Differential Propositional Calculus • 7

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 of a universe of discourse 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

Singular Propositions May Be Written As Products

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.} ~~ \text{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{)}.

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This entry was posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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