also be called “coordinate propositions”
or “simple propositions”.

Among the 2^(2ⁿ) propositions in [a₁, …, aₙ] are several families
numbering 2ⁿ propositions each which take on special forms with
respect to the basis {a₁, …, aₙ}. Three of those 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 a₁, …, aₙ
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 (a₁)(a₂)(a₃), that is,
¬a₁ ∧ ¬a₂ ∧ ¬a₃.

The basic propositions a_i : Bⁿ → B are both linear and positive.
So those two families of propositions, the linear and the positive,
may be viewed as two different ways of generalizing the class of
basic propositions.

It is important to note that all the above distinctions are relative to
the choice of a particular logical basis †A† = {a₁, …, aₙ}. 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 that entire determination is tantamount to the purely conventional
choice of a cell as origin.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/VjKj3b
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 10
• https://inquiryintoinquiry.com/2024/12/09/differential-propositional-calculus-10-b/

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/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/Lpk02K
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 11
• https://inquiryintoinquiry.com/2024/12/10/differential-propositional-calculus-11-b/

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/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/VoqgZr
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 12
• https://inquiryintoinquiry.com/2024/12/10/differential-propositional-calculus-12-b/

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)

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/l8DXYE
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 13
• https://inquiryintoinquiry.com/2024/12/11/differential-propositional-calculus-13-b/

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 the construction of
the first order extended universe EA° and the first order differential
propositions f : EA→B we arrive at the foothills of differential logic.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/laYDqx
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 14
• https://inquiryintoinquiry.com/2024/12/12/differential-propositional-calculus-14-b/

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/

Survey of Differential Logic
• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7

Regards,

Jon

cc: https://www.academia.edu/community/L2P68r
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 15
• https://inquiryintoinquiry.com/2024/12/13/differential-propositional-calculus-15-b/

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/V3KyY4
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 16
• https://inquiryintoinquiry.com/2024/12/14/differential-propositional-calculus-16-b/

Differential Propositions • Qualitative Analogues of Differential Equations —

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

To give the new symbols a name, we define the “differential alphabet”
or “tangent alphabet” d‡A‡ = {“da₁”, …, “daₙ”}, in principle just an
arbitrary set 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/V3K4OE
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 17
• https://inquiryintoinquiry.com/2024/12/15/differential-propositional-calculus-17-b/

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/lQ1nxp
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 18
• https://inquiryintoinquiry.com/2024/12/16/differential-propositional-calculus-18-b/

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/l76pGk
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 19
• https://inquiryintoinquiry.com/2024/12/17/differential-propositional-calculus-19-b/

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 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/LxkGPy
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 20
• https://inquiryintoinquiry.com/2024/12/18/differential-propositional-calculus-20-b/

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 examine how the
various concepts apply in the simplest possible concrete cases, where
the initial dimension is only 1 or 2. In spite of the simplicity of
those cases it is possible to observe how central difficulties of
the subject begin to arise already at that 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/5wQO83
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 21
• https://inquiryintoinquiry.com/2024/12/18/differential-propositional-calculus-21-b/

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/laY4w3
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 22
• https://inquiryintoinquiry.com/2024/12/19/differential-propositional-calculus-22-b/

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 “differential inference rules”.

For example, relative to a frame of observation to be elaborated more fully in time,
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/546DOv
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 23
• https://inquiryintoinquiry.com/2024/12/20/differential-propositional-calculus-23-b/

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 realize 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.

That raises questions 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. Inquiries of that order
serve but to wrap up our present puzzles in further riddles and are
far too involved to be handled at our current level of approximation.
We'll return to them another time.

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/5AqmbK
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 24
• https://inquiryintoinquiry.com/2024/12/21/differential-propositional-calculus-24-b/

❝Urge and urge and urge,
Always the procreant urge of the world.❞

— Walt Whitman • Leaves of Grass

Example 1. A Square Rigging —

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.

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/lJAmrE
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 25
• https://inquiryintoinquiry.com/2024/12/22/differential-propositional-calculus-25-b/

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/5NAKA3
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 26
• https://inquiryintoinquiry.com/2024/12/22/differential-propositional-calculus-26-b/

Example 1. A Square Rigging (concl.)

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/lnYp70
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 27
• https://inquiryintoinquiry.com/2024/12/23/differential-propositional-calculus-27-b/

Commentary On Small Models 1 —

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.

The generative complexity of formal and natural languages tends to lead
investigators to adopt the strategy of focusing on a single aspect of the
domain, abandoning hope of understanding the whole, whether it is the still
living natural language or the dynamics of inquiry crystallized in formal logic.

In the perspective adopted here, a language is a syntactic system evolved
or designed to express a set of descriptions. If the explicit symbols of
a language have extensions in its object world which are actually infinite,
or if 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 the symbols and functions of that
language, all across the spectrum from object language to metalinguistic reflection.

In the aphorism of Wilhelm 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/lnYpm1
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 28
• https://inquiryintoinquiry.com/2024/12/24/differential-propositional-calculus-28-b/

Commentary On Small Models 2 —

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 that because the progress of inquiry is driven by the
manifest discrepancies occurring 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 that is what
the differential extension of propositional calculus is intended to supply. But why
is that 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 that 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/VvWJME
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 29
• https://inquiryintoinquiry.com/2024/12/25/differential-propositional-calculus-29-b/

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 to 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 construct 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.

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/VBqD3R
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[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/5wQD11
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 31
• https://inquiryintoinquiry.com/2024/12/27/differential-propositional-calculus-31-b/

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 fₖ = eₖ∙τ, where τ is a logical tautology using
all the variables in †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}.

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/Lg70pv
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 32
• https://inquiryintoinquiry.com/2024/12/28/differential-propositional-calculus-32-b/

❝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

Example 2. Drives and Their Vicissitudes —

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 affords 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 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/VBqpez
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 33
• https://inquiryintoinquiry.com/2024/12/28/differential-propositional-calculus-33-b/

Example 2. Drives and Their Vicissitudes (cont.)

Expressed in the language of “drives” and “gears” our next Example
may be described as the family of fourth‑gear curves through 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 order differences are
equal to 0.

Because 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 the orbits
and show how they partition the points of E³X. It turns out there are just
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/lP14eK
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 34
• https://inquiryintoinquiry.com/2024/12/29/differential-propositional-calculus-34-b/

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 the state q are the values d^k (q)
for k = 0 to m, where d⁰A is defined as 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 terms, 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/lJAaKx
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 35
• https://inquiryintoinquiry.com/2024/12/29/differential-propositional-calculus-35-b/

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 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/V96Dbj
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 36
• https://inquiryintoinquiry.com/2024/12/30/differential-propositional-calculus-36-b/

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/lnY2aN
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus

[Jon Awbrey]

Differential Propositional Calculus • 37
• https://inquiryintoinquiry.com/2024/12/31/differential-propositional-calculus-37-b/

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/LGvKA9
cc: https://www.researchgate.net/post/Differential_Propositional_Calculus