Author: Jon Awbrey
Transformations of Discourse (cont.)
Transformations of Type B² → B¹
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition $J(u,v)=u\!\cdot \!v,$ regarded either as the functional product of the maps $u$ and $v$ or as the logical conjunction of the features $u$ and $v,$ a map whose fiber of truth $J^{1}(1)$ picks out the single cell of that logical description in the universe of discourse $U^{\bullet }.$ Thus $J,$ or $u\!\cdot \!v,$ may be treated as another name for the point whose coordinates are $(1,1)$ in $U^{\bullet }.$
Analytic Expansion of Conjunction

In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a soul.
What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.


— Robert Musil, The Man Without Qualities, [Mus, 118]

Figure 35 pictures the form of conjunction $J:\mathbb {B} ^{2}\to \mathbb {B}$ as a transformation from the $2$dimensional universe $[u,v]$ to the $1$dimensional universe $[x].$ This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition $J:\langle u,v\rangle \to \mathbb {B}$ is being recast into the thematized role of a transformation $J:[u,v]\to [x],$ where the new variable $x$ takes the part of a thematic variable ${\check {J}}.$

${\text{Figure 35.}}~~{\text{Conjunction as Transformation}}$

Tacit Extension of Conjunction

I teach straying from me, yet who can stray from me?
I follow you whoever you are from the present hour;
My words itch at your ears till you understand them.


— Walt Whitman, Leaves of Grass, [Whi, 83]

Earlier we defined the tacit extension operators ${\boldsymbol {\varepsilon }}:X^{\bullet }\to Y^{\bullet }$ as maps embedding each proposition of a given universe $X^{\bullet }$ in a more generously given universe $Y^{\bullet }\supset X^{\bullet }.$ Of immediate interest are the tacit extensions ${\boldsymbol {\varepsilon }}:U^{\bullet }\to \mathrm {E} U^{\bullet },$ that locate each proposition of $U^{\bullet }$ in the enlarged context of $\mathrm {E} U^{\bullet }.$ In its application to the propositional conjunction $J=u\!\cdot \!v$ in $[u,v],$ the tacit extension operator ${\boldsymbol {\varepsilon }}$ yields the proposition ${\boldsymbol {\varepsilon }}J$ in $\mathrm {E} U^{\bullet }=[u,v,\mathrm {d} u,\mathrm {d} v].$ The extended proposition ${\boldsymbol {\varepsilon }}J$ may be computed according to the scheme in Table 36, in effect doing nothing more that conjoining a tautology of $[\mathrm {d} u,\mathrm {d} v]$ to $J$ in $U^{\bullet }.$
${\text{Table 36.}}~~{\text{Computation of}}~{\boldsymbol {\varepsilon }}J$
${\begin{array}{*{9}{l}}{\boldsymbol {\varepsilon }}J&=&J{}_{^{\langle }}u,v{}_{^{\rangle }}\\[4pt]&=&u\cdot v\\[4pt]&=&u\cdot v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} v{\texttt {)}}&+&u\cdot v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\cdot {\texttt {}}\mathrm {d} v{\texttt {}}&+&u\cdot v\cdot {\texttt {}}\mathrm {d} u{\texttt {}}\cdot {\texttt {(}}\mathrm {d} v{\texttt {)}}&+&u\cdot v\cdot {\texttt {}}\mathrm {d} u{\texttt {}}\cdot {\texttt {}}\mathrm {d} v{\texttt {}}\end{array}}$

${\begin{array}{*{4}{l}}{\boldsymbol {\varepsilon }}J&=&&u\cdot v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&u\cdot v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\cdot {\texttt {~}}\mathrm {d} v{\texttt {~}}\\[4pt]&&+&u\cdot v\cdot {\texttt {~}}\mathrm {d} u{\texttt {~}}\cdot {\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&+&u\cdot v\cdot {\texttt {~}}\mathrm {d} u{\texttt {~}}\cdot {\texttt {~}}\mathrm {d} v{\texttt {~}}\end{array}}$

The lower portion of the Table contains the dispositional features of ${\boldsymbol {\varepsilon }}J$ arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function $F$ being employed in a differential context is equivalent to ${\boldsymbol {\varepsilon }}F$ for a suitable ${\boldsymbol {\varepsilon }}.$
Figures 37a through 37d present several pictures of the proposition $J$ and its tacit extension ${\boldsymbol {\varepsilon }}J.$ Notice in these Figures how ${\boldsymbol {\varepsilon }}J$ in $\mathrm {E} U^{\bullet }$ visibly extends $J$ in $U^{\bullet }$ by annexing to the indicated cells of $J$ all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.

${\text{Figure 37a.}}~~{\text{Tacit Extension of}}~J~{\text{(Areal)}}$


${\text{Figure 37b.}}~~{\text{Tacit Extension of}}~J~{\text{(Bundle)}}$


${\text{Figure 37c.}}~~{\text{Tacit Extension of}}~J~{\text{(Compact)}}$


${\text{Figure 37d.}}~~{\text{Tacit Extension of}}~J~{\text{(Digraph)}}$

The computational scheme shown in Table 36 treated $J$ as a proposition in $U^{\bullet }$ and formed ${\boldsymbol {\varepsilon }}J$ as a proposition in $\mathrm {E} U^{\bullet }.$ When $J$ is regarded as a mapping $J:U^{\bullet }\to X^{\bullet }$ then ${\boldsymbol {\varepsilon }}J$ must be obtained as a mapping ${\boldsymbol {\varepsilon }}J:\mathrm {E} U^{\bullet }\to X^{\bullet }.$ By default, the tacit extension of the map $J:[u,v]\to [x]$ is naturally taken to be a particular map,
${\boldsymbol {\varepsilon }}J:[u,v,\mathrm {d} u,\mathrm {d} v]\to [x]\subseteq [x,\mathrm {d} x],$

namely, the one that looks like $J$ when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that $J$ already takes.
But the choice of a particular thematic variable, for example $x$ for ${\check {J}},$ is a shade more arbitrary than the choice of original variable names $\{u,v\},$ so the map we are calling the trope extension,
$\eta J:[u,v,\mathrm {d} u,\mathrm {d} v]\to [\mathrm {d} x]\subseteq [x,\mathrm {d} x],$

since it looks just the same as ${\boldsymbol {\varepsilon }}J$ in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.
These considerations have the practical consequence that all of our computations and illustrations of ${\boldsymbol {\varepsilon }}J$ perform the double duty of capturing $\eta J$ as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension $\eta J,$ because it would be identical to the work already done for ${\boldsymbol {\varepsilon }}J.$ Since the computations given for ${\boldsymbol {\varepsilon }}J$ are expressed solely in terms of the variables $\{u,v,\mathrm {d} u,\mathrm {d} v\},$ they work equally well for finding $\eta J.$ Further, since each of the above Figures shows only how the level sets of ${\boldsymbol {\varepsilon }}J$ partition the extended source universe $\mathrm {E} U^{\bullet }=[u,v,\mathrm {d} u,\mathrm {d} v],$ all of them serve equally well as portraits of $\eta J.$
Enlargement Map of Conjunction

No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.


— Robert Musil, The Man Without Qualities, [Mus, 62]

The enlargement map $\mathrm {E} J$ is computed from the proposition $J$ by making a particular class of formal substitutions for its variables, in this case $u+\mathrm {d} u$ for $u$ and $v+\mathrm {d} v$ for $v,$ and afterwards expanding the result in whatever way is found convenient.
Table 38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing $\mathrm {E} J$ over the cells of $[u,v].$ The critical step of this procedure uses the facts that ${\texttt {(}}0,x{\texttt {)}}=0+x=x$ and ${\texttt {(}}1,x{\texttt {)}}=1+x={\texttt {(}}x{\texttt {)}}$ for any boolean variable $x.$
${\text{Table 38.}}~~{\text{Computation of}}~\mathrm {E} J~{\text{(Method 1)}}$
${\begin{array}{*{9}{l}}\mathrm {E} J&=&J_{(u+\mathrm {d} u,v+\mathrm {d} v)}\\[4pt]&=&{\texttt {(}}u{\texttt {,}}\mathrm {d} u{\texttt {)}}\cdot {\texttt {(}}v{\texttt {,}}\mathrm {d} v{\texttt {)}}\\[4pt]&=&{\texttt {}}u{\texttt {}}{\texttt {}}v{\texttt {}}\cdot J_{(1+\mathrm {d} u,1+\mathrm {d} v)}&+&{\texttt {}}u{\texttt {}}{\texttt {(}}v{\texttt {)}}\cdot J_{(1+\mathrm {d} u,\mathrm {d} v)}&+&{\texttt {(}}u{\texttt {)}}{\texttt {}}v{\texttt {}}\cdot J_{(\mathrm {d} u,1+\mathrm {d} v)}&+&{\texttt {(}}u{\texttt {)}}{\texttt {(}}v{\texttt {)}}\cdot J_{(\mathrm {d} u,\mathrm {d} v)}\\[4pt]&=&{\texttt {}}u{\texttt {}}{\texttt {}}v{\texttt {}}\cdot J_{({\texttt {(}}\mathrm {d} u{\texttt {)}},{\texttt {(}}\mathrm {d} v{\texttt {)}})}&+&{\texttt {}}u{\texttt {}}{\texttt {(}}v{\texttt {)}}\cdot J_{({\texttt {(}}\mathrm {d} u{\texttt {)}},\mathrm {d} v)}&+&{\texttt {(}}u{\texttt {)}}{\texttt {}}v{\texttt {}}\cdot J_{(\mathrm {d} u,{\texttt {(}}\mathrm {d} v{\texttt {)}})}&+&{\texttt {(}}u{\texttt {)}}{\texttt {(}}v{\texttt {)}}\cdot J_{(\mathrm {d} u,\mathrm {d} v)}\end{array}}$

