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==Propositions and Sentences==
 
==Propositions and Sentences==
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<pre>
+
'''Residual Remarks'''
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The "fibers" of truth and falsity under a proposition f : X -> %B%
  −
are subsets of X that are variously described as follows:
     −
1.  The fiber of %1% under f  =  [| f |]  =  f^(-1)(%1%)
+
Where are we?  We just defined the concept of a functional fiber in several of the most excruciating ways possible, but that's just because this method of refining functional fibers is intended partly for machine consumputation, so its schemata must be rendered free of all admixture of animate intuition. However, just between us, a single picture may suffice to sum up the notion:
−
 
  −
                              =  {x in X  :  f(x) = %1%}
  −
 
  −
                              =  {x in X  :  f(x) }.
  −
 
  −
2.  The fiber of %0% under f  =  ~[| f |]  =  f^(-1)(%0%)
  −
 
  −
                              =  {x in X  :  f(x) = %0%}
  −
 
  −
                              =  {x in X  :  (f(x)) }.
  −
 
  −
Perhaps this looks like a lot of work for the sake of what seems to be
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such a trivial form of syntactic transformation, but it is an important
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step in loosening up the syntactic privileges that are held by the sign
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of logical equivalence "<=>", as written between logical sentences, and
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by the sign of equality "=", as written between their logical values, or
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else between propositions and their boolean values.  Doing this removes
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a longstanding but wholly unnecessary conceptual confound between the
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idea of an "assertion" and notion of an "equation", and it allows one
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to treat logical equality on a par with the other logical operations.
  −
 
  −
----
  −
 
  −
Where are we?  We just defined the concept of a functional fiber in several
  −
of the most excruciating ways possible, but that's just because this method
  −
of refining functional fibers is intended partly for machine consumputation,
  −
so its schemata must be rendered free of all admixture of animate intuition.
  −
However, just between us, a single picture may suffice to sum up the notion:
      +
<pre>
 
|  X-[| f |] ,  [| f |]  c  X
 
|  X-[| f |] ,  [| f |]  c  X
 
|  o      o  o  o  o      |
 
|  o      o  o  o  o      |
Line 114: Line 85:  
|      o          o          v
 
|      o          o          v
 
|  {  %0%    ,    %1%  }  =  %B%
 
|  {  %0%    ,    %1%  }  =  %B%
 +
</pre>
   −
For the sake of current reference:
+
Why are we doing this?  The immediate reason &mdash; whose critique I defer &mdash; has to do with finding a ''modus vivendi'', whether a working compromise or a genuine integration, between the assertive-declarative languages and the functional-procedural languages that we have available for the sake of conceptual-logical-ontological analysis, clarification, description, inference, problem-solving, programming, representation, or whatever.
   −
| The "fibers" of truth and falsity in a proposition f : X -> %B%
+
In the next few installments, I will be working toward the definition of an operation called the ''stretch''.  This is related to the concept from category theory that is called a ''pullback''.  As a few will know the uses of that already, maybe there's hope of stretching the number.
−
| are the subsets [| f |] and X - [| f |] of X that are variously
  −
| described as follows:
  −
|
  −
| The fiber of %1% under f
  −
|
  −
| =  [| f |]  =  f^(-1)(%1%)
  −
|
  −
| =  {x in X  :  f(x) = %1%}
  −
|
  −
| =  {x in X  :  f(x) }.
  −
|
  −
| The fiber of %0% under f
  −
|
  −
| =  ~[| f |]  =  f^(-1)(%0%)
  −
|
  −
| =  {x in X  :  f(x) = %0%}
  −
|
  −
| =  {x in X  :  (f(x)) }.
  −
 
  −
Oh, by the way, the outer parentheses in "(f(g))" signify negation.
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I did not have here the "stricken parentheses" that I normally use.
  −
 
  −
Why are we doing this?  The immediate reason -- whose critique I defer --
  −
has to do with finding a modus vivendi, whether a working compromise or
  −
a genuine integration, between the assertive-declarative languages and
  −
the functional-procedural languages that we have available for the sake
  −
of conceptual-logical-ontological analysis, clarification, description,
  −
inference, problem-solving, programming, representation, or whatever.
  −
 
  −
In the next few installments, I will be working toward the definition
  −
of an operation called the "stretch".  This is related to the concept
  −
from category theory that is called a "pullback".  As a few will know
  −
the uses of that already, maybe there's hope of stretching the number.
  −
 
  −
----
  −
 
  −
In this episode, I compile a collection of definitions,
  −
leading up to the particular conception of a "sentence"
  −
that I'll be using throughout the rest of this inquiry.
      +
<pre>
 
1.3.10.3  Propositions & Sentences (cont.)
 
1.3.10.3  Propositions & Sentences (cont.)
  
Jon Awbrey
12,080

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