Talk:Differential Logic : Introduction

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Frankl 5

Putting all thought of the Frankl Conjecture out of our minds for the moment, let's return to the proposition in Example 1 and work through its differential analysis from scratch.

Example 1

Venn Diagram PQR ALT.jpg (1)

Consider the proposition p \land q \land r,\! in boolean terms the function f : \mathbb{B}^3 \to \mathbb{B}\! such that f(p, q, r) = pqr,\! as illustrated by the venn diagram in Figure 1.

The enlargement \mathrm{E}f\! of f\! is the boolean function \mathrm{E}f : \mathbb{B}^3 \times \mathbb{D}^3 \to \mathbb{B}\! defined by the following equation:

\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) ~=~ f(p + \mathrm{d}p, q + \mathrm{d}q, r + \mathrm{d}r).\!

Given that f\! is the boolean product of its three arguments, \mathrm{E}f\! may be written as follows:

\mathrm{E}f ~=~ (p + \mathrm{d}p)(q + \mathrm{d}q)(r + \mathrm{d}r).\!

Difficulties of notation in differential logic are greatly eased by introducing the family of minimal negation operators on finite numbers of boolean variables. For our immediate purposes we need only the minimal negation operators on one and two variables.

  • The minimal negation operator on one variable x\! is notated with monospace parentheses as \texttt{(} x \texttt{)}\! and is simply another notation for the logical negation \lnot x.\!
  • The minimal negation operator on two variables x, y\! is notated with monospace delimiters as \texttt{(} x \texttt{,} y \texttt{)}\! and is simply another notation for the exclusive disjunction x + y.\!

In this notation the previous expression for \mathrm{E}f\! takes the following form:

\mathrm{E}f ~=~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)(} r \texttt{,} \mathrm{d}r \texttt{)}.\!

A canonical form for \mathrm{E}f\! may be derived by means of a boolean expansion, in effect, a case analysis that evaluates \mathrm{E}f\! at each triple of values for the base variables p, q, r\! and forms the disjunction of the partial evaluations. Each term of the boolean expansion corresponds to a cell of the venn diagram and is formed by multiplying the value of that cell by a coefficient that amounts to the value of \mathrm{E}f\! on that cell.

For example, in the case pqr\! where all three base variables are true, the corresponding coefficient is computed as follows:

 

\begin{array}{lll}
\mathrm{E}f(1, 1, 1, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r)
& = & (1 + \mathrm{d}p)(1 + \mathrm{d}q)(1 + \mathrm{d}r)
\\[4pt]
& = & \lnot\mathrm{d}p \land \lnot\mathrm{d}q \land \lnot\mathrm{d}r
\\[4pt]
& = & \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\end{array}\!

Collecting the cases yields the boolean expansion of \mathrm{E}f\! via the following computation:

Step 1

  \mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{smallmatrix}
&
p q r \cdot f(1 + \mathrm{d}p, 1 + \mathrm{d}q, 1 + \mathrm{d}r)
& + &
p q \tilde{r} \cdot f(1 + \mathrm{d}p, 1 + \mathrm{d}q, 0 + \mathrm{d}r)
& + &
p \tilde{q} r \cdot f(1 + \mathrm{d}p, 0 + \mathrm{d}q, 1 + \mathrm{d}r)
& + &
p \tilde{q} \tilde{r} \cdot f(1 + \mathrm{d}p, 0 + \mathrm{d}q, 0 + \mathrm{d}q)
\\[4pt]
+ &
\tilde{p} q r \cdot f(0 + \mathrm{d}p, 1 + \mathrm{d}q, 1 + \mathrm{d}r)
& + &
\tilde{p} q \tilde{r} \cdot f(0 + \mathrm{d}p, 1 + \mathrm{d}q, 0 + \mathrm{d}r)
& + &
\tilde{p} \tilde{q} r \cdot f(0 + \mathrm{d}p, 0 + \mathrm{d}q, 1 + \mathrm{d}r)
& + &
\tilde{p} \tilde{q} \tilde{r} \cdot f(0 + \mathrm{d}p, 0 + \mathrm{d}q, 0 + \mathrm{d}q)
\end{smallmatrix}\!

Step 2

  \mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{matrix}
&
p q r \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p q \texttt{(} r \texttt{)} \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r
& + &
p \texttt{(} q \texttt{)} r \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
& + &
p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r
\\[4pt]
+ &
\texttt{(} p \texttt{)} q r \,\cdot\,
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\,
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r
& + &
\texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\,
\mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\,
\mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r
\end{matrix}\!

Frankl 6

Venn Diagram Frankl Figure 3 ALT.jpg (3)

Figure 3 shows the eight terms of the enlarged proposition \mathrm{E}f\! as arcs, arrows, or directed edges in the venn diagram of the original proposition f(p, q, r) = pqr.\! Each term of the enlargement \mathrm{E}f\! corresponds to an arc into the cell where f\! is true from one of the eight cells of the venn diagram.

For ease of reference, here is the expansion of \mathrm{E}f\! from the previous post:

  \mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{matrix}
&
p q r \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p q \texttt{(} r \texttt{)} \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r
& + &
p \texttt{(} q \texttt{)} r \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
& + &
p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r
\\[4pt]
+ &
\texttt{(} p \texttt{)} q r \,\cdot\,
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\,
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r
& + &
\texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\,
\mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\,
\mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r
\end{matrix}\!

Two examples suffice to convey the general idea of the enlarged venn diagram:

  • The term p q r \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}\! is shown as a looped arc starting in the cell where p q r\! is true and returning back to it. The differential factor \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}\! corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
  • The term \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r\! is shown as an arc from the cell where \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}\! is true to the cell where p q r\! is true. The differential factor \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r\! corresponds to the fact that the arc crosses all three logical feature boundaries from its source to its target.

Frankl 7

We continue with the differential analysis of the proposition in Example 1.

Example 1

Venn Diagram PQR ALT.jpg (1)

A proposition defined on one universe of discourse has natural extensions to larger universes of discourse. As a matter of course in a given context of discussion, some of these extensions come to be taken for granted as the most natural extensions to make in passing from one universe to the next and they tend to be assumed automatically, by default, in the absence of explicit notice to the contrary. These are the tacit extensions that apply in that context.

Differential logic, at the first order of analysis, treats extensions from boolean spaces of type \mathbb{B}^k\! to enlarged boolean spaces of type \mathbb{B}^k \times \mathbb{D}^k.\! In this setting \mathbb{B} \cong \mathbb{D} \cong \{ 0, 1 \}\! but we use different letters merely to distinguish base and differential features.

In our present example, the tacit extension \boldsymbol\varepsilon f\! of f\! is the boolean function \boldsymbol\varepsilon f : \mathbb{B}^3 \times \mathbb{D}^3 \to \mathbb{B}\! defined by the following equation:

\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) ~=~ f(p, q, r).\!

The boolean expansion of \boldsymbol\varepsilon f\! takes the following form:

  \boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{*{8}{l}}
&
p q r ~
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
p q r ~
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

In other words, \boldsymbol\varepsilon f\! is simply f\! on the base variables p, q, r\! extended by a tautology — commonly known as a “Don't Care” condition — on the differential variables \mathrm{d}p, \mathrm{d}q, \mathrm{d}r.\!

Frankl 8

Venn Diagram Frankl Figure 4 ALT.jpg (4)

Figure 4 shows the eight terms of the tacit extension \boldsymbol\varepsilon f\! as arcs, arrows, or directed edges in the venn diagram of the original proposition f(p, q, r) = pqr.\! Each term of the tacit extension \boldsymbol\varepsilon f\! corresponds to an arc that starts from the cell where f\! is true and ends in one of the eight cells of the venn diagram.