${\begin{array}{*{9}{l}}\mathrm {E} J&=&{\texttt {}}u{\texttt {}}{\texttt {}}v{\texttt {}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&&+&{\texttt {}}u{\texttt {}}{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}{\texttt {}}\mathrm {d} v{\texttt {}}\\[4pt]&&&&&+&{\texttt {(}}u{\texttt {)}}{\texttt {}}v{\texttt {}}\cdot {\texttt {}}\mathrm {d} u{\texttt {}}{\texttt {(}}\mathrm {d} v{\texttt {)}}\\[4pt]&&&&&&&+&{\texttt {(}}u{\texttt {)}}{\texttt {(}}v{\texttt {)}}\cdot {\texttt {}}\mathrm {d} u{\texttt {}}~{\texttt {}}\mathrm {d} v{\texttt {}}\end{array}}$

Table 39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.
${\text{Table 39.}}~~{\text{Computation of}}~\mathrm {E} J~{\text{(Method 2)}}$
${\begin{array}{*{9}{c}}\mathrm {E} J&=&(u+\mathrm {d} u)\cdot (v+\mathrm {d} v)\\[6pt]&=&u\cdot v&+&u\cdot \mathrm {d} v&+&v\cdot \mathrm {d} u&+&\mathrm {d} u\cdot \mathrm {d} v\\[6pt]\mathrm {E} J&=&{\texttt {}}u{\texttt {}}{\texttt {}}v{\texttt {}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {}}u{\texttt {}}{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}{\texttt {}}\mathrm {d} v{\texttt {}}&+&{\texttt {(}}u{\texttt {)}}{\texttt {}}v{\texttt {}}\cdot {\texttt {}}\mathrm {d} u{\texttt {}}{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}{\texttt {(}}v{\texttt {)}}\cdot {\texttt {}}\mathrm {d} u{\texttt {}}~{\texttt {}}\mathrm {d} v{\texttt {}}\end{array}}$

Figures 40a through 40d present several views of the enlarged proposition $\mathrm {E} J.$

${\text{Figure 40a.}}~~{\text{Enlargement of}}~J~{\text{(Areal)}}$


${\text{Figure 40b.}}~~{\text{Enlargement of}}~J~{\text{(Bundle)}}$


${\text{Figure 40c.}}~~{\text{Enlargement of}}~J~{\text{(Compact)}}$


${\text{Figure 40d.}}~~{\text{Enlargement of}}~J~{\text{(Digraph)}}$

An intuitive reading of the proposition $\mathrm {E} J$ becomes available at this point. Recall that propositions in the extended universe $\mathrm {E} U^{\bullet }$ express the dispositions of a system and the constraints that are placed on them. In other words, a differential proposition in $\mathrm {E} U^{\bullet }$ can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand $\mathrm {E} J$ as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the truth of $J,$ that is, the region of the universe where $J$ is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator $\mathrm {E} .$ Namely, $\mathrm {E} J$ can be read as a proposition that enlarges on the meaning of $J,$ in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects — the available options for differential action and the consequential effects that result from each choice.
Read this way, the enlargement $\mathrm {E} J$ has strong ties to the normal use of $J,$ no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of $J,$ in effect, pointing to the interpretive elements in its fiber of truth $J^{1}(1).$ It is this kind of “use” that is often contrasted with the “mention” of a proposition, and thereby hangs a tale.
Digression : Reflection on Use and Mention

Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.


— John Dewey, How We Think, [Dew, 57]

The contrast drawn in logic between the use and the mention of a proposition corresponds to the difference that we observe in functional terms between using ${}^{\backprime \backprime }J\,{}^{\prime \prime }$ to indicate the region $J^{1}(1)$ and using ${}^{\backprime \backprime }J\,{}^{\prime \prime }$ to indicate the function $J.$ You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name ${}^{\backprime \backprime }J\,{}^{\prime \prime }$ is used as a sign of the function $J,$ and if the function $J$ has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not $J,$ by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would be a color.
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of fullfledged sign relations.

The wellknown capacity that thoughts have — as doctors have discovered — for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.


— Robert Musil, The Man Without Qualities, [Mus, 130]

Difference Map of Conjunction

“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 dressingroom, he passed a punchingball 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, [Mus, 8]

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 44a through 44d illustrate the difference proposition $\mathrm {D} J.$

${\text{Figure 44a.}}~~{\text{Difference Map of}}~J~{\text{(Areal)}}$


${\text{Figure 44b.}}~~{\text{Difference Map of}}~J~{\text{(Bundle)}}$


${\text{Figure 44c.}}~~{\text{Difference Map of}}~J~{\text{(Compact)}}$


${\text{Figure 44d.}}~~{\text{Difference Map of}}~J~{\text{(Digraph)}}$

Differential of Conjunction

By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.


— Michel Foucault, The Archaeology of Knowledge, [Fou, 143]

Finally, at long last, the differential proposition $\mathrm {d} J$ can be gleaned from the difference proposition $\mathrm {D} J$ by ranging over the cells of $[u,v]$ and picking out the linear proposition of $[\mathrm {d} u,\mathrm {d} v]$ that is “closest” to the portion of $\mathrm {D} J$ that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.

He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.


