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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems (view source)
Revision as of 18:41, 26 June 2007
, 18:41, 26 June 2007→Terminological Interlude: wiki table
Table 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'''<sup>1</sup> or '''D'''<sup>1</sup>), can be regarded either as propositions W''J'' : E''U'' → '''B''' or as logical transformations W''J'' : E''U''<sup> •</sup> → ''X''<sup> •</sup>. 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'' : E''U'' → '''D''', does not distinguish it from the general run of differential propositions ''G'' : E''U'' → '''B''', it is usual to single out d''J'' as the ''tangent proposition'' of ''J''.
Table 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'''<sup>1</sup> or '''D'''<sup>1</sup>), can be regarded either as propositions W''J'' : E''U'' → '''B''' or as logical transformations W''J'' : E''U''<sup> •</sup> → ''X''<sup> •</sup>. 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'' : E''U'' → '''D''', does not distinguish it from the general run of differential propositions ''G'' : E''U'' → '''B''', it is usual to single out d''J'' as the ''tangent proposition'' of ''J''.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes
|+ '''Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes'''
|- style="background:paleturquoise"
| | Operator | Proposition | Map |
!
! Operator
| | | | |
! Proposition
| Tacit | !e! : | !e!J : | !e!J : |
! Map
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x] |
|-
| | (U%->X%)->(EU%->X%) | B^2 x D^2 -> B | [B^2 x D^2]->[B^1] |
|
| | | | |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tacit
| | | | |
|-
| Trope | !h! : | !h!J : | !h!J : |
| Extension
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
|}
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
|
| | | | |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\epsilon</math> :
| | | | |
|-
| Enlargement | E : | EJ : | EJ : |
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
|-
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
| | | | |
|}
|
| | | | |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Difference | D : | DJ : | DJ : |
| <math>\epsilon</math>''J'' :
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
|-
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
| | | | |
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''B'''
|}
| Differential | d : | dJ : | dJ : |
|
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| <math>\epsilon</math>''J'' :
| | | | |
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'']
| | | | |
|-
| Remainder | r : | rJ : | rJ : |
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
|}
| | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
|-
| | | | |
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| | | | |
| Trope
| Radius | $e$ = <!e!, !h!> : | | $e$J : |
|-
| Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
| Extension
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math> :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math>''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <math>\eta</math>''J'' :
|-
</pre>
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Enlargement
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| E''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Difference
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| D''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Differential
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Remainder
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| r''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Radius
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
|-
|
|-
|
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''e'''</font>''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Secant
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
|-
|
|-
|
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''E'''</font>''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Chord
|-
| Operator
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
|
|-
|
|-
|
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''D'''</font>''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|-
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| Tangent
|-
| Functor
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
|-
| ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| d''J'' :
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
|}
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
| <font face=georgia>'''T'''</font>''J'' :
|-
| [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
|-
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
|}
|}<br>
====End of Perfunctory Chatter : Time to Roll the Clip!====
====End of Perfunctory Chatter : Time to Roll the Clip!====