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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, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \}.$  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, $\mathcal{A} = \{ a_1, \ldots, a_n \}.$

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, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \}.$  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 $\mathbf{n}.$

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 \PMlinkname{categorical}{Category} 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 4 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.

Table 4 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.