Field Picture
Let us summarize the outlook on differential logic we’ve reached so far. We’ve been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse 
The operators we’ve examined so far are the enlargement or shift operator 
At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and help us keep our wits about us as we venture into ever more rarefied airs of abstraction.
One good picture comes to us by way of the field concept. Given a space 
In the field picture a proposition
For example, consider the logical conjunction 
Each of the operators 
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are shown below. Table A5 shows the expansion of 
over the set 
of ordinary variables and Table A6 shows the expansion of 
of differential variables.
over the set 
and contemplated the value of that proposition in various other states, as determined by the differential propositions 
and 
describing the steps we might take to change our state.
to the boolean function of four variables 
as shown in the entry for 
in the first three columns of Table A3.
and examined how the differential operators 
Each operator takes the boolean function of two variables 
and gives back a boolean function of four variables, 
and 
respectively.
in the above Tables.
and abstract type 
Our inquiry into the differential aspects of logical conjunction will pay dividends as we study the actions of 
![\begin{array}{cccl} \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}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bcccl%7D++%5Ctext%7BLet%7D+%26+X+%26+%3D+%26+X_1+%5Ctimes+%5Cldots+%5Ctimes+X_k.++%5C%5C%5B6pt%5D++%5Ctext%7BLet%7D+%26+%5Cmathrm%7Bd%7DX+%26+%3D+%26+%5Cmathrm%7Bd%7DX_1+%5Ctimes+%5Cldots+%5Ctimes+%5Cmathrm%7Bd%7DX_k.++%5C%5C%5B6pt%5D++%5Ctext%7BThen%7D+%26+%5Cmathrm%7BE%7DX+%26+%3D+%26+X+%5Ctimes+%5Cmathrm%7Bd%7DX++%5C%5C%5B6pt%5D++%26+%26+%3D+%26+X_1+%5Ctimes+%5Cldots+%5Ctimes+X_k+%7E%5Ctimes%7E+%5Cmathrm%7Bd%7DX_1+%5Ctimes+%5Cldots+%5Ctimes+%5Cmathrm%7Bd%7DX_k++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
the (first order) enlargement of 
defined by the following equation.
are boolean variables of the same type as the ordinary variables 
Although it is conventional to distinguish the (first order) differential variables with the operational prefix 
that way of notating differential variables is entirely optional. It is their existence in particular relations to the initial variables, not their names, which defines them as differential variables.
the enlargement 
To that end, the value of 
at each 
may be computed in graphical fashion as shown below.
is drawn as a loop at the point 
![\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \mathrm{E}f & = & p & \cdot & q & \cdot & \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p \texttt{~~} \mathrm{d}q \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcccccc%7D++f+%26+%3D+%26+p++%26+%5Ccdot+%26+q++%5C%5C%5B4pt%5D++%5Cmathrm%7BE%7Df+%26+%3D+%26+p++%26+%5Ccdot+%26++q++%26+%5Ccdot+%26++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++p++%26+%5Ccdot+%26+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D++%26+%5Ccdot+%26++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7D+%5Ctexttt%7B%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26++q++%26+%5Ccdot+%26++%5Ctexttt%7B%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%7D+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D++%26+%5Ccdot+%26+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dq++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
of the proposition 
at the point 
where 
and 
the four propositions 
can be taken to indicate the so‑called “cells” or smallest distinguished regions of the universe, otherwise indicated by their coordinates as the “points” 
respectively. In that regard the four propositions are called singular propositions because they serve to single out the minimal regions of the universe of discourse.
so long as we know the frame of reference in force.
indicated by the difference map 
namely, the following six points or singular propositions.
that is, the move you have to make in order to reach a point where the value of the proposition 
is different from what it is where you started.
as illustrated by the example 
and 
The resulting difference map 
Abstracting from that picture, the difference map 
and whose arrows are labeled with the elements of 
as shown in the following Figure.
![\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \mathrm{D}f & = & p & \cdot & q & \cdot & \texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcccccc%7D++f+%26+%3D+%26+p+%26+%5Ccdot+%26+q++%5C%5C%5B4pt%5D++%5Cmathrm%7BD%7Df+%26+%3D+%26++p+%26+%5Ccdot+%26+q+%26+%5Ccdot+%26++%5Ctexttt%7B%28%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%29%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++p+%26+%5Ccdot+%26+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26++%5Ctexttt%7B%7E%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%7E%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26+q+%26+%5Ccdot+%26++%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7E%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26++%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%7E%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
The concrete type takes into account the qualitative dimensions or “units” of the case, which can be explained as follows.
be the set of values 
be the set of values 
as functions of the concrete type 
is a function which takes one function 
written here as 
its (first order) differential extension 
is constructed according to the following specifications.
![\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}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcc%7D++X+%26+%3D+%26+P+%5Ctimes+Q++%5C%5C%5B4pt%5D++%5Cmathrm%7Bd%7DX+%26+%3D+%26+%5Cmathrm%7Bd%7DP+%5Ctimes+%5Cmathrm%7Bd%7DQ++%5C%5C%5B4pt%5D++%5Cmathrm%7Bd%7DP+%26+%3D+%26+%5C%7B+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7D%2C+%7E+%5Cmathrm%7Bd%7Dp+%5C%7D++%5C%5C%5B4pt%5D++%5Cmathrm%7Bd%7DQ+%26+%3D+%26+%5C%7B+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D%2C+%7E+%5Cmathrm%7Bd%7Dq+%5C%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
”.
requires four logical dimensions, 
but it is possible to project a suggestion of what the differential features 
in either direction and reading an arrow as 
in either direction, as indicated in the following figure.
says the same thing as 
in other words, there is no change in 
or, written out in full:
at a single location in the universe of discourse, namely, at the point picked out by the singular proposition 
in terms of coordinates, at the place where 
or 
and we arrived at the locally applicable law which may be stated and illustrated as follows.
into the following exclusive disjunction.
Inquiry Into Inquiry 