Differential Logic

[Jon Awbrey]

Differential Logic • Overview
• https://inquiryintoinquiry.com/2026/02/03/differential-logic-overview-b/

A reader once told me “venn diagrams are obsolete” and of course
we all know how unwieldy they become as our universes of discourse
expand beyond four or five dimensions. Indeed, one of the first
lessons I learned when I set about implementing Peirce’s graphs
and Spencer Brown’s forms on the computer is that 2‑dimensional
representations of logic quickly become death traps on numerous
conceptual and computational counts.

Still, venn diagrams do us good service at the outset in visualizing the
relationships among extensional, functional, and intensional aspects of
logic. A facility with those connections is critical to the computational
applications and statistical generalizations of propositional logic commonly
used in mathematical and empirical practice.

All things considered, then, it is useful to make the links between
various styles of imagery in logical representation as visible as
possible. The first few steps in that direction are set out in
the sketch of Differential Logic to follow.

Resources —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/5Ab4X6
cc: https://mathstodon.xyz/@Inquiry/116006851988905276
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 1
• https://inquiryintoinquiry.com/2026/02/05/differential-logic-1-b/

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 —

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/VorrZG
cc: https://mathstodon.xyz/@Inquiry/116019443087919778
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 2
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/

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” and is “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” and is “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 —

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

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/V9AARQ
cc: https://mathstodon.xyz/@Inquiry/116024091698022180
cc: https://mathstodon.xyz/@Inquiry/116024421133432490
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 3
• https://inquiryintoinquiry.com/2026/02/07/differential-logic-3-b/

Cactus Language for Propositional Logic (cont.)

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.com/wp-content/uploads/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/

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/L6nnzj
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 4
• https://inquiryintoinquiry.com/2026/02/08/differential-logic-4-b/

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/

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/V0GGNA
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 5
• https://inquiryintoinquiry.com/2026/02/09/differential-logic-5-b/

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.

Diff Log 5.1 • Venn Diagram f = pq
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-f-p-and-q.jpg

Diff Log 5.2 • 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.

Diff Log 5.3 • 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.

Diff Log 5.4 • 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.

Diff Log 5.5 • 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

Diff Log 5.6 • Venn Diagram Difference pq @ pq
• https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-difference-pq-40-pq.jpg

Diff Log 5.7 • 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/

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/VWW0zA
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 6
• https://inquiryintoinquiry.com/2026/02/12/differential-logic-6-b/

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 those 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/

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/VB2eg3
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 7
• https://inquiryintoinquiry.com/2026/02/15/differential-logic-7-b/

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/

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/LYeKRG
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 8
• https://inquiryintoinquiry.com/2026/02/16/differential-logic-8-b/

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.com/wp-content/uploads/2024/11/propositional-forms-on-two-variables-index-order-1.0.png

Resources —

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

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/V1o03y
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 9
• https://inquiryintoinquiry.com/2026/02/19/differential-logic-9-b/

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.com/wp-content/uploads/2024/11/propositional-forms-on-two-variables-orbit-order-1.0.png

Regards,

Jon

cc: https://www.academia.edu/community/LE2jMB
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements

[Jon Awbrey]

Differential Logic • 10
• https://inquiryintoinquiry.com/2026/02/22/differential-logic-10-b/

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.com/wp-content/uploads/2024/11/propositional-forms-on-two-variables-index-order-1.0.png

Table A2. Propositional Forms on Two Variables (Orbit Order)
• https://inquiryintoinquiry.com/wp-content/uploads/2024/11/propositional-forms-on-two-variables-orbit-order-1.0.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/

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

Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Regards,

Jon

cc: https://www.academia.edu/community/5wWa9N
cc: https://www.researchgate.net/post/Differential_Logic_First_Elements