— Robert Musil, The Man Without Qualities, [Mus, 144]

Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form — the limitary concept of a selfcorrective process and the coefficient concept of a completable product — are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.
Awaiting that determination, I proceed with what seems like the obvious course, and compute $\mathrm {d} J$ according to the pattern in Table 45.
${\text{Table 45.}}~~{\text{Computation of}}~\mathrm {d} J$
${\begin{array}{c*{8}{l}}\mathrm {D} J&=&u\!\cdot \!v\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&u\,{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\,\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}\,v\cdot \mathrm {d} u\,{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u\!\cdot \!\mathrm {d} v{\texttt {~}}\\[6pt]\Downarrow \\[6pt]\mathrm {d} J&=&u\!\cdot \!v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u\,{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}\,v\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\end{array}}$

Figures 46a through 46d 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 $J.$

${\text{Figure 46a.}}~~{\text{Differential of}}~J~{\text{(Areal)}}$


${\text{Figure 46b.}}~~{\text{Differential of}}~J~{\text{(Bundle)}}$


${\text{Figure 46c.}}~~{\text{Differential of}}~J~{\text{(Compact)}}$


${\text{Figure 46d.}}~~{\text{Differential of}}~J~{\text{(Digraph)}}$

Remainder of Conjunction

I bequeath myself to the dirt to grow from the grass I love,
If you want me again look for me under your bootsoles.
You will hardly know who I am or what I mean,
But I shall be good health to you nevertheless,
And filter and fibre your blood.
Failing to fetch me at first keep encouraged,
Missing me one place search another,
I stop some where waiting for you


— Walt Whitman, Leaves of Grass, [Whi, 88]

Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition $\mathrm {E} J$ in a series of stages. First, we considered the equation $\mathrm {E} J={\boldsymbol {\varepsilon }}J+\mathrm {D} J,$ which was involved in the definition of $\mathrm {D} J$ as the difference $\mathrm {E} J{\boldsymbol {\varepsilon }}J.$ Next, we contemplated the equation $\mathrm {D} J=\mathrm {d} J+\mathrm {r} J,$ which expresses $\mathrm {D} J$ in terms of two components, the differential $\mathrm {d} J$ that was just extracted and the residual component $\mathrm {r} J=\mathrm {D} J\mathrm {d} J.$ This remaining proposition $\mathrm {r} J$ can be computed as shown in Table 47.
${\text{Table 47.}}~~{\text{Computation of}}~\mathrm {r} J$
${\begin{array}{*{5}{l}}\mathrm {r} J&=&\mathrm {D} J&+&\mathrm {d} J\end{array}}$

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

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

As it happens, the remainder $\mathrm {r} J$ falls under the description of a second order differential $\mathrm {r} J=\mathrm {d} ^{2}J.$ This means that the expansion of $\mathrm {E} J$ in the form:
${\begin{array}{*{7}{l}}\mathrm {E} J&=&{\boldsymbol {\varepsilon }}J&+&\mathrm {D} J\\[6pt]&=&{\boldsymbol {\varepsilon }}J&+&\mathrm {d} J&+&\mathrm {r} J\\[6pt]&=&\mathrm {d} ^{0}J&+&\mathrm {d} ^{1}J&+&\mathrm {d} ^{2}J\end{array}}$

which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.
Figures 48a through 48d illustrate the proposition $\mathrm {r} J=\mathrm {d} ^{2}J,$ which forms the remainder map of $J$ and also, in this instance, the second order differential of $J.$

${\text{Figure 48a.}}~~{\text{Remainder of}}~J~{\text{(Areal)}}$


${\text{Figure 48b.}}~~{\text{Remainder of}}~J~{\text{(Bundle)}}$


${\text{Figure 48c.}}~~{\text{Remainder of}}~J~{\text{(Compact)}}$


${\text{Figure 48d.}}~~{\text{Remainder of}}~J~{\text{(Digraph)}}$

Summary of Conjunction
To establish a convenient reference point for further discussion, Table 49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition $J.$
${\text{Table 49.}}~~{\text{Computation Summary for}}~J$
${\begin{array}{c*{8}{l}}{\boldsymbol {\varepsilon }}J&=&u\!\cdot \!v\cdot 1&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\\[6pt]\mathrm {E} J&=&u\!\cdot \!v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u\cdot \mathrm {d} v\\[6pt]\mathrm {D} J&=&u\!\cdot \!v\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u\cdot \mathrm {d} v\\[6pt]\mathrm {d} J&=&u\!\cdot \!v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\\[6pt]\mathrm {r} J&=&u\!\cdot \!v\cdot \mathrm {d} u\cdot \mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u\cdot \mathrm {d} v\end{array}}$

Analytic Series : Coordinate Method

And if he is told that something is the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could “just as easily” be, and to attach no more importance to what is than to what is not.


— Robert Musil, The Man Without Qualities, [Mus, 12]

Table 50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.
The first six columns of the Table, taken as a whole, represent the variables of a construct called the contingent universe $[u,v,\mathrm {d} u,\mathrm {d} v,u',v'],$ or the bundle of contingency spaces $[\mathrm {d} u,\mathrm {d} v,u',v']$ over the universe $[u,v].$ Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a codependency among them, as described by the following equations:
${\begin{matrix}u'&=&u+\mathrm {d} u&=&{\texttt {(}}u{\texttt {,}}\mathrm {d} u{\texttt {)}}\\[8pt]v'&=&v+\mathrm {d} v&=&{\texttt {(}}v{\texttt {,}}\mathrm {d} v{\texttt {)}}\end{matrix}}$

These relations correspond to the formal substitutions that are made in defining $\mathrm {E} J$ and $\mathrm {D} J.$ For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.
The five columns to the right of the double bar in Table 50 contain the values of the dependent variables $\{{\boldsymbol {\varepsilon }}J,~\mathrm {E} J,~\mathrm {D} J,~\mathrm {d} J,~\mathrm {d} ^{2}\!J\}.$ These are normally interpreted as values of functions $\mathrm {W} J:\mathrm {E} U\to \mathbb {B}$ or as values of propositions in the extended universe $[u,v,\mathrm {d} u,\mathrm {d} v]$ but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set $\{u,v,u',v'\}.$
The column for ${\boldsymbol {\varepsilon }}J$ is computed as $J(u,v)=uv$ and together with the columns for $u$ and $v$ illustrates how we “share structure” in the Table by listing only the first entries of each constant block.
The column for $\mathrm {E} J$ is computed by means of the following chain of identities, where the contingent variables $u'$ and $v'$ are defined as $u'=u+\mathrm {d} u$ and $v'=v+\mathrm {d} v.$
${\begin{matrix}\mathrm {E} J(u,v,\mathrm {d} u,\mathrm {d} v)&=&J(u+\mathrm {d} u,v+\mathrm {d} v)&=&J(u',v')\end{matrix}}$

This makes it easy to determine $\mathrm {E} J$ by inspection, computing the conjunction $J(u',v')=u'v'$ from the columns headed $u'$ and $v'.$ Since each of these forms expresses the same proposition $\mathrm {E} J$ in $\mathrm {E} U^{\bullet },$ the dependence on $\mathrm {d} u$ and $\mathrm {d} v$ is still present but merely left implicit in the final variant $J(u',v').$
 Note. On occasion, it is tempting to use the further notation $J'(u,v)=J(u',v'),$ especially to suggest a transformation that acts on whole propositions, for example, taking the proposition $J$ into the proposition $J'=\mathrm {E} J.$ The prime $({}^{\prime })$ then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.
Given the values of ${\boldsymbol {\varepsilon }}J$ and $\mathrm {E} J,$ the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation $\mathrm {D} J={\boldsymbol {\varepsilon }}J+\mathrm {E} J.$ The first order differential $\mathrm {d} J$ is found by looking in each block of constant argument pairs $u,v$ and choosing the linear function of $\mathrm {d} u,\mathrm {d} v$ that best approximates $\mathrm {D} J$ in that block. Finally, the remainder is computed as $\mathrm {r} J=\mathrm {D} J+\mathrm {d} J,$ in this case yielding the second order differential $\mathrm {d} ^{2}\!J.$
Analytic Series : Recap
Let us now summarize the results of Table 50 by writing down for each column and for each block of constant argument pairs $u,v$ a reasonably canonical symbolic expression for the function of $\mathrm {d} u,\mathrm {d} v$ that appears there. The synopsis formed in this way is presented in Table 51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.
Figures 52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of $\mathrm {E} J=J+\mathrm {D} J$ and $\mathrm {D} J=\mathrm {d} J+\mathrm {r} J$ in two different styles of diagram.

${\text{Figure 52.}}~~{\text{Decomposition of}}~\mathrm {E} J$


${\text{Figure 53.}}~~{\text{Decomposition of}}~\mathrm {D} J$

Terminological Interlude

Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been “starred”, in spite of their solidity …


— Gaston Leroux, The Phantom of the Opera, [Ler, 230]

