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

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A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.
      
==Casual introduction==
 
==Casual introduction==
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<br>
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'''&hellip;'''
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Suppose we discover or begin to suspect that something we had been treating as a simple quality, <math>q,\!</math> is actually compounded of other qualities, <math>u\!</math> and <math>v\!</math>, according to a propositional formula <math>q = q(u, v).\!</math>
    
'''&hellip;'''
 
'''&hellip;'''
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The preceding discussion outlined the ideas leading to the differential extension of propositional logic.  The next task is to lay out the concepts and terminology that are needed to describe various orders of differential propositional calculi.
 
The preceding discussion outlined the ideas leading to the differential extension of propositional logic.  The next task is to lay out the concepts and terminology that are needed to describe various orders of differential propositional calculi.
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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 form a finite alphabet, <math>\mathfrak{A} = \lbrace\!</math>&nbsp;“<math>a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>a_n\!</math>”&nbsp;<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>
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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>&nbsp;“<math>a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>a_n\!</math>”&nbsp;<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>
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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 unified 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>
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'''&hellip;'''
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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>
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Table 4 summarizes the basic notations that are needed to describe ordinary propositional calculi in a parametric fashion.  The notations <math>[n]\!</math> or <math>\mathbf{n}</math> denote the data type of a finite set on <math>n\!</math> elements.
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Table 4 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="background:lightcyan; text-align:left; width:96%"
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
 
|+ '''Table 4.  Propositional Calculus : Basic Notation'''
 
|+ '''Table 4.  Propositional Calculus : Basic Notation'''
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Table&nbsp;5 summarizes the basic notations that are needed to describe the (first order) differential extensions of propositional calculi in a corresponding manner.
 
Table&nbsp;5 summarizes the basic notations that are needed to describe the (first order) differential extensions of propositional calculi in a corresponding manner.
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
 
|+ '''Table 5.  Differential Extension : Basic Notation'''
 
|+ '''Table 5.  Differential Extension : Basic Notation'''
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==Expository examples==
 
==Expository examples==
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'''&hellip;'''
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Consider the logical proposition represented by the following venn diagram:
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<center><pre>
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o-----------------------------------------------------------o
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'''Figure 1.  Proposition''' <math>q : X \to \mathbb{B}</math>
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The following language is useful in describing the facts represented by the venn diagram.
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* The universe of discourse is a set, <math>X,\!</math> represented by the area inside the large rectangle.
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* The boolean domain is a set of two elements, <math>\mathbb{B} = \{ 0, 1 \},</math> represented by the two distinct shadings of the regions inside the rectangle.
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* According to the conventions observed in this context, the algebraic value 0 is interpreted as the logical value <math>\operatorname{false}</math> and represented by the lighter shading, while the algebraic value 1 is interpreted as the logical value <math>\operatorname{true}</math> and represented by the darker shading.
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* The universe of discourse <math>X\!</math> is the domain of three functions <math>u, v, w : X \to \mathbb{B}</math> called ''basic'', ''coordinate'', or ''simple'' propositions.
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* As with any proposition, <math>p : X \to \mathbb{B},</math> a simple proposition partitions <math>X\!</math> into two fibers, the fiber of 0 under <math>p,\!</math> defined as <math>p^{-1}(0) \subseteq X,</math> and the fiber of 1 under <math>p,\!</math> defined as <math>p^{-1}(1) \subseteq X.</math>
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* Each coordinate proposition is represented by a "circle", or a simple closed curve, that divides the rectangular region into the region exterior to the circle, representing the fiber of 0 under <math>p,\!</math> and the region interior to the circle, representing the fiber of 1 under <math>p.\!</math>
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* The fibers of 1 under the propositions <math>u, v, w\!</math> are the respective subsets <math>U, V, W \subseteq X.</math>
    
'''&hellip;'''
 
'''&hellip;'''
Jon Awbrey
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