Theme One • A Program Of Inquiry : 12

Re: Laws Of Form Discussions • (1)(2)(3)
Re: Peirce List Discussions • (1)(2)(3)

Logical Cacti (cont.)

The main things to take away from the previous post are the following two ideas, one syntactic and one semantic:

  • The compositional structures of cactus graphs and cactus expressions are constructed from two kinds of connective operations.
  • There are two ways of mapping these compositional structures into the compositional structures of propositional sentences.

The two kinds of connective operations are described as follows:

  • The node connective joins a number of component cacti C_1, \ldots, C_k to a node:

    node connective
  • The lobe connective joins a number of component cacti C_1, \ldots, C_k to a lobe:

    lobe connective

The two ways of mapping cactus structures to logical meanings are summarized in Table 3, which compares the existential and entitative interpretations of the basic cactus structures, in effect, the graphical constants and connectives.

\text{Table 3.} ~~ \text{Logical Interpretations of Cactus Structures}
\text{Graph} \text{Expression} \begin{matrix}  \text{Existential} \\ \text{Interpretation}  \end{matrix} \begin{matrix}  \text{Entitative} \\ \text{Interpretation}  \end{matrix}

“ ”
~ \mathrm{true} \mathrm{false}

( )
\texttt{(} ~ \texttt{)} \mathrm{false} \mathrm{true}

node connective
C_1 \ldots C_k C_1 \land \ldots \land C_k C_1 \lor \ldots \lor C_k

lobe connective
\texttt{(} C_1 \texttt{,} \ldots \texttt{,} C_k \texttt{)} \begin{matrix}  \text{just one of}  \\[6px]  C_1, \ldots, C_k  \\[6px]  \text{is false}  \end{matrix} \begin{matrix}  \text{not just one of}  \\[6px]  C_1, \ldots,  C_k  \\[6px]  \text{is true}  \end{matrix}

Resources

This entry was posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Computation, Computational Complexity, Cybernetics, Data Structures, Differential Logic, Form, Formal Languages, Graph Theory, Inquiry, Inquiry Driven Systems, Intelligent Systems, Laws of Form, Learning, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Pragmatics, Programming, Propositional Calculus, Propositional Equation Reasoning Systems, Reasoning, Semantics, Semiotics, Spencer Brown, Syntax, Theorem Proving and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

One Response to Theme One • A Program Of Inquiry : 12

  1. Pingback: Survey of Theme One Program • 2 | Inquiry Into Inquiry

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