For ease of reference, here is the expansion of \boldsymbol\varepsilon f\! from the previous post:

  \boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{*{8}{l}}
&
p q r ~
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
p q r ~
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

Two examples suffice to convey the general idea of the extended venn diagram:

  • The term pqr \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}\! is shown as a looped arc starting in the cell where pqr\! is true and returning back to it. The differential factor \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}\! corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
  • The term pqr \cdot \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r\! is shown as an arc going from the cell where pqr\! is true to the cell where \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}\! is true. The differential factor \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r\! corresponds to the fact that the arc crosses all three logical feature boundaries from its source to its target.

Frankl 9

“It doesn't matter what one does,” the Man Without Qualities said to himself, shrugging his shoulders. “In a tangle of forces like this it doesn't make a scrap of difference.” He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.

Robert Musil • The Man Without Qualities

We continue with the differential analysis of the proposition in Example 1.

Example 1

Venn Diagram PQR ALT.jpg (1)

The difference operator \mathrm{D}\! is defined as the difference \mathrm{E} - \boldsymbol\varepsilon\! between the enlargement operator \mathrm{E}\! and the tacit extension operator \boldsymbol\varepsilon.\!

The difference map \mathrm{D}f~\! is the result of applying the difference operator \mathrm{D}\! to the function f.\! When the sense is clear, we may refer to \mathrm{D}f~\! simply as the difference of f.\!

In boolean spaces there is no difference between the sum (+)\! and the difference (-)\! so the difference operator \mathrm{D}\! is equally well expressed as the exclusive disjunction or symmetric difference \mathrm{E} + \boldsymbol\varepsilon.\! In this case the difference map \mathrm{D}f~\! can be computed according to the formula \mathrm{D}f = (\mathrm{E} + \boldsymbol\varepsilon)f = \mathrm{E}f + \boldsymbol\varepsilon f.\!

The action of \mathrm{D}\! on our present example, f(p, q, r) = pqr,\! can be computed from the data on hand according to the following prescription.

The enlargement map \mathrm{E}f,\! computed in Post 5 and graphed in Post 6, is shown again here:

  \mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{matrix}
&
p q r \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p q \texttt{(} r \texttt{)} \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r
& + &
p \texttt{(} q \texttt{)} r \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
& + &
p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r
\\[4pt]
+ &
\texttt{(} p \texttt{)} q r \,\cdot\,
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\,
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r
& + &
\texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\,
\mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\,
\mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r
\end{matrix}\!

The tacit extension \boldsymbol\varepsilon f,\! computed in Post 7 and graphed in Post 8, is shown again here:

  \boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{*{8}{l}}
&
p q r ~
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
p q r ~
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
p q r ~
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

The difference map \mathrm{D}f~\! is the sum of the enlargement map \mathrm{E}f\! and the tacit extension \boldsymbol\varepsilon f.\!

Here we adopt a paradigm of computation for \mathrm{D}f~\! that aids not only in organizing the stages of the work but also in highlighting the diverse facets of logical meaning that may be read off the result.

The terms of the enlargement map \mathrm{E}f\! are obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.

Table 1.0 PQR Enlargement Map.png

The terms of the tacit extension \boldsymbol\varepsilon f\! are obtained from the next table below by multiplying the base factor at the head of the first column by each of the differential factors that appear beneath it in the body of the table.

Table 1.0 PQR Tacit Extension ISW.png

Finally, the terms of the difference map \mathrm{D}f~\! are obtained by overlaying the displays for \mathrm{E}f\! and \boldsymbol\varepsilon f\! and taking their boolean sum entry by entry.

Table 1.0 PQR Difference Map.png

Notice that the “loop” or “no change” term p q r \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}\! cancels out, leaving 14 terms in the end.

Frankl 10

Venn Diagram Frankl Figure 5 ALT.jpg (5)

Figure 5 shows the 14 terms of the difference map \mathrm{D}f~\! as arcs, arrows, or directed edges in the venn diagram of the original proposition f(p, q, r) = pqr.\! The arcs of \mathrm{E}f\! are directed into the cell where f\! is true from each of the other cells. The arcs of \boldsymbol\varepsilon f\! are directed from the cell where f\! is true into each of the other cells.

The expansion of \mathrm{D}f~\! computed in the previous post is shown again below with the terms arranged by number of positive differential features, from lowest to highest.

\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

Frankl 11

Let's take a moment from the differential analysis of the proposition in Example 1 to form a handy compendium of the results obtained so far.

Example 1

Venn Diagram PQR ALT.jpg (1)

Enlargement Map E(pqr) of the Conjunction pqr

Venn Diagram Frankl Figure 3 ALT.jpg (3)
Table -- PQR Enlargement Map.png

Tacit Extension ε(pqr) of the Conjunction pqr

Venn Diagram Frankl Figure 4 ALT.jpg (4)
Table -- PQR Tacit Extension.png

Difference Map D(pqr) of the Conjunction pqr

Venn Diagram Frankl Figure 5 ALT.jpg (5)
Table -- PQR Difference Map.png

Frankl 12

It is one of the rules of my system of general harmony, that the present is big with the future, and that he who sees all sees in that which is that which shall be.

Leibniz • Theodicy

We continue with the differential analysis of the proposition in Example 1.

Example 1

Venn Diagram PQR ALT.jpg (1)

Like any moderately complex proposition, the difference map of a proposition has many equivalent logical expressions and can be read in many different ways.

Venn Diagram Frankl Figure 5 ALT.jpg (5)

The expansion of \mathrm{D}f~\! computed in Post 9 and further discussed in Post 10 is shown again below with the terms arranged by number of positive differential features, from lowest to highest.

\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

The terms of the difference map \mathrm{D}f~\! may be obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.

Table -- PQR Difference Map.png

The full boolean expansion of \mathrm{D}f~\! may be condensed to a degree by collecting terms that share the same base factors, as shown in the following display:

  \mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{cccc}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
& \cdot &
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))}
\\[4pt]
+ &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)~}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~}
& \cdot &
\texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)}
& \cdot &
\texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~}
\\[4pt]
+ &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)}
& \cdot &
\texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~}
\end{array}\!

This amounts to summing terms along columns of the previous table, as shown at the bottom margin of the next table:

Table 4.0 PQR Difference Map Col Sum.png

Collecting terms with the same differential factors produces the following expression:

  \mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{cccc}
&
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& \cdot &
q \texttt{~} r
\\[4pt]
+ &
\texttt{~} \mathrm{d}q \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}r \texttt{)}
& \cdot &
p \texttt{~} r
\\[4pt]
+ &
\texttt{~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
& \cdot &
p \texttt{~} q
\\[4pt]
+ &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& \cdot &
\texttt{((} p \texttt{,} q \texttt{))} ~ r
\\[4pt]
+ &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}q \texttt{)}
& \cdot &
\texttt{((} p \texttt{,} r \texttt{))} ~ q
\\[4pt]
+ &
\texttt{~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)}
& \cdot &
\texttt{((} q \texttt{,} r \texttt{))} ~ p
\\[4pt]
+ &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& \cdot &
p \texttt{~} q \texttt{~} r
\\[4pt]
+ &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& \cdot &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
\end{array}\!

This is roughly what one would get by summing along rows of the previous tables.

Frankl 13

We have come to a fork in the road. I wish I could take the wit's advice and Take It! but the fork has more tines than I can count and the horse knows the way Not! to ride off in all directions at once. But I can see two classes of choices clearly enough to consider the possibilities of at least those two.

The work we've been doing so far amounts to applying the logical analogue of the finite difference calculus to the proposition in Example 1.

