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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

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A set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> affords a basis for generating an <math>n\!</math>-dimensional universe of discourse, written <math>A^\circ = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].</math>  It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points <math>A = \langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>A^\uparrow = \{ f : A \to \mathbb{B} \}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features.  Accordingly, the universe of discourse <math>A^\circ</math> may be regarded as an ordered pair <math>(A, A^\uparrow)</math> having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> and this last type designation may be abbreviated as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[ \mathbb{B}^n ].</math>  For convenience, the data type of a finite set on <math>n\!</math> elements may be indicated by either one of the equivalent notations, <math>[n]\!</math> or <math>\mathbf{n}.</math>
 
A set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> affords a basis for generating an <math>n\!</math>-dimensional universe of discourse, written <math>A^\circ = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].</math>  It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points <math>A = \langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>A^\uparrow = \{ f : A \to \mathbb{B} \}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features.  Accordingly, the universe of discourse <math>A^\circ</math> may be regarded as an ordered pair <math>(A, A^\uparrow)</math> having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> and this last type designation may be abbreviated as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[ \mathbb{B}^n ].</math>  For convenience, the data type of a finite set on <math>n\!</math> elements may be indicated by either one of the equivalent notations, <math>[n]\!</math> or <math>\mathbf{n}.</math>
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Table 5 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.
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Table&nbsp;5 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
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<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:left; width:90%"
 
|+ '''Table 5.  Propositional Calculus : Basic Notation'''
 
|+ '''Table 5.  Propositional Calculus : Basic Notation'''
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|- style="background:ghostwhite"
+
|- style="background:#e6e6ff"
 
! Symbol
 
! Symbol
 
! Notation
 
! Notation
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|-
 
|-
 
| <math>A\!</math>
 
| <math>A\!</math>
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| <math>\langle \mathcal{A} \rangle</math><br>
+
|
 +
<math>\langle \mathcal{A} \rangle</math><br>
 
<math>\langle a_1, \ldots, a_n \rangle</math><br>
 
<math>\langle a_1, \ldots, a_n \rangle</math><br>
 
<math>\{ (a_1, \ldots, a_n) \}\!</math>
 
<math>\{ (a_1, \ldots, a_n) \}\!</math>
 
<math>A_1 \times \ldots \times A_n</math><br>
 
<math>A_1 \times \ldots \times A_n</math><br>
 
<math>\textstyle \prod_i A_i\!</math>
 
<math>\textstyle \prod_i A_i\!</math>
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| Set of cells,<br>
+
|
 +
Set of cells,<br>
 
coordinate tuples,<br>
 
coordinate tuples,<br>
 
points, or vectors<br>
 
points, or vectors<br>
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|-
 
|-
 
| <math>A^\circ</math>
 
| <math>A^\circ</math>
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| <math>[ \mathcal{A} ]</math><br>
+
|
 +
<math>[ \mathcal{A} ]</math><br>
 
<math>(A, A^\uparrow)</math><br>
 
<math>(A, A^\uparrow)</math><br>
 
<math>(A\ +\!\to \mathbb{B})</math><br>
 
<math>(A\ +\!\to \mathbb{B})</math><br>
 
<math>(A, (A \to \mathbb{B}))</math><br>
 
<math>(A, (A \to \mathbb{B}))</math><br>
 
<math>[ a_1, \ldots, a_n ]</math>
 
<math>[ a_1, \ldots, a_n ]</math>
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| Universe of discourse<br>
+
|
 +
Universe of discourse<br>
 
based on the features<br>
 
based on the features<br>
 
<math>\{ a_1, \ldots, a_n \}</math>
 
<math>\{ a_1, \ldots, a_n \}</math>
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| <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math><br>
+
|
 +
<math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math><br>
 
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
 
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
 
<math>[\mathbb{B}^n]</math>
 
<math>[\mathbb{B}^n]</math>
 
|}
 
|}
 +
 
<br>
 
<br>
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A proposition in a differential extension of a universe of discourse is called a ''differential proposition'' and forms the analogue of a system of differential equations in ordinary calculus.  With these constructions, the first order extended universe <math>\operatorname{E}A^\circ</math> and the first order differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at the foothills of [[differential logic]].
 
