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, 20:02, 30 November 2015→Quick Overview: + rest of section
It is important to note that the last expressions are not equivalent to the 3-place parenthesis <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}.\!</math>
It is important to note that the last expressions are not equivalent to the 3-place parenthesis <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}.\!</math>
===Differential Expansions of Propositions===
====Bird's Eye View====
An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it feasible to compute the finite differences and the differentials of those functions and propositions.
For example, consider a proposition of the form <math>{}^{\backprime\backprime} \, p ~\mathrm{and}~ q \, {}^{\prime\prime}\!</math> that is graphed as two letters attached to a root node:
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Existential P And Q.jpg|500px]]
|}
Written as a string, this is just the concatenation <math>p~q\!</math>.
The proposition <math>pq\!</math> may be taken as a boolean function <math>f(p, q)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!</math> where <math>\mathbb{B} = \{ 0, 1 \}~\!</math> is read in such a way that <math>0\!</math> means <math>\mathrm{false}\!</math> and <math>1\!</math> means <math>\mathrm{true}.\!</math>
Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown in the following Figure:
{| align="center" cellpadding="10"
| [[Image:Venn Diagram P And Q.jpg|500px]]
|}
Now ask yourself: What is the value of the proposition <math>pq\!</math> at a distance of <math>\mathrm{d}p\!</math> and <math>\mathrm{d}q\!</math> from the cell <math>pq\!</math> where you are standing?
Don't think about it — just compute:
{| align="center" cellpadding="10"
| [[Image:Cactus Graph (P,dP)(Q,dQ).jpg|500px]]
|}
The cactus formula <math>\texttt{(p, dp)(q, dq)}\!</math> and its corresponding graph arise by substituting <math>p + \mathrm{d}p\!</math> for <math>p\!</math> and <math>q + \mathrm{d}q\!</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum <math>p + \mathrm{d}p\!</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:
{| align="center" cellpadding="10"
| [[Image:Cactus Graph (P,dP) ISW.jpg|500px]]
|}
Next question: What is the difference between the value of the proposition <math>pq\!</math> over there, at a distance of <math>\mathrm{d}p\!</math> and <math>\mathrm{d}q,\!</math> and the value of the proposition <math>pq\!</math> where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:
{| align="center" cellpadding="10"
| [[Image:Cactus Graph ((P,dP)(Q,dQ),PQ).jpg|500px]]
|}
There is one thing that I ought to mention at this point: Computed over <math>\mathbb{B},\!</math> plus and minus are identical operations. This will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger than usual, but we will get into that later.
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where <math>pq\!</math> is true? Well, substituting <math>1\!</math> for <math>p\!</math> and <math>1\!</math> for <math>q\!</math> in the graph amounts to erasing the labels <math>p\!</math> and <math>q\!,\!</math> as shown here:
{| align="center" cellpadding="10"
| [[Image:Cactus Graph (( ,dP)( ,dQ), ).jpg|500px]]
|}
And this is equivalent to the following graph:
{| align="center" cellpadding="10"
| [[Image:Cactus Graph ((dP)(dQ)) ISW.jpg|500px]]
|}
We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
<math>\begin{matrix}
p ~\mathrm{and}~ q
& \quad &
\xrightarrow{\quad\mathrm{Diff}\quad}
& \quad &
\mathrm{d}p ~\mathrm{or}~ \mathrm{d}q
\end{matrix}\!</math>
|}
{| align="center" cellpadding="10"
| [[Image:Cactus Graph PQ Diff ((dP)(dQ)).jpg|500px]]
|}
It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.
If the form of the above statement reminds you of De Morgan's rule, it is no accident, as differentiation and negation turn out to be closely related operations. Indeed, one can find discussions of logical difference calculus in the Boole–De Morgan correspondence and Peirce also made use of differential operators in a logical context, but the exploration of these ideas has been hampered by a number of factors, not the least of which has been the lack of a syntax that was adequate to handle the complexity of expressions that evolve.
