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Directory:Jon Awbrey/Papers/Cactus Language (view source)

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→‎The Cactus Language : Semantics: set table
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Tables 14.1 and 14.2 summarize the relations that serve to connect the formal language of sentences with the logical language of propositions.  Between these two realms of expression there is a family of graphical data structures that arise in parsing the sentences and that serve to facilitate the performance of computations on the indicator functions.  The graphical language supplies an intermediate form of representation between the formal sentences and the indicator functions, and the form of mediation that it provides is very useful in rendering the possible connections between the other two languages conceivable in fact, not to mention in carrying out the necessary translations on a practical basis.  These Tables include this intermediate domain in their Central Columns.  Between their First and Middle Columns they illustrate the mechanics of parsing the abstract sentences of the cactus language into the graphical data structures of the corresponding species.  Between their Middle and Final Columns they summarize the semantics of interpreting the graphical forms of representation for the purposes of reasoning with propositions.
 
Tables 14.1 and 14.2 summarize the relations that serve to connect the formal language of sentences with the logical language of propositions.  Between these two realms of expression there is a family of graphical data structures that arise in parsing the sentences and that serve to facilitate the performance of computations on the indicator functions.  The graphical language supplies an intermediate form of representation between the formal sentences and the indicator functions, and the form of mediation that it provides is very useful in rendering the possible connections between the other two languages conceivable in fact, not to mention in carrying out the necessary translations on a practical basis.  These Tables include this intermediate domain in their Central Columns.  Between their First and Middle Columns they illustrate the mechanics of parsing the abstract sentences of the cactus language into the graphical data structures of the corresponding species.  Between their Middle and Final Columns they summarize the semantics of interpreting the graphical forms of representation for the purposes of reasoning with propositions.
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<br>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
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|+ '''Table 14.1  Semantic Translation : Functional Form'''
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|- style="background:whitesmoke"
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|
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{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:whitesmoke; width:100%"
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| width="20%" | <math>\operatorname{Sentence}</math>
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| width="20%" | <math>\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Parse}}</math>
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| width="20%" | <math>\operatorname{Graph}</math>
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| width="20%" | <math>\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Denotation}}</math>
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| width="20%" | <math>\operatorname{Proposition}</math>
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|}
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|-
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|
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{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
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| width="20%" | <math>s_j\!</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>C_j\!</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>q_j\!</math>
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|}
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|-
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|
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{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
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| width="20%" | <math>\operatorname{Conc}^0</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>\operatorname{Node}^0</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>\underline{1}</math>
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|-
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| width="20%" | <math>\operatorname{Conc}^k_j s_j</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>\operatorname{Node}^k_j c_j</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>\operatorname{Conj}^k_j q_j</math>
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|}
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|-
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|
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{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
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| width="20%" | <math>\operatorname{Surc}^0</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>\operatorname{Lobe}^0</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>\underline{0}</math>
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|-
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| width="20%" | <math>\operatorname{Surc}^k_j s_j</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>\operatorname{Lobe}^k_j c_j</math>
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| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
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| width="20%" | <math>\operatorname{Surj}^k_j q_j</math>
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|}
 +
|}
 +
 +
<br>
    
<pre>
 
<pre>
−
Table 14.1  Semantic Translations : Functional Form
  −
o-------------------o-----o-------------------o-----o-------------------o
  −
|                  | Par |                  | Den |                  |
  −
| Sentence          | --> | Graph            | --> | Proposition      |
  −
o-------------------o-----o-------------------o-----o-------------------o
  −
|                  |    |                  |    |                  |
  −
| S_j              | --> | C_j              | --> | Q_j              |
  −
|                  |    |                  |    |                  |
  −
o-------------------o-----o-------------------o-----o-------------------o
  −
|                  |    |                  |    |                  |
  −
| Conc^0            | --> | Node^0            | --> | %1%              |
  −
|                  |    |                  |    |                  |
  −
| Conc^k_j  S_j    | --> | Node^k_j  C_j    | --> | Conj^k_j  Q_j    |
  −
|                  |    |                  |    |                  |
  −
o-------------------o-----o-------------------o-----o-------------------o
  −
|                  |    |                  |    |                  |
  −
| Surc^0            | --> | Lobe^0            | --> | %0%              |
  −
|                  |    |                  |    |                  |
  −
| Surc^k_j  S_j    | --> | Lobe^k_j  C_j    | --> | Surj^k_j  Q_j    |
  −
|                  |    |                  |    |                  |
  −
o-------------------o-----o-------------------o-----o-------------------o
  −
   
Table 14.2  Semantic Translations : Equational Form
 
Table 14.2  Semantic Translations : Equational Form
 
o-------------------o-----o-------------------o-----o-------------------o
 
o-------------------o-----o-------------------o-----o-------------------o
Jon Awbrey
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