Directory talk:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0

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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, \(\mathfrak{A} = \lbrace\!\) “\(a_1\!\)” \(, \ldots,\!\) “\(a_n\!\)” \(\rbrace.\!\) 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 \(\mathfrak{A}\) there is then a set of logical features, \(\mathcal{A} = \{ a_1, \ldots, a_n \}.\)

A set of logical features, \(\mathcal{A} = \{ a_1, \ldots, a_n \},\) affords a basis for generating an \(n\!\)-dimensional universe of discourse, written \(A^\circ = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].\) It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points \(A = \langle a_1, \ldots, a_n \rangle\) and the set of propositions \(A^\uparrow = \{ f : A \to \mathbb{B} \}\) that are implicit with the ordinary picture of a venn diagram on \(n\!\) features. Accordingly, the universe of discourse \(A^\circ\) may be regarded as an ordered pair \((A, A^\uparrow)\) having the type \((\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),\) and this last type designation may be abbreviated as \(\mathbb{B}^n\ +\!\to \mathbb{B},\) or even more succinctly as \([ \mathbb{B}^n ].\) For convenience, the data type of a finite set on \(n\!\) elements may be indicated by either one of the equivalent notations, \([n]\!\) or \(\mathbf{n}.\)

Table 2 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.

Table 4. Propositional Calculus : Basic Notation

Symbol	Notation	Description	Type
\(\mathfrak{A}\)	\(\lbrace\!\) “\(a_1\!\)” \(, \ldots,\!\) “\(a_n\!\)” \(\rbrace\!\)	Alphabet	\([n] = \mathbf{n}\)
\(\mathcal{A}\)	\(\{ a_1, \ldots, a_n \}\)	Basis	\([n] = \mathbf{n}\)
\(A_i\!\)	\(\{ \overline{a_i}, a_i \}\!\)	Dimension \(i\!\)	\(\mathbb{B}\)
\(A\!\)	\(\langle \mathcal{A} \rangle\) \(\langle a_1, \ldots, a_n \rangle\) \(\{ (a_1, \ldots, a_n) \}\!\) \(A_1 \times \ldots \times A_n\) \(\textstyle \prod_i A_i\!\)	Set of cells, coordinate tuples, points, or vectors in the universe of discourse	\(\mathbb{B}^n\)
\(A^*\!\)	\((\operatorname{hom} : A \to \mathbb{B})\)	Linear functions	\((\mathbb{B}^n)^* \cong \mathbb{B}^n\)
\(A^\uparrow\)	\((A \to \mathbb{B})\)	Boolean functions	\(\mathbb{B}^n \to \mathbb{B}\)
\(A^\circ\)	\([ \mathcal{A} ]\) \((A, A^\uparrow)\) \((A\ +\!\to \mathbb{B})\) \((A, (A \to \mathbb{B}))\) \([ a_1, \ldots, a_n ]\)	Universe of discourse based on the features \(\{ a_1, \ldots, a_n \}\)	\((\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))\) \((\mathbb{B}^n\ +\!\to \mathbb{B})\) \([\mathbb{B}^n]\)