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We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.
===The Extended Universe of Discourse===
<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
|+ '''Table 8. Notation for the Differential Extension of Propositional Calculus'''
|- style="background:ghostwhite"
! 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>]
|}
</font><br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
|+ '''Table 2. Propositional Calculus : Basic Notation'''
|- style="background:ghostwhite"
! Symbol
! Notation
! Description
! Type
|-
| <math>\mathfrak{A}</math>
| <math>\lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace\!</math>
| <math>\operatorname{Alphabet}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>\mathcal{A}</math>
| <math>\{a_1, \ldots, a_n\}</math>
| <math>\operatorname{Basis}</math>
| <math>[n] = \mathbf{n}</math>
|-
| <math>A_i\!</math>
| <math>\{(a_i), a_i\}\!</math>
| <math>\operatorname{Dimension}\ i</math>
| <math>\mathbb{B}</math>
|-
| <math>A\!</math>
|
<p><math>\langle \mathcal{A} \rangle</math></p>
<p><math>\langle a_1, \ldots, a_n \rangle</math></p>
<p><math>\{(a_1, \ldots, a_n)\}</math></p>
<p><math>A_1 \times \ldots \times A_n</math></p>
<p><math>\textstyle \prod_{i=1}^n A_i</math></p>
|
<p><math>\operatorname{Set~of~cells},</math></p>
<p><math>\operatorname{coordinate~tuples},</math></p>
<p><math>\operatorname{points,~or~vectors}</math></p>
<p><math>\operatorname{in~the~universe}</math></p>
<p><math>\operatorname{of~discourse}</math></p>
| <math>\mathbb{B}^n</math>
|-
| <math>A^*\!</math>
| <math>(\operatorname{hom} : A \to \mathbb{B})</math>
| <math>\operatorname{Linear~functions}</math>
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
|-
| <math>A^\uparrow</math>
| <math>(A \to \mathbb{B})</math>
| <math>\operatorname{Boolean~functions}</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>A^\circ</math>
|
<p><math>[\mathcal{A}]</math></p>
<p><math>(A, A^\uparrow)</math></p>
<p><math>(A\ +\!\to \mathbb{B})</math></p>
<p><math>(A, (A \to \mathbb{B}))</math></p>
<p><math>[a_1, \ldots, a_n]</math></p>
|
<p><math>\operatorname{Universe~of~discourse}</math></p>
<p><math>\operatorname{based~on~the~features}</math></p>
<p><math>\{a_1, \ldots, a_n\}</math></p>
|
<p><math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math></p>
<p><math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math></p>
<p><math>[\mathbb{B}^n]</math></p>
|}<br>