At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we restress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per se, their operands, and their results. Furthermore, in working with composite forms of operators $\mathrm {W} =(\mathrm {W} _{1},\ldots ,\mathrm {W} _{n}),$ transformations $\mathrm {F} =(\mathrm {F} _{1},\ldots ,\mathrm {F} _{n}),$ and target domains $X^{\bullet }=[x_{1},\ldots ,x_{n}],$ we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts operator and operand, that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like opus, opera, and operant, but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive map as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase tangent map to denote the end product of the tangent functor acting on its operand map.
 Scholium. See [JGH, 69] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have $1$dimensional ranges, we are free to shift between the native form of a proposition $J:U\to \mathbb {B}$ and the thematized form of a mapping $J:U^{\bullet }\to [x]$ without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example $J,$ and for each operator $\mathrm {W}$ in the set $\{{\boldsymbol {\varepsilon }},\eta ,\mathrm {E} ,\mathrm {D} ,\mathrm {d} ,\mathrm {r} \},$ both the operand $J$ and the result $\mathrm {W} J$ could be viewed in either one of two ways. On one hand we may treat them as propositions $J:U\to \mathbb {B}$ and $\mathrm {W} J:\mathrm {E} U\to \mathbb {B} ,$ ignoring the distinction between the range $[x]\cong \mathbb {B}$ of ${\boldsymbol {\varepsilon }}J$ and the range $[\mathrm {d} x]\cong \mathbb {D}$ of the other types of $\mathrm {W} J.$ This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps $J:U^{\bullet }\to [x]=X^{\bullet }$ and ${\boldsymbol {\varepsilon }}J:\mathrm {E} U^{\bullet }\to [x]\subseteq \mathrm {E} X^{\bullet }$ or $\mathrm {W} J:\mathrm {E} U^{\bullet }\to [\mathrm {d} x]\subseteq \mathrm {E} X^{\bullet },$ in which case the qualitative characters of the output features are not ignored.
At the beginning of this Section we recast the natural form of a proposition $J:U\to \mathbb {B}$ into the thematic role of a transformation $J:U^{\bullet }\to [x],$ where $x$ was a variable recruited to express the newly independent ${\check {J}}.$ However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe $\mathrm {E} U^{\bullet },$ in other words, as propositions $\mathrm {W} J:\mathrm {E} U\to \mathbb {B} ,$ where $\mathrm {W}$ ranged over the set $\{{\boldsymbol {\varepsilon }},\mathrm {E} ,\mathrm {D} ,\mathrm {d} ,\mathrm {r} \}.$ That is as it should be. We have worked hard to devise a language that gives us these advantages — the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables 54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case $J=uv.$ The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.
Table 54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans serif operators ${\mathsf {W}}\in \{{\mathsf {e}},{\mathsf {E}},{\mathsf {D}},{\mathsf {d}},{\mathsf {r}}\}$ and their components $\mathrm {W} \in \{{\boldsymbol {\varepsilon }},\eta ,\mathrm {E} ,\mathrm {D} ,\mathrm {d} ,\mathrm {r} \}$ both have the same broad type ${\mathsf {W}},\mathrm {W} :(U^{\bullet }\to X^{\bullet })\to (\mathrm {E} U^{\bullet }\to \mathrm {E} X^{\bullet }),$ as appropriate to operators that map transformations $J:U^{\bullet }\to X^{\bullet }$ to extended transformations ${\mathsf {W}}J,\mathrm {W} J:\mathrm {E} U^{\bullet }\to \mathrm {E} X^{\bullet }.$
${\text{Table 54.}}~~{\text{Cast of Characters : Expansive Subtypes of Objects and Operators}}$
${\text{Symbol}}$

${\text{Notation}}$

${\text{Description}}$

${\text{Type}}$

$U^{\bullet }$

$=[u,v]$

${\text{Source universe}}$

$[\mathbb {B} ^{2}]$

$X^{\bullet }$

$=[x]$

${\text{Target universe}}$

$[\mathbb {B} ^{1}]$

$\mathrm {E} U^{\bullet }$

$=[u,v,\mathrm {d} u,\mathrm {d} v]$

${\text{Extended source universe}}$

$[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]$

$\mathrm {E} X^{\bullet }$

$=[x,\mathrm {d} x]$

${\text{Extended target universe}}$

$[\mathbb {B} ^{1}\!\times \!\mathbb {D} ^{1}]$

$J$

$J:U\!\to \!\mathbb {B}$

${\text{Proposition}}$

$(\mathbb {B} ^{2}\!\to \!\mathbb {B} )\in [\mathbb {B} ^{2}]$

$J$

$J:U^{\bullet }\!\to \!X^{\bullet }$

${\text{Transformation or Map}}$

$[\mathbb {B} ^{2}]\!\to \![\mathbb {B} ^{1}]$

${\begin{matrix}{\boldsymbol {\varepsilon }}\\\eta \\\mathrm {E} \\\mathrm {D} \\\mathrm {d} \end{matrix}}$

${\begin{array}{l}\mathrm {W} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },\\\mathrm {W} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet },\\\mathrm {W} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\\{\text{for each}}~\mathrm {W} ~{\text{in the set:}}\\\{{\boldsymbol {\varepsilon }},\eta ,\mathrm {E} ,\mathrm {D} ,\mathrm {d} \}\end{array}}$

${\begin{array}{ll}{\text{Tacit extension operator}}&{\boldsymbol {\varepsilon }}\\{\text{Trope extension operator}}&\eta \\{\text{Enlargement operator}}&\mathrm {E} \\{\text{Difference operator}}&\mathrm {D} \\{\text{Differential operator}}&\mathrm {d} \end{array}}$

${\begin{array}{l}{[\mathbb {B} ^{2}]\!\to \![\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]},\\{[\mathbb {B} ^{1}]\!\to \![\mathbb {B} ^{1}\!\times \!\mathbb {D} ^{1}]},\\\\([\mathbb {B} ^{2}]\!\to \![\mathbb {B} ^{1}])\!\to \!\\([\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {B} ^{1}\!\times \!\mathbb {D} ^{1}])\end{array}}$

${\begin{matrix}{\mathsf {e}}\\{\mathsf {E}}\\{\mathsf {D}}\\{\mathsf {T}}\end{matrix}}$

