Differential Logic

[Jon Awbrey]

Differential Logic • 1
• https://inquiryintoinquiry.com/2024/10/30/differential-logic-1-a/

Introduction —

Differential logic is the component of logic whose object is
the description of variation — focusing on the aspects of change,
difference, distribution, and diversity — in universes of discourse
subject to logical description. A definition that broad naturally
incorporates any study of variation by way of mathematical models,
but differential logic is especially charged with the qualitative
aspects of variation pervading or preceding quantitative models.
To the extent a logical inquiry makes use of a formal system, its
differential component governs the use of a “differential logical
calculus”, that is, a formal system with the expressive capacity
to describe change and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by
“differential propositional calculi”. A differential propositional
calculus is a propositional calculus extended by a set of terms for
describing aspects of change and difference, for example, processes
taking place in a universe of discourse or transformations mapping
a source universe to a target universe. Such a calculus augments
ordinary propositional calculus in the same way the differential
calculus of Leibniz and Newton augments the analytic geometry of
Descartes.

Resources —

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

Regards,

Jon

cc: https://www.academia.edu/community/lJX2qa
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 2
• https://inquiryintoinquiry.com/2024/10/31/differential-logic-2-a/

Cactus Language for Propositional Logic —

The development of differential logic is facilitated by having a
moderately efficient calculus in place at the level of boolean-valued
functions and elementary logical propositions. One very efficient
calculus on both conceptual and computational grounds is based on
just two types of logical connectives, both of variable k-ary scope.
The syntactic formulas of that calculus map into a family of graph-
theoretic structures called “painted and rooted cacti” which lend
visual representation to the functional structures of propositions
and smooth the path to efficient computation.

The first kind of connective is a parenthesized sequence of propositional expressions,
written (e₁, e₂, …, eₖ₋₁, eₖ) to mean exactly one of the propositions e₁, e₂, …, eₖ₋₁, eₖ
is false, in short, their “minimal negation” is true. An expression of that form is
associated with a cactus structure called a “lobe”, in this case, “painted” with the
colors e₁, e₂, …, eₖ₋₁, eₖ as shown below.

Lobe Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-lobe-connective.jpg

The second kind of connective is a concatenated sequence of propositional expressions,
written e₁ e₂ … eₖ₋₁ eₖ to mean all the propositions e₁, e₂, …, eₖ₋₁, eₖ are true, in
short, their “logical conjunction” is true. An expression of that form is associated
with a cactus structure called a “node”, in this case, “painted” with the colors
e₁, e₂, …, eₖ₋₁, eₖ as shown below.

Node Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-node-connective.jpg

All other propositional connectives can be obtained through combinations of
the above two forms. As it happens, the parenthesized form is sufficient to
define the concatenated form, making the latter formally dispensable, but
it's convenient to maintain it as a concise way of expressing more complicated
combinations of parenthesized forms. While working with expressions solely
in propositional calculus, it's easiest to use plain parentheses for logical
connectives. In contexts where ordinary parentheses are needed for other
purposes an alternate typeface ( … ) may be used for the logical operators.

Resources —

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

Regards,

Jon

cc: https://www.academia.edu/community/VDnEpr
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 3
• https://inquiryintoinquiry.com/2024/11/02/differential-logic-3-a/

Table 1 shows the cactus graphs, the corresponding cactus expressions,
their logical meanings under the so‑called “existential interpretation”,
and their translations into conventional notations for a sample of basic
propositional forms.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic
• https://inquiryintoinquiry.files.wordpress.com/2022/10/syntax-and-semantics-of-a-calculus-for-propositional-logic-4.0.png

The simplest expression for logical truth is the empty word, typically denoted
by ε or λ in formal languages, where it is the identity element for concatenation.
To make it visible in context, it may be denoted by the equivalent expression “(())”
or, especially if operating in an algebraic context, by a simple “1”. Also when
working in an algebraic mode, the plus sign “+” may be used for exclusive disjunction.
Thus we have the following translations of algebraic expressions into cactus expressions.

• a + b = (a, b)

• a + b + c = (a, (b, c)) = ((a, b), c)

It is important to note the last expressions are not equivalent to
the 3‑place form (a, b, c).

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/l8D7KG
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 4
• https://inquiryintoinquiry.com/2024/11/03/differential-logic-4-a/

Differential Expansions of Propositions —

Bird's Eye View —

An efficient calculus for the realm of logic represented by boolean
functions and elementary propositions makes it feasible to compute
the finite differences and the differentials of those functions
and propositions.

