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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
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, 11:58, 16 May 2008add excerpts for reuse in this context
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=Materials from "Dif Log Dyn Sys" for Reuse Here=
==A Functional Conception of Propositional Calculus==
<blockquote>
<p>Out of the dimness opposite equals advance . . . .<br>
Always substance and increase,<br>
Always a knit of identity . . . . always distinction . . . .<br>
always a breed of life.</p>
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 28]</p>
</blockquote>
In the general case, we start with a set of logical features {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>} that represent properties of objects or propositions about the world. In concrete examples the features {''a''<sub>''i''</sub>} commonly appear as capital letters from an ''alphabet'' like {''A'', ''B'', ''C'', …} or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters {''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>} as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.
The set of logical features {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>} provides a basis for generating an ''n''-dimensional ''universe of discourse'' that I denote as [''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>]. It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points 〈''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>〉 and the set of propositions ''f'' : 〈''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>〉 → '''B''' that are implicit with the ordinary picture of a venn diagram on ''n'' features. Thus, we may regard the universe of discourse [''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>] as an ordered pair having the type ('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''), and we may abbreviate this last type designation as '''B'''<sup>''n''</sup> +→ '''B''', or even more succinctly as ['''B'''<sup>''n''</sup>]. (Used this way, the angle brackets 〈…〉 are referred to as ''generator brackets''.)
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations [''n''] or '''''n''''' to denote the data type of a finite set on n elements.
<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 2. Fundamental Notations for Propositional Calculus'''
|- style="background:paleturquoise"
! Symbol
! Notation
! Description
! Type
|-
| <font face="lucida calligraphy">A<font>
| {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}
| Alphabet
| [''n''] = '''n'''
|-
| ''A''<sub>''i''</sub>
| {(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}
| Dimension ''i''
| '''B'''
|-
| ''A''
|
〈<font face="lucida calligraphy">A</font>〉<br>
〈''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>〉<br>
{‹''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>›}<br>
''A''<sub>1</sub> × … × ''A''<sub>''n''</sub><br>
∏<sub>''i''</sub> ''A''<sub>''i''</sub>
|
Set of cells,<br>
coordinate tuples,<br>
points, or vectors<br>
in the universe<br>
of discourse
| '''B'''<sup>''n''</sup>
|-
| ''A''*
| (hom : ''A'' → '''B''')
| Linear functions
| ('''B'''<sup>''n''</sup>)* = '''B'''<sup>''n''</sup>
|-
| ''A''^
| (''A'' → '''B''')
| Boolean functions
| '''B'''<sup>''n''</sup> → '''B'''
|-
| ''A''<sup>•</sup>
|
[<font face="lucida calligraphy">A</font>]<br>
(''A'', ''A''^)<br>
(''A'' +→ '''B''')<br>
(''A'', (''A'' → '''B'''))<br>
[''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>]
|
Universe of discourse<br>
based on the features<br>
{''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}
|
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
('''B'''<sup>''n''</sup> +→ '''B''')<br>
['''B'''<sup>''n''</sup>]
|}
</font><br>