Differential Logic and Dynamic Systems • Part 5

Author: Jon Awbrey

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• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Appendices • References • Document History •

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Transformations of Discourse (concl.)

Taking Aim at Higher Dimensional Targets

	The past and present wilt . . . . I have filled them and      emptied them, And proceed to fill my next fold of the future.
	— Walt Whitman, Leaves of Grass, [Whi, 87]

In the next Section we consider a transformation $F$ of concrete type $F:[u,v]\to [x,y]$ and abstract type $F:[\mathbb {B} ^{2}]\to [\mathbb {B} ^{2}].$ From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with $1$-dimensional ranges, as follows:

${\begin{array}{ccccccl}F&=&(F_{1},F_{2})&=&(f,g)&:&[u,v]\to [x,y],\\[6pt]&&F_{1}&=&f&:&[u,v]\to [x],\\[6pt]&&F_{2}&=&g&:&[u,v]\to [y].\end{array}}$

Then we tackle the separate components, now viewed as propositions $F_{i}:U\to \mathbb {B} ,$ one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)

Let us now refer to the dimension of the target space or codomain as the toll (or tole) of a transformation, as distinguished from the dimension of the range or image that is customarily called the rank. When we keep to transformations with a toll of $1,$ as $J:[u,v]\to [x],$ we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.

Well, perhaps we can carry it a little further. After all, the operator result $\mathrm {W} J:\mathrm {E} U^{\bullet }\to \mathrm {E} X^{\bullet }$ is a map of toll $2,$ and cannot be unfolded in one piece as a proposition. But when a map has rank $1,$ like ${\boldsymbol {\varepsilon }}J:\mathrm {E} U\to X\subseteq \mathrm {E} X$ or $\mathrm {d} J:\mathrm {E} U\to \mathrm {d} X\subseteq \mathrm {E} X,$ we naturally choose to concentrate on the $1$-dimensional range of the operator result $\mathrm {W} J,$ ignoring the final difference in quality between the spaces $X$ and $\mathrm {d} X,$ and view $\mathrm {W} J$ as a proposition about $\mathrm {E} U.$

In this way, an initial ambivalence about the role of the operand $J$ conveys a double duty to the result $\mathrm {W} J.$ The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of $\mathrm {W} J.$ This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results $\mathrm {W} J$ as propositions or as transformations, indifferently.

But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map $F:[\mathbb {B} ^{2}]\to [\mathbb {B} ^{2}],$ and begin to pave the way to some extent for discussing any transformation of the form $F:[\mathbb {B} ^{n}]\to [\mathbb {B} ^{k}].$

${\text{Table 58.}}~~{\text{Cast of Characters : Expansive Subtypes of Objects and Operators}}$

${\text{Symbol}}$	${\text{Notation}}$	${\text{Description}}$	${\text{Type}}$
$U^{\bullet }$	$=[u,v]$	${\text{Source universe}}$	$[\mathbb {B} ^{n}]$
$X^{\bullet }$	${\begin{array}{l}=[x,y]\\=[f,g]\end{array}}$	${\text{Target universe}}$	$[\mathbb {B} ^{k}]$
$\mathrm {E} U^{\bullet }$	$=[u,v,\mathrm {d} u,\mathrm {d} v]$	${\text{Extended source universe}}$	$[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]$
$\mathrm {E} X^{\bullet }$	${\begin{array}{l}=[x,y,\mathrm {d} x,\mathrm {d} y]\\=[f,g,\mathrm {d} f,\mathrm {d} g]\end{array}}$	${\text{Extended target universe}}$	$[\mathbb {B} ^{k}\!\times \!\mathbb {D} ^{k}]$
${\begin{matrix}f\\g\end{matrix}}$	${\begin{array}{ll}f:U\!\to \![x]\cong \mathbb {B} \\g:U\!\to \![y]\cong \mathbb {B} \end{array}}$	${\text{Proposition}}$	${\begin{array}{l}\mathbb {B} ^{n}\!\to \!\mathbb {B} \\\in (\mathbb {B} ^{n},\mathbb {B} ^{n}\!\to \!\mathbb {B} )=[\mathbb {B} ^{n}]\end{array}}$
$F$	$F=(f,g):U^{\bullet }\!\to \!X^{\bullet }$	${\text{Transformation of Map}}$	$[\mathbb {B} ^{n}]\!\to \![\mathbb {B} ^{k}]$
${\begin{matrix}{\boldsymbol {\varepsilon }}\\\eta \\\mathrm {E} \\\mathrm {D} \\\mathrm {d} \end{matrix}}$	${\begin{array}{l}\mathrm {W} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },\\\mathrm {W} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet },\\\mathrm {W} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\\{\text{for each}}~\mathrm {W} ~{\text{in the set:}}\\\{{\boldsymbol {\varepsilon }},\eta ,\mathrm {E} ,\mathrm {D} ,\mathrm {d} \}\end{array}}$	${\begin{array}{ll}{\text{Tacit extension operator}}&{\boldsymbol {\varepsilon }}\\{\text{Trope extension operator}}&\eta \\{\text{Enlargement operator}}&\mathrm {E} \\{\text{Difference operator}}&\mathrm {D} \\{\text{Differential operator}}&\mathrm {d} \end{array}}$	${\begin{array}{l}{[\mathbb {B} ^{n}]\!\to \![\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]},\\{[\mathbb {B} ^{k}]\!\to \![\mathbb {B} ^{k}\!\times \!\mathbb {D} ^{k}]},\\\\([\mathbb {B} ^{n}]\!\to \![\mathbb {B} ^{k}])\!\to \!\\([\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {B} ^{k}\!\times \!\mathbb {D} ^{k}])\end{array}}$
${\begin{matrix}{\mathsf {e}}\\{\mathsf {E}}\\{\mathsf {D}}\\{\mathsf {T}}\end{matrix}}$	${\begin{array}{l}{\mathsf {W}}:U^{\bullet }\!\to \!{\mathsf {T}}U^{\bullet }=\mathrm {E} U^{\bullet },\\{\mathsf {W}}:X^{\bullet }\!\to \!{\mathsf {T}}X^{\bullet }=\mathrm {E} X^{\bullet },\\{\mathsf {W}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!({\mathsf {T}}U^{\bullet }\!\to \!{\mathsf {T}}X^{\bullet })\\{\text{for each}}~{\mathsf {W}}~{\text{in the set:}}\\\{{\mathsf {e}},{\mathsf {E}},{\mathsf {D}},{\mathsf {T}}\}\end{array}}$	${\begin{array}{lll}{\text{Radius operator}}&{\mathsf {e}}&=({\boldsymbol {\varepsilon }},\eta )\\{\text{Secant operator}}&{\mathsf {E}}&=({\boldsymbol {\varepsilon }},\mathrm {E} )\\{\text{Chord operator}}&{\mathsf {D}}&=({\boldsymbol {\varepsilon }},\mathrm {D} )\\{\text{Tangent functor}}&{\mathsf {T}}&=({\boldsymbol {\varepsilon }},\mathrm {d} )\end{array}}$	${\begin{array}{l}{[\mathbb {B} ^{n}]\!\to \![\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]},\\{[\mathbb {B} ^{k}]\!\to \![\mathbb {B} ^{k}\!\times \!\mathbb {D} ^{k}]},\\\\([\mathbb {B} ^{n}]\!\to \![\mathbb {B} ^{k}])\!\to \!\\([\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {B} ^{k}\!\times \!\mathbb {D} ^{k}])\end{array}}$

