## Differential Propositional Calculus • 2

### Cactus Calculus

Table 6 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable $k$-ary scope.

• A bracketed list of propositional expressions in the form $\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}$ indicates exactly one of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ is false.
• A concatenation of propositional expressions in the form $e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k$ indicates all the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ are true, in other words, their logical conjunction is true.

Table 6.  Syntax and Semantics of a Calculus for Propositional Logic

All other propositional connectives can be obtained through combinations of these two forms.  Strictly speaking, the concatenation form is dispensable in light of the bracket form, but it is convenient to maintain it as an abbreviation for more complicated bracket expressions.  While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives.  In contexts where parentheses are needed for other purposes “teletype” parentheses $\texttt{(} \ldots \texttt{)}$ or barred parentheses $(\!| \ldots |\!)$ may be used for logical operators.

The briefest expression for logical truth is the empty word, abstractly denoted $\boldsymbol\varepsilon$ or $\boldsymbol\lambda$ in formal languages, where it forms the identity element for concatenation.  It may be given visible expression in this context by means of the logically equivalent form $\texttt{((} ~ \texttt{))},$ or, especially if operating in an algebraic context, by a simple $1.$  Also when working in an algebraic mode, the plus sign ${+}$ may be used for exclusive disjunction.  For example, we have the following paraphrases of algebraic expressions:

$\begin{matrix} x + y ~=~ \texttt{(} x \texttt{,} y \texttt{)} \\[6pt] x + y + z ~=~ \texttt{((} x \texttt{,} y \texttt{),} z \texttt{)} ~=~ \texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))} \end{matrix}$

It is important to note the last expressions are not equivalent to the triple bracket $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}.$