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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
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A proposition in the tangent universe [E<font face="lucida calligraphy">A</font>] is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
With these constructions, to be specific, the differential extension E''A'' and the differential proposition ''h'' : E''A'' → '''B''', we have arrived, in concept at least, at one of the major subgoals of this study. At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 8).
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|+ '''Table 8. Notation for the Differential Extension of Propositional Calculus'''
|- style="background:paleturquoise"
! Symbol
! Notation
! Description
! Type
|-
| d<font face="lucida calligraphy">A<font>
| {d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
|
Alphabet of<br>
differential<br>
features
| [''n''] = '''n'''
|-
| d''A''<sub>''i''</sub>
| {(d''a''<sub>''i''</sub>), d''a''<sub>''i''</sub>}
|
Differential<br>
dimension ''i''
| '''D'''
|-
| d''A''
|
〈d<font face="lucida calligraphy">A</font>〉<br>
〈d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>〉<br>
{‹d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>›}<br>
d''A''<sub>1</sub> × … × d''A''<sub>''n''</sub><br>
∏<sub>''i''</sub> d''A''<sub>''i''</sub>
|
Tangent space<br>
at a point:<br>
Set of changes,<br>
motions, steps,<br>
tangent vectors<br>
at a point
| '''D'''<sup>''n''</sup>
|-
| d''A''*
| (hom : d''A'' → '''B''')
|
Linear functions<br>
on d''A''
| ('''D'''<sup>''n''</sup>)* = '''D'''<sup>''n''</sup>
|-
| d''A''^
| (d''A'' → '''B''')
|
Boolean functions<br>
on d''A''
| '''D'''<sup>''n''</sup> → '''B'''
|-
| d''A''<sup>•</sup>
|
[d<font face="lucida calligraphy">A</font>]<br>
(d''A'', d''A''^)<br>
(d''A'' +→ '''B''')<br>
(d''A'', (d''A'' → '''B'''))<br>
[d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>]
|
Tangent universe<br>
at a point of ''A''<sup>•</sup>,<br>
based on the<br>
tangent features<br>
{d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
|
('''D'''<sup>''n''</sup>, ('''D'''<sup>''n''</sup> → '''B'''))<br>
('''D'''<sup>''n''</sup> +→ '''B''')<br>
['''D'''<sup>''n''</sup>]
|}
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