Changes

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''.)

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''.)

Table 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 [''n''] or '''''n''''' to denote the data type of a finite set on n elements.

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.

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

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