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| The first kind of propositional expression is a parenthesized sequence of propositional expressions, written as <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}\!</math> and read to say that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> is false, in other words, that their [[minimal negation]] is true.  A clause of this form maps into a PARC structure called a ''lobe'', in this case, one that is ''painted'' with the colors <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> as shown below.

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<p>The first kind of propositional expression is a parenthesized sequence of propositional expressions, written as <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}\!</math> and read to say that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> is false, in other words, that their [[minimal negation]] is true.  A clause of this form maps into a PARC structure called a ''lobe'', in this case, one that is ''painted'' with the colors <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> as shown below.<p>

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| The second kind of propositional expression is a concatenated sequence of propositional expressions, written as <math>e_1\ e_2\ \ldots\ e_{k-1}\ e_k\!</math> and read to say that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> are true, in other words, that their [[logical conjunction]] is true.  A clause of this form maps into a PARC structure called a ''node'', in this case, one that is ''painted'' with the colors <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> as shown below.

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<p>The second kind of propositional expression is a concatenated sequence of propositional expressions, written as <math>e_1\ e_2\ \ldots\ e_{k-1}\ e_k\!</math> and read to say that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> are true, in other words, that their [[logical conjunction]] is true.  A clause of this form maps into a PARC structure called a ''node'', in this case, one that is ''painted'' with the colors <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> as shown below.</p>

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Table&nbsp;1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.

Table&nbsp;1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.

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\end{matrix}~\!</math>

\end{matrix}~\!</math>

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The simplest expression for logical truth is the empty word, usually denoted by <math>\boldsymbol\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by means of parenthesized expressions:

The simplest expression for logical truth is the empty word, usually denoted by <math>\boldsymbol\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by means of parenthesized expressions: