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

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Elementary notions

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 \(\{ a_1, \ldots, a_n \}\)	\((\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))\) \((\mathbb{B}^n\ +\!\to \mathbb{B})\) \([\mathbb{B}^n]\)

Special classes of propositions

A basic proposition, coordinate proposition, or simple proposition in the universe of discourse \([a_1, \ldots, a_n]\) is one of the propositions in the set \(\{ a_1, \ldots, a_n \}.\)

Among the \(2^{2^n}\) propositions in \([a_1, \ldots, a_n]\) are several families of \(2^n\!\) propositions each that take on special forms with respect to the basis \(\{ a_1, \ldots, a_n \}.\) Three of these families are especially prominent in the present context, the linear, the positive, and the singular propositions. Each family is naturally parameterized by the coordinate \(n\!\)-tuples in \(\mathbb{B}^n\) and falls into \(n + 1\!\) ranks, with a binomial coefficient \(\tbinom{n}{k}\) giving the number of propositions that have rank or weight \(k.\!\)

• The linear propositions, \(\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\) may be written as sums:

  \(\sum_{i=1}^n e_i = e_1 + \ldots + e_n\)  where  \(e_i = a_i\!\)  or  \(e_i = 0\!\)  for  \(i = 1\!\)  to  \(n.\!\)

• The positive propositions, \(\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\) may be written as products:

  \(\prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n\)  where  \(e_i = a_i\!\)  or  \(e_i = 1\!\)  for  \(i = 1\!\)  to  \(n.\!\)

• The singular propositions, \(\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\) may be written as products:

  \(\prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n\)  where  \(e_i = a_i\!\)  or  \(e_i = (a_i)\!\)  for  \(i = 1\!\)  to  \(n.\!\)

In each case the rank \(k\!\) ranges from \(0\!\) to \(n\!\) and counts the number of positive appearances of the coordinate propositions \(a_1, \ldots, a_n\!\) in the resulting expression. For example, for \(n = 3,\!\) the linear proposition of rank \(0\!\) is \(0,\!\) the positive proposition of rank \(0\!\) is \(1,\!\) and the singular proposition of rank \(0\!\) is \((a_1)(a_2)(a_3).\!\)

The basic propositions \(a_i : \mathbb{B}^n \to \mathbb{B}\) are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.

Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis \(\mathcal{A} = \{ a_1, \ldots, a_n \}.\) For example, a singular proposition with respect to the basis \(\mathcal{A}\) will not remain singular if \(\mathcal{A}\) is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options \(\{ a_i \} \cup \{ (a_i) \}\) to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.