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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. | + | | |
| + | <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. | + | | |
| + | <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 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives. | | Table 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: |