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Directory talk:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 18:38, 5 July 2008
, 18:38, 5 July 2008add [work area]
=Work Area=
Logical description of a universe of discourse begins with a set of logical signs. For the sake of simplicity in a first approach, assume that these logical signs are collected in the form of a finite alphabet, <math>\mathfrak{A} = \lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace.\!</math> Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse. Corresponding to the alphabet <math>\mathfrak{A}</math> there is then a set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_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>
Table 2 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
|+ '''Table 4. 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>
| Alphabet
| <math>[n] = \mathbf{n}</math>
|-
| <math>\mathcal{A}</math>
| <math>\{ a_1, \ldots, a_n \}</math>
| Basis
| <math>[n] = \mathbf{n}</math>
|-
| <math>A_i\!</math>
| <math>\{ \overline{a_i}, a_i \}\!</math>
| Dimension <math>i\!</math>
| <math>\mathbb{B}</math>
|-
| <math>A\!</math>
| <math>\langle \mathcal{A} \rangle</math><br>
<math>\langle a_1, \ldots, a_n \rangle</math><br>
<math>\{ (a_1, \ldots, a_n) \}\!</math>
<math>A_1 \times \ldots \times A_n</math><br>
<math>\textstyle \prod_i A_i\!</math>
| Set of cells,<br>
coordinate tuples,<br>
points, or vectors<br>
in the universe<br>
of discourse
| <math>\mathbb{B}^n</math>
|-
| <math>A^*\!</math>
| <math>(\operatorname{hom} : A \to \mathbb{B})</math>
| Linear functions
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
|-
| <math>A^\uparrow</math>
| <math>(A \to \mathbb{B})</math>
| Boolean functions
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>A^\circ</math>
| <math>[ \mathcal{A} ]</math><br>
<math>(A, A^\uparrow)</math><br>
<math>(A\ +\!\to \mathbb{B})</math><br>
<math>(A, (A \to \mathbb{B}))</math><br>
<math>[ a_1, \ldots, a_n ]</math>
| Universe of discourse<br>
based on the features<br>
<math>\{ a_1, \ldots, a_n \}</math>
| <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]</math>
|}
Logical description of a universe of discourse begins with a set of logical signs. For the sake of simplicity in a first approach, assume that these logical signs are collected in the form of a finite alphabet, <math>\mathfrak{A} = \lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace.\!</math> Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse. Corresponding to the alphabet <math>\mathfrak{A}</math> there is then a set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_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>
Table 2 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
|+ '''Table 4. 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>
| Alphabet
| <math>[n] = \mathbf{n}</math>
|-
| <math>\mathcal{A}</math>
| <math>\{ a_1, \ldots, a_n \}</math>
| Basis
| <math>[n] = \mathbf{n}</math>
|-
| <math>A_i\!</math>
| <math>\{ \overline{a_i}, a_i \}\!</math>
| Dimension <math>i\!</math>
| <math>\mathbb{B}</math>
|-
| <math>A\!</math>
| <math>\langle \mathcal{A} \rangle</math><br>
<math>\langle a_1, \ldots, a_n \rangle</math><br>
<math>\{ (a_1, \ldots, a_n) \}\!</math>
<math>A_1 \times \ldots \times A_n</math><br>
<math>\textstyle \prod_i A_i\!</math>
| Set of cells,<br>
coordinate tuples,<br>
points, or vectors<br>
in the universe<br>
of discourse
| <math>\mathbb{B}^n</math>
|-
| <math>A^*\!</math>
| <math>(\operatorname{hom} : A \to \mathbb{B})</math>
| Linear functions
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
|-
| <math>A^\uparrow</math>
| <math>(A \to \mathbb{B})</math>
| Boolean functions
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>A^\circ</math>
| <math>[ \mathcal{A} ]</math><br>
<math>(A, A^\uparrow)</math><br>
<math>(A\ +\!\to \mathbb{B})</math><br>
<math>(A, (A \to \mathbb{B}))</math><br>
<math>[ a_1, \ldots, a_n ]</math>
| Universe of discourse<br>
based on the features<br>
<math>\{ a_1, \ldots, a_n \}</math>
| <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]</math>
|}