Differential Logic • 3

Cactus Language for Propositional Logic

Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so-called existential interpretation, and their translations into conventional notations for a sample of basic propositional forms.

Table 1.  Syntax and Semantics of a Calculus for Propositional Logic

Table 1.  Syntax and Semantics of a Calculus for Propositional Logic

The simplest expression for logical truth is the empty word, typically denoted by \boldsymbol\varepsilon or \lambda in formal languages, where it is the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression {}^{\backprime\backprime} \texttt{((}~\texttt{))} {}^{\prime\prime}, or, especially if operating in an algebraic context, by a simple {}^{\backprime\backprime} 1 {}^{\prime\prime}.  Also when working in an algebraic mode, the plus sign {}^{\backprime\backprime} + {}^{\prime\prime} may be used for exclusive disjunction.  Thus we have the following translations of algebraic expressions into cactus expressions.

\begin{matrix}  a + b \quad = \quad \texttt{(} a \texttt{,} b \texttt{)}  \\[8pt]  a + b + c  \quad = \quad \texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}  \quad = \quad \texttt{((} a \texttt{,} b \texttt{),} c \texttt{)}  \end{matrix}

It is important to note the last expressions are not equivalent to the 3-place form \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}.

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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, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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