Differential Propositional Calculus • 5

Special Classes of Propositions (cont.)

Let’s pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general.  We can do this by recruiting our visual imaginations and drawing up a sufficient budget of venn diagrams for each family of propositions.  The case for 3 variables is exemplary enough for a start.

Linear Propositions

Linear Propositions May Be Written As Sums

One thing to keep in mind about these sums is that the values in \mathbb{B} = \{ 0, 1 \} are added “modulo 2”, that is, in such a way that 1 + 1 = 0.

In a universe of discourse based on three boolean variables, p, q, r, the linear propositions take the shapes shown in Figure 8.

Linear Propositions on Three Variables

\text{Figure 8.} ~~ \text{Linear Propositions} : \mathbb{B}^3 \to \mathbb{B}

At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following three forms:

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

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

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

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

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

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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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