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The enlargement map E''J'' is computed from the proposition ''J'' by making a particular class of formal substitutions for its variables, in this case ''u'' + d''u'' for ''u'' and ''v'' + d''v'' for ''v'', and subsequently expanding the result in whatever way happens to be convenient for the end in view.

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&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables, and ultimately developing E''J'' over the cells of [''u'',&nbsp;''v''].  The critical step of this procedure uses the facts that (0,&nbsp;''x'')&nbsp;=&nbsp;0&nbsp;+&nbsp;''x''&nbsp;=&nbsp;''x'' and (1,&nbsp;''x'')&nbsp;=&nbsp;1&nbsp;+&nbsp;''x''&nbsp;=&nbsp;(''x'') for any boolean variable ''x''.

Table&nbsp;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>

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Figures&nbsp;40-a through 40-d present several views of the enlarged proposition E''J''.

Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math>

 

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<p>[[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]</p>

 

<p><center><font size="+1">'''Figure 40-a. Enlargement of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p>

| [[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>

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<p>[[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]</p>

 

<p><center><font size="+1">'''Figure 40-b. Enlargement of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p>

| [[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>

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<p>[[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]

<p><center><font size="+1">'''Figure 40-d. Enlargement of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math>

An intuitive reading of the proposition E''J'' becomes available at this point, and may be useful.  Recall that propositions in the extended universe E''U''^&nbsp;&bull; express the ''dispositions'' of system and the constraints that are placed on them.  In other words, a differential proposition in E''U''^&nbsp;&bull; can be read as referring to various changes that a system might undergo in and from its various states.  In particular, we can understand E''J'' as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of ''J'', that is, the region of the universe where ''J'' is true.  This interpretation is visibly clear in the Figures above, and appeals to the imagination in a satisfying way, but it has the added benefit of giving fresh meaning to the original name of the shift operator E.  Namely, E''J'' can be read as a proposition that ''enlarges'' on the meaning of ''J'', in the sense of explaining its practical bearings and clarifying what it means in terms of the available options for differential action and the consequential effects that result from each choice.

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 &mdash; the available options for differential action and the consequential effects that result from each choice.

Treated this way, the enlargement E''J'' has strong ties to the normal use of ''J'', no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of ''J'', in effect, pointing to the interpretive elements in its fiber of truth ''J''^&ndash;1(1).  It is this kind of ''use'' that is often compared with the ''mention'' of a proposition, and thereby hangs a tale.

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 &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.

=====Digression : Reflection on Use and Mention=====

=====Digression : Reflection on Use and Mention=====

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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 "''J''&nbsp;" to indicate the region ''J''^&ndash;1(1) and using "''J''&nbsp;" to indicate the function ''J''.  You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion.  But there seems to be no likelihood in practice that their interactions can be avoided.  If the name "''J''&nbsp;" is used as a sign of the function ''J'', and if the function ''J'' has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not "''J''&nbsp;" 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 we have an inference like the following:  If a buffalo is white, and white is a color, then a buffalo is a color.  But a buffalo is not, only buff is.

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.

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.

Table&nbsp;54 provides basic notation and descriptive information for the objects and operators that are used used in this Example, giving the generic type (or broadest defined type) for each entity.  Here, the operators <font face=georgia>'''W'''</font> in {<font face=georgia>'''e'''</font>,&nbsp;<font face=georgia>'''E'''</font>,&nbsp;<font face=georgia>'''D'''</font>,&nbsp;<font face=georgia>'''d'''</font>,&nbsp;<font face=georgia>'''r'''</font>} and their components W in {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d,&nbsp;r} both have the same broad type <font face=georgia>'''W'''</font>,&nbsp;W&nbsp;:&nbsp;(''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull;)&nbsp;&rarr;&nbsp;(E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''X''^&nbsp;&bull;), as would be expected of operators that map transformations ''J''&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull; to extended transformations <font face=georgia>'''W'''</font>''J'',&nbsp;W''J''&nbsp;:&nbsp;E''U^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''X''^&nbsp;&bull;.

Table&nbsp;54 provides basic notation and descriptive information for the objects and operators that are used used in this Example, giving the generic type (or broadest defined type) for each entity.  Here, the operators <font face=georgia>'''W'''</font> in {<font face=georgia>'''e'''</font>,&nbsp;<font face=georgia>'''E'''</font>,&nbsp;<font face=georgia>'''D'''</font>,&nbsp;<font face=georgia>'''d'''</font>,&nbsp;<font face=georgia>'''r'''</font>} and their components W in {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d,&nbsp;r} both have the same broad type <font face=georgia>'''W'''</font>,&nbsp;W&nbsp;:&nbsp;(''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull;)&nbsp;&rarr;&nbsp;(E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''X''^&nbsp;&bull;), as would be expected of operators that map transformations ''J''&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull; to extended transformations <font face=georgia>'''W'''</font>''J'',&nbsp;W''J''&nbsp;:&nbsp;E''U^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''X''^&nbsp;&bull;.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:70%"

|+ '''Table 54.  Cast of Characters:  Expansive Subtypes of Objects and Operators'''

|+ '''Table 54.  Cast of Characters:  Expansive Subtypes of Objects and Operators'''

|- style="background:ghostwhite"

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| Tangent Functor || <font face=georgia>'''T'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›

| Tangent Functor || <font face=georgia>'''T'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›

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Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results.  Here, I list the restrictive subtype (or narrowest defined subtype) that applies to each entity, and I indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol.  Accordingly, each of the component operator maps W''J'', since their ranges are 1-dimensional (of type '''B'''¹ or '''D'''¹), can be regarded either as propositions W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''' or as logical transformations W''J''&nbsp;:&nbsp;E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull;.  As a rule, the plan of the Table allows us to name each entry by detaching the 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 d''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''D''', does not distinguish it from the general run of differential propositions ''G''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''', it is usual to single out d''J'' as the ''tangent proposition'' of ''J''.

Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results.  Here, I list the restrictive subtype (or narrowest defined subtype) that applies to each entity, and I indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol.  Accordingly, each of the component operator maps W''J'', since their ranges are 1-dimensional (of type '''B'''¹ or '''D'''¹), can be regarded either as propositions W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''' or as logical transformations W''J''&nbsp;:&nbsp;E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull;.  As a rule, the plan of the Table allows us to name each entry by detaching the 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 d''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''D''', does not distinguish it from the general run of differential propositions ''G''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''', it is usual to single out d''J'' as the ''tangent proposition'' of ''J''.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:70%"

|+ '''Table 55.  Synopsis of Terminology:  Restrictive and Alternative Subtypes'''

|+ '''Table 55.  Synopsis of Terminology:  Restrictive and Alternative Subtypes'''

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| ['''B'''²&nbsp;&times;&nbsp;'''D'''²]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']

| ['''B'''²&nbsp;&times;&nbsp;'''D'''²]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']

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====End of Perfunctory Chatter : Time to Roll the Clip!====

====End of Perfunctory Chatter : Time to Roll the Clip!====