For example, consider a proposition of the form “p and q”
graphed as two letters attached to a root node, as shown below.

Cactus Graph Existential p and q
• https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-existential-p-and-q.jpg

Written as a string, this is just the concatenation p q.

The proposition pq may be taken as a boolean function f(p, q)
having the abstract type f : B × B → B, where B = {0, 1} is
read in such a way that 0 means false and 1 means true.

Imagine yourself standing in a fixed cell of the corresponding
venn diagram, say, the cell where the proposition pq is true,
as shown in the following Figure.

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

Now ask yourself: What is the value of the proposition pq at
a distance of dp and dq from the cell pq where you are standing?

Don't think about it — just compute:

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

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

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

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

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

There is one thing I ought to mention at this point: Computed over B,
plus and minus are identical operations. This will make the relation
between the differential and the integral parts of the appropriate
calculus slightly stranger than usual, but we will get into that later.

Last question, for now: What is the value of this expression from your
current standpoint, that is, evaluated at the point where pq is true?
Well, substituting 1 for p and 1 for q in the graph amounts to erasing
the labels p and q, as shown below.

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

And this is equivalent to the following graph.

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

We have just met with the fact that the differential
of the “and” is the “or” of the differentials.

• p and q →Diff→ dp or dq

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

It will be necessary to develop a more refined analysis of
that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan's rule, it is
no accident, as differentiation and negation turn out to be closely related
operations. Indeed, one can find discussion of logical difference calculus
in the personal correspondence between Boole and De Morgan and Peirce, too,
made use of differential operators in a logical context, but the exploration
of those ideas has been hampered by a number of factors, not the least of which
has been the lack of a syntax adequate to handle the complexity of expressions
evolving in the process.

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/lP1xWp
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 5
• https://inquiryintoinquiry.com/2024/11/04/differential-logic-5-a/

Differential Expansions of Propositions —

Worm's Eye View —

Let's run through the initial example again, keeping an eye on the
meanings of the formulas which develop along the way. We begin with
a proposition or a boolean function f(p, q) = pq whose venn diagram
and cactus graph are shown below.

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

Cactus Graph f = pq
• https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-f-p-and-q.jpg

A function like f has an abstract type and a concrete type. The abstract type is
what we invoke when we write things like f : B×B→B or f : B²→B. The concrete type
takes into account the qualitative dimensions or the “units” of the case, which can
be explained as follows.

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

Then interpret the usual propositions about p, q
as functions of the concrete type f : P×Q→B.

We are going to consider various operators on these functions.
An operator F is a function which takes one function f into
another function Ff.

The first couple of operators we need are logical analogues of two which
play a founding role in the classical finite difference calculus, namely,
the following.

• The “difference operator” ∆, written here as D.
• The “enlargement operator”, written here as E.

These days, E is more often called the “shift operator”.

In order to describe the universe in which these operators operate, it is
necessary to enlarge the original universe of discourse. Starting from
the initial space X = P×Q, its “(first order) differential extension” EX
is constructed according to the following specifications.

• EX = X×dX

where:

• X = P×Q
• dX = dP×dQ
• dP = {(dp), dp}
• dQ = {(dq), dq}

The interpretations of these new symbols can be diverse, but the easiest
option for now is just to say dp means “change p” and dq means “change q”.

Drawing a venn diagram for the differential extension EX = X×dX requires four
logical dimensions, P, Q, dP, dQ, but it is possible to project a suggestion
of what the differential features dp and dq are about on the 2‑dimensional
base space X = P×Q by drawing arrows that cross the boundaries of the basic
circles in the venn diagram for X, reading an arrow as dp if it crosses
the boundary between p and (p) in either direction and reading an arrow
as dq if it crosses the boundary between q and (q) in either direction,
as indicated in the following figure.

Venn Diagram p q dp dq
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-q-dp-dq.jpg

Propositions are formed on differential variables, or any combination of
ordinary logical variables and differential logical variables, in the
same ways propositions are formed on ordinary logical variables alone.
For example, the proposition (dp (dq)) says the same thing as dp ⇒ dq,
in other words, there is no change in p without a change in q.

Given the proposition f(p, q) over the space X = P×Q, the “(first order)
enlargement of f” is the proposition Ef over the differential extension EX
defined by the following formula.

• Ef(p, q, dp, dq)
= f(p + dp, q + dq)
= f(p xor dp, q xor dq)

In the example f(p, q) = pq, the enlargement Ef is computed as follows.

