## Differential Propositional Calculus • 6

### Special Classes of Propositions (cont.)

Next we take up the family of positive propositions and follow the same plan as before, tracing the rule of their formation in the case of a 3-dimensional universe of discourse.

#### Positive Propositions In a universe of discourse based on three boolean variables, $p, q, r,$ there are $2^3 = 8$ positive propositions.  Their venn diagrams are shown in Figure 9. $\text{Figure 9.} ~~ \text{Positive Propositions} : \mathbb{B}^3 \to \mathbb{B}$

At the top is the venn diagram for the positive proposition of rank 3, corresponding to the boolean product or logical conjunction $pqr.$

Next are the venn diagrams for the three positive propositions of rank 2, corresponding to the three boolean products, $pr, qr, pq,$ respectively.

Next are the three positive propositions of rank 1, which are none other than the three basic propositions, $p, q, r.$

At the bottom is the positive proposition of rank 0, the everywhere true proposition or the constant $1$ function, which may be expressed by the form $\texttt{((}~\texttt{))}$ or by a simple $1.$

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