====Worm's Eye View====
Let's run through the initial example again, this time attempting to interpret the formulas that develop at each stage along the way. We begin with a proposition or a boolean function <math>f(p, q) = pq.\!</math>
{| align="center" cellpadding="10"
| [[Image:Venn Diagram F = P And Q ISW.jpg|500px]]
|-
| [[Image:Cactus Graph F = P And Q ISW.jpg|500px]]
|}
A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\!</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.\!</math> The concrete type takes into account the qualitative dimensions or the “units” of the case, which can be explained as follows.
{| align="center" cellpadding="10" width="90%"
| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \mathrm{not}~ p,~ p \} ~\cong~ \mathbb{B}.\!</math>
|-
| Let <math>Q\!</math> be the set of values <math>\{ \texttt{(} q \texttt{)},~ q \} ~=~ \{ \mathrm{not}~ q,~ q \} ~\cong~ \mathbb{B}.\!</math>
|}
Then interpret the usual propositions about <math>p, q\!</math> as functions of the concrete type <math>f : P \times Q \to \mathbb{B}.\!</math>
We are going to consider various ''operators'' on these functions. Here, an operator <math>\mathrm{F}\!</math> is a function that takes one function <math>f\!</math> into another function <math>\mathrm{F}f.\!</math>
The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:
{| align="center" cellpadding="10" width="90%"
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\mathrm{D}.\!</math>
|-
| The ''enlargement operator'' <math>\Epsilon,\!</math> written here as <math>\mathrm{E}.\!</math>
|}
These days, <math>\mathrm{E}\!</math> is more often called the ''shift operator''.
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space <math>X = P \times Q,\!</math> its ''(first order) differential extension'' <math>\mathrm{E}X\!</math> is constructed according to the following specifications:
{| align="center" cellpadding="10" width="90%"
|
<math>\begin{array}{rcc}
\mathrm{E}X & = & X \times \mathrm{d}X
\end{array}\!</math>
|}
where:
{| align="center" cellpadding="10" width="90%"
|
<math>\begin{array}{rcc}
X
& = &
P \times Q
\\[4pt]
\mathrm{d}X
& = &
\mathrm{d}P \times \mathrm{d}Q
\\[4pt]
\mathrm{d}P
& = &
\{ \texttt{(} \mathrm{d}p \texttt{)},~ \mathrm{d}p \}
\\[4pt]
\mathrm{d}Q
& = &
\{ \texttt{(} \mathrm{d}q \texttt{)},~ \mathrm{d}q \}
\end{array}\!</math>
|}
The interpretations of these new symbols can be diverse, but the easiest option for now is just to say that <math>\mathrm{d}p\!</math> means “change <math>p\!</math>” and <math>\mathrm{d}q\!</math> means “change <math>q\!</math>”.
Drawing a venn diagram for the differential extension <math>\mathrm{E}X = X \times \mathrm{d}X\!</math> requires four logical dimensions, <math>P, Q, \mathrm{d}P, \mathrm{d}Q,\!</math> but it is possible to project a suggestion of what the differential features <math>\mathrm{d}p\!</math> and <math>\mathrm{d}q\!</math> are about on the 2-dimensional base space <math>X = P \times Q\!</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X,\!</math> reading an arrow as <math>\mathrm{d}p\!</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}\!</math> in either direction and reading an arrow as <math>\mathrm{d}q\!</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}\!</math> in either direction.