${\text{Table 59.}}~~{\text{Synopsis of Terminology : Restrictive and Alternative Subtypes}}$

	${\begin{matrix}{\text{Operator}}\\{\text{or}}\\{\text{Operand}}\end{matrix}}$	${\begin{matrix}{\text{Proposition}}\\{\text{or}}\\{\text{Component}}\end{matrix}}$	${\begin{matrix}{\text{Transformation}}\\{\text{or}}\\{\text{Map}}\end{matrix}}$
${\underline {\text{Operand}}}$	${\begin{array}{l}F=(F_{1},F_{2})\\F=(f,g):U\!\to \!X\end{array}}$	${\begin{array}{l}F_{i}:\langle u,v\rangle \!\to \!\mathbb {B} \\F_{i}:\mathbb {B} ^{n}\!\to \!\mathbb {B} \end{array}}$	${\begin{array}{l}F:[u,v]\!\to \![x,y]\\F:[\mathbb {B} ^{n}]\!\to \![\mathbb {B} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Tacit}}}\\{\text{extension}}\end{matrix}}$	${\begin{array}{l}{\boldsymbol {\varepsilon }}:U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~{\boldsymbol {\varepsilon }}:X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\{\boldsymbol {\varepsilon }}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!X^{\bullet })\end{array}}$	${\begin{array}{l}{\boldsymbol {\varepsilon }}F_{i}:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {B} \\{\boldsymbol {\varepsilon }}F_{i}:\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}\!\to \!\mathbb {B} \end{array}}$	${\begin{array}{l}{\boldsymbol {\varepsilon }}F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x,y]\\{\boldsymbol {\varepsilon }}F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {B} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Trope}}}\\{\text{extension}}\end{matrix}}$	${\begin{array}{l}\eta :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\eta :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\eta :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$	${\begin{array}{l}\eta F_{i}:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\eta F_{i}:\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}\!\to \!\mathbb {D} \end{array}}$	${\begin{array}{l}\eta F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x,\mathrm {d} y]\\\eta F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {D} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Enlargement}}}\\{\text{operator}}\end{matrix}}$	${\begin{array}{l}\mathrm {E} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\mathrm {E} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\mathrm {E} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$	${\begin{array}{l}\mathrm {E} F_{i}:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {E} F_{i}:\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}\!\to \!\mathbb {D} \end{array}}$	${\begin{array}{l}\mathrm {E} F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x,\mathrm {d} y]\\\mathrm {E} F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {D} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Difference}}}\\{\text{operator}}\end{matrix}}$	${\begin{array}{l}\mathrm {D} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\mathrm {D} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\mathrm {D} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$	${\begin{array}{l}\mathrm {D} F_{i}:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {D} F_{i}:\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}\!\to \!\mathbb {D} \end{array}}$	${\begin{array}{l}\mathrm {D} F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x,\mathrm {d} y]\\\mathrm {D} F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {D} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Differential}}}\\{\text{operator}}\end{matrix}}$	${\begin{array}{l}\mathrm {d} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\mathrm {d} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\mathrm {d} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$	${\begin{array}{l}\mathrm {d} F_{i}:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {d} F_{i}:\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}\!\to \!\mathbb {D} \end{array}}$	${\begin{array}{l}\mathrm {d} F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x,\mathrm {d} y]\\\mathrm {d} F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {D} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Remainder}}}\\{\text{operator}}\end{matrix}}$	${\begin{array}{l}\mathrm {r} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\mathrm {r} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\mathrm {r} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$	${\begin{array}{l}\mathrm {r} F_{i}:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {r} F_{i}:\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}\!