Differential Propositional Calculus • 10

Posted on December 9, 2024 by Jon Awbrey

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

The linear propositions, \{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}), may be written as sums:

\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n ~\text{where}~ \left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\} ~\text{for}~ i = 1 ~\text{to}~ n.

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. 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.

Resources

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

This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Functional Logic, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Propositional Calculus, Time, Topology, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

6 Responses to Differential Propositional Calculus • 10

  1. Pingback: Survey of Differential Logic • 7 | Inquiry Into Inquiry

  2. Unknown's avatar Tom says:
    December 9, 2024 at 8:32 am

    Are there analogues of differential equations in this framework?

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    • Jon Awbrey's avatar Jon Awbrey says:
      January 3, 2025 at 9:00 am

      Yes, Post 13 introduces the concept of a differential proposition —

      “A proposition in a differential extension of a universe of discourse is called a differential proposition and forms the analogue of a system of differential equations in ordinary calculus.”

      And this idea is developed more fully throughout the rest of the series.

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  3. Pingback: Survey of Differential Logic • 7 | Inquiry Into Inquiry

  4. Pingback: Survey of Differential Logic • 8 | Inquiry Into Inquiry

  5. Pingback: Survey of Differential Logic • 8 | Systems Community of Inquiry

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