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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
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Here I now delve into subject matters that are more specifically logical in the character of their interpretation.
Here I now delve into subject matters that are more specifically logical in the character of their interpretation.
'''Working Note.''' Need transition to explain the use of [[Cactus Language]].
'''Working Note.''' Need segue here to explain the use of [[Cactus Language]].
Imagine that we are sitting in one of the cells of a venn diagram, contemplating the walls. There are <math>k\!</math> of them, one for each positive feature <math>x_1, \ldots, x_k</math> in our universe of discourse. Our particular cell is described by a concatenation of <math>k\!</math> signed assertions, positive or negative, regarding each of these features, and this description of our position amounts to what is called an "interpretation" of whatever proposition may rule the space, or reign on the universe of discourse. But are we locked into this interpretation?
Imagine that we are sitting in one of the cells of a venn diagram, contemplating the walls. There are <math>k\!</math> of them, one for each positive feature <math>x_1, \ldots, x_k</math> in our universe of discourse. Our particular cell is described by a concatenation of <math>k\!</math> signed assertions, positive or negative, regarding each of these features, and this description of our position amounts to what is called an "interpretation" of whatever proposition may rule the space, or reign on the universe of discourse. But are we locked into this interpretation?
o-----------------------------------------------------------o
o-----------------------------------------------------------o
Figure 1. Polymorphous Set Q
Figure 1. Polymorphous Set Q
</pre>
The proposition or the truth-function q that describes Q is:
The proposition or the truth-function <math>q\!</math> that describes <math>Q\!</math>is:
: <code>(( u v )( u w )( v w ))</code>
Conjoining the query that specifies the center cell gives:
Conjoining the query that specifies the center cell gives:
: <code>(( u v )( u w )( v w )) u v w</code>
And we know the value of the interpretation by
And we know the value of the interpretation by whether this last expression issues in a model.
whether this last expression issues in a model.
Applying the enlargement operator E
Applying the enlargement operator <math>\operatorname{E}</math> to the initial proposition <math>q\!</math> yields:
to the initial proposition q yields:
<code>
(( ( u , du )( v , dv )
(( ( u , du )( v , dv )
)( ( u , du )( w , dw )
)( ( u , du )( w , dw )
)( ( v , dv )( w , dw )
)( ( v , dv )( w , dw )
))
))
</code>
Conjoining a query on the center cell yields:
Conjoining a query on the center cell yields:
<code>
(( ( u , du )( v , dv )
(( ( u , du )( v , dv )
)( ( u , du )( w , dw )
)( ( u , du )( w , dw )
)( ( v , dv )( w , dw )
)( ( v , dv )( w , dw )
))
))
u v w
u v w
</code>
<pre>
The models of this last expression tell us which combinations of
The models of this last expression tell us which combinations of
feature changes among the set {du, dv, dw} will take us from our
feature changes among the set {du, dv, dw} will take us from our