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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.

In the general case, we start with a set of logical features <math>\{ a_1, \ldots, a_n \}</math> that represent properties of objects or propositions about the world.  In concrete examples the features <math>\{ a_i \}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{ A, B, C, \ldots \}</math> or as meaningful words from a linguistic ''vocabulary'' of codes.  This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation.  In the application to dynamic systems we tend to use the letters <math>\{ x_1, \ldots, x_n \}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space.  Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.

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>

The set of logical features <math>\{ a_1, \ldots, a_n \}</math> provides a basis for generating an <math>n\!</math>-dimensional ''universe of discourse'' that I denote as <math>[ a_1, \ldots, a_n ].</math>  It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features.  Thus, we may regard the universe of discourse <math>[ a_1, \ldots, a_n ]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle \ldots \rangle</math> are referred to as ''generator brackets''.)

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>

Table&nbsp;4 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams.  Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.

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Table&nbsp;4 summarizes the basic notations that are needed to describe ordinary propositional calculi in a parametric fashion.

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.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"