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| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math>
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math>
| <math>\begin{matrix}\underline\mathcal{X} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math>
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math>
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math>
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math>
|-
|-
\underline{X}
\underline{X}
\\
\\
= & \langle \underline\mathcal{X} \rangle
= & \langle \underline{\mathcal{X}} \rangle
\\
\\
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle
\underline{X}^\bullet
\underline{X}^\bullet
\\
\\
= & [\underline\mathcal{X}]
= & [\underline{\mathcal{X}}]
\\
\\
= & [\underline{x}_1, \ldots, \underline{x}_n]
= & [\underline{x}_1, \ldots, \underline{x}_n]
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline\mathcal{X} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a “hurdle” for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a “hurdle” for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math>
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math>
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline\mathbf{x},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.
<br>
<br>
==Transformations of Discourse==
{| width="100%" cellpadding="0" cellspacing="0"
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It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 39]
|}
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.
===Foreshadowing Transformations : Extensions and Projections of Discourse===
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.
| width="4%" |
|-
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]
|}
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.
====Extension from 1 to 2 Dimensions====
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math>
|}
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math>
|}
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math>
|}
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math>
|}
====Extension from 2 to 4 Dimensions====
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math>
|}
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math>
|}
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math>
|}
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math>
|}
===Thematization of Functions : And a Declaration of Independence for Variables===
{| width="100%"
| align="left" |
''And as imagination bodies forth''<br>
''The forms of things unknown, the poet's pen''<br>
''Turns them to shapes, and gives to airy nothing''<br>
''A local habitation and a name.''
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18
|}
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.
====Thematization : Venn Diagrams====
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The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.
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|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 11–12]
|}
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math>
|}
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math>
|}
{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ / \ /
\ / \ /
\ / \ J /
\ / \ /
\ / \ /
o----------\---------/----------o o----------\---------/----------o
| \ / | | \ / |
| \ / | | \ / |
| o-----@-----o | | o-----@-----o |
| /`````````````\ | | /`````````````\ |
| /```````````````\ | | /```````````````\ |
| /`````````````````\ | | /`````````````````\ |
| o```````````````````o | | o```````````````````o |
| |```````````````````| | | |```````````````````| |
| |```````` J ````````| | | |```````` x ````````| |
| |```````````````````| | | |```````````````````| |
| o```````````````````o | | o```````````````````o |
| \`````````````````/ | | \`````````````````/ |
| \```````````````/ | | \```````````````/ |
| \`````````````/ | | \`````````````/ |
| o-----------o | | o-----------o |
| | | |
| | | |
o-------------------------------o o-------------------------------o
J = u v x = J<u, v>
Figure 20-ii. Thematization of Conjunction (Stage 2)
</pre>
|}
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math>
|}
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math>
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math>
|}
====Thematization : Truth Tables====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.
| width="4%" |
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 19]
|}
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.
A preliminary step, as illustrated in Table 22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math>
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math>
|- style="height:35px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| style="border-left:1px solid black" | <math>f\!</math>
| <math>g\!</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\end{matrix}\!</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\end{matrix}\!</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}\!</math>
|}
<br>
Next, each propositional form is individually represented in the fashion shown in Tables 23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math>
<br>
{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math>
| <math>x\!</math>
| <math>\varphi\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math>
| <math>y\!</math>
| <math>\gamma\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
|}
|}
<br>
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables 24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.
<br>
{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>f\!</math>
| <math>x\!</math>
| style="border-left:1px solid black" | <math>\varphi\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <font size="+2"> <br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>g\!</math>
| <math>y\!</math>
| style="border-left:1px solid black" | <math>\gamma\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|}
|}
<br>
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables 25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math>
<br>
{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>f\!</math>
| <math>x\!</math>
| style="border-left:1px solid black" | <math>\varphi\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2"> <br> <br> <br></font>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <font size="+2"> <br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>g\!</math>
| <math>y\!</math>
| style="border-left:1px solid black" | <math>\gamma\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|}
|}
<br>
Finally, Tables 26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.
<br>
{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>x\!</math>
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math>
| <math>\theta f\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>y\!</math>
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math>
| <math>\theta g\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|}
|}
<br>
Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column 5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math>
<br>
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math>
|- style="height:30px; background:ghostwhite"
|
|
| <math>{f}\!</math>
| <math>\theta f\!</math>
| <math>\theta f\!</math>
|- style="background:ghostwhite"
| align="right" | <math>u\colon\!</math>
| <math>1~1~0~0\!</math>
|
|
|
|- style="background:ghostwhite"
| align="right" | <math>v\colon\!</math>
| <math>1~0~1~0\!</math>
|
|
|
|-
| <math>f_{0}\!</math>
| <math>0~0~0~0\!</math>
| <math>\texttt{(~)}\!</math>
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math>
| align="left" | <math>\check{f} + 1\!</math>
|-
|
<math>\begin{matrix}
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{4}
\\[4pt]
f_{8}
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~1~0~0
\\[4pt]
1~0~0~0
\end{matrix}\!</math>
|
<math>\begin{matrix}
\texttt{(} u \texttt{)(} v \texttt{)}
\\[4pt]
\texttt{(} u \texttt{)~} v \texttt{~}
\\[4pt]
\texttt{~} u \texttt{~(} v \texttt{)}
\\[4pt]
\texttt{~} u \texttt{~~} v \texttt{~}
\end{matrix}\!</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(u)(v)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~(u)~v~~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~u~(v)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~u~~v~~))}
\end{array}</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + u + v + uv
\\[4pt]
\check{f} + v + uv + 1
\\[4pt]
\check{f} + u + uv + 1
\\[4pt]
\check{f} + uv + 1
\end{array}\!</math>
|-
|
<math>\begin{matrix}
f_{3}
\\[4pt]
f_{12}
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~0~1~1
\\[4pt]
1~1~0~0
\end{matrix}\!</math>
|
<math>\begin{matrix}
\texttt{(} u \texttt{)}
\\[4pt]
\texttt{~} u \texttt{~}
\end{matrix}\!</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(u)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~u~~))}
\end{array}\!</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + u
\\[4pt]
\check{f} + u + 1
\end{array}\!</math>
|-
|
<math>\begin{matrix}
f_{6}
\\[4pt]
f_{9}
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~1~1~0
\\[4pt]
1~0~0~1
\end{matrix}\!</math>
|
<math>\begin{matrix}
\texttt{(} u \texttt{,} v \texttt{)}
\\[4pt]
\texttt{((} u \texttt{,} v \texttt{))}
\end{matrix}\!</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}
\end{array}\!</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + u + v + 1
\\[4pt]
\check{f} + u + v
\end{array}\!</math>
|-
|
<math>\begin{matrix}
f_{5}
\\[4pt]
f_{10}
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~1~0~1
\\[4pt]
1~0~1~0
\end{matrix}\!</math>
|
<math>\begin{matrix}
\texttt{(} v \texttt{)}
\\[4pt]
\texttt{~} v \texttt{~}
\end{matrix}</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}
\end{array}\!</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + v
\\[4pt]
\check{f} + v + 1
\end{array}\!</math>
|-
|
<math>\begin{matrix}
f_{7}
\\[4pt]
f_{11}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~1~1~1
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\end{matrix}\!</math>
|
<math>\begin{matrix}
\texttt{(~} u \texttt{~~} v \texttt{~)}
\\[4pt]
\texttt{(~} u \texttt{~(} v \texttt{))}
\\[4pt]
\texttt{((} u \texttt{)~} v \texttt{~)}
\\[4pt]
\texttt{((} u \texttt{)(} v \texttt{))}
\end{matrix}\!</math>
| align="left" |
<math>\begin{array}{l}
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}
\\[4pt]