Algebraically,

\begin{array}{lll}
\mathrm{E}(pqr) & = & (p + \mathrm{d}p)(q + \mathrm{d}q)(r + \mathrm{d}r)
\\[6pt]
& = & pqr
\quad + \quad pq\,\mathrm{d}r + pr\,\mathrm{d}q + qr\,\mathrm{d}p
\quad + \quad p\,\mathrm{d}q\,\mathrm{d}r + q\,\mathrm{d}p\,\mathrm{d}r + r\,\mathrm{d}p\,\mathrm{d}q
\quad + \quad \mathrm{d}p\,\mathrm{d}q\,\mathrm{d}r
\end{array}~\!

Consequently,

\begin{array}{lll}
\mathrm{D}(pqr) & = & (p + \mathrm{d}p)(q + \mathrm{d}q)(r + \mathrm{d}r) ~ - ~ pqr
\\[6pt]
& = & pq\,\mathrm{d}r + pr\,\mathrm{d}q + qr\,\mathrm{d}p
\quad + \quad p\,\mathrm{d}q\,\mathrm{d}r + q\,\mathrm{d}p\,\mathrm{d}r + r\,\mathrm{d}p\,\mathrm{d}q
\quad + \quad \mathrm{d}p\,\mathrm{d}q\,\mathrm{d}r
\end{array}\!

Consequently,

\begin{matrix}
\mathrm{d}(pqr)
& = & pq\,\mathrm{d}r
& + & pr\,\mathrm{d}q
& + & qr\,\mathrm{d}p
\\[6pt]
& = & \partial_r(pqr)\,\mathrm{d}r
& + & \partial_q(pqr)\,\mathrm{d}q
& + & \partial_p(pqr)\,\mathrm{d}p
\end{matrix}\!


\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!

\begin{array}{c*{8}{l}}
\mathrm{D}J
& = &
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}
& + &
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}J
& = &
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u
& + &
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}

Figures 46-a through 46-d illustrate the proposition {\mathrm{d}J},\! rounded out in our usual array of prospects. This proposition of \mathrm{E}U^\bullet\! is what we refer to as the (first order) differential of J,\! and normally regard as the differential proposition corresponding to J.\!

Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif
\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!
Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif
\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!
Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif
\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!
Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif
\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!
\text{Difference Map} ~ \mathrm{D}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&&
\\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
\\[8pt]
\hline &&&&&&&
\\
0 &&&&&&&
\\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&
\\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&&&
\\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&&
\\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&
\mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&
\\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r &&
\\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r &
\\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&&
\mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r
\\[8pt]
\hline &&&&&&&
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &
\mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r &
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r &
\mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r
\\[8pt]
\hline
\end{array}\!

Option 1. Tangent Map as Local Linear Approximation


\text{Tangent Map} ~ \mathrm{d}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&&
\\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
\\[8pt]
\hline &&&&&&&
\\
0 &&&&&&&
\\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&
\\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&&&
\\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&&
\\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&
\mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&
\\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r &&
\\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r &
\\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&&
\mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r
\\[8pt]
\hline &&&&&&&
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &
\mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r &
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r &
\mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r
\\[8pt]
\hline &&&&&&&
\\
\texttt{(} \mathrm{d}p \texttt{,(} \mathrm{d}q \texttt{,} \mathrm{d}r \texttt{))} &
\mathrm{d}p &
\mathrm{d}q &
\mathrm{d}r &
0 &
0 &
0 &
0
\\[8pt]
\hline
\end{array}\!


Table A9. Tangent Proposition as Pointwise Linear Approximation


\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!
f\!

\begin{matrix}
\mathrm{d}f =
\\[2pt]
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
\end{matrix}

\begin{matrix}
\mathrm{d}^2\!f =
\\[2pt]
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}

\mathrm{d}f|_{x \, y} \mathrm{d}f|_{x \, \texttt{(} y \texttt{)}} \mathrm{d}f|_{\texttt{(} x \texttt{)} \, y} \mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}
f_0\! 0\! 0\! 0\! 0\! 0\! 0\!

\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!

\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\end{matrix}

\begin{matrix}
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\end{matrix}

\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix} \begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix} \begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix} \begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}

\begin{matrix}f_{3}\\f_{12}\end{matrix}

\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}

\begin{matrix}0\\0\end{matrix} \begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix} \begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix} \begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix} \begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}

\begin{matrix}f_{6}\\f_{9}\end{matrix}

\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}

\begin{matrix}0\\0\end{matrix} \begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix} \begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix} \begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix} \begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}

\begin{matrix}f_{5}\\f_{10}\end{matrix}\!

\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!

\begin{matrix}0\\0\end{matrix} \begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\! \begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\! \begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\! \begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!

\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}

\begin{matrix}
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\end{matrix}\!

\begin{matrix}
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\end{matrix} \begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix} \begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix} \begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix} \begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}
f_{15}\! 0\! 0\! 0\! 0\! 0\! 0\!


Differential Operator Tables • Conjunction PQR

Version 1

See Frankl 9


\text{Enlargement Map} ~ \mathrm{E}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \\
\hline &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&& \\
& \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&&&& \\
&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&&& \\
&&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r &&&& \\
&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&& \\
&&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r && \\
&&&&&& \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & \\
&&&&&&& \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r  \\
\hline
\end{array}\!


\text{Tacit Extension} ~ \boldsymbol\varepsilon (pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \\
\hline &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&& \\
\hline
\end{array}\!


\text{Difference Map} ~ \mathrm{D}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \\
\hline &&&&&&& \\
0 &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
&&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r &&&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
&&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r && \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
&&&&&& \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
&&&&&&& \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \\
\hline
\end{array}\!


\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!


Version 2

See Frankl 10


\text{Enlargement Map} ~ \mathrm{E}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \\
\hline &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&& \\
&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&& \\
&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&&& \\
& \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&&&& \\
&&&&&& \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & \\
&&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r && \\
&&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r &&&& \\
&&&&&&& \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r  \\
\hline
\end{array}\!


\text{Tacit Extension} ~ \boldsymbol\varepsilon (pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \\
\hline &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&& \\
\hline
\end{array}\!


\text{Difference Map} ~ \mathrm{D}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \\
\hline &&&&&&& \\
0 &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
&&&&&& \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
&&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r && \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
&&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r &&&& \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
&&&&&&& \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \\
\hline
\end{array}\!


\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!


Version 3


\text{Enlargement Map} ~ \mathrm{E}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \\
\hline &&&&&&& \\
&&&&&&& \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} \\
&&&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & \\
&&&&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} && \\
&&&& \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&& \\
&&& \mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&& \\
&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&&& \\
& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r &&&&&& \\
\mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r &&&&&&& \\
\hline
\end{array}\!


\text{Tacit Extension} ~ \boldsymbol\varepsilon (pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \\
\hline &&&&&&& \\
&&&&&&& \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} \\
&&&&&&& \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} \\
&&&&&&& \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\
&&&&&&& \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \\
&&&&&&& \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\
&&&&&&& \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \\
&&&&&&& \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\
&&&&&&& \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\
\hline
\end{array}\!


\text{Difference Map} ~ \mathrm{D}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \\
\hline &&&&&&& \\
&&&&&&& 0 \\
&&&&&&
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
&
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} \\
&&&&&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
&&
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\
&&&&
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r
&&&
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \\
&&&
\mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
&&&&
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\
&&
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r
&&&&&
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \\
&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r
&&&&&&
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\
\mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r
&&&&&&&
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\
\hline
\end{array}\!


\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!


Version 4


\text{Enlargement Map} ~ \mathrm{E}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \\
\hline &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&& \\
& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&& \\
&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&&& \\
&&& \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&& \\
&&&& \mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&& \\
&&&&& \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r && \\
&&&&&& \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r & \\
&&&&&&& \mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r \\
\hline
\end{array}\!