{| align="center" cellpadding="10"
| [[Image:Venn Diagram P Q dP dQ.jpg|500px]]
|}
Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways that propositions are formed on ordinary logical variables alone. For example, the proposition <math>\texttt{(} \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{))}\!</math> says the same thing as <math>\mathrm{d}p \Rightarrow \mathrm{d}q,\!</math> in other words, that there is no change in <math>p\!</math> without a change in <math>q.\!</math>
Given the proposition <math>f(p, q)\!</math> over the space <math>X = P \times Q,\!</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\mathrm{E}f\!</math> over the differential extension <math>\mathrm{E}X\!</math> that is defined by the
following formula:
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
& = &
f(p + \mathrm{d}p,~ q + \mathrm{d}q)
& = &
f( \texttt{(} p, \mathrm{d}p \texttt{)},~ \texttt{(} q, \mathrm{d}q \texttt{)} )
\end{matrix}\!</math>
|}
In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\mathrm{E}f\!</math> is computed as follows:
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
& = &
(p + \mathrm{d}p)(q + \mathrm{d}q)
& = &
\texttt{(} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)}
\end{matrix}\!</math>
|-
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
|}
Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,\!</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\mathrm{D}f~\!</math> over <math>\mathrm{E}X\!</math> that is defined by the formula <math>\mathrm{D}f = \mathrm{E}f - f,\!</math> or, written out in full:
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
\mathrm{D}f(p, q, \mathrm{d}p, \mathrm{d}q)
& = &
f(p + \mathrm{d}p,~ q + \mathrm{d}q) - f(p, q)
& = &
\texttt{(} f( \texttt{(} p, \mathrm{d}p \texttt{)},~ \texttt{(} q, \mathrm{d}q \texttt{)} ),~ f(p, q) \texttt{)}
\end{matrix}\!</math>
|}
In the example <math>f(p, q) = pq,\!</math> the difference <math>\mathrm{D}f~\!</math> is computed as follows:
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
\mathrm{D}f(p, q, \mathrm{d}p, \mathrm{d}q)
& = &
(p + \mathrm{d}p)(q + \mathrm{d}q) - pq
& = &
\texttt{((} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)}, pq \texttt{)}
\end{matrix}\!</math>
|-
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
|}
We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math> This evaluation is written in the form <math>\mathrm{D}f|_{pq}\!</math> or <math>\mathrm{D}f|_{(1, 1)},\!</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
{| align="center" cellpadding="10" style="text-align:center"
|
<math>f(p, q) ~=~ pq ~=~ p ~\mathrm{and}~ q \quad \Rightarrow \quad \mathrm{D}f|_{pq} ~=~ \texttt{((} \mathrm{dp} \texttt{)(} \mathrm{d}q \texttt{))} ~=~ \mathrm{d}p ~\mathrm{or}~ \mathrm{d}q\!</math>
|-
| [[Image:Venn Diagram PQ Difference Conj At Conj.jpg|500px]]
|-
| [[Image:Cactus Graph PQ Difference Conj At Conj.jpg|500px]]
|}
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}\!</math> into the following exclusive disjunction:
{| align="center" cellpadding="10" style="text-align:center"
|
<math>\begin{matrix}
\mathrm{d}p ~\texttt{(} \mathrm{d}q \texttt{)}
& + &
\texttt{(} \mathrm{d}p \texttt{)}~ \mathrm{d}q
& + &
\mathrm{d}p ~\mathrm{d}q
\end{matrix}\!</math>
|}
The differential proposition that results may be interpreted to say “change <math>p\!</math> or change <math>q\!</math> or both”. And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.
Last time we computed what is variously called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\mathrm{D}f_x\!</math> of the proposition <math>f(p, q) = pq\!</math> at the point <math>x\!</math> where <math>p = 1\!</math> and <math>q = 1.\!</math>
In the universe <math>X = P \times Q,\!</math> the four propositions <math>pq,~ p \texttt{(} q \texttt{)},~ \texttt{(} p \texttt{)} q,~ \texttt{(} p \texttt{)(} q \texttt{)}\!</math> that indicate the “cells”, or the smallest regions of the venn diagram, are called ''singular propositions''. These serve as an alternative notation for naming the points <math>(1, 1),~ (1, 0),~ (0, 1),~ (0, 0),\!</math> respectively.
Thus we can write <math>\mathrm{D}f_x = \mathrm{D}f|x = \mathrm{D}f|(1, 1) = \mathrm{D}f|pq,\!</math> so long as we know the frame of reference in force.