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}
\end{array}\!</math>
| align="left" |
<math>\begin{array}{l}
\check{f} + uv
\\[4pt]
\check{f} + u + uv
\\[4pt]
\check{f} + v + uv
\\[4pt]
\check{f} + u + v + uv + 1
\end{array}\!</math>
|-
| <math>f_{15}\!</math>
| <math>1~1~1~1\!</math>
| <math>\texttt{((~))}\!</math>
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math>
| align="left" | <math>\check{f}\!</math>
|}
<br>
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables 28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math>
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math>
|- style="height:35px; background:ghostwhite"
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math>
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math>
|-
| <math>0\!</math>
| <math>0\!</math>
| style="border-left:1px solid black" |
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|-
| <math>0\!</math>
| <math>1\!</math>
| style="border-left:1px solid black" |
|
| <math>1\!</math>
| <math>1\!</math>
|
|
| <math>1\!</math>
| <math>1\!</math>
|
|
| <math>1\!</math>
| <math>1\!</math>
|
|
| <math>1\!</math>
| <math>1\!</math>
|-
| <math>1\!</math>
| <math>0\!</math>
| style="border-left:1px solid black" |
|
|
|
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
|
|
|
|
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
|-
| <math>1\!</math>
| <math>1\!</math>
| style="border-left:1px solid black" |
|
|
|
|
|
|
|
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
|}
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math>
|- style="height:35px; background:ghostwhite"
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math>
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math>
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math>
|-
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| style="border-left:1px solid black" | <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
|-
| <math>0\!</math>
| <math>0\!</math>
| <math>1\!</math>
| style="border-left:1px solid black" |
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|
| <math>1\!</math>
|-
| <math>0\!</math>
| <math>1\!</math>
| <math>0\!</math>
| style="border-left:1px solid black" | <math>1\!</math>
| <math>1\!</math>
|
|
| <math>1\!</math>
| <math>1\!</math>
|
|
| <math>1\!</math>
| <math>1\!</math>
|
|
| <math>1\!</math>
| <math>1\!</math>
|
|
|-
| <math>0\!</math>
| <math>1\!</math>
| <math>1\!</math>
| style="border-left:1px solid black" |
|
| <math>1\!</math>
| <math>1\!</math>
|
|
| <math>1\!</math>
| <math>1\!</math>
|
|
| <math>1\!</math>
| <math>1\!</math>
|
|
| <math>1\!</math>
| <math>1\!</math>
|-
| <math>1\!</math>
| <math>0\!</math>
| <math>0\!</math>
| style="border-left:1px solid black" | <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
|
|
|
|
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
|
|
|
|
|-
| <math>1\!</math>
| <math>0\!</math>
| <math>1\!</math>
| style="border-left:1px solid black" |
|
|
|
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
|
|
|
|
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
|-
| <math>1\!</math>
| <math>1\!</math>
| <math>0\!</math>
| style="border-left:1px solid black" | <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
|
|
|
|
|
|
|
|
|-
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| style="border-left:1px solid black" |
|
|
|
|
|
|
|
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
| <math>1\!</math>
|}
<br>
===Propositional Transformations===
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
If only the word ‘artificial’ were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.
| width="4%" |
|-
| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 56–57]
|}
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.
====Alias and Alibi Transformations====
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.
====Transformations of General Type====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
''Es ist passiert'', “it just sort of happened”, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 34]
|}
Consider the situation illustrated in Figure 30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math>
{| align="center" border="0" cellpadding="10"
|
<pre>
o-------------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| 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 \ |
| / / \ \ | | / / \ \ | | / / \ \ |
| o o o o | | o o o o | | o o o o |
| | u | | v | | | | u | | v | | | | u | | v | |
| 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
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ g | \ f / | h /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ o----------|-----------\-----/-----------|----------o /
\ | X | \ / | | /
\ | | \ / | | /
\ | | o-----o-----o | | /
\| | / \ | |/
\ | / \ | /
|\ | / \ | /|
| \ | / \ | / |
| \ | / \ | / |
| \ | o x o | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \| | | |/ |
| o--o--------o o--------o--o |
| / \ \ / / \ |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o--o-----o--o o |
| | | | | |
| | | | | |
| | | | | |
| | y | | z | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------------------------------o
\ /
\ /
\ /
\ /
\ /
\ p , q /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
o
Figure 30. Generic Frame of a Logical Transformation
</pre>
|}
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math>
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"
|
<math>\begin{matrix}
x & = & f(u, v)
\\[10pt]
y & = & g(u, v)
\\[10pt]
z & = & h(u, v)
\end{matrix}</math>
|}
<br>
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.
===Analytic Expansions : Operators and Functors===
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.
| width="4%" |
|-
| align="right" colspan="3" | — C.S. Peirce, “The Maxim of Pragmatism”, CP 5.438
|}
Given the barest idea of a logical transformation, as suggested by the sketch in Figure 30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.
====Operators on Propositions and Transformations====
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to “get the drift” of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure 31 illustrates the typical situation.
{| align="center" border="0" cellpadding="20"
|
<pre>
o---------------------------------------o
| |
| |
| U% F X% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| !W! | | !W! |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| !W!U% !W!F !W!X% |
| |
| |
o---------------------------------------o
Figure 31. Operator Diagram (1)
</pre>
|}
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math>
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.
In Figure 31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure 32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math>
{| align="center" border="0" cellpadding="20"
|
<pre>
o---------------------------------------o
| |
| |
| U% !W! !W!U% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| F | | !W!F |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| X% !W! !W!X% |
| |
| |
o---------------------------------------o
Figure 32. Operator Diagram (2)
</pre>
|}
The latter arrangement, as exhibited in Figure 32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure 30.
====Differential Analysis of Propositions and Transformations====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''
| width="4%" |
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 43]
|}
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\mathsf{W}
& : &
( U^\bullet \to X^\bullet )
& \to &
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )
\end{matrix}\!</math>
|}
<br>
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\mathsf{W}
& : &
( [\mathbb{B}^n] \to [\mathbb{B}^k] )
& \to &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )
\end{matrix}\!</math>
|}
<br>
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{array}{lccccc}
\text{Concrete type}
& \boldsymbol\varepsilon
& : &
( U^\bullet \to X^\bullet )
& \to &
( \mathrm{E}U^\bullet \to X^\bullet )
\\[10pt]
\text{Abstract type}
& \boldsymbol\varepsilon
& : &
( [\mathbb{B}^n] \to [\mathbb{B}^k] )
& \to &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )
\end{array}</math>
|}
<br>
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{array}{lccccc}
\text{Concrete type}
& W
& : &
( U^\bullet \to X^\bullet )
& \to &
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )
\\[10pt]
\text{Abstract type}
& W
& : &
( [\mathbb{B}^n] \to [\mathbb{B}^k] )
& \to &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )
\end{array}</math>
|}
<br>
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\boldsymbol\varepsilon F
& : &
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )
& \cong &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )
\\[10pt]
WF
& : &
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )
& \cong &
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )
\end{matrix}</math>
|}
<br>
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the “sans serif” operators <math>\mathsf{W}\!</math> and their “serified” components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet “map”, in conformity with an already standard practice.
=====The Secant Operator : '''E'''=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.
| width="4%" |
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
|}
Figures 33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)
{| align="center" border="0" cellpadding="10"
|
<pre>
U% $E$ $E$U% $E$U% $E$U%
o------------------>o============o============o
| | | |
| | | |
| | | |
| | | |
F | | $E$F = | $d$^0.F + | $r$^0.F
| | | |
| | | |
| | | |
v v v v
o------------------>o============o============o
X% $E$ $E$X% $E$X% $E$X%
Figure 33-i. Analytic Diagram (1)
</pre>
|}
{| align="center" border="0" cellpadding="10"
|
<pre>
U% $E$ $E$U% $E$U% $E$U% $E$U%
o------------------>o============o============o============o
| | | | |
| | | | |
| | | | |
| | | | |
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F
| | | | |
| | | | |
| | | | |
v v v v v
o------------------>o============o============o============o
X% $E$ $E$X% $E$X% $E$X% $E$X%
Figure 33-ii. Analytic Diagram (2)
</pre>
|}
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure 30.
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the “big picture”, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full “parts diagram” of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure 30.
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.