\text{Tacit Extension} ~ \boldsymbol\varepsilon (pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \\
\hline &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&& \\
\hline
\end{array}\!


\text{Difference Map} ~ \mathrm{D}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&& \\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \\
\hline &&&&&&& \\
0 &&&&&&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&& \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&&& \\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&& \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&
\mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&& \\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r && \\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r & \\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&&
\mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r \\
\hline
\end{array}\!


\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!


Venn Diagrams • Three Variables P Q R

Venn Diagram • Empty Frame P Q R

Venn Diagram P Q R Empty Frame ALT.jpg

Venn Diagram • Conjunction PQR

Venn Diagram PQR ALT.jpg

Venn Diagram • Frankl Figure 2

Venn Diagram Frankl Figure 2 ALT.jpg

Work Area • 2 Variables

Difference Map of Conjunction

D(uv) • Version 1

\text{Computation of}~ \mathrm{D}(uv)\!

\begin{array}{|*{4}{c}|}
\hline &&&
\\
\texttt{~} u \texttt{~~} v \texttt{~} &
\texttt{~} u \texttt{~(} v \texttt{)} &
\texttt{(} u \texttt{)~} v \texttt{~} &
\texttt{(} u \texttt{)(} v \texttt{)}
\\[8pt]
\hline &&&
\\
0 &&&
\\
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} &
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} &&
\\[6pt]
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} &&
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} &
\\[6pt]
\texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} &&&
\texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\\[8pt]
\hline &&&
\\
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} &
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} &
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} &
\texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\\[8pt]
\hline
\end{array}~\!


With the tacit extension map \boldsymbol\varepsilon J\! and the enlargement map \mathrm{E}J\! well in place, the difference map \mathrm{D}J\! can be computed along the lines displayed in Table 41, ending up with an expansion of \mathrm{D}J\! over the cells of [u, v].\!


\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!

\begin{array}{*{9}{l}}
\mathrm{D}J
& = & \mathrm{E}J
& + & \boldsymbol\varepsilon J
\\[6pt]
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}
& + & J_{(u, v)}
\\[6pt]
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
& + & u \cdot v
\end{array}

\begin{array}{*{9}{l}}
\mathrm{D}J
& = &
u \cdot v \cdot \qquad 0
\\[6pt]
& + &
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
& + &
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
\\[6pt]
& + &
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
&&& + &
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
& + &
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}
&&&&& + &
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}
\end{array}

\begin{array}{*{9}{l}}
\mathrm{D}J
& = &
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
& + &
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}
\end{array}


Alternatively, the difference map \mathrm{D}J\! can be expanded over the cells of [\mathrm{d}u, \mathrm{d}v]\! to arrive at the formulation shown in Table 42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.


\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!

\begin{array}{*{9}{l}}
\mathrm{D}J
& = & \boldsymbol\varepsilon J
& + & \mathrm{E}J
\\[6pt]
& = & J_{(u, v)}
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}
\\[6pt]
& = & u \cdot v
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
& = & 0
& + & u \cdot \mathrm{d}v
& + & v \cdot \mathrm{d}u
& + & \mathrm{d}u \cdot \mathrm{d}v
\\[6pt]
\mathrm{D}J
& = & 0
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v
\end{array}


Even more simply, the same result is reached by matching up the propositional coefficients of \boldsymbol\varepsilon J and \mathrm{E}J\! along the cells of [\mathrm{d}u, \mathrm{d}v]\! and adding the pairs under boolean addition, that is, “mod 2”, where 1 + 1 = 0, as shown in Table 43.


\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!

\begin{array}{*{5}{l}}
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J
\end{array}

\begin{array}{*{9}{l}}
\boldsymbol\varepsilon J
& = &   u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + &   u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{E}J
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + & u ~ \texttt{(} v \texttt{)}   \cdot   \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v
\end{array}

\begin{array}{*{9}{l}}
\mathrm{D}J
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
& + & ~~ u ~  \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & ~~~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}\!


The difference map \mathrm{D}J\! can also be given a dispositional interpretation. First, recall that \boldsymbol\varepsilon J\! exhibits the dispositions to change from anywhere in J\! to anywhere at all in the universe of discourse and \mathrm{E}J\! exhibits the dispositions to change from anywhere in the universe to anywhere in J.\! Next, observe that each of these classes of dispositions may be divided in accordance with the case of J\! versus \texttt{(} J \texttt{)}\! that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to \boldsymbol\varepsilon J and \mathrm{E}J\! have in common the dispositions to preserve J,\! their symmetric difference \texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\! is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of J\! in one direction or the other. In other words, we may conclude that \mathrm{D}J\! expresses the collective disposition to make a definite change with respect to J,\! no matter what value it holds in the current state of affairs.


\begin{array}{lllll}
\boldsymbol\varepsilon J
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}
\\[6pt]
\mathrm{E}J
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}
\\[6pt]
\mathrm{D}J
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}
\end{array}


Figures 44-a through 44-d illustrate the difference proposition \mathrm{D}J.\!

Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif
\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!


Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif
\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!


Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif
\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!


Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif
\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!


D(uv) • Version 2

\text{Computation of}~ \mathrm{D}(uv)\!

\begin{array}{|*{4}{c}|}
\hline &&&
\\
\texttt{~} u \texttt{~~} v \texttt{~} &
\texttt{(} u \texttt{)~} v \texttt{~} &
\texttt{~} u \texttt{~(} v \texttt{)} &
\texttt{(} u \texttt{)(} v \texttt{)}
\\[8pt]
\hline &&&
\\
0 &&&
\\
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} &
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} &&
\\
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} &&
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} &
\\
\texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} &&&
\texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\\[8pt]
\hline &&&
\\
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} &
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} &
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} &
\texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\\[8pt]
\hline
\end{array}~\!


With the enlargement map \mathrm{E}J\! and the tacit extension map \boldsymbol\varepsilon J\! well in place, the difference map \mathrm{D}J\! can be computed along the lines displayed in Table 41, ending up with an expansion of \mathrm{D}J\! over the cells of [u, v].\!


\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!

\begin{array}{*{9}{l}}
\mathrm{D}J
& = & \mathrm{E}J
& + & \boldsymbol\varepsilon J
\\[6pt]
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}
& + & J_{(u, v)}
\\[6pt]
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
& + & u \cdot v
\end{array}

\begin{array}{*{9}{l}}
\mathrm{D}J
& = &
u \cdot v \cdot \qquad 0
\\[6pt]
& + &
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
& + &
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
&&& + &
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
\\[6pt]
& + &
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}
&&&&& + &
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}
\end{array}

\begin{array}{*{9}{l}}
\mathrm{D}J
& = &
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}
& + &
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v
& + &
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}
\end{array}


Alternatively, the difference map \mathrm{D}J\! can be expanded over the cells of [\mathrm{d}u, \mathrm{d}v]\! to arrive at the formulation shown in Table 42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.


\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!

\begin{array}{*{9}{l}}
\mathrm{D}J
& = & \boldsymbol\varepsilon J
& + & \mathrm{E}J
\\[6pt]
& = & J_{(u, v)}
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}
\\[6pt]
& = & u \cdot v
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
& = & 0
& + & u \cdot \mathrm{d}v
& + & v \cdot \mathrm{d}u
& + & \mathrm{d}u \cdot \mathrm{d}v
\\[6pt]
\mathrm{D}J
& = & 0
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v
\end{array}


Differential of Conjunction

\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!

\begin{array}{c*{8}{l}}
\mathrm{D}J
& = &
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + &
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v
& + &
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}J
& = &
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + &
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v
& + &
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u
& + &
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}


Figures 46-a through 46-d illustrate the proposition {\mathrm{d}J},\! rounded out in our usual array of prospects. This proposition of \mathrm{E}U^\bullet\! is what we refer to as the (first order) differential of J,\! and normally regard as the differential proposition corresponding to J.\!

Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif
\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!


Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif
\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!


Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif
\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!


Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif
\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!

Work Area • 3 Variables

Difference Map of Conjunction

Version 0

\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{lc*{6}{l}}
& 0 & +
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& +
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& +
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[8pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& +
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& +
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& +
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[8pt]
+ & 0 & +
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)}
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& +
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~}
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& +
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)}
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[8pt]
+ &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~}
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& +
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)}
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& +
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~}
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& +
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

Version 1 (a)

\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

Version 1 (b)

\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\end{array}\!

Version 2

\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

Version 3

\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

Version 4

\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!

\begin{array}{*{4}{l}}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& + &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& + &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

Summary Tables


\text{Difference Map} ~ \mathrm{D}(pqr)\!

\begin{array}{|*{8}{c}|}
\hline &&&&&&&
\\
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
\\[8pt]
\hline &&&&&&&
\\
0 &&&&&&&
\\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &&&&&&
\\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&&&
\\
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &&&&
\\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} &&&&
\mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &&&
\\
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} &&&&&
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r &&
\\
\texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r &
\\
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} &&&&&&&
\mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r
\\[8pt]
\hline &&&&&&&
\\
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} &
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r &
\mathrm{d}p ~ \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r &
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q ~ \mathrm{d}r &
\mathrm{d}p ~ \mathrm{d}q ~ \mathrm{d}r
\\[8pt]
\hline
\end{array}\!


Various Arrays

  \mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{*{8}{l}}
&
p q r \,\cdot\,
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))}
& + &
\texttt{(} p \texttt{)} q r \,\cdot\,
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& + &
p \texttt{(} q \texttt{)} r \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
& + &
p q \texttt{(} r \texttt{)} \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r
\\[6pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\,
\mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r
& + &
p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\,
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r
& + &
\texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\,
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r
& + &
\texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\,
\mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
\end{array}\!

  \mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{cccc}
&
p q r
& \cdot &
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))}
\\[4pt]
+ &
\texttt{(} p \texttt{)} q r
& \cdot &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
p \texttt{(} q \texttt{)} r
& \cdot &
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
p q \texttt{(} r \texttt{)}
& \cdot &
\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)} r
& \cdot &
\mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{(} p \texttt{)} q \texttt{(} r \texttt{)}
& \cdot &
\mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r
\\[4pt]
+ &
p \texttt{(} q \texttt{)(} r \texttt{)}
& \cdot &
\texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
& \cdot &
\mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r
\end{array}\!

  \mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{cccc}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
& \cdot &
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))}
\\[4pt]
+ &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)~}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~}
& \cdot &
\texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)}
& \cdot &
\texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~}
\\[4pt]
+ &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)}
& \cdot &
\texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~}
\end{array}\!

  \mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{cccc}
&
p \texttt{~} q \texttt{~} r
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
p \texttt{~} q
& \cdot &
\texttt{~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\\[4pt]
+ &
p \texttt{~} r
& \cdot &
\texttt{~} \mathrm{d}q \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
q \texttt{~} r
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{((} p \texttt{,} q \texttt{))} ~ r
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{((} p \texttt{,} r \texttt{))} ~ q
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}q \texttt{)}
\\[4pt]
+ &
\texttt{((} q \texttt{,} r \texttt{))} ~ p
& \cdot &
\texttt{~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!

Column Sum

  \mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{cccc}
&
\texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~}
& \cdot &
\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))}
\\[4pt]
+ &
\texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)~}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~}
& \cdot &
\texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~}
\\[4pt]
+ &
\texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)}
& \cdot &
\texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~}
\\[4pt]
+ &
\texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~}
\\[4pt]
+ &
\texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)}
& \cdot &
\texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
& \cdot &
\texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~}
\end{array}\!

Row Sum

  \mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{cccc}
&
p \texttt{~} q \texttt{~} r
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\\[4pt]
+ &
p \texttt{~} q
& \cdot &
\texttt{~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
\\[4pt]
+ &
p \texttt{~} r
& \cdot &
\texttt{~} \mathrm{d}q \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
q \texttt{~} r
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{((} p \texttt{,} q \texttt{))} ~ r
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
\\[4pt]
+ &
\texttt{((} p \texttt{,} r \texttt{))} ~ q
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}q \texttt{)}
\\[4pt]
+ &
\texttt{((} q \texttt{,} r \texttt{))} ~ p
& \cdot &
\texttt{~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)}
\\[4pt]
+ &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
& \cdot &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
\end{array}\!


  \mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =\!
 

\begin{array}{cccc}
&
\texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}
& \cdot &
q \texttt{~} r
\\[4pt]
+ &
\texttt{~} \mathrm{d}q \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}r \texttt{)}
& \cdot &
p \texttt{~} r
\\[4pt]
+ &
\texttt{~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}
& \cdot &
p \texttt{~} q
\\[4pt]
+ &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}
& \cdot &
\texttt{((} p \texttt{,} q \texttt{))} ~ r
\\[4pt]
+ &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}q \texttt{)}
& \cdot &
\texttt{((} p \texttt{,} r \texttt{))} ~ q
\\[4pt]
+ &
\texttt{~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)}
& \cdot &
\texttt{((} q \texttt{,} r \texttt{))} ~ p
\\[4pt]
+ &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& \cdot &
p \texttt{~} q \texttt{~} r
\\[4pt]
+ &
\texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}
& \cdot &
\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}
\end{array}\!

Work Area • Table Images

Version 1.0

Table 1.0 PQR Enlargement Map.png
Table 1.0 PQR Tacit Extension ISW.png
Table 1.0 PQR Difference Map.png

Version 2.0

Table 2.0 PQR Enlargement Map.jpg
Table 2.0 PQR Tacit Extension ALT.jpg
Table 2.0 PQR Tacit Extension.jpg
Table 2.0 PQR Difference Map.jpg

Version 3.0

Table 3.0 PQR Enlargement Map.png
Table 3.0 PQR Tacit Extension.png
Table 3.0 PQR Difference Map.png

Version 4.0

Table -- PQR Enlargement Map.png
Table -- PQR Tacit Extension.png
Table -- PQR Difference Map.png
Table 4.0 PQR Difference Map Col Sum.png

Work Area • Dispositions

The difference map \mathrm{D}f~\! may be given a dispositional interpretation. The enlargement map \mathrm{E}f\! exhibits the dispositions to change from anywhere at all in the universe of discourse to anywhere in the region where f\! is true and the tacit extension \boldsymbol\varepsilon f\! exhibits the dispositions to change from anywhere in the region where f\! is true to anywhere at all in the universe of discourse.

In the interest of a compact notation for concepts like “the region where f\! is true” and “the region where f\! is false” let's turn the game symbol \Game\! to the purpose in a way that makes the following formulas equivalent:

Definition. For a boolean-valued function f : X \to \mathbb{B},\! the following are equivalent:

  • \Game f\!
  • f^{-1}(1)\!
  • \{ x \in X : f(x) = 1 \}\!

Thus we have that \Game f\! is the region where f\! is true and \Game \texttt{(} f \texttt{)}\! is the region where f\! is false.

