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

Positive Propositions May Be Written As Products

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.

Positive Propositions on Three Variables

\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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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.

3 Responses to Differential Propositional Calculus • 6

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