Changes

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;.

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

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

o------o-------------------------o------------------o----------------------------o

|- style="background:paleturquoise"

| Item | Notation                | Description      | Type                      |

o------o-------------------------o------------------o----------------------------o

|     |                        |                  |                            |

| U%  | = [u, v]                | Source Universe  | [B^2]                      |

|     |                        |                  |                            |

|-

o------o-------------------------o------------------o----------------------------o

|     |                         |                  |                            |

| X%  | = [x]                  | Target Universe | [B^1]                      |

|     |                         |                 |                           |

o------o-------------------------o------------------o----------------------------o

|-

|     |                        |                  |                            |

| EU%  | = [u, v, du, dv]        | Extended        | [B^2 x D^2]                |

|     |                         | Source Universe  |                            |

|     |                         |                 |                           |

o------o-------------------------o------------------o----------------------------o

|-

|     |                        |                  |                            |

| EX%  | = [x, dx]              | Extended        | [B^1 x D^1]                |

|     |                        | Target Universe  |                            |

|     |                         |                  |                            |

o------o-------------------------o------------------o----------------------------o

|-

|     |                        |                  |                            |

| E''X''^&nbsp;&bull;

| J    | J : U -> B              | Proposition      | (B^2 -> B) c [B^2]        |

| = [''x'', d''x'']

| Extended Target Universe

| ['''B'''¹ &times; '''D'''¹]

|-

|-

|-

|-

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{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

|-

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| &rarr;

|-

| (E''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;E''X''^&nbsp;&bull;)&nbsp;,

|-

| for each W in the set:

|-

| {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d}

{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"

</pre>

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| d

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| Tacit Extension Operator || <math>\epsilon</math>

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| Trope Extension Operator || <math>\eta</math>

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| for each <font face=georgia>'''W'''</font> in the set:

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| {<font face=georgia>'''e'''</font>,&nbsp;<font face=georgia>'''E'''</font>,&nbsp;<font face=georgia>'''D'''</font>,&nbsp;<font face=georgia>'''T'''</font>}

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| Operator

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| &nbsp;

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| <font face=georgia>'''D'''</font>

| <font face=georgia>'''T'''</font>

| valign="top" | &nbsp;

| colspan="2"  |

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| Radius Operator || <font face=georgia>'''e'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;<math>\eta</math>›

|-

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

|}<br>

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''.