The sets of dispositions to and from the region where f~\! is true may each be divided in accordance with the case of f\! versus \texttt{(} f \texttt{)}\! that applies to their points of destination and departure. Since the dispositions corresponding to \mathrm{E}f\! and \boldsymbol\varepsilon f have in common the dispositions to preserve f,\! their symmetric difference \texttt{(} \mathrm{E}f \texttt{,} \boldsymbol\varepsilon f \texttt{)}\! is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of f\! in one direction or the other. In other words, we may conclude that \mathrm{D}f~\! expresses the collective disposition to make a definite change with respect to f,\! no matter what value it holds in the current state of affairs.


\begin{array}{*{5}{l}}
\mathrm{E}f
& = & \{ \text{Dispositions from}~ \Game \texttt{(} f \texttt{)} ~\text{to}~ \Game f \}
& + & \{ \text{Dispositions from}~ \Game f ~\text{to}~ \Game f \}
\\[6pt]
\boldsymbol\varepsilon f
& = & \{ \text{Dispositions from}~ \Game f ~\text{to}~ \Game f \}
& + & \{ \text{Dispositions from}~ \Game f ~\text{to}~ \Game \texttt{(} f \texttt{)} \}
\\[6pt]
\mathrm{D}f
& = & \{ \text{Dispositions from}~ \Game \texttt{(} f \texttt{)} ~\text{to}~ \Game f \}
& + & \{ \text{Dispositions from}~ \Game f ~\text{to}~ \Game \texttt{(} f \texttt{)} \}
\end{array}\!

Work Area • Exposition

What is the function of a boolean function?

To answer this question let's begin by recalling a couple of basic definitions:

  • A boolean-valued function is a function of the form f : X \to \mathbb{B},\! where X\! is an arbitrary set and where \mathbb{B}\! is a boolean domain, typically, \mathbb{B} = \{ 0, 1 \}.\!.
  • A boolean function is a function of the form f : \mathbb{B}^k \to \mathbb{B},\! where k\! is a nonnegative integer. In the case where k = 0,\! a boolean function is simply a constant element of \mathbb{B}.\!

For most intents and purposes, its function is to indicate a portion of the universe of discourse, namely, the portion where its value is 1, and by contrast to counterindicate the portion where its value is 0. That is why boolean functions are often called indicator functions. There are two types of situation where indicators functions come into play.

  • In the more general type of situation we have a boolean-valued function f : X \to \mathbb{B}\! that indicates the subset of X\! where f(x) = 1.~\! In formal terms, the indicated subset is described as f^{-1}(1) = \{ x \in X : f(x) = 1 \}.\!
  • In the more special type of situation we have a boolean function f : \mathbb{B}^k \to \mathbb{B}\! ...

The function of f\! is to indicate a portion of the universe of discourse X,\! that portion where its value is 1,\! and by contrast to counterindicate that portion where its value is 0.\!

In many situations the universe of discourse X\! is described in terms of a finite boolean coordinate basis. This amounts to a finite set of boolean coordinate functions \bigstar = \{ x_j : X \to \mathbb{B} \}_{j=1}^k.\! Relative to this coordinates basis the indicative function of f\! may be relegated to the boolean function f_\bigstar : \mathbb{B}^k \to \mathbb{B}.\!

Work Area • Logical Cacti

Theme One Program — Logical Cacti
http://stderr.org/pipermail/inquiry/2005-February/thread.html#2348
http://stderr.org/pipermail/inquiry/2005-February/002360.html
http://stderr.org/pipermail/inquiry/2005-February/002361.html

Original Version

Up till now we've been working to hammer out a two-edged sword of syntax, honing the syntax of painted and rooted cacti and expressions (PARCAE), and turning it to use in taming the syntax of two-level formal languages.

But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs that can denote objective propositions about some universe of objects.

One of the difficulties that we face in this discussion is that the words interpretation, meaning, semantics, and so on will have so many different meanings from one moment to the next of their use. A dedicated neologician might be able to think up distinctive names for all of the aspects of meaning and all of the approaches to them that will concern us here, but I will just have to do the best that I can with the common lot of ambiguous terms, leaving it to context and the intelligent interpreter to sort it out as much as possible.

As it happens, the language of cacti is so abstract that it can bear at least two different interpretations as logical sentences denoting logical propositions. The two interpretations that I know about are descended from the ones that Charles Sanders Peirce called the entitative and the existential interpretations of his systems of graphical logics. For our present aims, I shall briefly introduce the alternatives and then quickly move to the existential interpretation of logical cacti.

Table A illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.


\text{Table A.} ~~ \text{Existential Interpretation}\!
\text{Cactus Graph}\! \text{Cactus Expression}\! \text{Interpretation}\!
Cactus Node Big Fat.jpg ~\! \operatorname{true}\!
Cactus Spike Big Fat.jpg \texttt{(}~\texttt{)}\! \operatorname{false}\!
Cactus A Big.jpg a\! a\!
Cactus (A) Big.jpg \texttt{(} a \texttt{)}\!

\begin{matrix}
\tilde{a}
\\[2pt]
a^\prime
\\[2pt]
\lnot a
\\[2pt]
\operatorname{not}~ a
\end{matrix}\!

Cactus ABC Big.jpg a~b~c\!

\begin{matrix}
a \land b \land c
\\[6pt]
a ~\operatorname{and}~ b ~\operatorname{and}~ c
\end{matrix}\!

Cactus ((A)(B)(C)) Big.jpg \texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\!

\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\operatorname{or}~ b ~\operatorname{or}~ c
\end{matrix}\!

Cactus (A(B)) Big.jpg \texttt{(} a \texttt{(} b \texttt{))}\!

\begin{matrix}
a \Rightarrow b
\\[2pt]
a ~\operatorname{implies}~ b
\\[2pt]
\operatorname{if}~ a ~\operatorname{then}~ b
\\[2pt]
\operatorname{not}~ a ~\operatorname{without}~ b
\end{matrix}\!

Cactus (A,B) Big ISW.jpg \texttt{(} a \texttt{,} b \texttt{)}\!

\begin{matrix}
a + b
\\[2pt]
a \neq b
\\[2pt]
a ~\operatorname{exclusive-or}~ b
\\[2pt]
a ~\operatorname{not~equal~to}~ b
\end{matrix}\!

Cactus ((A,B)) Big.jpg \texttt{((} a \texttt{,} b \texttt{))}\!

\begin{matrix}
a = b
\\[2pt]
a \iff b
\\[2pt]
a ~\operatorname{equals}~ b
\\[2pt]
a ~\operatorname{if~and~only~if}~ b
\end{matrix}\!

Cactus (A,B,C) Big.jpg \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}\!

\begin{matrix}
\operatorname{just~one~of}
\\
a, b, c
\\
\operatorname{is~false}
\end{matrix}\!

Cactus ((A),(B),(C)) Big.jpg \texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}\!

\begin{matrix}
\operatorname{just~one~of}
\\
a, b, c
\\
\operatorname{is~true}
\end{matrix}\!

Cactus (A,(B),(C)) Big.jpg \texttt{(} a \texttt{,(} b \texttt{),(} c \texttt{))}\!

\begin{matrix}
\operatorname{genus}~ a ~\operatorname{of~species}~ b, c
\\[6pt]
\operatorname{partition}~ a ~\operatorname{into}~ b, c
\\[6pt]
\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c
\end{matrix}\!


Table B illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.


\text{Table B.} ~~ \text{Entitative Interpretation}\!
\text{Cactus Graph}\! \text{Cactus Expression}\! \text{Interpretation}\!
Cactus Node Big Fat.jpg ~\! \operatorname{false}\!
Cactus Spike Big Fat.jpg \texttt{(}~\texttt{)}\! \operatorname{true}\!
Cactus A Big.jpg a\! a\!
Cactus (A) Big.jpg \texttt{(} a \texttt{)}\!

\begin{matrix}
\tilde{a}
\\[2pt]
a^\prime
\\[2pt]
\lnot a
\\[2pt]
\operatorname{not}~ a
\end{matrix}\!