Given a transformation
{| align="center" cellpadding="6" width="90%"
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math>
|}
of concrete type
{| align="center" cellpadding="6" width="90%"
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math>
|}
the transformation
{| align="center" cellpadding="6" width="90%"
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math>
|}
of concrete type
{| align="center" cellpadding="6" width="90%"
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math>
|}
is defined by means of the following system of logical equations:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\\[16pt]
\mathrm{d}x_1
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
\mathrm{d}x_k
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)
\end{matrix}</math>
|}
<br>
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:
{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]
& = &
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].
\end{matrix}</math>
|}
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math>
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:
{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\mathsf{E}F
& = &
(\boldsymbol\varepsilon, \mathrm{E})F
& = &
(\boldsymbol\varepsilon F, \mathrm{E}F)
& = &
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).
\end{matrix}</math>
|}
Quite a lot of “thematic infrastructure” or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math>
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure 33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math>
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.
=====The Radius Operator : '''e'''=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.
| width="4%" |
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
|}
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
\mathsf{e}F
& = &
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F
\\[4pt]
& = &
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)
\\[4pt]
& = &
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).
\end{array}</math>
|}
which is tantamount to the system of equations below.
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\\[16pt]
\mathrm{d}x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
\mathrm{d}x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\end{matrix}</math>
|}
<br>
=====The Phantom of the Operators : '''η'''=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!
| width="4%" |
|-
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]
|}
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:
{| align="center" cellpadding="8" width="90%"
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math>
|}
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:
{| align="center" cellpadding="8" width="90%"
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math>
|}
which is defined by the following equations:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\begin{matrix}
\mathrm{d}x_1
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_1 (u_1, \ldots, u_n)
\\[4pt]
\cdots && \cdots && \cdots
\\[4pt]
\mathrm{d}x_k
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)
& = & F_k (u_1, \ldots, u_n)
\end{matrix}</math>
|}
<br>
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.
=====The Chord Operator : '''D'''=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.
| width="4%" |
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 45]
|}
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure 33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure 33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math>
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\begin{array}{lll}
\mathsf{D}
& = & \mathsf{E} - \mathsf{e}
\\[6pt]
& = & \mathsf{r}^0
\\[6pt]
& = & \mathsf{d}^1 + \mathsf{r}^1
\\[6pt]
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m
\\[6pt]
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m
\end{array}</math>
|}
<br>
=====The Tangent Operator : '''T'''=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.
| width="4%" |
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 300]
|}
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure 33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.
Figure 34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.
{| align="center" border="0" cellpadding="10"
|
<pre>
U% $T$ $T$U% $T$U%
o------------------>o============o
| | |
| | |
| | |
| | |
F | | $T$F = | <!e!, d> F
| | |
| | |
| | |
v v v
o------------------>o============o
X% $T$ $T$X% $T$X%
Figure 34. Tangent Functor Diagram
</pre>
|}
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.
===Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>1</sup>===
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 <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math>
====Analytic Expansion of Conjunction====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
<p>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.</p>
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p>
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 118]
|}
Figure 35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> 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 <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math>
|}
=====Tacit Extension of Conjunction=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
I teach straying from me, yet who can stray from me?<br>
I follow you whoever you are from the present hour;<br>
My words itch at your ears till you understand them.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 83]
|}
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table 36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math>
<br>
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math>
|
<math>\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}\!</math>
|-
|
<math>\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}\!</math>
|}
<br>
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> 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 <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math>
Figures 37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> 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.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math>
|}
The computational scheme shown in Table 36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,
{| align="center" cellpadding="6" width="90%"
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math>
|}
namely, the one that looks like <math>J\!</math> 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 <math>J\!</math> already takes.
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',
{| align="center" cellpadding="6" width="90%"
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math>
|}
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> 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 <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math>
=====Enlargement Map of Conjunction=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
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.
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 62]
|}
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> 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 <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math>
<br>
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math>
|
<math>\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}\!</math>
|-
|
<math>\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}</math>
|}
<br>
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.
<br>
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math>
|
<math>\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}\!</math>
|}
<br>
Figures 40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math>
|}
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> 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 <math>J,\!</math> that is, the region of the universe where <math>J\!</math> 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 <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> 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 <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> 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=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
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.
| width="4%" |
|-
| align="right" colspan="3" | — 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 <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> 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 <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> 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 <math>J,\!</math> 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 full-fledged sign relations.
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
The well-known 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.
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 130]
|}
=====Difference Map of Conjunction=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
“It doesn't matter what one does,” the Man Without Qualities said to himself, shrugging his shoulders. “In a tangle of forces like this it doesn't make a scrap of difference.” He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.
| width="4%" |
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 8]
|}
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table 41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math>
<br>
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math>
|
<math>\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}</math>
|-
|
<math>\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}</math>
|-
|
<math>\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}</math>
|}
<br>
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> 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.
<br>
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math>
|
<math>\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}</math>
|}
<br>
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, “mod 2”, where 1 + 1 = 0, as shown in Table 43.
<br>
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math>
|
<math>\begin{array}{*{5}{l}}
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J
\end{array}</math>
|-
|
<math>\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}</math>
|-
|
<math>\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}\!</math>
|}
<br>
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\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}</math>
|}
<br>
Figures 44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math>
|}
=====Differential of Conjunction=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
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.
| width="4%" |
|-
| align="right" colspan="3" | — Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]
|}
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is “closest” to the portion of <math>\mathrm{D}J\!</math> 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.
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.
| width="4%" |
|-
| align="right" colspan="3" | — 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 self-corrective 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 <math>\mathrm{d}J\!</math> according to the pattern in Table 45.
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math>
|
<math>\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}</math>
|}
<br>
Figures 46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math>
|}
=====Remainder of Conjunction=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
<p>I bequeath myself to the dirt to grow from the grass I love,<br>
If you want me again look for me under your bootsoles.</p>
<p>You will hardly know who I am or what I mean,<br>
But I shall be good health to you nevertheless,<br>
And filter and fibre your blood.</p>
<p>Failing to fetch me at first keep encouraged,<br>
Missing me one place search another,<br>
I stop some where waiting for you</p>
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 88]
|}
<br>
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table 47.
<br>
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math>
|
<math>\begin{array}{*{5}{l}}
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J
\end{array}\!</math>
|-
|
<math>\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}</math>
|-
|
<math>\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}\!</math>
|}
<br>
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|
<math>\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}</math>
|}
<br>
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.
Figures 48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math>
|}
=====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 <math>J.\!</math>
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math>
|
<math>\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}</math>
|}
<br>
====Analytic Series : Coordinate Method====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
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.
| width="4%" |
|-
| align="right" colspan="3" | — 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.
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math>
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math>
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math>
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math>
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math>
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math>
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math>
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math>
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math>
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math>
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math>
|-
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
|-
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
|-
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
|-
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
|}
<br>
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\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}</math>
|}
<br>
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> 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 <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> 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 <math>\{ u, v, u', v' \}.\!</math>
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we “share structure” in the Table by listing only the first entries of each constant block.
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math>
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
|
<math>\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}</math>
|}
<br>
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math>
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> 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 <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math>
====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 <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> 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.