In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\mathrm{D}f_x\!</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
|-
| [[Image:Cactus Graph Df@PQ = ((dP)(dQ)).jpg|500px]]
|-
| [[Image:Cactus Graph Df@P(Q) = (dP)dQ.jpg|500px]]
|-
| [[Image:Cactus Graph Df@(P)Q = dP(dQ) ISW Alt.jpg|500px]]
|-
| [[Image:Cactus Graph Df@(P)(Q) = dP dQ.jpg|500px]]
|}
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
{| align="center" cellpadding="10"
| [[Image:Cactus Graph Lobe Rule.jpg|500px]]
|-
| [[Image:Cactus Graph Spike Rule.jpg|500px]]
|}
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
{| align="center" cellpadding="10"
| [[Image:Venn Diagram PQ Difference Conj.jpg|500px]]
|}
The Figure shows the points of the extended universe <math>\mathrm{E}X = P \times Q \times \mathrm{d}P \times \mathrm{d}Q\!</math> that are indicated by the difference map <math>\mathrm{D}f : \mathrm{E}X \to \mathbb{B},\!</math> namely, the following six points or singular propositions::
{| align="center" cellpadding="10"
|
<math>\begin{array}{rcccc}
1. & p & q & \mathrm{d}p & \mathrm{d}q
\\
2. & p & q & \mathrm{d}p & (\mathrm{d}q)
\\
3. & p & q & (\mathrm{d}p) & \mathrm{d}q
\\
4. & p & (q) & (\mathrm{d}p) & \mathrm{d}q
\\
5. & (p) & q & \mathrm{d}p & (\mathrm{d}q)
\\
6. & (p) & (q) & \mathrm{d}p & \mathrm{d}q
\end{array}\!</math>
|}
The information borne by <math>\mathrm{D}f~\!</math> should be clear enough from a survey of these six points — they tell you what you have to do from each point of <math>X\!</math> in order to change the value borne by <math>f(p, q),\!</math> that is, the move you have to make in order to reach a point where the value of the proposition <math>f(p, q)\!</math> is different from what it is where you started.
We have been studying the action of the difference operator <math>\mathrm{D}\!</math> on propositions of the form <math>f : P \times Q \to \mathbb{B},\!</math> as illustrated by the example <math>f(p, q) = pq\!</math> that is known in logic as the conjunction of <math>p\!</math> and <math>q.\!</math> The resulting difference map <math>\mathrm{D}f~\!</math> is a (first order) differential proposition, that is, a proposition of the form <math>\mathrm{D}f : P \times Q \times \mathrm{d}P \times \mathrm{d}Q \to \mathbb{B}.\!</math>
Abstracting from the augmented venn diagram that shows how the ''models'' or ''satisfying interpretations'' of <math>\mathrm{D}f~\!</math> distribute over the extended universe of discourse <math>\mathrm{E}X = P \times Q \times \mathrm{d}P \times \mathrm{d}Q,\!</math> the difference map <math>\mathrm{D}f~\!</math> can be represented in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>X = P \times Q\!</math> and whose arrows are labeled with the elements of <math>\mathrm{d}X = \mathrm{d}P \times \mathrm{d}Q,\!</math> as shown in the following Figure.
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Directed Graph PQ Difference Conj.jpg|500px]]
|-
|
<math>\begin{array}{rcccccc}
f
& = & p & \cdot & q
\\[4pt]
\mathrm{D}f
& = & p & \cdot & q & \cdot & ((\mathrm{d}p)(\mathrm{d}q))
\\[4pt]
& + & p & \cdot & (q) & \cdot & ~(\mathrm{d}p)~\mathrm{d}q~~
\\[4pt]
& + & (p) & \cdot & q & \cdot & ~~\mathrm{d}p~(\mathrm{d}q)~
\\[4pt]
& + & (p) & \cdot & (q) & \cdot & ~~\mathrm{d}p~~\mathrm{d}q~~
\end{array}\!</math>
|}
Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal an unsuspected aspect of the proposition's meaning. We will encounter more and more of these alternative readings as we go.
The ''enlargement'' or ''shift'' operator <math>\mathrm{E}\!</math> exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, <math>f(p, q) = pq.\!</math>
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
{| align="center" cellpadding="10" width="90%"
|
<math>\begin{array}{lccl}
\text{Let} &
X
& = &
X_1 \times \ldots \times X_k.