Cactus ABC Big.jpg a~b~c\!

\begin{matrix}
a \lor b \lor c
\\[6pt]
a ~\operatorname{or}~ b ~\operatorname{or}~ c
\end{matrix}\!

Cactus ((A)(B)(C)) Big.jpg \texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\!

\begin{matrix}
a \land b \land c
\\[6pt]
a ~\operatorname{and}~ b ~\operatorname{and}~ c
\end{matrix}\!

Cactus (A)B Big.jpg \texttt{(} a \texttt{)} b\!

\begin{matrix}
a \Rightarrow b
\\[2pt]
a ~\operatorname{implies}~ b
\\[2pt]
\operatorname{if}~ a ~\operatorname{then}~ b
\\[2pt]
\operatorname{not}~ a, ~\operatorname{or}~ b
\end{matrix}\!

Cactus (A,B) Big ISW.jpg \texttt{(} a \texttt{,} b \texttt{)}\!

\begin{matrix}
a = b
\\[2pt]
a \iff b
\\[2pt]
a ~\operatorname{equals}~ b
\\[2pt]
a ~\operatorname{if~and~only~if}~ b
\end{matrix}\!

Cactus ((A,B)) Big.jpg \texttt{((} a \texttt{,} b \texttt{))}\!

\begin{matrix}
a + b
\\[2pt]
a \neq b
\\[2pt]
a ~\operatorname{exclusive-or}~ b
\\[2pt]
a ~\operatorname{not~equal~to}~ b
\end{matrix}\!

Cactus (A,B,C) Big.jpg \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}\!

\begin{matrix}
\operatorname{not~just~one~of}
\\
a, b, c
\\
\operatorname{is~true}
\end{matrix}\!

Cactus ((A,B,C)) Big.jpg \texttt{((} a \texttt{,} b \texttt{,} c \texttt{))}\!

\begin{matrix}
\operatorname{just~one~of}
\\
a, b, c
\\
\operatorname{is~true}
\end{matrix}\!

Cactus (((A),B,C)) Big.jpg \texttt{(((} a \texttt{),} b \texttt{,} c \texttt{))}\!

\begin{matrix}
\operatorname{genus}~ a ~\operatorname{of~species}~ b, c
\\[6pt]
\operatorname{partition}~ a ~\operatorname{into}~ b, c
\\[6pt]
\operatorname{pie}~ a ~\operatorname{of~slices}~ b, c
\end{matrix}\!


For the time being, the main things to take away from Tables A and B are the ideas that the compositional structure of cactus graphs and expressions can be articulated in terms of two different kinds of connective operations, and that there are two distinct ways of mapping this compositional structure into the compositional structure of propositional sentences, say, in English:

1. The node connective joins a number of component cacti C_1, \ldots, C_k\! at a node:
 
    C_1 ... C_k
         @
2. The lobe connective joins a number of component cacti C_1, \ldots, C_k\! to a lobe:
 
    C_1 C_2   C_k
     o---o-...-o
      \       /
       \     /
        \   /
         \ /
          @

Table 15 summarizes the existential and entitative interpretations of the primitive cactus structures, in effect, the graphical constants and connectives.

Table 15.  Existential & Entitative Interpretations of Cactus Structures
o-----------------o-----------------o-----------------o-----------------o
|  Cactus Graph   |  Cactus String  |  Existential    |   Entitative    |
|                 |                 | Interpretation  | Interpretation  |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        @        |       " "       |      true       |      false      |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|        o        |                 |                 |                 |
|        |        |                 |                 |                 |
|        @        |       ( )       |      false      |      true       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|   C_1 ... C_k   |                 |                 |                 |
|        @        |   C_1 ... C_k   | C_1 & ... & C_k | C_1 v ... v C_k |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o
|                 |                 |                 |                 |
|  C_1 C_2   C_k  |                 |  Just one       |  Not just one   |
|   o---o-...-o   |                 |                 |                 |
|    \       /    |                 |  of the C_j,    |  of the C_j,    |
|     \     /     |                 |                 |                 |
|      \   /      |                 |  j = 1 to k,    |  j = 1 to k,    |
|       \ /       |                 |                 |                 |
|        @        | (C_1, ..., C_k) |  is not true.   |  is true.       |
|                 |                 |                 |                 |
o-----------------o-----------------o-----------------o-----------------o

It is possible to specify abstract rules of equivalence (AROEs) between cacti, rules for transforming one cactus into another that are formal in the sense of being indifferent to the above choices for logical or semantic interpretations, and that partition the set of cacti into formal equivalence classes.

A reduction is an equivalence transformation that is applied in the direction of decreasing graphical complexity.

A basic reduction is a reduction that applies to one of the two families of basic connectives.

Table 16 schematizes the two types of basic reductions in a purely formal, interpretation-independent fashion.

Table 16.  Basic Reductions
o---------------------------------------o
|                                       |
|    C_1 ... C_k                        |
|         @         =         @         |
|                                       |
|    if and only if                     |
|                                       |
|    C_j = @ for all j = 1 to k         |
|                                       |
o---------------------------------------o
|                                       |
|   C_1 C_2   C_k                       |
|    o---o-...-o                        |
|     \       /                         |
|      \     /                          |
|       \   /                           |
|        \ /                            |
|         @         =         @         |
|                                       |
|   if and only if                      |
|                                       |
|         o                             |
|         |                             |
|   C_j = @ for exactly one j in [1, k] |
|                                       |
o---------------------------------------o

The careful reader will have noticed that we have begun to use graphical paints like "a", "b", "c" and schematic proxies like "C_1", "C_j", "C_k" in a variety of novel and unjustified ways.

The careful writer would have already introduced a whole bevy of technical concepts and proved a whole crew of formal theorems to justify their use before contemplating this stage of development, but I have been hurrying to proceed with the informal exposition, and this expedition must leave steps to the reader's imagination.

Of course I mean the active imagination. So let me assist the prospective exercise with a few hints of what it would take to guarantee that these practices make sense.

Partial Rewrites

Table 13 illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

Even though I do most of my thinking in the existential interpretation, I will continue to speak of these forms as logical graphs, because I think it is an important fact about them that the formal validity of the axioms and theorems is not dependent on the choice between the entitative and the existential interpretations.

The first extension is the reflective extension of logical graphs (RefLog). It is obtained by generalizing the negation operator "\texttt{(~)}\!" in a certain way, calling "\texttt{(~)}\!" the controlled, moderated, or reflective negation operator of order 1, then adding another such operator for each finite k = 2, 3, \ldots .\!

In sum, these operators are symbolized by bracketed argument lists as follows: "\texttt{(~)}\!", "\texttt{(~,~)}\!", "\texttt{(~,~,~)}\!", …, where the number of slots is the order of the reflective negation operator in question.

The cactus graph and the cactus expression shown here are both described as a spike.

o---------------------------------------o
|                                       |
|                   o                   |
|                   |                   |
|                   @                   |
|                                       |
o---------------------------------------o
|                  ( )                  |
o---------------------------------------o

The rule of reduction for a lobe is:

o---------------------------------------o
|                                       |
|  x_1   x_2   ...   x_k                |
|   o-----o--- ... ---o                 |
|    \               /                  |
|     \             /                   |
|      \           /                    |
|       \         /                     |
|        \       /                      |
|         \     /                       |
|          \   /                        |
|           \ /                         |
|            @      =      @            |
|                                       |
o---------------------------------------o

if and only if exactly one of the x_j\! is a spike.