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

The corresponding cactus graph is shown below.

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

Given the proposition f(p, q) over X = P×Q, the “(first order) difference of f”
is the proposition Df over EX defined by the formula Df = Ef - f, or, written out
in full:

• Df(p, q, dp, dq)
= f(p + dp, q + dq) - f(p, q)
= f(p xor dp, q xor dq) xor f(p, q)

In the example f(p, q) = pq, the difference Df is computed as follows.

• Df(p, q, dp, dq)
= (p + dp)(q + dq) - pq
= (p xor dp)(q xor dq) xor pq

The corresponding cactus graph is shown below.

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

This brings us by the road meticulous to the point
we reached at the end of the previous post.

Differential Logic • 4
• https://inquiryintoinquiry.com/2024/11/03/differential-logic-4-a/

There we evaluated the above proposition, the “first order difference
of conjunction” Df, at a single location in the universe of discourse,
namely, at the point picked out by the singular proposition pq, in terms
of coordinates, at the place where p = 1 and q = 1. This evaluation is
written in the form Df|{pq} or Df|{(1, 1)}, and we arrived at the locally
applicable law which may be stated and illustrated as follows.

• f(p, q) = pq = p and q ⇒ Df|{pq} = ((dp)(dq)) = dp or dq

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

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

The venn diagram shows the analysis of the inclusive disjunction “dp or dq”
into the following exclusive disjunction.

• (dp and not dq) xor (dq and not dp) xor (dp and dq)

The resultant differential proposition may be read to say
“change p or change q or both”. And this can be recognized
as just what you need to do if you happen to find yourself in
the center cell and require a complete and detailed description
of ways to escape it.

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/lnYWY0
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 6
• https://inquiryintoinquiry.com/2024/11/05/differential-logic-6-a/

Differential Expansions of Propositions —

Panoptic View • Difference Maps —

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

In the universe of discourse X = P × Q the four propositions
pq, p(q), (p)q, (p)(q) can be taken to indicate the so‑called
“cells” or smallest distinguished regions of the universe,
otherwise indicated by their coordinates as the “points”
(1,1), (1,0), (0,1), (0,0), respectively. In that regard
the four propositions are called “singular propositions”
because they serve to single out the minimal regions of
the universe of discourse.

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

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

Cactus Graph Df = ((p,dp)(q,dq),pq)

• https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-df-pdpqdqpq-1.jpg

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

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

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

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

The easy way to visualize the values of the above graphical expressions
is just to notice the following graphical equations.

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

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

Adding the arrows to the venn diagram gives us
the picture of a “differential vector field”.

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

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

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

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

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

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

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

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

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

Any proposition worth its salt can be analyzed from many different points of view,
any one of which has the potential to reveal previously unsuspected aspects of
the proposition's meaning. We will encounter more and more such alternative
readings as we go.

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/lOQPpB
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 7
• https://inquiryintoinquiry.com/2024/11/06/differential-logic-7-a/

Differential Expansions of Propositions —

Panoptic View • Enlargement Maps —

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

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

• Let X = X₁ × … × Xₖ.
• Let dX = dX₁ × … × dXₖ.
• Then EX = X × dX
= X₁ × … × Xₖ × dX₁ × … × dXₖ

For a proposition of the form f : X₁ × … × Xₖ → B, the

“(first order) enlargement” of f is the proposition

Ef : EX→B defined by the following equation.

• Ef(x₁, …, xₖ, dx₁, …, dxₖ)
= f(x₁ + dx₁, …, xₖ + dxₖ)
= f(x₁ xor dx₁, …, xₖ xor dxₖ)

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

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

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

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

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

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

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

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

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

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

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

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

• Ef = p q · Ef| p q
+ p (q) · Ef| p (q)
+ (p) q · Ef|(p) q}
+ (p)(q) · Ef|{p)(q)

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

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

Logical Formula Enlargement pq
• https://inquiryintoinquiry.files.wordpress.com/2024/11/logical-formula-enlargement-pq.png

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

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/5NAZ9q
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 8
• https://inquiryintoinquiry.com/2024/11/07/differential-logic-8-a/

Propositional Forms on Two Variables —

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

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

Table A1. Propositional Forms on Two Variables
• https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a1.-propositional-forms-on-two-variables.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/lQ19dr
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 9
• https://inquiryintoinquiry.com/2024/11/08/differential-logic-9-a/

Propositional Forms on Two Variables —

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

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

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/LYpp4B
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 10
• https://inquiryintoinquiry.com/2024/11/09/differential-logic-10-a/

Propositional Forms on Two Variables —

Tables A1 and A2 showed two ways of organizing the sixteen boolean functions
or propositional forms on two variables, as expressed in several notations.
In future discussions the two Tables will be described as the “Index Order”
and the “Orbit Order” of propositions, respectively, “orbits” being the usual
term in mathematics for similarity classes under a group action. For ease of
comparison, here are fresh copies of both Tables on the same page.

