## 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

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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