Line 1,944: |
Line 1,944: |
| <p>The conjunction <math>\operatorname{Conj}_j^J q_j</math> can be represented by a sentence that is constructed by concatenating the <math>s_j\!</math> in the following fashion:</p> | | <p>The conjunction <math>\operatorname{Conj}_j^J q_j</math> can be represented by a sentence that is constructed by concatenating the <math>s_j\!</math> in the following fashion:</p> |
| | | |
− | <p><math>\operatorname{Conj}_j^J q_j ~\leftarrowtail~ s_1 s_2 \ldots s_k.</math></p></li> | + | <p><math>\operatorname{Conj}_j^J q_j ~\leftrightsquigarrow~ s_1 s_2 \ldots s_k.</math></p></li> |
| | | |
| <li> | | <li> |
| <p>The surjunction <math>\operatorname{Surj}_j^J q_j</math> can be represented by a sentence that is constructed by surcatenating the <math>s_j\!</math> in the following fashion:</p> | | <p>The surjunction <math>\operatorname{Surj}_j^J q_j</math> can be represented by a sentence that is constructed by surcatenating the <math>s_j\!</math> in the following fashion:</p> |
| | | |
− | <p><math>\operatorname{Surj}_j^J q_j ~\leftarrowtail~ \underline{(} s_1, s_2, \ldots, s_k \underline{)}.</math></p></li> | + | <p><math>\operatorname{Surj}_j^J q_j ~\leftrightsquigarrow~ \underline{(} s_1, s_2, \ldots, s_k \underline{)}.</math></p></li> |
| | | |
| </ol> | | </ol> |
Line 1,955: |
Line 1,955: |
| If one opts for a mode of interpretation that moves more directly from the parse graph of a sentence to the potential logical meaning of both the PARC and the PARCE, then the following specifications are in order: | | If one opts for a mode of interpretation that moves more directly from the parse graph of a sentence to the potential logical meaning of both the PARC and the PARCE, then the following specifications are in order: |
| | | |
− | <pre>
| + | A cactus rooted at a particular node is taken to represent what that node denotes, its logical denotation or its logical interpretation. |
− | A cactus rooted at a particular node is taken to represent what that | |
− | node denotes, its logical denotation or its logical interpretation. | |
| | | |
− | 1. The logical denotation of a node is the logical conjunction of that node's
| + | # The logical denotation of a node is the logical conjunction of that node's arguments, which are defined as the logical denotations of that node's attachments. The logical denotation of either a blank symbol or an empty node is the boolean value <math>\underline{1} = \operatorname{true}.</math> The logical denotation of the paint <math>\mathfrak{p}_j\!</math> is the proposition <math>p_j,\!</math> a proposition that is regarded as ''primitive'', at least, with respect to the level of analysis that is represented in the current instance of <math>\mathfrak{C} (\mathfrak{P}).</math> |
− | "arguments", which are defined as the logical denotations of that node's
| + | # The logical denotation of a lobe is the logical surjunction of that lobe's arguments, which are defined as the logical denotations of that lobe's accoutrements. As a corollary, the logical denotation of the parse graph of <math>\underline{(} \underline{)},</math> otherwise called a ''needle'', is the boolean value <math>\underline{0} = \operatorname{false}.</math> |
− | attachments. The logical denotation of either a blank symbol or an empty
| |
− | node is the boolean value %1% = "true". The logical denotation of the
| |
− | paint p_j is the proposition P_j, a proposition that is regarded as
| |
− | "primitive", at least, with respect to the level of analysis that
| |
− | is represented in the current instance of !C!(!P!).
| |
− | | |
− | 2. The logical denotation of a lobe is the logical surjunction of that lobe's
| |
− | "arguments", which are defined as the logical denotations of that lobe's
| |
− | accoutrements. As a corollary, the logical denotation of the parse graph
| |
− | of "-()-", otherwise called a "needle", is the boolean value %0% = "false".
| |
| | | |
| + | <pre> |
| If one takes the point of view that PARC's and PARCE's amount to a | | If one takes the point of view that PARC's and PARCE's amount to a |
| pair of intertranslatable languages for the same domain of objects, | | pair of intertranslatable languages for the same domain of objects, |