\to \!\mathbb {D} \end{array}}$	${\begin{array}{l}\mathrm {r} F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x,\mathrm {d} y]\\\mathrm {r} F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {D} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Radius}}}\\{\text{operator}}\end{matrix}}$	${\begin{array}{l}{\mathsf {e}}=({\boldsymbol {\varepsilon }},\eta )\\{\mathsf {e}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\end{array}}$		${\begin{array}{l}{\mathsf {e}}F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x,y,\mathrm {d} x,\mathrm {d} y]\\{\mathsf {e}}F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {B} ^{k}\!\times \!\mathbb {D} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Secant}}}\\{\text{operator}}\end{matrix}}$	${\begin{array}{l}{\mathsf {E}}=({\boldsymbol {\varepsilon }},\mathrm {E} )\\{\mathsf {E}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\end{array}}$		${\begin{array}{l}{\mathsf {E}}F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x,y,\mathrm {d} x,\mathrm {d} y]\\{\mathsf {E}}F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {B} ^{k}\!\times \!\mathbb {D} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Chord}}}\\{\text{operator}}\end{matrix}}$	${\begin{array}{l}{\mathsf {D}}=({\boldsymbol {\varepsilon }},\mathrm {D} )\\{\mathsf {D}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\end{array}}$		${\begin{array}{l}{\mathsf {D}}F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x,y,\mathrm {d} x,\mathrm {d} y]\\{\mathsf {D}}F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {B} ^{k}\!\times \!\mathbb {D} ^{k}]\end{array}}$
${\begin{matrix}{\underline {\text{Tangent}}}\\{\text{functor}}\end{matrix}}$	${\begin{array}{l}{\mathsf {T}}=({\boldsymbol {\varepsilon }},\mathrm {d} )\\{\mathsf {T}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\end{array}}$	${\begin{array}{l}\mathrm {d} F_{i}:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {d} F_{i}:\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}\!\to \!\mathbb {D} \end{array}}$	${\begin{array}{l}{\mathsf {T}}F:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x,y,\mathrm {d} x,\mathrm {d} y]\\{\mathsf {T}}F:[\mathbb {B} ^{n}\!\times \!\mathbb {D} ^{n}]\!\to \![\mathbb {B} ^{k}\!\times \!\mathbb {D} ^{k}]\end{array}}$

Transformations of Type B² → B²

To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from $U^{\bullet }=[u,v]$ to $X^{\bullet }=[x,y]$ that is defined by the following system of equations:

${\begin{array}{lllll}x&=&f(u,v)&=&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\[8pt]y&=&g(u,v)&=&{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\end{array}}$

The component notation $F=(F_{1},F_{2})=(f,g):U^{\bullet }\to X^{\bullet }$ allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:

${\begin{array}{lllll}(x,y)&=&F(u,v)&=&(\;{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\;,\;{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\;)\end{array}}$

Logical Transformations

The information that defines the logical transformation $F$ can be represented in the form of a truth table, as shown in Table 60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.

${\text{Table 60.}}~~{\text{A Propositional Transformation}}$

$u$	$v$	$f$	$g$
${\begin{matrix}0\\[4pt]0\\[4pt]1\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]1\\[4pt]0\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]1\\[4pt]1\\[4pt]1\end{matrix}}$	${\begin{matrix}1\\[4pt]0\\[4pt]0\\[4pt]1\end{matrix}}$
$u$	$v$	${\texttt {((}}u{\texttt {)(}}v{\texttt {))}}$	${\texttt {((}}u{\texttt {,}}v{\texttt {))}}$

Figure 61 shows how we might paint a picture of the transformation $F$ in the manner of Figure 30.

${\text{Figure 61.}}~~{\text{A Propositional Transformation}}$

Figure 62 extracts the gist of Figure 61, exhibiting a style of diagram that is adequate for most purposes.

${\text{Figure 62.}}~~{\text{A Propositional Transformation (Short Form)}}$

Local Transformations

Figure 63 gives a more complete picture of the transformation $F,$ showing how the points of $U^{\bullet }$ are transformed into points of $X^{\bullet }.$ The bold lines crossing from one universe to the other trace the action that $F$ induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.

${\text{Figure 63.}}~~{\text{A Transformation of Positions}}$

Table 64 shows how the action of $F$ on cells or points can be computed in terms of coordinates.