\\[6pt]
\text{Let} &
\mathrm{d}X
& = &
\mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k.
\\[6pt]
\text{Then} &
\mathrm{E}X
& = &
X \times \mathrm{d}X
\\[6pt]
&
& = & X_1 \times \ldots \times X_k ~\times~ \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k
\end{array}\!</math>
|}
For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},\!</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\mathrm{E}f : \mathrm{E}X \to \mathbb{B}\!</math> that is defined by the following equation:
{| align="center" cellpadding="10" width="90%"
|
<math>\begin{array}{l}
\mathrm{E}f(x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k)
\\[6pt]
= \quad f(x_1 + \mathrm{d}x_1, \ldots, x_k + \mathrm{d}x_k)
\\[6pt]
= \quad f( \texttt{(} x_1, \mathrm{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k, \mathrm{d}x_k \texttt{)} )
\end{array}\!</math>
|}
The ''differential variables'' <math>\mathrm{d}x_j\!</math> are boolean variables of the same basic type as the ordinary variables <math>x_j.\!</math> Although it is conventional to distinguish the (first order) differential variables with the operative prefix “<math>\mathrm{d}\!</math>” this way of notating differential variables is entirely optional. It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.
In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\mathrm{E}f\!</math> is formulated as follows:
{| align="center" cellpadding="10" width="90%"
|
<math>\begin{array}{l}
\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
\\[6pt]
= \quad (p + \mathrm{d}p)(q + \mathrm{d}q)
\\[6pt]
= \quad \texttt{(} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)}
\end{array}\!</math>
|}
Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is permissible to “multiply things out” in the usual manner to arrive at the following result:
{| align="center" cellpadding="10" width="90%"
|
<math>\begin{matrix}
\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)
& = &
p~q
& + &
p~\mathrm{d}q
& + &
q~\mathrm{d}p
& + &
\mathrm{d}p~\mathrm{d}q
\end{matrix}\!</math>
|}
To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\mathrm{E}f\!</math> in the same way that we did for <math>\mathrm{D}f.\!</math> Toward that end, the value of <math>\mathrm{E}f_x\!</math> at each <math>x \in X\!</math> may be computed in graphical fashion as shown below:
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
|-
| [[Image:Cactus Graph Ef@PQ = (dP)(dQ).jpg|500px]]
|-
| [[Image:Cactus Graph Ef@P(Q) = (dP)dQ.jpg|500px]]
|-
| [[Image:Cactus Graph Ef@(P)Q = dP(dQ).jpg|500px]]
|-
| [[Image:Cactus Graph Ef@(P)(Q) = dP dQ.jpg|500px]]
|}
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\mathrm{E}f.\!</math>
{| align="center" cellpadding="10" width="90%"
|
<math>\begin{matrix}
\mathrm{E}f
& = &
pq \cdot \mathrm{E}f_{pq}
& + &
p(q) \cdot \mathrm{E}f_{p(q)}
& + &
(p)q \cdot \mathrm{E}f_{(p)q}
& + &
(p)(q) \cdot \mathrm{E}f_{(p)(q)}
\end{matrix}\!</math>
|}
Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element <math>(\mathrm{d}p)(\mathrm{d}q)\!</math> is drawn as a loop at the point <math>p~q.\!</math>
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
|-
|
<math>\begin{array}{rcccccc}
f
& = & p & \cdot & q
\\[4pt]
\mathrm{E}f
& = & p & \cdot & q & \cdot & (\mathrm{d}p)(\mathrm{d}q)
\\[4pt]
& + & p & \cdot & (q) & \cdot & (\mathrm{d}p)~\mathrm{d}q~
\\[4pt]
& + & (p) & \cdot & q & \cdot & ~\mathrm{d}p~(\mathrm{d}q)
\\[4pt]
& + & (p) & \cdot & (q) & \cdot & ~\mathrm{d}p~~\mathrm{d}q~\end{array}\!</math>
|}
We may understand the enlarged proposition <math>\mathrm{E}f\!</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</math>
==Logical Cacti==
==Logical Cacti==