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math>
|- style="height:35px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| style="border-left:1px solid black" | <math>J\!</math>
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math>
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math>
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math>
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \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}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \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}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
\mathrm{d}u
\\[4pt]
\mathrm{d}v
\\[4pt]
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\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}</math>
|}
<br>
Figures 52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math>
|}
====Terminological Interlude====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
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 …
| width="4%" |
|-
| align="right" colspan="3" | — 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 re-stress 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 <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> 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, 6-9] 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 <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> 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 <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> 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 <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> 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 <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> 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 <math>J = uv.\!</math> 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 <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math>
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math>
|- style="height:40px; background:ghostwhite"
| align="center" | <math>\text{Symbol}\!</math>
| align="center" | <math>\text{Notation}\!</math>
| align="center" | <math>\text{Description}\!</math>
| align="center" | <math>\text{Type}\!</math>
|-
| align="center" | <math>U^\bullet\!</math>
| <math>= [u, v]\!</math>
| <math>\text{Source universe}\!</math>
| <math>[\mathbb{B}^2]\!</math>
|-
| align="center" | <math>X^\bullet~\!</math>
| <math>= [x]\!</math>
| <math>\text{Target universe}\!</math>
| <math>[\mathbb{B}^1]~\!</math>
|-
| align="center" | <math>\mathrm{E}U^\bullet\!</math>
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math>
| <math>\text{Extended source universe}\!</math>
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math>
|-
| align="center" | <math>\mathrm{E}X^\bullet\!</math>
| <math>= [x, \mathrm{d}x]~\!</math>
| <math>\text{Extended target universe}\!</math>
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math>
|-
| align="center" | <math>J\!</math>
| <math>J : U \!\to\! \mathbb{B}\!</math>
| <math>\text{Proposition}\!</math>
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math>
|-
| align="center" | <math>J\!</math>
| <math>J : U^\bullet \!\to\! X^\bullet\!</math>
| <math>\text{Transformation or Map}\!</math>
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math>
|-
| align="center" |
<math>\begin{matrix}
\boldsymbol\varepsilon
\\
\eta
\\
\mathrm{E}
\\
\mathrm{D}
\\
\mathrm{d}
\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" |
<math>\begin{matrix}
\mathsf{e}
\\
\mathsf{E}
\\
\mathsf{D}
\\
\mathsf{T}
\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|}
<br>
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 <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> 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 <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math>
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math>
|- style="height:40px; background:ghostwhite"
|
| align="center" | <math>\text{Operator}\!</math>
| align="center" | <math>\text{Proposition}\!</math>
| align="center" | <math>\text{Map}\!</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}\!</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|}
<br>
====End of Perfunctory Chatter : Time to Roll the Clip!====
Two steps remain to finish the analysis of <math>J\!</math> 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 <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures 56-a and the ''box views'' in Figures 56-b. Finally, in Figures 57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> 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=====
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math>
|}
=====Operator Maps : Box Views=====
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math>
|}
=====Operator Diagrams for the Conjunction J = uv=====
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math>
|}
===Taking Aim at Higher Dimensional Targets===
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
The past and present wilt . . . . I have filled them and<br>
emptied them,<br>
And proceed to fill my next fold of the future.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 87]
|}
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\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}</math>
|}
<br>
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> 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 <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> 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 <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math>
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> 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 <math>\mathrm{W}J.\!</math> 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 <math>\mathrm{W}J\!</math> 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 <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math>
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math>
|- style="height:40px; background:ghostwhite"
| align="center" | <math>\text{Symbol}\!</math>
| align="center" | <math>\text{Notation}\!</math>
| align="center" | <math>\text{Description}\!</math>
| align="center" | <math>\text{Type}\!</math>
|-
| align="center" | <math>U^\bullet\!</math>
| <math>= [u, v]\!</math>
| <math>\text{Source universe}\!</math>
| <math>[\mathbb{B}^n]\!</math>
|-
| align="center" | <math>X^\bullet~\!</math>
| <math>\begin{array}{l}
= [x, y] \\
= [f, g]
\end{array}</math>
| <math>\text{Target universe}\!</math>
| <math>[\mathbb{B}^k]\!</math>
|-
| align="center" | <math>\mathrm{E}U^\bullet\!</math>
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math>
| <math>\text{Extended source universe}\!</math>
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math>
|-
| align="center" | <math>\mathrm{E}X^\bullet\!</math>
| <math>\begin{array}{l}
= [x, y, \mathrm{d}x, \mathrm{d}y] \\
= [f, g, \mathrm{d}f, \mathrm{d}g]
\end{array}</math>
| <math>\text{Extended target universe}\!</math>
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math>
|-
| align="center" |
<math>\begin{matrix}
f \\ g
\end{matrix}</math>
|
<math>\begin{array}{ll}
f : U \!\to\! [x] \cong \mathbb{B} \\
g : U \!\to\! [y] \cong \mathbb{B}
\end{array}</math>
| <math>\text{Proposition}\!</math>
|
<math>\begin{array}{l}
\mathbb{B}^n \!\to\! \mathbb{B} \\
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]
\end{array}</math>
|-
| align="center" | <math>F\!</math>
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math>
| <math>\text{Transformation of Map}\!</math>
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math>
|-
| align="center" |
<math>\begin{matrix}
\boldsymbol\varepsilon
\\
\eta
\\
\mathrm{E}
\\
\mathrm{D}
\\
\mathrm{d}
\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" |
<math>\begin{matrix}
\mathsf{e}
\\
\mathsf{E}
\\
\mathsf{D}
\\
\mathsf{T}
\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math>
|- style="height:40px; background:ghostwhite"
|
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math>
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math>
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math>
|-
| align="center" | <math>\underline{\text{Operand}}\!</math>
|
<math>\begin{array}{l}
F = (F_1, F_2) \\
F = (f, g) : U \!\to\! X
\end{array}</math>
|
<math>\begin{array}{l}
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\
F_i : \mathbb{B}^n \!\to\! \mathbb{B}
\end{array}</math>
|
<math>\begin{array}{l}
F : [u, v] \!\to\! [x, y] \\
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]
\end{array}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}\!</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math>
|
<math>\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}</math>
|
|
<math>\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}</math>
|-
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|
<math>\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}</math>
|}
<br>
===Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>===
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 <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\begin{array}{lllll}
x
& = & f(u, v)
& = & \texttt{((} u \texttt{)(} v \texttt{))}
\\[8pt]
y
& = & g(u, v)
& = & \texttt{((} u \texttt{,} v \texttt{))}
\end{array}</math>
|}
<br>
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\begin{array}{lllll}
(x, y)
& = & F(u, v)
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)
\end{array}</math>
|}
<br>
====Logical Transformations====
The information that defines the logical transformation <math>F\!</math> 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.
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math>
|- style="height:40px; background:ghostwhite; width:100%"
| style="width:25%" | <math>u\!</math>
| style="width:25%" | <math>v\!</math>
| style="width:25%; border-left:1px solid black" | <math>f\!</math>
| style="width:25%" | <math>g\!</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
|- style="height:40px; background:ghostwhite"
| style="border-top:1px solid black" | <math>u\!</math>
| style="border-top:1px solid black" | <math>v\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math>
| style="border-top:1px solid black" |
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math>
|}
<br>
Figure 61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure 30.
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math>
|}
<br>
Figure 62 extracts the gist of Figure 61, exhibiting a style of diagram that is adequate for most purposes.
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math>
|}
<br>
====Local Transformations====
Figure 63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> 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.
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math>
|}
<br>
Table 64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math>
|- style="height:40px; background:ghostwhite; width:100%"
| style="width:8%" | <math>u\!</math>
| style="width:8%" | <math>v\!</math>
| style="width:12%; border-left:1px solid black" | <math>x\!</math>
| style="width:12%" | <math>y\!</math>
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math>
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math>
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math>
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math>
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
0
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
0
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
0
\\[4pt]
0
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\uparrow
\\[4pt]
F =
\\[4pt]
(f, g)
\\[4pt]
\uparrow
\end{matrix}</math>
|- style="height:40px; background:ghostwhite"
| style="border-top:1px solid black" | <math>u\!</math>
| style="border-top:1px solid black" | <math>v\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math>
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math>
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math>
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math>
| style="border-top:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math>
|}
<br>
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.