${\begin{array}{l}{\mathsf {W}}:U^{\bullet }\!\to \!{\mathsf {T}}U^{\bullet }=\mathrm {E} U^{\bullet },\\{\mathsf {W}}:X^{\bullet }\!\to \!{\mathsf {T}}X^{\bullet }=\mathrm {E} X^{\bullet },\\{\mathsf {W}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!({\mathsf {T}}U^{\bullet }\!\to \!{\mathsf {T}}X^{\bullet })\\{\text{for each}}~{\mathsf {W}}~{\text{in the set:}}\\\{{\mathsf {e}},{\mathsf {E}},{\mathsf {D}},{\mathsf {T}}\}\end{array}}$

${\begin{array}{lll}{\text{Radius operator}}&{\mathsf {e}}&=({\boldsymbol {\varepsilon }},\eta )\\{\text{Secant operator}}&{\mathsf {E}}&=({\boldsymbol {\varepsilon }},\mathrm {E} )\\{\text{Chord operator}}&{\mathsf {D}}&=({\boldsymbol {\varepsilon }},\mathrm {D} )\\{\text{Tangent functor}}&{\mathsf {T}}&=({\boldsymbol {\varepsilon }},\mathrm {d} )\end{array}}$

${\begin{array}{l}{[\mathbb {B} ^{2}]\!\to \![\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]},\\{[\mathbb {B} ^{1}]\!\to \![\mathbb {B} ^{1}\!\times \!\mathbb {D} ^{1}]},\\\\([\mathbb {B} ^{2}]\!\to \![\mathbb {B} ^{1}])\!\to \!\\([\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {B} ^{1}\!\times \!\mathbb {D} ^{1}])\end{array}}$

Table 55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps $\mathrm {W} J$ have $1$dimensional ranges, either $\mathbb {B} ^{1}$ or $\mathbb {D} ^{1},$ and so they can be viewed either as propositions $\mathrm {W} J:\mathrm {E} U\to \mathbb {B}$ or as logical transformations $\mathrm {W} J:\mathrm {E} U^{\bullet }\to X^{\bullet }.$ As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase differential proposition, applied to the result $\mathrm {d} J:\mathrm {E} U\to \mathbb {D} ,$ does not distinguish it from the general run of differential propositions $\mathrm {G} :\mathrm {E} U\to \mathbb {B} ,$ it is usual to single out $\mathrm {d} J$ as the tangent proposition of $J.$
${\text{Table 55.}}~~{\text{Synopsis of Terminology : Restrictive and Alternative Subtypes}}$

${\text{Operator}}$

${\text{Proposition}}$

${\text{Map}}$

${\begin{matrix}{\underline {\text{Tacit}}}\\{\text{extension}}\end{matrix}}$

${\begin{array}{l}{\boldsymbol {\varepsilon }}:U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~{\boldsymbol {\varepsilon }}:X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\{\boldsymbol {\varepsilon }}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!X^{\bullet })\end{array}}$

${\begin{array}{l}{\boldsymbol {\varepsilon }}J:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {B} \\{\boldsymbol {\varepsilon }}J:\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}\!\to \!\mathbb {B} \end{array}}$

${\begin{array}{l}{\boldsymbol {\varepsilon }}J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x]\\{\boldsymbol {\varepsilon }}J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {B} ^{1}]\end{array}}$

${\begin{matrix}{\underline {\text{Trope}}}\\{\text{extension}}\end{matrix}}$

${\begin{array}{l}\eta :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\eta :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\eta :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$

${\begin{array}{l}\eta J:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\eta J:\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}\!\to \!\mathbb {D} \end{array}}$

${\begin{array}{l}\eta J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x]\\\eta J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {D} ^{1}]\end{array}}$

${\begin{matrix}{\underline {\text{Enlargement}}}\\{\text{operator}}\end{matrix}}$

${\begin{array}{l}\mathrm {E} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\mathrm {E} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\mathrm {E} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$

${\begin{array}{l}\mathrm {E} J:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {E} J:\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}\!\to \!\mathbb {D} \end{array}}$

${\begin{array}{l}\mathrm {E} J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x]\\\mathrm {E} J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {D} ^{1}]\end{array}}$

${\begin{matrix}{\underline {\text{Difference}}}\\{\text{operator}}\end{matrix}}$

${\begin{array}{l}\mathrm {D} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\mathrm {D} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\mathrm {D} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$

${\begin{array}{l}\mathrm {D} J:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {D} J:\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}\!\to \!\mathbb {D} \end{array}}$

${\begin{array}{l}\mathrm {D} J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x]\\\mathrm {D} J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {D} ^{1}]\end{array}}$

${\begin{matrix}{\underline {\text{Differential}}}\\{\text{operator}}\end{matrix}}$

${\begin{array}{l}\mathrm {d} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\mathrm {d} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\mathrm {d} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$

${\begin{array}{l}\mathrm {d} J:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {d} J:\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}\!\to \!\mathbb {D} \end{array}}$

${\begin{array}{l}\mathrm {d} J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x]\\\mathrm {d} J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {D} ^{1}]\end{array}}$