${\text{Table 64.}}~~{\text{A Transformation of Positions}}$

$u$	$v$	$x$	$y$	$x~y$	$x{\texttt {(}}y{\texttt {)}}$	${\texttt {(}}x{\texttt {)}}y$	${\texttt {(}}x{\texttt {)(}}y{\texttt {)}}$	$X^{\bullet }=[x,y]$
${\begin{matrix}0\\[4pt]0\\[4pt]1\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]1\\[4pt]0\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]1\\[4pt]1\\[4pt]1\end{matrix}}$	${\begin{matrix}1\\[4pt]0\\[4pt]0\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]0\\[4pt]0\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]1\\[4pt]1\\[4pt]0\end{matrix}}$	${\begin{matrix}1\\[4pt]0\\[4pt]0\\[4pt]0\end{matrix}}$	${\begin{matrix}0\\[4pt]0\\[4pt]0\\[4pt]0\end{matrix}}$	${\begin{matrix}\uparrow \\[4pt]F=\\[4pt](f,g)\\[4pt]\uparrow \end{matrix}}$
$u$	$v$	${\texttt {((}}u{\texttt {)(}}v{\texttt {))}}$	${\texttt {((}}u{\texttt {,}}v{\texttt {))}}$	$u~v$	${\texttt {(}}u{\texttt {,}}v{\texttt {)}}$	${\texttt {(}}u{\texttt {)(}}v{\texttt {)}}$	$0$	$U^{\bullet }=[u,v]$

Table 65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the induced action of the transformation on the structures in question.

${\text{Table 65-a.}}~~{\text{An Induced Transformation on Propositions}}$

$X^{\bullet }$	$\longleftarrow$	$F=(f,g)$	$\longleftarrow$	$U^{\bullet }$
$f_{i}(x,y)$	${\begin{matrix}u=\\v=\end{matrix}}$	${\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}}$	${\begin{matrix}=u\\=v\end{matrix}}$	$f_{j}(u,v)$
	${\begin{matrix}x=\\y=\end{matrix}}$	${\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}}$	${\begin{matrix}=f(u,v)\\=g(u,v)\end{matrix}}$
${\begin{matrix}f_{0}\\[2pt]f_{1}\\[2pt]f_{2}\\[2pt]f_{3}\\[2pt]f_{4}\\[2pt]f_{5}\\[2pt]f_{6}\\[2pt]f_{7}\end{matrix}}$	${\begin{matrix}{\texttt {(~)}}\\[2pt]{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\[2pt]{\texttt {(}}x{\texttt {)~}}y{\texttt {~}}\\[2pt]{\texttt {(}}x{\texttt {)~~}}\\[2pt]{\texttt {~}}x{\texttt {~(}}y{\texttt {)}}\\[2pt]{\texttt {~~(}}y{\texttt {)}}\\[2pt]{\texttt {(}}x{\texttt {,~}}y{\texttt {)}}\\[2pt]{\texttt {(}}x{\texttt {~~}}y{\texttt {)}}\end{matrix}}$	${\begin{matrix}0~0~0~0\\[2pt]0~0~0~0\\[2pt]0~0~0~1\\[2pt]0~0~0~1\\[2pt]0~1~1~0\\[2pt]0~1~1~0\\[2pt]0~1~1~1\\[2pt]0~1~1~1\end{matrix}}$	${\begin{matrix}{\texttt {(~)}}\\[2pt]{\texttt {(~)}}\\[2pt]{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\\[2pt]{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\\[2pt]{\texttt {(}}u{\texttt {,~}}v{\texttt {)}}\\[2pt]{\texttt {(}}u{\texttt {,~}}v{\texttt {)}}\\[2pt]{\texttt {(}}u{\texttt {~~}}v{\texttt {)}}\\[2pt]{\texttt {(}}u{\texttt {~~}}v{\texttt {)}}\end{matrix}}$	${\begin{matrix}f_{0}\\[2pt]f_{0}\\[2pt]f_{1}\\[2pt]f_{1}\\[2pt]f_{6}\\[2pt]f_{6}\\[2pt]f_{7}\\[2pt]f_{7}\end{matrix}}$
${\begin{matrix}f_{8}\\[2pt]f_{9}\\[2pt]f_{10}\\[2pt]f_{11}\\[2pt]f_{12}\\[2pt]f_{13}\\[2pt]f_{14}\\[2pt]f_{15}\end{matrix}}$	${\begin{matrix}{\texttt {~~}}x{\texttt {~~}}y{\texttt {~~}}\\[2pt]{\texttt {((}}x{\texttt {,~}}y{\texttt {))}}\\[2pt]{\texttt {~~~~}}y{\texttt {~~}}\\[2pt]{\texttt {~(}}x{\texttt {~(}}y{\texttt {))}}\\[2pt]{\texttt {~~}}x{\texttt {~~~~}}\\[2pt]{\texttt {((}}x{\texttt {)~}}y{\texttt {)~}}\\[2pt]{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\\[2pt]{\texttt {((~))}}\end{matrix}}$	${\begin{matrix}1~0~0~0\\[2pt]1~0~0~0\\[2pt]1~0~0~1\\[2pt]1~0~0~1\\[2pt]1~1~1~0\\[2pt]1~1~1~0\\[2pt]1~1~1~1\\[2pt]1~1~1~1\end{matrix}}$	${\begin{matrix}{\texttt {~~}}u{\texttt {~~}}v{\texttt {~~}}\\[2pt]{\texttt {~~}}u{\texttt {~~}}v{\texttt {~~}}\\[2pt]{\texttt {((}}u{\texttt {,~}}v{\texttt {))}}\\[2pt]{\texttt {((}}u{\texttt {,~}}v{\texttt {))}}\\[2pt]{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\[2pt]{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\[2pt]{\texttt {((~))}}\\[2pt]{\texttt {((~))}}\end{matrix}}$	${\begin{matrix}f_{8}\\[2pt]f_{8}\\[2pt]f_{9}\\[2pt]f_{9}\\[2pt]f_{14}\\[2pt]f_{14}\\[2pt]f_{15}\\[2pt]f_{15}\end{matrix}}$