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math>
|- style="height:50px; background:ghostwhite"
| style="width:20%" | <math>X^\bullet~\!</math>
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math>
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math>
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math>
| style="width:20%" | <math>U^\bullet~\!</math>
|- style="background:ghostwhite"
| rowspan="2" | <math>f_i (x, y)\!</math>
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math>
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math>
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math>
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math>
|- style="background:ghostwhite"
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math>
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math>
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}\!</math>
| valign="bottom" |
<math>\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}</math>
|-
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math>
|- style="height:50px; background:ghostwhite"
| style="width:20%" | <math>X^\bullet~\!</math>
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math>
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math>
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math>
| style="width:20%" | <math>U^\bullet~\!</math>
|- style="background:ghostwhite"
| rowspan="2" | <math>f_i (x, y)\!</math>
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math>
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math>
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math>
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math>
|- style="background:ghostwhite"
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math>
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math>
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math>
|-
| <math>f_{0}\!</math>
| <math>\texttt{(~)}\!</math>
| <math>0~0~0~0\!</math>
| <math>\texttt{(~)}\!</math>
| <math>f_{0}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{1}
\\[2pt]
f_{2}
\\[2pt]
f_{4}
\\[2pt]
f_{8}
\end{matrix}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~0
\\[2pt]
0~0~0~1
\\[2pt]
0~1~1~0
\\[2pt]
1~0~0~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(~)}
\\[2pt]
\texttt{(} u \texttt{)(} v \texttt{)}
\\[2pt]
\texttt{(} u \texttt{,~} v \texttt{)}
\\[2pt]
\texttt{~} u \texttt{~~} v \texttt{~}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0}
\\[2pt]
f_{1}
\\[2pt]
f_{6}
\\[2pt]
f_{8}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{3}
\\[2pt]
f_{12}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\[2pt]
\texttt{~} x \texttt{~}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~1
\\[2pt]
1~1~1~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{~(} u \texttt{)(} v \texttt{)~}
\\[2pt]
\texttt{((} u \texttt{)(} v \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{1}
\\[2pt]
f_{14}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{6}
\\[2pt]
f_{9}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\[2pt]
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~1
\\[2pt]
1~0~0~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} u \texttt{~~} v \texttt{)}
\\[2pt]
\texttt{~} u \texttt{~~} v \texttt{~}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{7}
\\[2pt]
f_{8}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{5}
\\[2pt]
f_{10}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\[2pt]
\texttt{~} y \texttt{~}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~0
\\[2pt]
1~0~0~1
\end{matrix}~\!</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{~(} u \texttt{,~} v \texttt{)~}
\\[2pt]
\texttt{((} u \texttt{,~} v \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{6}
\\[2pt]
f_{9}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{7}
\\[2pt]
f_{11}
\\[2pt]
f_{13}
\\[2pt]
f_{14}
\end{matrix}</math>
| valign="bottom" |
<math>\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}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~1
\\[2pt]
1~0~0~1
\\[2pt]
1~1~1~0
\\[2pt]
1~1~1~1
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{~(} u \texttt{~~} v \texttt{)~}
\\[2pt]
\texttt{((} u \texttt{,~} v \texttt{))}
\\[2pt]
\texttt{((} u \texttt{)(} v \texttt{))}
\\[2pt]
\texttt{((~))}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{7}
\\[2pt]
f_{9}
\\[2pt]
f_{14}
\\[2pt]
f_{15}
\end{matrix}</math>
|-
| <math>f_{15}\!</math>
| <math>\texttt{((~))}\!</math>
| <math>1~1~1~1\!</math>
| <math>\texttt{((~))}\!</math>
| <math>f_{15}\!</math>
|}
<br>
====Difference Operators and Tangent Functors====
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math>
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\begin{array}{lll}
\mathrm{E}G_i
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)
\end{array}</math>
|}
<br>
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\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}</math>
|}
<br>
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> 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 <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> 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 <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
|
<math>\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}</math>
|}
<br>
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 <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix 3 and a summary of the results is presented in Tables 66-i and 66-ii.
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math>
|
<math>\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}</math>
|}
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math>
|
<math>\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}</math>
|}
<br>
Table 67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table 68 recaps these results in symbolic terms, agreeing with earlier derivations.
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math>
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math>
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math>
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math>
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math>
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math>
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math>
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math>
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math>
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math>
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math>
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math>
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math>
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math>
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math>
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math>
|-
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
|-
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
|-
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
|-
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math>
| style="vertical-align:top; border-top:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
|}
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math>
|- style="height:40px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| style="border-left:1px solid black" | <math>f\!</math>
| <math>g\!</math>
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math>
| <math>{\mathrm{D}g}\!</math>
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math>
| <math>{\mathrm{d}g}\!</math>
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math>
| <math>{\mathrm{d}^2\!g}\!</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\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}</math>
| style="border-top:1px solid black" |
<math>\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}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[4pt]
\mathrm{d}v
\\[4pt]
\mathrm{d}u
\\[4pt]
0
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\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}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\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}\!</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
0
\\[4pt]
0
\end{matrix}</math>
|}
<br>
Figure 69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables 66-i and 66-ii, for ease of reference repeated below.
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|
<math>\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}</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math>
|}
<br>
Figure 70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables 66-i and 66-ii, the corresponding rows of which are repeated below.
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|
<math>\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}</math>
|}
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math>
|}
<br>
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math>
<br>
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math>
|}
<br>
* '''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.
{| align="center" border="0" cellpadding="10"
|
<pre>
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))>
</pre>
|}
==Epilogue, Enchoiry, Exodus==
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
It is time to explain myself . . . . let us stand up.
| width="4%" |
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 79]
|}
==Appendices==
===Appendix 1. Propositional Forms and Differential Expansions===
====Table A1. Propositional Forms on Two Variables====
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math>
|- style="background:ghostwhite"
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
|- style="background:ghostwhite"
|
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0\!</math>
| || ||
|- style="background:ghostwhite"
|
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0\!</math>
| || ||
|-
|
<math>\begin{matrix}
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1
\end{matrix}\!</math>
|
<math>\begin{matrix}
\texttt{(~)}
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{(} x \texttt{)~ ~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~ ~(} y \texttt{)}
\\
\texttt{(} x \texttt{,~} y \texttt{)}
\\
\texttt{(} x \texttt{~~} y \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{false}
\\
\text{neither}~ x ~\text{nor}~ y
\\
y ~\text{without}~ x
\\
\text{not}~ x
\\
x ~\text{without}~ y
\\
\text{not}~ y
\\
x ~\text{not equal to}~ y
\\
\text{not both}~ x ~\text{and}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
0
\\
\lnot x \land \lnot y
\\
\lnot x \land y
\\
\lnot x
\\
x \land \lnot y
\\
\lnot y
\\
x \ne y
\\
\lnot x \lor \lnot y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}
\end{matrix}\!</math>
|
<math>\begin{matrix}
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{~~} x \texttt{~~} y \texttt{~~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\\
\texttt{~ ~ ~ ~} y \texttt{~~}
\\
\texttt{~(} x \texttt{~(} y \texttt{))}
\\
\texttt{~~} x \texttt{~ ~ ~ ~}
\\
\texttt{((} x \texttt{)~} y \texttt{)~}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{((~))}
\end{matrix}</math>
|
<math>\begin{matrix}
x ~\text{and}~ y
\\
x ~\text{equal to}~ y
\\
y
\\
\text{not}~ x ~\text{without}~ y
\\
x
\\
\text{not}~ y ~\text{without}~ x
\\
x ~\text{or}~ y
\\
\text{true}
\end{matrix}</math>
|
<math>\begin{matrix}
x \land y
\\
x = y
\\
y
\\
x \Rightarrow y
\\
x
\\
x \Leftarrow y
\\
x \lor y
\\
1
\end{matrix}</math>
|}
<br>
====Table A2. Propositional Forms on Two Variables====
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math>
|- style="background:ghostwhite"
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
|- style="background:ghostwhite"
|
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0\!</math>
| || ||
|- style="background:ghostwhite"
|
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0\!</math>
| || ||
|-
| <math>f_{0}\!</math>
| <math>f_{0000}\!</math>
| <math>0~0~0~0</math>
| <math>\texttt{(~)}\!</math>
| <math>\text{false}\!</math>
| <math>0\!</math>
|-
|
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{neither}~ x ~\text{nor}~ y
\\
y ~\text{without}~ x
\\
x ~\text{without}~ y
\\
x ~\text{and}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot x \land \lnot y
\\
\lnot x \land y
\\
x \land \lnot y
\\
x \land y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0011}\\f_{1100}
\end{matrix}</math>
|
<math>\begin{matrix}
0~0~1~1\\1~1~0~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not}~ x
\\
x
\end{matrix}\!</math>
|
<math>\begin{matrix}
\lnot x
\\
x
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0110}\\f_{1001}
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~1~1~0\\1~0~0~1
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
x ~\text{not equal to}~ y
\\
x ~\text{equal to}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
x \ne y
\\
x = y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0101}\\f_{1010}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~0~1\\1~0~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not}~ y
\\
y
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot y
\\
y
\end{matrix}</math>
|-
|
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
|
<math>\begin{matrix}
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}
\end{matrix}</math>
|
<math>\begin{matrix}
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{~(} x \texttt{~~} y \texttt{)~}
\\