In Ref Log, an expression of the form \texttt{((}~ e_1 ~\texttt{),(}~ e_2 ~\texttt{),(}~ \ldots ~\texttt{),(}~ e_k ~\texttt{))}\! expresses the fact that exactly one of the e_j\! is true. Expressions of this form are called universal partition expressions, and they parse into a type of graph called a painted and rooted cactus (PARC):

o---------------------------------------o
|                                       |
|  e_1   e_2   ...   e_k                |
|   o     o           o                 |
|   |     |           |                 |
|   o-----o--- ... ---o                 |
|    \               /                  |
|     \             /                   |
|      \           /                    |
|       \         /                     |
|        \       /                      |
|         \     /                       |
|          \   /                        |
|           \ /                         |
|            @                          |
|                                       |
o---------------------------------------o
o---------------------------------------o
|                                       |
| ( x1, x2, ..., xk )  =  [blank]       |
|                                       |
| iff                                   |
|                                       |
| Just one of the arguments             |
| x1, x2, ..., xk  =  ()                |
|                                       |
o---------------------------------------o

The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:

Existential Interpretation: Just one of the k argument is false.
Entitative Interpretation: Not just one of the k arguments is true.

Tables

\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!
\text{Graph}\! \text{Expression}~\! \text{Interpretation}\! \text{Other Notations}\!
Cactus Node Big Fat.jpg ~\! \operatorname{true}\! 1\!
Cactus Spike Big Fat.jpg \texttt{(~)}\! \operatorname{false}\! 0\!
Cactus A Big.jpg a\! a\! a\!
Cactus (A) Big.jpg \texttt{(} a \texttt{)}\! \operatorname{not}~ a\! \lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime~\!
Cactus ABC Big.jpg a ~ b ~ c\! a ~\operatorname{and}~ b ~\operatorname{and}~ c\! a \land b \land c\!
Cactus ((A)(B)(C)) Big.jpg \texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\! a ~\operatorname{or}~ b ~\operatorname{or}~ c\! a \lor b \lor c\!
Cactus (A(B)) Big.jpg \texttt{(} a \texttt{(} b \texttt{))}\!

\begin{matrix}
a ~\operatorname{implies}~ b
\\[6pt]
\operatorname{if}~ a ~\operatorname{then}~ b
\end{matrix}\!

a \Rightarrow b\!
Cactus (A,B) Big ISW.jpg \texttt{(} a \texttt{,} b \texttt{)}\!

\begin{matrix}
a ~\operatorname{not~equal~to}~ b
\\[6pt]
a ~\operatorname{exclusive~or}~ b
\end{matrix}\!

\begin{matrix}
a \neq b
\\[6pt]
a + b
\end{matrix}\!

Cactus ((A,B)) Big.jpg \texttt{((} a \texttt{,} b \texttt{))}\!

\begin{matrix}
a ~\operatorname{is~equal~to}~ b
\\[6pt]
a ~\operatorname{if~and~only~if}~ b
\end{matrix}\!

\begin{matrix}
a = b
\\[6pt]
a \Leftrightarrow b
\end{matrix}\!

Cactus (A,B,C) Big.jpg \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}\!

\begin{matrix}
\operatorname{just~one~of}
\\
a, b, c
\\
\operatorname{is~false}
\end{matrix}\!

\begin{matrix}
& \bar{a} ~ b ~ c
\\
\lor & a ~ \bar{b} ~ c
\\
\lor & a ~ b ~ \bar{c}
\end{matrix}\!

Cactus ((A),(B),(C)) Big.jpg \texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}\!

\begin{matrix}
\operatorname{just~one~of}
\\
a, b, c
\\
\operatorname{is~true}
\\[6pt]
\operatorname{partition~all}
\\
\operatorname{into}~ a, b, c
\end{matrix}\!

\begin{matrix}
& a ~ \bar{b} ~ \bar{c}
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\lor & \bar{a} ~ \bar{b} ~ c
\end{matrix}\!

Cactus (A,(B,C)) Big.jpg \texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}\!

\begin{matrix}
\operatorname{oddly~many~of}
\\
a, b, c
\\
\operatorname{are~true}
\end{matrix}\!

a + b + c\!


\begin{matrix}
& a ~ b ~ c
\\
\lor & a ~ \bar{b} ~ \bar{c}
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\lor & \bar{a} ~ \bar{b} ~ c
\end{matrix}\!

Cactus (X,(A),(B),(C)) Big.jpg \texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}\!

\begin{matrix}
\operatorname{partition}~ x
\\
\operatorname{into}~ a, b, c
\\[6pt]
\operatorname{genus}~ x ~\operatorname{comprises}
\\
\operatorname{species}~ a, b, c
\end{matrix}\!

\begin{matrix}
& \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
\\
\lor & x ~ a ~ \bar{b} ~ \bar{c}
\\
\lor & x ~ \bar{a} ~ b ~ \bar{c}
\\
\lor & x ~ \bar{a} ~ \bar{b} ~ c
\end{matrix}~\!


\text{Table C.} ~~ \text{Dualing Interpretations}\!
\text{Graph}\! \text{String}\! \text{Existential}\! \text{Entitative}\!
Cactus Node Big Fat.jpg \texttt{~}\! \operatorname{true}\! \operatorname{false}\!
Cactus Spike Big Fat.jpg \texttt{(~)}\! \operatorname{false}\! \operatorname{true}\!
Cactus A Big.jpg a\! a\! a\!
Cactus (A) Big.jpg \texttt{(} a \texttt{)}\! \lnot a\! \lnot a\!
Cactus ABC Big.jpg a~b~c\! a \land b \land c\! a \lor  b \lor  c\!
Cactus ((A)(B)(C)) Big.jpg \texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\! a \lor  b \lor  c\! a \land b \land c\!
Cactus (A(B)) Big.jpg \texttt{(} a \texttt{(} b \texttt{))}\! a \Rightarrow b\!  
Cactus (A)B Big.jpg \texttt{(} a \texttt{)} b\!   a \Rightarrow b\!
Cactus (A,B) Big ISW.jpg \texttt{(} a \texttt{,} b \texttt{)}\! a \neq b\! a = b\!
Cactus ((A,B)) Big.jpg \texttt{((} a \texttt{,} b \texttt{))}\! a = b\! a \neq b\!
Cactus (A,B,C) Big.jpg \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}\!

\begin{matrix}
\operatorname{just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~false}
\end{matrix}\!

\begin{matrix}
\operatorname{not~just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~true}
\end{matrix}\!

Cactus ((A),(B),(C)) Big.jpg \texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}\!

\begin{matrix}
\operatorname{just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~true}
\end{matrix}\!

\begin{matrix}
\operatorname{not~just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~false}
\end{matrix}\!

Cactus ((A,B,C)) Big.jpg \texttt{((} a \texttt{,} b \texttt{,} c \texttt{))}\!

\begin{matrix}
\operatorname{not~just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~false}
\end{matrix}\!

\begin{matrix}
\operatorname{just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~true}
\end{matrix}\!

Cactus (((A),(B),(C))) Big.jpg \texttt{(((} a \texttt{),(} b \texttt{),(} c \texttt{)))}\!

\begin{matrix}
\operatorname{not~just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~true}
\end{matrix}\!

\begin{matrix}
\operatorname{just~one}
\\
\operatorname{of}~ a, b, c
\\
\operatorname{is~false}
\end{matrix}\!

Cactus (A,(B),(C)) Big.jpg \texttt{(} a \texttt{,(} b \texttt{),(} c \texttt{))}\!

\begin{matrix}
\operatorname{partition}~ a
\\
\operatorname{into}~ b, c
\end{matrix}\!

 
Cactus (((A),B,C)) Big.jpg \texttt{(((} a \texttt{),} b \texttt{,} c \texttt{))}\!  

\begin{matrix}
\operatorname{partition}~ a
\\
\operatorname{into}~ b, c
\end{matrix}\!