Differential Propositional Calculus • 12

Posted on November 27, 2023 by Jon Awbrey

Special Classes of Propositions (concl.)

Last and literally least in extent, we examine the family of singular propositions in a 3‑dimensional universe of discourse.

In our model of propositions as mappings from a universe of discourse X to a set of two values, in other words, indicator functions of the form f : X \to \mathbb{B}, singular propositions are those singling out the minimal distinct regions of the universe, represented by single cells of the corresponding venn diagram.

Singular Propositions

The singular propositions, \{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \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 = \texttt{(} a_i \texttt{)} \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 singular propositions.  Their venn diagrams are shown in Figure 10.

Singular Propositions on Three Variables
\text{Figure 10. Singular Propositions} : \mathbb{B}^3 \to \mathbb{B}

At the top is the venn diagram for the singular proposition of rank 3, corresponding to the boolean product pqr and identical with the positive proposition of rank 3.

Next are the venn diagrams for the three singular propositions of rank 2, which may be expressed by the following three forms, respectively.

pr \texttt{(} q \texttt{)}, \qquad qr \texttt{(} p \texttt{)}, \qquad pq \texttt{(} r \texttt{)}.

Next are the three singular propositions of rank 1, which may be expressed by the following three forms, respectively.

q \texttt{(} p \texttt{)(} r \texttt{)}, \qquad p \texttt{(} q \texttt{)(} r \texttt{)}, \qquad r \texttt{(} p \texttt{)(} q \texttt{)}.

At the bottom is the singular proposition of rank 0, which may be expressed by the following form.

\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}.

Resources

cc: FB | Differential Logic • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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