\texttt{~(} x \texttt{~(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{)~}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
\text{not both}~ x ~\text{and}~ y
\\
\text{not}~ x ~\text{without}~ y
\\
\text{not}~ y ~\text{without}~ x
\\
x ~\text{or}~ y
\end{matrix}</math>
|
<math>\begin{matrix}
\lnot x \lor \lnot y
\\
x \Rightarrow y
\\
x \Leftarrow y
\\
x \lor y
\end{matrix}</math>
|-
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| <math>1~1~1~1\!</math>
| <math>\texttt{((~))}\!</math>
| <math>\text{true}\!</math>
| <math>1\!</math>
|}
<br>
====Table A3. E''f'' Expanded Over Differential Features====
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{0}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} x \texttt{~~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} x \texttt{~}
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} x \texttt{~}
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,~} y \texttt{))}
\\
\texttt{~(} x \texttt{,~} y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,~} y \texttt{))}
\\
\texttt{~(} x \texttt{,~} y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{(~} x \texttt{~~} y \texttt{~)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
|- style="background:ghostwhite"
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math>
|}
<br>
====Table A4. D''f'' Expanded Over Differential Features====
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math>
|-
| style="border-top:1px solid black" | <math>f_{0}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,~} y \texttt{))}
\\
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\\
y
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\\
x
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
x
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}1\\1\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} x \texttt{,~} y \texttt{))}
\\
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
y
\\
\texttt{(} y \texttt{)}
\\
y
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
x
\\
x
\\
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
|}
<br>
====Table A5. E''f'' Expanded Over Ordinary Features====
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{E}f|_{xy}\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math>
|-
| style="border-top:1px solid black" | <math>f_{0}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{1}\\f_{2}\\f_{4}\\f_{8}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\\
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{3}\\f_{12}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~}
\\
\texttt{(} \mathrm{d}x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~}
\\
\texttt{(} \mathrm{d}x \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{6}\\f_{9}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
\\
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
\\
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
\\
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
\\
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{5}\\f_{10}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}y \texttt{~}
\\
\texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}y \texttt{~}
\\
\texttt{(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}y \texttt{~}
\end{matrix}</math>
|-
| style="border-top:1px solid black" |
<math>\begin{matrix}
f_{7}\\f_{11}\\f_{13}\\f_{14}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
\\
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
\\
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
\\
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
\\
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
\\
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
\\
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
|}
<br>
====Table A6. D''f'' Expanded Over Ordinary Features====
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math>
|- style="background:ghostwhite"
| style="width:10%; border-bottom:1px solid black" |
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
<math>\mathrm{D}f|_{xy}\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math>
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math>
|-
| style="border-top:1px solid black" | <math>f_{0}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
\\
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
\\
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
\\
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{15}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math>
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math>
|}
<br>
===Appendix 2. Differential Forms===
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables A7 and A8.
Table A7 expands the differential forms that result over a ''logical basis'':
{| align="center" cellpadding="6" style="text-align:center"
|
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math>
|}
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:
{| align="center" cellpadding="6" style="text-align:center"
|
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> and <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math>
|}
Table A8 expands the differential forms that result over an ''algebraic basis'':
{| align="center" cellpadding="6" style="text-align:center"
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math>
|}
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.
====Table A7. Differential Forms Expanded on a Logical Basis====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math>
|- style="background:ghostwhite; height:40px"
|
| style="border-right:none" | <math>f\!</math>
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math>
| <math>\mathrm{d}f~\!</math>
|-
| <math>f_{0}\!</math>
| style="border-right:none" | <math>\texttt{(~)}\!</math>
| style="border-left:4px double black" | <math>0\!</math>
| <math>0\!</math>
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\\
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
\texttt{~} y \texttt{~} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{~} x \texttt{~} ~\partial y
\\
\texttt{~} y \texttt{~} ~\partial x
& + &
\texttt{~} x \texttt{~} ~\partial y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
\\
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}\!</math>
|
<math>\begin{matrix}
\partial x
\\
\partial x
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
\\
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\partial x & + & \partial y
\\
\partial x & + & \partial y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\partial y
\\
\partial y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
\\
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
& + &
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
& + &
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{~} y \texttt{~} ~\partial x
& + &
\texttt{~} x \texttt{~} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{~} x \texttt{~} ~\partial y
\\
\texttt{~} y \texttt{~} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\\
\texttt{(} y \texttt{)} ~\partial x
& + &
\texttt{(} x \texttt{)} ~\partial y
\end{matrix}</math>
|-
| <math>f_{15}\!</math>
| style="border-right:none" | <math>\texttt{((~))}\!</math>
| style="border-left:4px double black" | <math>0\!</math>
| <math>0\!</math>
|}
<br>
====Table A8. Differential Forms Expanded on an Algebraic Basis====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math>
|- style="background:ghostwhite; height:40px"
|
| style="border-right:none" | <math>f\!</math>
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math>
| <math>\mathrm{d}f~\!</math>
|-
| <math>f_{0}\!</math>
| style="border-right:none" | <math>\texttt{(~)}\!</math>
| style="border-left:4px double black" | <math>0\!</math>
| <math>0\!</math>
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
\\
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
\\
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
\\
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
\\
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
\\
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
\\
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}\!</math>
| <math>\begin{matrix}
\mathrm{d}x
\\
\mathrm{d}x
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}x & + & \mathrm{d}y
\\
\mathrm{d}x & + & \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x & + & \mathrm{d}y
\\
\mathrm{d}x & + & \mathrm{d}y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y
\\
\mathrm{d}y
\end{matrix}</math>
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-right:none" |
<math>\begin{matrix}
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
\\
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
\\
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
\\
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
\\
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
\\
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
\\
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
\end{matrix}</math>
|-
| <math>f_{15}\!</math>
| style="border-right:none" | <math>\texttt{((~))}\!</math>
| style="border-left:4px double black" | <math>0\!</math>
| <math>0\!</math>
|}
<br>
====Table A9. Tangent Proposition as Pointwise Linear Approximation====
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math>
|- style="background:ghostwhite; height:40px"
| style="border-right:none" | <math>f\!</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{d}f =
\\[2pt]
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}^2\!f =
\\[2pt]
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\mathrm{d}f|_{x \, y}</math>
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-right:none" | <math>f_0\!</math>
| style="border-left:4px double black" | <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math>
| style="border-left:4px double black" |
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
|-
| style="border-right:none" |
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\end{matrix}\!</math>
| <math>\begin{matrix}
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\\
\mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
|-
| style="border-right:none" | <math>f_{15}\!</math>
| style="border-left:4px double black" | <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
|}
<br>
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" |
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math>
|- style="background:ghostwhite; height:40px"
| style="border-right:none" | <math>f\!</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\mathrm{D}f
\\
= & \mathrm{d}f & + & \mathrm{d}^2\!f
\\
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
| <math>\mathrm{d}f|_{x \, y}</math>
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
|-
| style="border-right:none" | <math>f_0\!</math>
| style="border-left:4px double black" | <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\\mathrm{d}x
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|-
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
| style="border-left:4px double black" |
<math>\begin{matrix}
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
\end{matrix}</math>
|
<math>\begin{matrix}
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
\end{matrix}</math>
|
<math>\begin{matrix}
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
\end{matrix}</math>
|-
| style="border-right:none" | <math>f_{15}\!</math>
| style="border-left:4px double black" | <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
|}
<br>
====Table A11. Partial Differentials and Relative Differentials====
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math>
|- style="background:ghostwhite; height:50px"
|
| <math>f\!</math>
| <math>\frac{\partial f}{\partial x}\!</math>
| <math>\frac{\partial f}{\partial y}\!</math>
|
<math>\begin{matrix}
\mathrm{d}f =
\\[2pt]
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math>
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math>
|-
| <math>f_0\!</math>
| <math>\texttt{(~)}\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
|-
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)(} y \texttt{)}
\\
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\texttt{~} x \texttt{~~} y \texttt{~}
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\
\texttt{~} x \texttt{~}
\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\end{matrix}</math>
| <math>\begin{matrix}0\\0\end{matrix}</math>
| <math>\begin{matrix}1\\1\end{matrix}</math>
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
|-
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{(~} x \texttt{~~} y \texttt{~)}
\\
\texttt{(~} x \texttt{~(} y \texttt{))}
\\
\texttt{((} x \texttt{)~} y \texttt{~)}
\\
\texttt{((} x \texttt{)(} y \texttt{))}
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\\
\texttt{~} y \texttt{~}
\\
\texttt{(} y \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{~} x \texttt{~}
\\
\texttt{~} x \texttt{~}
\\
\texttt{(} x \texttt{)}
\\
\texttt{(} x \texttt{)}
\end{matrix}</math>
|
<math>\begin{matrix}
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
\\
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\\
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
|-
| <math>f_{15}\!</math>
| <math>\texttt{((~))}\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
| <math>0\!</math>
|}
<br>
====Table A12. Detail of Calculation for the Difference Map====
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math>
|- style="background:ghostwhite"
| style="width:6%" |