${\text{Table 65-b.}}~~{\text{An Induced Transformation on Propositions}}$

$X^{\bullet }$	$\longleftarrow$	$F=(f,g)$	$\longleftarrow$	$U^{\bullet }$
$f_{i}(x,y)$	${\begin{matrix}u=\\v=\end{matrix}}$	${\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}}$	${\begin{matrix}=u\\=v\end{matrix}}$	$f_{j}(u,v)$
	${\begin{matrix}x=\\y=\end{matrix}}$	${\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}}$	${\begin{matrix}=f(u,v)\\=g(u,v)\end{matrix}}$
$f_{0}$	${\texttt {(~)}}$	$0~0~0~0$	${\texttt {(~)}}$	$f_{0}$
${\begin{matrix}f_{1}\\[2pt]f_{2}\\[2pt]f_{4}\\[2pt]f_{8}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)(}}y{\texttt {)}}\\[2pt]{\texttt {(}}x{\texttt {)~}}y{\texttt {~}}\\[2pt]{\texttt {~}}x{\texttt {~(}}y{\texttt {)}}\\[2pt]{\texttt {~}}x{\texttt {~~}}y{\texttt {~}}\end{matrix}}$	${\begin{matrix}0~0~0~0\\[2pt]0~0~0~1\\[2pt]0~1~1~0\\[2pt]1~0~0~0\end{matrix}}$	${\begin{matrix}{\texttt {(~)}}\\[2pt]{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\\[2pt]{\texttt {(}}u{\texttt {,~}}v{\texttt {)}}\\[2pt]{\texttt {~}}u{\texttt {~~}}v{\texttt {~}}\end{matrix}}$	${\begin{matrix}f_{0}\\[2pt]f_{1}\\[2pt]f_{6}\\[2pt]f_{8}\end{matrix}}$
${\begin{matrix}f_{3}\\[2pt]f_{12}\end{matrix}}$	${\begin{matrix}{\texttt {(}}x{\texttt {)}}\\[2pt]{\texttt {~}}x{\texttt {~}}\end{matrix}}$	${\begin{matrix}0~0~0~1\\[2pt]1~1~1~0\end{matrix}}$	${\begin{matrix}{\texttt {~(}}u{\texttt {)(}}v{\texttt {)~}}\\[2pt]{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\end{matrix}}$	${\begin{matrix}f_{1}\\[2pt]f_{14}\end{matrix}}$
${\begin{matrix}f_{6}\\[2pt]f_{9}\end{matrix}}$	${\begin{matrix}{\texttt {~(}}x{\texttt {,~}}y{\texttt {)~}}\\[2pt]{\texttt {((}}x{\texttt {,~}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}0~1~1~1\\[2pt]1~0~0~0\end{matrix}}$	${\begin{matrix}{\texttt {(}}u{\texttt {~~}}v{\texttt {)}}\\[2pt]{\texttt {~}}u{\texttt {~~}}v{\texttt {~}}\end{matrix}}$	${\begin{matrix}f_{7}\\[2pt]f_{8}\end{matrix}}$
${\begin{matrix}f_{5}\\[2pt]f_{10}\end{matrix}}$	${\begin{matrix}{\texttt {(}}y{\texttt {)}}\\[2pt]{\texttt {~}}y{\texttt {~}}\end{matrix}}$	${\begin{matrix}0~1~1~0\\[2pt]1~0~0~1\end{matrix}}$	${\begin{matrix}{\texttt {~(}}u{\texttt {,~}}v{\texttt {)~}}\\[2pt]{\texttt {((}}u{\texttt {,~}}v{\texttt {))}}\end{matrix}}$	${\begin{matrix}f_{6}\\[2pt]f_{9}\end{matrix}}$
${\begin{matrix}f_{7}\\[2pt]f_{11}\\[2pt]f_{13}\\[2pt]f_{14}\end{matrix}}$	${\begin{matrix}{\texttt {~(}}x{\texttt {~~}}y{\texttt {)~}}\\[2pt]{\texttt {~(}}x{\texttt {~(}}y{\texttt {))}}\\[2pt]{\texttt {((}}x{\texttt {)~}}y{\texttt {)~}}\\[2pt]{\texttt {((}}x{\texttt {)(}}y{\texttt {))}}\end{matrix}}$	${\begin{matrix}0~1~1~1\\[2pt]1~0~0~1\\[2pt]1~1~1~0\\[2pt]1~1~1~1\end{matrix}}$	${\begin{matrix}{\texttt {~(}}u{\texttt {~~}}v{\texttt {)~}}\\[2pt]{\texttt {((}}u{\texttt {,~}}v{\texttt {))}}\\[2pt]{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\[2pt]{\texttt {((~))}}\end{matrix}}$	${\begin{matrix}f_{7}\\[2pt]f_{9}\\[2pt]f_{14}\\[2pt]f_{15}\end{matrix}}$
$f_{15}$	${\texttt {((~))}}$	$1~1~1~1$	${\texttt {((~))}}$	$f_{15}$

