Differential Propositional Calculus • 11

Posted on December 10, 2024 by Jon Awbrey

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

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 ~\text{where}~ \left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\} ~\text{for}~ i = 1 ~\text{to}~ n.

In a universe of discourse based on three boolean variables, p, q, r, there are 2^3 = 8 positive propositions, taking the shapes shown in Figure 9.

Positive Propositions on Three Variables
\text{Figure 9. 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.

Resources

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

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