A proposition in a differential extension of a universe of discourse is called a ''differential proposition'' and forms the analogue of a system of differential equations in ordinary calculus.  With these constructions, the first order extended universe <math>\operatorname{E}A^\circ</math> and the first order differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at the foothills of [[differential logic]].
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Table 6 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.
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Table&nbsp;6 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.
 +
 
 +
<br>
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:left; width:90%"
 
|+ '''Table 6.  Differential Extension : Basic Notation'''
 
|+ '''Table 6.  Differential Extension : Basic Notation'''
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|- style="background:ghostwhite"
+
|- style="background:#e6e6ff"
 
! Symbol
 
! Symbol
 
! Notation
 
! Notation
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| <math>\operatorname{d}\mathfrak{A}</math>
 
| <math>\operatorname{d}\mathfrak{A}</math>
 
| <math>\lbrace\!</math>&nbsp;“<math>\operatorname{d}a_1</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>\operatorname{d}a_n</math>”&nbsp;<math>\rbrace\!</math>
 
| <math>\lbrace\!</math>&nbsp;“<math>\operatorname{d}a_1</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>\operatorname{d}a_n</math>”&nbsp;<math>\rbrace\!</math>
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| Alphabet of<br>
+
|
 +
Alphabet of<br>
 
differential<br>
 
differential<br>
 
symbols
 
symbols
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| <math>\operatorname{d}\mathcal{A}</math>
 
| <math>\operatorname{d}\mathcal{A}</math>
 
| <math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
 
| <math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
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| Basis of<br>
+
|
 +
Basis of<br>
 
differential<br>
 
differential<br>
 
features
 
features
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| <math>\operatorname{d}A_i</math>
 
| <math>\operatorname{d}A_i</math>
 
| <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}</math>
 
| <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}</math>
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| Differential<br>
+
|
 +
Differential<br>
 
dimension <math>i\!</math>
 
dimension <math>i\!</math>
 
| <math>\mathbb{D}</math>
 
| <math>\mathbb{D}</math>
 
|-
 
|-
 
| <math>\operatorname{d}A</math>
 
| <math>\operatorname{d}A</math>
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| <math>\langle \operatorname{d}\mathcal{A} \rangle</math><br>
+
|
 +
<math>\langle \operatorname{d}\mathcal{A} \rangle</math><br>
 
<math>\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle</math><br>
 
<math>\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle</math><br>
 
<math>\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}</math><br>
 
<math>\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}</math><br>
 
<math>\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n</math><br>
 
<math>\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n</math><br>
 
<math>\textstyle \prod_i \operatorname{d}A_i</math>
 
<math>\textstyle \prod_i \operatorname{d}A_i</math>
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| Tangent space<br>
+
|
 +
Tangent space<br>
 
at a point:<br>
 
at a point:<br>
 
Set of changes,<br>
 
Set of changes,<br>
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| <math>\operatorname{d}A^*</math>
 
| <math>\operatorname{d}A^*</math>
 
| <math>(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})</math>
 
| <math>(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})</math>
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| Linear functions<br>
+
|
 +
Linear functions<br>
 
on <math>\operatorname{d}A</math>
 
on <math>\operatorname{d}A</math>
 
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>
 
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>
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| <math>\operatorname{d}A^\uparrow</math>
 
| <math>\operatorname{d}A^\uparrow</math>
 
| <math>(\operatorname{d}A \to \mathbb{B})</math>
 
| <math>(\operatorname{d}A \to \mathbb{B})</math>
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| Boolean functions<br>
+
|
 +
Boolean functions<br>
 
on <math>\operatorname{d}A</math>
 
on <math>\operatorname{d}A</math>
 
| <math>\mathbb{D}^n \to \mathbb{B}</math>
 
| <math>\mathbb{D}^n \to \mathbb{B}</math>
 
|-
 
|-
 
| <math>\operatorname{d}A^\circ</math>
 
| <math>\operatorname{d}A^\circ</math>
−
| <math>[\operatorname{d}\mathcal{A}]</math><br>
+
|
 +
<math>[\operatorname{d}\mathcal{A}]</math><br>
 
<math>(\operatorname{d}A, \operatorname{d}A^\uparrow)</math><br>
 
<math>(\operatorname{d}A, \operatorname{d}A^\uparrow)</math><br>
 
<math>(\operatorname{d}A\ +\!\to \mathbb{B})</math><br>
 
<math>(\operatorname{d}A\ +\!\to \mathbb{B})</math><br>
 
<math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))</math><br>
 
<math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))</math><br>
 
<math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]</math>
 
<math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]</math>
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| Tangent universe<br>
+
|
 +
Tangent universe<br>
 
at a point of <math>A^\circ,</math><br>
 
at a point of <math>A^\circ,</math><br>
 
based on the<br>
 
based on the<br>
 
tangent features<br>
 
tangent features<br>
 
<math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
 
<math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
−
| <math>(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))</math><br>
+
|
 +
<math>(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))</math><br>
 
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
 
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
 
<math>[\mathbb{D}^n]</math>
 
<math>[\mathbb{D}^n]</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
Jon Awbrey
12,080

edits

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