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

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

Recalling the discussion up to this point, we took as our first example the
boolean function f₈(p, q) = pq corresponding to the logical conjunction p∧q
and examined how the differential operators E and D act on f₈. Each operator
takes the boolean function of two variables f₈(p, q) and gives back a boolean
function of four variables, Ef₈(p, q, dp, dq) and Df₈(p, q, dp, dq), respectively.

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

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/LGvvRw
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 11
• https://inquiryintoinquiry.com/2024/11/10/differential-logic-11-a/

Transforms Expanded over Ordinary and Differential Variables —

As promised last time, in the next several posts we'll extend

our scope to the full set of boolean functions on two variables
and examine how the differential operators E and D act on that set.
There being some advantage to singling out the enlargement or shift

operator E in its own right, we'll begin by computing Ef for each of
the functions f : B×B → B.

Enlargement Map Expanded over Ordinary Variables —

We first encountered the shift operator when we imagined ourselves being in
a state described by the truth of a certain proposition and contemplated the
value of that proposition in various other states, as determined by a collection
of differential propositions describing the steps we might take to change our state.

Restated in terms of our initial example, we imagined ourselves being in
a state described by the truth of the proposition pq and contemplated the
value of that proposition in various other states, as determined by the
differential propositions dp and dq describing the steps we might take
to change our state.

Those thoughts led us from the boolean function of two variables f₈(p, q) = pq
to the boolean function of four variables Ef₈(p, q, dp, dq) = (p , dp)(q , dq),
as shown in the entry for f₈ in the first three columns of Table A3.

Table A3. Ef Expanded over Ordinary Variables {p, q}
• https://inquiryintoinquiry.files.wordpress.com/2020/04/ef-expanded-over-ordinary-variables-p-q.png

Let's catch a breath here and discuss the full Table next time.

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/5AqoEK
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 12
• https://inquiryintoinquiry.com/2024/11/12/differential-logic-12-a/

Transforms Expanded over Ordinary and Differential Variables —

A first view of how the shift operator E acts on the set
of sixteen functions f : B×B → B was provided by Table A3
in the previous post, expanding the expressions of Ef over
the set {p, q} of ordinary variables.

A complementary view of the same material is provided by
Table 4 below, this time expanding the expressions of Ef
over the set {dp, dq} of differential variables.

Enlargement Map Expanded over Differential Variables —

Table A4. Ef Expanded over Differential Variables {dp, dq}
• https://inquiryintoinquiry.files.wordpress.com/2020/04/ef-expanded-over-differential-variables-dp-dq.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/L2PO9d
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 13
• https://inquiryintoinquiry.com/2024/11/14/differential-logic-13-a/

Transforms Expanded over Ordinary and Differential Variables —

Two views of how the difference operator D acts on the set of
sixteen functions f : B×B → B are shown below. Table A5 shows
the expansion of Df over the set {p, q} of ordinary variables
and Table A6 shows the expansion of Df over the set {dp, dq}
of differential variables.

Difference Map Expanded over Ordinary Variables —

Table A5. Df Expanded over Ordinary Variables {p, q}
• https://inquiryintoinquiry.files.wordpress.com/2024/11/df-expanded-over-ordinary-variables-p-q-1.0.png

Difference Map Expanded over Differential Variables —

Table A6. Df Expanded over Differential Variables {dp, dq}
• https://inquiryintoinquiry.files.wordpress.com/2024/11/df-expanded-over-differential-variables-dp-dq-1.0.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/546XW3
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 14
• https://inquiryintoinquiry.com/2024/11/16/differential-logic-14-a/

Field Picture —

Let us summarize the outlook on differential logic we've reached so far.
We've been considering a class of operators on universes of discourse,
each of which takes us from considering one universe of discourse Xº to
considering a larger universe of discourse EXº. An operator W of that
general type, namely, W : Xº → EXº, acts on each proposition f : X → B
of the source universe Xº to produce a proposition Wf : EX → B of the
target universe EXº.