| style="width:14%; border-left:1px solid black" | <math>f\!</math>
| style="width:20%; border-left:4px double black" |
<math>\begin{array}{cr}
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}
\\[4pt]
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}
\\[4pt]
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}
\end{array}</math>
| style="width:20%; border-left:1px solid black" |
<math>\begin{array}{cr}
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
\\[4pt]
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
\\[4pt]
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
\end{array}</math>
| style="width:20%; border-left:1px solid black" |
<math>\begin{array}{cr}
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
\\[4pt]
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
\\[4pt]
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
\end{array}</math>
| style="width:20%; border-left:1px solid black" |
<math>\begin{array}{cr}
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
\\[4pt]
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
\\[4pt]
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
\end{array}</math>
|-
| style="border-top:4px double black" | <math>f_{0}\!</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math>
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math>
|-
| style="border-top:4px double black" | <math>f_{1}\!</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{2}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{4}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{8}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{3}\!</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{(} x \texttt{)}\!</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & x
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & x
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & \texttt{(} x \texttt{)}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{12}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>x\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & x
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} x \texttt{)}
\\[4pt]
+ & x
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & x
\\[4pt]
+ & x
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & x
\\[4pt]
+ & x
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{6}\!</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{9}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{,~} y \texttt{))}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{5}\!</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{(} y \texttt{)}\!</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & y
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & y
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & \texttt{(} y \texttt{)}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{10}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>y\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & y
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & y
\\[4pt]
+ & y
\\[4pt]
= & 0
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{(} y \texttt{)}
\\[4pt]
+ & y
\\[4pt]
= & 1
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & y
\\[4pt]
+ & y
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{7}\!</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math>
| style="border-top:4px double black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{matrix}</math>
| style="border-top:4px double black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{11}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{13}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:1px solid black" | <math>f_{14}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math>
| style="border-top:1px solid black; border-left:4px double black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & \texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
\end{matrix}</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
~ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
+ & \texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
= & 0
\end{matrix}</math>
|-
| style="border-top:4px double black" | <math>f_{15}\!</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math>
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math>
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math>
|}
<br>
===Appendix 3. Computational Details===
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====
=====Computation of ε''f''<sub>8</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{8}
& = && f_{8}(u, v)
\\[4pt]
& = && uv
\\[4pt]
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v
\\[20pt]
\boldsymbol\varepsilon f_{8}
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
\\[4pt]
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\end{array}\!</math>
|}
<br>
=====Computation of E''f''<sub>8</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{E}f_{8}
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[4pt]
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[4pt]
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)
\\[4pt]
& = & \texttt{ } u \texttt{ } v \texttt{ } \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{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\\[20pt]
\mathrm{E}f_{8}
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
\\[4pt]
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}\!</math>
|}
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math>
|
<math>\begin{array}{*{9}{c}}
\mathrm{E}f_{8}
& = & (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}f_{8}
& = & \texttt{ } u \texttt{ } v \texttt{ } \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{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}\!</math>
|}
<br>
=====Computation of D''f''<sub>8</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{8}
& = && \mathrm{E}f_{8}
& + & \boldsymbol\varepsilon f_{8}
\\[4pt]
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
& + & f_{8}(u, v)
\\[4pt]
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
& + & uv
\\[20pt]
\mathrm{D}f_{8}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \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{~}
& + & 0
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
& + & 0
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}
\\[20pt]
\mathrm{D}f_{8}
& = && \texttt{ } u \texttt{ } v \texttt{ } \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{~}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}\!</math>
|}
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{8}
& = & \boldsymbol\varepsilon f_{8}
& + & \mathrm{E}f_{8}
\\[6pt]
& = & f_{8}(u, v)
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[6pt]
& = & uv
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
& = & 0
& + & u \cdot \mathrm{d}v
& + & v \cdot \mathrm{d}u
& + & \mathrm{d}u ~ \mathrm{d}v
\\[6pt]
\mathrm{D}f_{8}
& = & 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 ~ \mathrm{d}v
\end{array}</math>
|}
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math>
|
<math>\begin{array}{c*{9}{l}}
\mathrm{D}f_{8}
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}
\\[20pt]
\boldsymbol\varepsilon f_{8}
& = & 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}f_{8}
& = & 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
\\[20pt]
\mathrm{D}f_{8}
& = & ~ ~ 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}\!</math>
|}
=====Computation of d''f''<sub>8</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{8}
& = & uv \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 ~ \mathrm{d}v
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{8}
& = & uv \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}</math>
|}
<br>
=====Computation of r''f''<sub>8</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}
\\[20pt]
\mathrm{D}f_{8}
& = & uv \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 ~ \mathrm{d}v
\\[6pt]
\mathrm{d}f_{8}
& = & uv \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
\\[20pt]
\mathrm{r}f_{8}
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}
<br>
=====Computation Summary for Conjunction=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{8}
& = & uv \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\mathrm{E}f_{8}
& = & uv \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 ~ \mathrm{d}v
\\[6pt]
\mathrm{D}f_{8}
& = & uv \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 ~ \mathrm{d}v
\\[6pt]
\mathrm{d}f_{8}
& = & uv \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}f_{8}
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}
<br>
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====
=====Computation of ε''f''<sub>9</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{9}
& = && f_{9}(u, v)
\\[4pt]
& = && \texttt{((} u \texttt{,~} v \texttt{))}
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)
\\[4pt]
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}
\\[20pt]
\boldsymbol\varepsilon f_{9}
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\end{array}</math>
|}
<br>
=====Computation of E''f''<sub>9</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{9}
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\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{) }
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\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{))}
\\[20pt]
\mathrm{E}f_{9}
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \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{~}
& + & 0
\\[4pt]
&& + & 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \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{)}
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\end{array}</math>
|}
<br>
=====Computation of D''f''<sub>9</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{9}
& = && \mathrm{E}f_{9}
& + & \boldsymbol\varepsilon f_{9}
\\[4pt]
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)
& + & f_{9}(u, v)
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
& + & \texttt{((} u \texttt{,} v \texttt{))}
\\[20pt]
\mathrm{D}f_{9}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\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{~}
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\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{~}
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\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{)}
\\[4pt]
&& + & 0
& + & 0
& + & 0
& + & 0
\\[20pt]
\mathrm{D}f_{9}
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\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}\!</math>
|}
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{9}
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}
<br>
=====Computation of d''f''<sub>9</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{9}
& = & \texttt{ } u \texttt{ } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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]
\Downarrow
\\[6pt]
\mathrm{d}f_{9}
& = & \texttt{ } u \texttt{ } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}</math>
|}
<br>
=====Computation of r''f''<sub>9</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}
\\[20pt]
\mathrm{D}f_{9}
& = & \texttt{ } u \texttt{ } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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_{9}
& = & \texttt{ } u \texttt{ } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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{)}
\\[20pt]
\mathrm{r}f_{9}
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
|}
<br>
=====Computation Summary for Equality=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{9}
& = & uv \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{9}
& = & uv \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}f_{9}
& = & uv \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}f_{9}
& = & uv \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}f_{9}
& = & uv \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
|}
<br>
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====
=====Computation of ε''f''<sub>11</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{11}
& = && f_{11}(u, v)
\\[4pt]
& = && \texttt{(} u \texttt{(} v \texttt{))}
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ }
& + & 0
& + & \texttt{(} u \texttt{) } v \texttt{ }
& + & \texttt{(} u \texttt{)(} v \texttt{)}
\\[20pt]
\boldsymbol\varepsilon f_{11}
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{) } v \texttt{ } \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{)}
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
& + & 0
& + & \texttt{(} u \texttt{) } v \texttt{ } \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{~}
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{) } v \texttt{ } \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{)}
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
& + & 0
& + & \texttt{(} u \texttt{) } v \texttt{ } \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}\!</math>
|}
<br>
=====Computation of E''f''<sub>11</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{11}
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[4pt]
& = &&
\texttt{(}
\\
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\\
&&& \texttt{(}
\\
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\
&&& \texttt{))}
\\[4pt]
& = &&
u 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{))}
\\[4pt]
& = &&
u 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 ~ \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)}
\!\cdot\!