${\begin{matrix}{\underline {\text{Remainder}}}\\{\text{operator}}\end{matrix}}$

${\begin{array}{l}\mathrm {r} :U^{\bullet }\!\to \!\mathrm {E} U^{\bullet },~\mathrm {r} :X^{\bullet }\!\to \!\mathrm {E} X^{\bullet }\\\mathrm {r} :(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {d} X^{\bullet })\end{array}}$

${\begin{array}{l}\mathrm {r} J:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {r} J:\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}\!\to \!\mathbb {D} \end{array}}$

${\begin{array}{l}\mathrm {r} J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![\mathrm {d} x]\\\mathrm {r} J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {D} ^{1}]\end{array}}$

${\begin{matrix}{\underline {\text{Radius}}}\\{\text{operator}}\end{matrix}}$

${\begin{array}{l}{\mathsf {e}}=({\boldsymbol {\varepsilon }},\eta )\\{\mathsf {e}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\end{array}}$


${\begin{array}{l}{\mathsf {e}}J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x,\mathrm {d} x]\\{\mathsf {e}}J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {B} ^{1}\!\times \!\mathbb {D} ^{1}]\end{array}}$

${\begin{matrix}{\underline {\text{Secant}}}\\{\text{operator}}\end{matrix}}$

${\begin{array}{l}{\mathsf {E}}=({\boldsymbol {\varepsilon }},\mathrm {E} )\\{\mathsf {E}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\end{array}}$


${\begin{array}{l}{\mathsf {E}}J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x,\mathrm {d} x]\\{\mathsf {E}}J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {B} ^{1}\!\times \!\mathbb {D} ^{1}]\end{array}}$

${\begin{matrix}{\underline {\text{Chord}}}\\{\text{operator}}\end{matrix}}$

${\begin{array}{l}{\mathsf {D}}=({\boldsymbol {\varepsilon }},\mathrm {D} )\\{\mathsf {D}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\end{array}}$


${\begin{array}{l}{\mathsf {D}}J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x,\mathrm {d} x]\\{\mathsf {D}}J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {B} ^{1}\!\times \!\mathbb {D} ^{1}]\end{array}}$

${\begin{matrix}{\underline {\text{Tangent}}}\\{\text{functor}}\end{matrix}}$

${\begin{array}{l}{\mathsf {T}}=({\boldsymbol {\varepsilon }},\mathrm {d} )\\{\mathsf {T}}:(U^{\bullet }\!\to \!X^{\bullet })\!\to \!(\mathrm {E} U^{\bullet }\!\to \!\mathrm {E} X^{\bullet })\end{array}}$

${\begin{array}{l}\mathrm {d} J:\langle u,v,\mathrm {d} u,\mathrm {d} v\rangle \!\to \!\mathbb {D} \\\mathrm {d} J:\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}\!\to \!\mathbb {D} \end{array}}$

${\begin{array}{l}{\mathsf {T}}J:[u,v,\mathrm {d} u,\mathrm {d} v]\!\to \![x,\mathrm {d} x]\\{\mathsf {T}}J:[\mathbb {B} ^{2}\!\times \!\mathbb {D} ^{2}]\!\to \![\mathbb {B} ^{1}\!\times \!\mathbb {D} ^{1}]\end{array}}$

End of Perfunctory Chatter : Time to Roll the Clip!
Two steps remain to finish the analysis of $J$ that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps ${\mathsf {W}}J:\mathrm {E} U^{\bullet }\to \mathrm {E} X^{\bullet }.$ This scheme is executed in two styles, using the areal views in Figures 56a and the box views in Figures 56b. Finally, in Figures 571 to 574 we put all the pieces together to construct the full operator diagrams for ${\mathsf {W}}:J\to {\mathsf {W}}J.$ There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.
Operator Maps : Areal Views

${\text{Figure 56a1.}}~~{\text{Radius Map of the Conjunction}}~J=uv$


${\text{Figure 56a2.}}~~{\text{Secant Map of the Conjunction}}~J=uv$


${\text{Figure 56a3.}}~~{\text{Chord Map of the Conjunction}}~J=uv$


${\text{Figure 56a4.}}~~{\text{Tangent Map of the Conjunction}}~J=uv$

Operator Maps : Box Views

${\text{Figure 56b1.}}~~{\text{Radius Map of the Conjunction}}~J=uv$


${\text{Figure 56b2.}}~~{\text{Secant Map of the Conjunction}}~J=uv$


${\text{Figure 56b3.}}~~{\text{Chord Map of the Conjunction}}~J=uv$


${\text{Figure 56b4.}}~~{\text{Tangent Map of the Conjunction}}~J=uv$

Operator Diagrams for the Conjunction J = uv

${\text{Figure 571.}}~~{\text{Radius Operator Diagram for the Conjunction}}~J=uv$


${\text{Figure 572.}}~~{\text{Secant Operator Diagram for the Conjunction}}~J=uv$


${\text{Figure 573.}}~~{\text{Chord Operator Diagram for the Conjunction}}~J=uv$


${\text{Figure 574.}}~~{\text{Tangent Functor Diagram for the Conjunction}}~J=uv$