The operators we've examined so far are the enlargement or shift operator
E : Xº → EXº and the difference operator D : Xº → EXº. The operators E and
D act on propositions in Xº, that is, propositions of the form f : X → B
which may be taken as being statements “about” the subject matter of X, and
they produce propositions of the forms Ef, Df : EX → B which may be taken
as being statements “about” specified collections of changes conceivably
occurring in X.

At this point we find ourselves in need of visual representations,
suitable arrays of concrete pictures to anchor our more earthy
intuitions and help us keep our wits about us as we venture
into ever more rarefied airs of abstraction.

One good picture comes to us by way of the “field” concept.
Given a space X, a “field” of a specified type Y over X is
formed by associating with each point of X an object of type Y.
If that sounds like the same thing as a function from X to the
space of things of type Y — it is nothing but — and yet it does
seem helpful to vary the mental images and take advantage of the
figures of speech most naturally springing to mind under the emblem
of the field idea.

In the field picture a proposition f : X → B becomes a “scalar field”,
that is, a field of values in B.

For example, consider the logical conjunction pq : X → B shown in the
following venn diagram.

Conjunction pq : X → B
• https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-conjunction-5.0.png

Each of the operators E, D : Xº → EXº takes us from considering
propositions f : X → B, here viewed as “scalar fields” over X,
to considering the corresponding “differential fields” over X,
analogous to what in real analysis are usually called “vector
fields” over X.

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/5k9wGN
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 15
• https://inquiryintoinquiry.com/2024/11/18/differential-logic-15-a/

Differential Fields —

The structure of a differential field may be described
as follows. With each point of X there is associated an
object of the following type: a proposition about changes
in X, that is, a proposition g : dX → B. In that frame of
reference, if Xº is the universe generated by the set of
coordinate propositions {p, q} then dXº is the differential
universe generated by the set of differential propositions
{dp, dq}. The differential propositions dp and dq may thus
be interpreted as indicating “change in p” and “change in q”,
respectively.

A differential operator W, of the first order type we are
currently considering, takes a proposition f : X → B and
gives back a differential proposition Wf : EX → B. In the
field view of the scene, we see the proposition f : X → B
as a scalar field and we see the differential proposition
Wf : EX → B as a vector field, specifically, a field of
propositions about contemplated changes in X.

The field of changes produced by E on pq
is shown in the following venn diagram.

Enlargement E(pq) : EX → B
• https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-enlargement-conjunction-5.0.png

E(pq)
= p ∙ q ∙ (dp)(dq)

+ p ∙ (q) ∙ (dp) dq
+ (p) ∙ q ∙ dp (dq)

+ (p) ∙ (q) ∙ dp dq

The differential field E(pq) specifies the changes which need
to be made from each point of X in order to reach one of the
models of the proposition pq, that is, in order to satisfy
the proposition pq.

The field of changes produced by D on pq
is shown in the following venn diagram.

Differential D(pq) : EX → B
• https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-difference-conjunction-5.0.png

D(pq)

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

+ (p) ∙ (q) ∙ dp dq

The differential field D(pq) specifies the changes which need
to be made from each point of X in order to feel a change in
the felt value of the field pq.

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/LGv6Q0
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 16
• https://inquiryintoinquiry.com/2024/11/19/differential-logic-16-a/

Propositions and Tacit Extensions —

Now that we've introduced the field picture as an aid to visualizing
propositions and their analytic series, a pleasing way to picture
the relationship of a proposition f : X → B to its enlargement or
shift map Ef : EX → B and its difference map Df : EX → B can now
be drawn.

To illustrate the possibilities, let's return to the differential
analysis of the conjunctive proposition f(p, q) = pq and give its
development a slightly different twist at the appropriate point.

The proposition pq : X → B is shown again in the venn diagram below.
In the field picture it may be seen as a scalar field — analogous to
a “potential hill” in physics but in logic amounting to a “potential
plateau” — where the shaded region indicates an elevation of 1 and
the unshaded region indicates an elevation of 0.

Proposition pq : X → B
• https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-conjunction-5.0.png

Given a proposition f : X → B, the “tacit extension” of f to EX is
denoted εf : EX → B and defined by the equation εf = f, so it's
really just the same proposition residing in a bigger universe.
Tacit extensions formalize the intuitive idea that a function
on a given set of variables can be extended to a function on
a superset of those variables in such a way that the new
function obeys the same constraints on the old variables,
with a “don't care” condition on the new variables.