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{E}f_{11}
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\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{)}
\\[4pt]
&& + & 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \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{~}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \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{~}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\end{array}</math>
|}
<br>
=====Computation of D''f''<sub>11</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{11}
& = && \mathrm{E}f_{11}
& + & \boldsymbol\varepsilon f_{11}
\\[4pt]
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
& + & f_{11}(u, v)
\\[4pt]
& = &&
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\texttt{))}
& + &
\texttt{(} u \texttt{(} v \texttt{))}
\\[20pt]
\mathrm{D}f_{11}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}
& + & 0
& + & 0
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
& + & 0
\\[20pt]
\mathrm{D}f_{11}
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math>
|
<math>\begin{array}{c*{9}{l}}
\mathrm{D}f_{11}
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}
\\[20pt]
\boldsymbol\varepsilon f_{11}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{11}
& = &
u 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 ~ \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)}
\cdot
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{D}f_{11}
& = &
u 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 ~ \mathrm{d}v \texttt{~}
& + &
\texttt{(} u \texttt{)(} v \texttt{)}
\cdot
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}
\end{array}</math>
|}
<br>
=====Computation of d''f''<sub>11</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\end{array}</math>
|}
<br>
=====Computation of r''f''<sub>11</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}
\\[20pt]
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\\[20pt]
\mathrm{r}f_{11}
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}
<br>
=====Computation Summary for Implication=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{11}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{11}
& = & u 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 ~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\\[6pt]
\mathrm{r}f_{11}
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}
<br>
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====
=====Computation of ε''f''<sub>14</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{14}
& = && f_{14}(u, v)
\\[4pt]
& = && \texttt{((} u \texttt{)(} v \texttt{))}
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ }
& + & \texttt{ } u \texttt{ (} v \texttt{)}
& + & \texttt{(} u \texttt{) } v \texttt{ }
& + & 0
\\[20pt]
\boldsymbol\varepsilon f_{14}
& = && \texttt{ } u \texttt{ } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \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{~}
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \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{~}
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
& + & 0
\end{array}</math>
|}
<br>
=====Computation of E''f''<sub>14</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{14}
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[4pt]
& = &&
\texttt{((}
\\
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\\
&&& \texttt{)(}
\\
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\
&&& \texttt{))}
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\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{))}
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\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{))}
\\[20pt]
\mathrm{E}f_{14}
& = && \texttt{ } u \texttt{ } v \texttt{ } \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{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
& + & 0
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \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{~}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
\\[4pt]
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{) } v \texttt{ } \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{)}
\\[4pt]
&& + & 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \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{~}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
\end{array}</math>
|}
<br>
=====Computation of D''f''<sub>14</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{14}
& = && \mathrm{E}f_{14}
& + & \boldsymbol\varepsilon f_{14}
\\[4pt]
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)
& + & f_{14}(u, v)
\\[4pt]
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}
& + & \texttt{((} u \texttt{)(} v \texttt{))}
\\[20pt]
\mathrm{D}f_{14}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & 0
& + & 0
& + & \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{~~}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
\\[4pt]
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
& + & 0
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
\\[20pt]
\mathrm{D}f_{14}
& = && uv \!\cdot\! \mathrm{d}u ~ \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{))}
\end{array}</math>
|}
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math>
|
<math>\begin{array}{*{9}{l}}
\mathrm{D}f_{14}
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
\end{array}</math>
|}
<br>
=====Computation of d''f''<sub>14</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{14}
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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]
\Downarrow
\\[6pt]
\mathrm{d}f_{14}
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\end{array}</math>
|}
<br>
=====Computation of r''f''<sub>14</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}
\\[20pt]
\mathrm{D}f_{14}
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{) } v \texttt{ } \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_{14}
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[20pt]
\mathrm{r}f_{14}
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}
<br>
=====Computation Summary for Disjunction=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{14}
& = & uv \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_{14}
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \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_{14}
& = & uv \cdot \mathrm{d}u ~ \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_{14}
& = & uv \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_{14}
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
\end{array}</math>
|}
<br>
===Appendix 4. Source Materials===
===Appendix 5. Various Definitions of the Tangent Vector===
==References==
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{| cellpadding=3
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|-
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|-
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|-
| valign=top | [Ed88]
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.
|-
| valign=top | [Fla]
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989.
|-
| valign=top | [Has]
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.
|-
| valign=top | [KoB]
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.
|-
| valign=top | [MaB]
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.
|-
| valign=top | [Mac]
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.
|-
| valign=top | [McC]
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.
|-
| valign=top | [Mc1]
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].
|-
| valign=top | [MiP]
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.
|-
| valign=top | [Rum]
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.
|}
===Incidental Works===
{| cellpadding=3
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|-
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| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.
|-
| valign=top | [Hom]
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|-
| valign=top | [Jam]
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.
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| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.
|-
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| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.
|-
| valign=top | [PlaR]
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.
|-
| valign=top | [PlaS]
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.
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| valign=top | [Qui]
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| valign=top | [Web]
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|}
==Document History==
<p align="center"><math>\begin{array}{lcr}
& \text{Differential Logic and Dynamic Systems} &
\\
\text{Author:} & \text{Jon Awbrey} & \text{October 20, 1994}
\\
\text{Course:} & \text{Engineering 690, Graduate Project} & \text{Winter Term 1994}
\\
\text{Supervisor:} & \text{M.A. Zohdy} & \text{Oakland University}
\\
\text{Created:} && \text{16 Dec 1993}
\\
\text{Relayed:} && \text{31 Oct 1994}
\\
\text{Revised:} && \text{03 Jun 2003}
\\
\text{Recoded:} && \text{03 Jun 2007}
\end{array}</math></p>
[[Category:Adaptive Systems]]
[[Category:Artificial Intelligence]]
[[Category:Boolean Algebra]]
[[Category:Boolean Functions]]
[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Cybernetics]]
[[Category:Differential Logic]]
[[Category:Discrete Systems]]
[[Category:Dynamical Systems]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Functional Logic]]
[[Category:Graph Theory]]
[[Category:Group Theory]]
[[Category:Inquiry]]
[[Category:Knowledge Representation]]
[[Category:Linguistics]]
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[[Category:Mathematics]]
[[Category:Mathematical Systems Theory]]
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[[Category:Systems Science]]
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