Difference Operators and Tangent Functors

Given the alphabets ${\mathcal {U}}=\{u,v\}$ and ${\mathcal {X}}=\{x,y\},$ along with the corresponding universes of discourse $U^{\bullet },X^{\bullet }\cong [\mathbb {B} ^{2}],$ how many logical transformations of the general form $G=(G_{1},G_{2}):U^{\bullet }\to X^{\bullet }$ are there? Since $G_{1}$ and $G_{2}$ can be any propositions of the type $\mathbb {B} ^{2}\to \mathbb {B} ,$ there are $2^{4}=16$ choices for each of the maps $G_{1}$ and $G_{2}$ and thus there are $2^{4}\cdot 2^{4}=2^{8}=256$ different mappings altogether of the form $G:U^{\bullet }\to X^{\bullet }.$ The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing $(U^{\bullet }\to X^{\bullet })=\{G:U^{\bullet }\to X^{\bullet }\},$ and so the cardinality of the function space $(U^{\bullet }\to X^{\bullet })$ is summed up by writing $|(U^{\bullet }\to X^{\bullet })|=|(\mathbb {B} ^{2}\to \mathbb {B} ^{2})|=4^{4}=256.$

Given a transformation $G=(G_{1},G_{2}):U^{\bullet }\to X^{\bullet }$ of this type, we proceed to define a pair of further transformations, related to $G,$ that operate between the extended universes, $\mathrm {E} U^{\bullet }$ and $\mathrm {E} X^{\bullet },$ of its source and target domains.

First, the enlargement map (or secant transformation) $\mathrm {E} G=(\mathrm {E} G_{1},\mathrm {E} G_{2}):\mathrm {E} U^{\bullet }\to \mathrm {E} X^{\bullet }$ is defined by the following set of component equations:

${\begin{array}{lll}\mathrm {E} G_{i}&=&G_{i}(u+\mathrm {d} u,v+\mathrm {d} v)\end{array}}$

Next, the difference map (or chordal transformation) $\mathrm {D} G=(\mathrm {D} G_{1},\mathrm {D} G_{2}):\mathrm {E} U^{\bullet }\to \mathrm {E} X^{\bullet }$ is defined in component-wise fashion as the boolean sum of the initial proposition $G_{i}$ and the enlarged proposition $\mathrm {E} G_{i},$ for $i=1,2,$ according to the following set of equations:

${\begin{array}{lllll}\mathrm {D} G_{i}&=&G_{i}(u,v)&+&\mathrm {E} G_{i}(u,v,\mathrm {d} u,\mathrm {d} v)\\[8pt]&=&G_{i}(u,v)&+&G_{i}(u+\mathrm {d} u,v+\mathrm {d} v)\end{array}}$

Maintaining a strict analogy with ordinary difference calculus would perhaps have us write $\mathrm {D} G_{i}=\mathrm {E} G_{i}-G_{i},$ but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition $q,$ then to compute the enlargement $\mathrm {E} q,$ and finally to determine the difference $\mathrm {D} q=q+\mathrm {E} q,$ so we let the variant order of terms reflect this sequence of considerations.

Viewed in this light the difference operator $\mathrm {D}$ is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation $G$ and its difference map $\mathrm {D} G,$ for example, taking the function space $(U^{\bullet }\to X^{\bullet })$ into $(\mathrm {E} U^{\bullet }\to \mathrm {E} X^{\bullet }).$ When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.

Acting on the logical transformation $F=(f,g)=(\;{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\;,\;{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\;),$ the operators $\mathrm {E}$ and $\mathrm {D}$ yield the enlarged map $\mathrm {E} F=(\mathrm {E} f,\mathrm {E} g)$ and the difference map $\mathrm {D} F=(\mathrm {D} f,\mathrm {D} g),$ respectively, whose components are given as follows.

${\begin{array}{lll}\mathrm {E} f&=&{\texttt {((}}u+\mathrm {d} u{\texttt {)(}}v+\mathrm {d} v{\texttt {))}}\\[8pt]\mathrm {E} g&=&{\texttt {((}}u+\mathrm {d} u{\texttt {,~}}v+\mathrm {d} v{\texttt {))}}\end{array}}$

${\begin{array}{lllll}\mathrm {D} f&=&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}&+&{\texttt {((}}u+\mathrm {d} u{\texttt {)(}}v+\mathrm {d} v{\texttt {))}}\\[8pt]\mathrm {D} g&=&{\texttt {((}}u{\texttt {,~}}v{\texttt {))}}&+&{\texttt {((}}u+\mathrm {d} u{\texttt {,~}}v+\mathrm {d} v{\texttt {))}}\end{array}}$

But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components $f$ and $g$ that we earlier used on $J.$ This work is recorded in Appendix 3 and a summary of the results is presented in Tables 66-i and 66-ii.

${\text{Table 66-i.}}~~{\text{Computation Summary for}}~f(u,v)={\texttt {((}}u{\texttt {)(}}v{\texttt {))}}$

${\begin{array}{c*{8}{l}}{\boldsymbol {\varepsilon }}f&=&u\!\cdot \!v\cdot 1&+&u{\texttt {(}}v{\texttt {)}}\cdot 1&+&{\texttt {(}}u{\texttt {)}}v\cdot 1&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\\[6pt]\mathrm {E} f&=&u\!\cdot \!v\cdot {\texttt {(}}\mathrm {d} u\cdot \mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {((}}\mathrm {d} u{\texttt {)}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\\[6pt]\mathrm {D} f&=&u\!\cdot \!v\cdot \mathrm {d} u\cdot \mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\\[6pt]\mathrm {d} f&=&u\!\cdot \!v\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[6pt]\mathrm {r} f&=&u\!\cdot \!v\cdot \mathrm {d} u\cdot \mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u\cdot \mathrm {d} v\end{array}}$

${\text{Table 66-ii.}}~~{\text{Computation Summary for}}~g(u,v)={\texttt {((}}u{\texttt {,}}v{\texttt {))}}$

${\begin{array}{c*{8}{l}}{\boldsymbol {\varepsilon }}g&=&u\!\cdot \!v\cdot 1&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 1\\[6pt]\mathrm {E} g&=&u\!\cdot \!v\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {))}}\\[6pt]\mathrm {D} g&=&u\!\cdot \!v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[6pt]\mathrm {d} g&=&u\!\cdot \!v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[6pt]\mathrm {r} g&=&u\!\cdot \!v\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\end{array}}$

Table 67 shows how to compute the analytic series for $F=(f,g)=(\;{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\;,\;{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\;)$ in terms of coordinates, and Table 68 recaps these results in symbolic terms, agreeing with earlier derivations.

${\text{Table 67.}}~~{\text{Computation of an Analytic Series in Terms of Coordinates}}$

$u$

$v$

$\mathrm {d} u$

$\mathrm {d} v$

$u'$

$v'$

$f$

$g$

${\mathrm {E} f}$

${\mathrm {E} g}$

${\mathrm {D} f}$

${\mathrm {D} g}$

${\mathrm {d} f}$

${\mathrm {d} g}$

${\mathrm {d} ^{2}\!f}$

${\mathrm {d} ^{2}\!g}$

$0$

$0$

${\begin{matrix}0\\0\\1\\1\end{matrix}}$

${\begin{matrix}0\\1\\0\\1\end{matrix}}$

${\begin{matrix}0\\0\\1\\1\end{matrix}}$

${\begin{matrix}0\\1\\0\\1\end{matrix}}$

$0$

$1$

${\begin{matrix}0\\1\\1\\1\end{matrix}}$

${\begin{matrix}1\\0\\0\\1\end{matrix}}$

${\begin{matrix}0\\1\\1\\1\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\0\\0\\1\end{matrix}}$

${\begin{matrix}0\\0\\0\\0\end{matrix}}$

$0$

$1$

${\begin{matrix}0\\0\\1\\1\end{matrix}}$

${\begin{matrix}0\\1\\0\\1\end{matrix}}$

${\begin{matrix}0\\0\\1\\1\end{matrix}}$

${\begin{matrix}1\\0\\1\\0\end{matrix}}$

$1$

$0$

${\begin{matrix}1\\0\\1\\1\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\1\\0\\0\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\1\\0\\1\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\0\\0\\1\end{matrix}}$

${\begin{matrix}0\\0\\0\\0\end{matrix}}$

$1$

$0$

${\begin{matrix}0\\0\\1\\1\end{matrix}}$

${\begin{matrix}0\\1\\0\\1\end{matrix}}$

${\begin{matrix}1\\1\\0\\0\end{matrix}}$

${\begin{matrix}0\\1\\0\\1\end{matrix}}$

$1$

$0$

${\begin{matrix}1\\1\\0\\1\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\0\\1\\0\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\0\\1\\1\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\0\\0\\1\end{matrix}}$

${\begin{matrix}0\\0\\0\\0\end{matrix}}$

$1$

$1$

${\begin{matrix}0\\0\\1\\1\end{matrix}}$

${\begin{matrix}0\\1\\0\\1\end{matrix}}$

${\begin{matrix}1\\1\\0\\0\end{matrix}}$

${\begin{matrix}1\\0\\1\\0\end{matrix}}$

$1$

$1$

${\begin{matrix}1\\1\\1\\0\end{matrix}}$

${\begin{matrix}1\\0\\0\\1\end{matrix}}$

${\begin{matrix}0\\0\\0\\1\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\0\\0\\0\end{matrix}}$

${\begin{matrix}0\\1\\1\\0\end{matrix}}$

${\begin{matrix}0\\0\\0\\1\end{matrix}}$

${\begin{matrix}0\\0\\0\\0\end{matrix}}$

${\text{Table 68.}}~~{\text{Computation of an Analytic Series in Symbolic Terms}}$

$u$	$v$	$f$	$g$	${\mathrm {D} f}$	${\mathrm {D} g}$	${\mathrm {d} f}$	${\mathrm {d} g}$	${\mathrm {d} ^{2}\!f}$	${\mathrm {d} ^{2}\!g}$
${\begin{matrix}0\\[4pt]0\\[4pt]1\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]1\\[4pt]0\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]1\\[4pt]1\\[4pt]1\end{matrix}}$	${\begin{matrix}1\\[4pt]0\\[4pt]0\\[4pt]1\end{matrix}}$	${\begin{matrix}{\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\\[4pt]{\texttt {~(}}\mathrm {d} u{\texttt {)~}}\mathrm {d} v{\texttt {~~}}\\[4pt]{\texttt {~~}}\mathrm {d} u{\texttt {~(}}\mathrm {d} v{\texttt {)~}}\\[4pt]{\texttt {~~}}\mathrm {d} u{\texttt {~~}}\mathrm {d} v{\texttt {~~}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[4pt]{\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[4pt]{\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[4pt]{\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[4pt]\mathrm {d} v\\[4pt]\mathrm {d} u\\[4pt]0\end{matrix}}$	${\begin{matrix}{\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[4pt]{\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[4pt]{\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[4pt]{\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\end{matrix}}$	${\begin{matrix}\mathrm {d} u\cdot \mathrm {d} v\\[4pt]\mathrm {d} u\cdot \mathrm {d} v\\[4pt]\mathrm {d} u\cdot \mathrm {d} v\\[4pt]\mathrm {d} u\cdot \mathrm {d} v\end{matrix}}$	${\begin{matrix}0\\[4pt]0\\[4pt]0\\[4pt]0\end{matrix}}$

Figure 69 gives a graphical picture of the difference map $\mathrm {D} F=(\mathrm {D} f,\mathrm {D} g)$ for the transformation $F=(f,g)=(\;{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\;,\;{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\;).$ This represents the same information about $\mathrm {D} f$ and $\mathrm {D} g$ that was given in the corresponding rows of Tables 66-i and 66-ii, for ease of reference repeated below.

${\begin{array}{c*{8}{l}}\mathrm {D} f&=&u\!\cdot \!v\cdot \mathrm {d} u\cdot \mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\\[8pt]\mathrm {D} g&=&u\!\cdot \!v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\end{array}}$

${\text{Figure 69.}}~~{\text{Difference Map of}}~F(u,v)=(\;{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\;,\;{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\;)$

Figure 70-a shows a way of visualizing the tangent functor map $\mathrm {d} F=(\mathrm {d} f,\mathrm {d} g)$ for the transformation $F=(f,g)=(\;{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\;,\;{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\;).$ This amounts to the same information about $\mathrm {d} f$ and $\mathrm {d} g$ that was given in Tables 66-i and 66-ii, the corresponding rows of which are repeated below.

${\begin{array}{c*{8}{l}}\mathrm {d} f&=&u\!\cdot \!v\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[8pt]\mathrm {d} g&=&u\!\cdot \!v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\end{array}}$

${\text{Figure 70-a.}}~~{\text{Tangent Functor Diagram for}}~F(u,v)=(\;{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\;,\;{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\;)$

Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation $F(u,v)=(\;{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\;,\;{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\;).$

${\text{Figure 70-b.}}~~{\text{Tangent Functor Ferris Wheel for}}~F(u,v)=(\;{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\;,\;{\texttt {((}}u{\texttt {,}}v{\texttt {))}}\;)$

• Note. The original Figure 70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.

o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ | | ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ | | //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ | | o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o | | |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| | | |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| | | |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| | | o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o | | \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ | | \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ | | \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ (u)(v) o-----------------------o dv' @ (u)(v) = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ | | / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ | | / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ | | o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o | | | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| | | | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| | | | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| | | o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o | | \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ | | \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ | | \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ (u) v o-----------------------o dv' @ (u) v = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ | | ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ | | /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ | | o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o | | |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| | | |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| | | |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| | | o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o | | \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ | | \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ | | \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ u (v) o-----------------------o dv' @ u (v) = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | dU | | dU | | dU | | o--o o--o | | o--o o--o | | o--o o--o | | / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ | | / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ | | / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ | | o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o | | | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| | | | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| | | | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| | | o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o | | \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ | | \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ | | \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ | | o--o o--o | | o--o o--o | | o--o o--o | | | | | | | o-----------------------o o-----------------------o o-----------------------o = du' @ u v o-----------------------o dv' @ u v = = | dU' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o o-----------------------o o-----------------------o o-----------------------o | U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\| | o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\| | /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\| | ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\| | /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\| | o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\| | |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\| | |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\| | |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\| | o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\| | \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\| | \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\| | \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\| | o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\| | | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\| o-----------------------o o-----------------------o o-----------------------o = u' o-----------------------o v' = = | U' | = = | o--o o--o | = = | /////\ /\\\\\ | = = | ///////o\\\\\\\ | = = | ////////X\\\\\\\\ | = = | o///////XXX\\\\\\\o | = = | |/////oXXXXXo\\\\\| | = = = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = = | |/////oXXXXXo\\\\\| | | o//////\XXX/\\\\\\o | | \//////\X/\\\\\\/ | | \//////o\\\\\\/ | | \///// \\\\\/ | | o--o o--o | | | o-----------------------o Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>

Epilogue, Enchoiry, Exodus

	It is time to explain myself . . . . let us stand up.
— Walt Whitman, Leaves of Grass, [Whi, 79]

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• Overview • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Appendices • References • Document History •

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