The tacit extension of the scalar field pq : X → B to the
differential field ε(pq) : EX → B is shown in the following
venn diagram.

Tacit Extension ε(pq) : EX → B
• https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-tacit-extension-conjunction-5.0.png

ε(pq)
= p ∙ q ∙ (dp)(dq)
+ p ∙ q ∙ (dp) dq
+ p ∙ q ∙ dp (dq)
+ p ∙ q ∙ dp 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/V00dPp
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 17
• https://inquiryintoinquiry.com/2024/11/22/differential-logic-17-a/

Enlargement and Difference Maps —

Continuing with the example pq : X → B, the following venn diagram
shows the enlargement or shift map E(pq) : EX → B in the same style
of field picture we drew for the tacit extension ε(pq) : EX → B.

Enlargement E(pq) : EX → B
• https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-enlargement-conjunction-5.0.png

E(pq)

= p ∙ q ∙ (dp)(dq)

+ p ∙ (q) ∙ (dp) dq
+ (p) ∙ q ∙ dp (dq)
+ (p) ∙ (q) ∙ dp dq

A very important conceptual transition has just occurred here,
almost tacitly, as it were. Generally speaking, having a set
of mathematical objects of compatible types, in this case the
two differential fields εf and Ef, both of the type EX → B, is
very useful, because it allows us to consider those fields as
integral mathematical objects which can be operated on and
combined in the ways we usually associate with algebras.

In the present case one notices the tacit extension εf
and the enlargement Ef are in a sense dual to each other.
The tacit extension εf indicates all the arrows out of the
region where f is true and the enlargement Ef indicates all
the arrows into the region where f is true. The only arc
they have in common is the no‑change loop (dp)(dq) at pq.
If we add the two sets of arcs in mod 2 fashion then the
loop of multiplicity 2 zeroes out, leaving the 6 arrows of
D(pq) = ε(pq) + E(pq) shown in the following venn diagram.

Differential D(pq) : EX → B
• https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-difference-conjunction-5.0.png

D(pq)
= p ∙ q ∙ ((dp)(dq))
+ p ∙ (q) ∙ (dp) dq
+ (p) ∙ q ∙ dp (dq)

+ (p) ∙ (q) ∙ dp 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/lP1a2M
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference

[Jon Awbrey]

Differential Logic • 18
• https://inquiryintoinquiry.com/2024/11/23/differential-logic-18-a/

Tangent and Remainder Maps —

If we follow the classical line which singles out linear functions
as ideals of simplicity then we may complete the analytic series
of the proposition f = pq : X → B in the following way.

The next venn diagram shows the differential proposition df =
d(pq) : EX → B we get by extracting the linear approximation
to the difference map Df = D(pq) : EX → B at each cell or point
of the universe X. What results is the logical analogue of what
would ordinarily be called “the differential” of pq but since the
adjective “differential” is being attached to just about everything
in sight the alternative name “tangent map” is commonly used for df
whenever it's necessary to single it out.

Tangent Map d(pq) : EX → B
• https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-differential-conjunction-5.0.png

To be clear about what's being indicated here,
it's a visual way of summarizing the following data.

d(pq)
= p ∙ q ∙ (dp , dq)
+ p ∙ (q) ∙ dq
+ (p) ∙ q ∙ dp
+ (p) ∙ (q) ∙ 0

To understand the extended interpretations, that is,
the conjunctions of basic and differential features
which are being indicated here, it may help to note
the following equivalences.

• (dp , dq) = dp ∙ (dq) + (dp) ∙ dq

• dp = dp ∙ dq + dp ∙ (dq)

• dq = dp ∙ dq + (dp) ∙ dq

Capping the analysis of the proposition pq in terms of succeeding
orders of linear propositions, the final venn diagram of the series
shows the “remainder map” r(pq) : EX → B, which happens to be linear
in pairs of variables.

Remainder r(pq) : EX → B
• https://inquiryintoinquiry.files.wordpress.com/2024/11/field-picture-pq-remainder-conjunction-5.0.png

Reading the arrows off the map produces the following data.

r(pq)
= p ∙ q ∙ dp ∙ dq
+ p ∙ (q) ∙ dp ∙ dq
+ (p) ∙ q ∙ dp ∙ dq

+ (p) ∙ (q) ∙ dp ∙ dq

In short, r(pq) is a constant field, having the value dp ∙ dq at each cell.

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/lypmDv
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference