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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)

Revision as of 16:47, 16 August 2009

158 bytes added ,  16:47, 16 August 2009
→‎Variations on a theme of transitivity: TeX markup
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Table 30-c.  Tacit Extensions of Projections of Syll
 
Table 30-c.  Tacit Extensions of Projections of Syll
 
o---------------o o---------------o o---------------o
 
o---------------o o---------------o o---------------o
−
|  TE(Syll_12)  | |  TE(Syll_13)  | |  TE(Syll_23)  |
+
|  te(Syll_12)  | |  te(Syll_13)  | |  te(Syll_23)  |
 
o---------------o o---------------o o---------------o
 
o---------------o o---------------o o---------------o
 
|  p  q  r  | |  p  q  r  | |  p  q  r  |
 
|  p  q  r  | |  p  q  r  | |  p  q  r  |
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|                                                |
 
|                                                |
 
o-------------------------------------------------o
 
o-------------------------------------------------o
−
Figure 30-d.  Tacit Extension TE_12_3 (Syll_12)
+
Figure 30-d.  Tacit Extension te_12_3 (Syll_12)
 
</pre>
 
</pre>
 
| (63)
 
| (63)
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|                                                |
 
|                                                |
 
o-------------------------------------------------o
 
o-------------------------------------------------o
−
Figure 30-e.  Tacit Extension TE_13_2 (Syll_13)
+
Figure 30-e.  Tacit Extension te_13_2 (Syll_13)
 
</pre>
 
</pre>
 
| (64)
 
| (64)
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|                                                |
 
|                                                |
 
o-------------------------------------------------o
 
o-------------------------------------------------o
−
Figure 30-f.  Tacit Extension TE_23_1 (Syll_23)
+
Figure 30-f.  Tacit Extension te_23_1 (Syll_23)
 
</pre>
 
</pre>
 
| (65)
 
| (65)
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The reader may wish to contemplate Figure&nbsp;31 and use it to verify the following two facts:
 
The reader may wish to contemplate Figure&nbsp;31 and use it to verify the following two facts:
   −
: ''Syll'' = ''TE''(''Syll''<sub>12</sub>) &cap; ''TE''(''Syll''<sub>23</sub>)
+
{| align="center" cellpadding="8" width="90%"
−
 
+
|
−
: ''Syll''<sub>13</sub> = ''Syll''<sub>12</sub> &omicron; ''Syll''<sub>23</sub>
+
<math>\begin{array}{lcc}
 +
\operatorname{Syll}
 +
& = &
 +
\operatorname{te}(\operatorname{Syll}_{12})
 +
\cap
 +
\operatorname{te}(\operatorname{Syll}_{23})
 +
\\[6pt]
 +
\operatorname{Syll}_{13}
 +
& = &
 +
\operatorname{Syll}_{12}
 +
\circ
 +
\operatorname{Syll}_{23}
 +
\end{array}</math>
 +
|}
    
{| align="center" cellpadding="10" style="text-align:center; width:90%"
 
{| align="center" cellpadding="10" style="text-align:center; width:90%"
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|                                                |
 
|                                                |
 
o-------------------------------------------------o
 
o-------------------------------------------------o
−
Figure 31.  Syll = TE(Syll_12) |^| TE(Syll_23)
+
Figure 31.  Syll = te(Syll_12) |^| te(Syll_23)
 
</pre>
 
</pre>
 
| (66)
 
| (66)
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Most of the customary names for this type of process have turned out to have misleading connotations, and so I will experiment with calling it the ''expressive'' aspect of the various rules for transitive inference, simply to emphasize the fact that rules can be given for it that operate solely on signs and expressions, without necessarily needing to look at the objects that are denoted by these signs and expressions.
 
Most of the customary names for this type of process have turned out to have misleading connotations, and so I will experiment with calling it the ''expressive'' aspect of the various rules for transitive inference, simply to emphasize the fact that rules can be given for it that operate solely on signs and expressions, without necessarily needing to look at the objects that are denoted by these signs and expressions.
   −
In the way of many experiments, the word ''expressive'' does not seem to work for what I wanted to say here, since we too often use it to suggest something that expresses an object or a purpose, and I wanted it to imply what is purely a matter of expression, shorn of consideration for anything objective.  Aside from coining a word like ''ennotative'', some other options would be ''connotative'', ''hermeneutic'', ''semiotic'', ''syntactic'' — each of which works in some range of interpretation but fails in others.  Trial 2.  Let's try ''formulaic''.
+
In the way of many experiments, the word ''expressive'' does not seem to work for what I wanted to say here, since we too often use it to suggest something that expresses an object or a purpose, and I wanted it to imply what is purely a matter of expression, shorn of consideration for anything objective.  Aside from coining a word like ''ennotative'', some other options would be ''connotative'', ''hermeneutic'', ''semiotic'', ''syntactic'' &mdash; each of which works in some range of interpretation but fails in others.  Trial 2.  Let's try ''formulaic''.
    
Despite how simple the formulaic aspects of transitive inference might appear on the surface, there are problems that wait for us just beneath the syntactic surface, as we quickly discover if we turn to considering the kinds of objects, abstract and concrete, that these formulas are meant to denote, and all the more so if we try to do this in a context of computational implementations, where the "interpreters" to be addressed take nothing on faith.  Thus we engage the ''denotative semantics'' or the ''model theory'' of these extremely simple programs that we call ''propositions''.
 
Despite how simple the formulaic aspects of transitive inference might appear on the surface, there are problems that wait for us just beneath the syntactic surface, as we quickly discover if we turn to considering the kinds of objects, abstract and concrete, that these formulas are meant to denote, and all the more so if we try to do this in a context of computational implementations, where the "interpreters" to be addressed take nothing on faith.  Thus we engage the ''denotative semantics'' or the ''model theory'' of these extremely simple programs that we call ''propositions''.
   −
Table 33 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference.  In it I use a number of abbreviated notations:
+
Table&nbsp;33 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference.  In it I use a number of abbreviated notations:
    
# I use the forms ''X''<b>:</b>''Y''<b>:</b>''Z'' and ''x''<b>:</b>''y''<b>:</b>''z'' as alternative notations for the cartesian product ''X'' &times; ''Y'' &times; ''Z'' and the tuple (''x'', ''y'', ''z''), respectively.
 
# I use the forms ''X''<b>:</b>''Y''<b>:</b>''Z'' and ''x''<b>:</b>''y''<b>:</b>''z'' as alternative notations for the cartesian product ''X'' &times; ''Y'' &times; ''Z'' and the tuple (''x'', ''y'', ''z''), respectively.
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|                  |        |                  |
 
|                  |        |                  |
 
o-------------------o        o-------------------o
 
o-------------------o        o-------------------o
−
|TE(Syll_12) c B:B:B|        |TE(Syll_23) c B:B:B|
+
|te(Syll_12) c B:B:B|        |te(Syll_23) c B:B:B|
 
o-------------------o        o-------------------o
 
o-------------------o        o-------------------o
 
|    [| f_207 |]    |        |    [| f_187 |]    |
 
|    [| f_207 |]    |        |    [| f_187 |]    |
Jon Awbrey
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