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

Revision as of 04:46, 27 January 2009

32 bytes removed ,  04:46, 27 January 2009
→‎Syntactic Transformations: math markup + copy edit
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| <math>\downharpoonleft x \in Q \downharpoonright ~=~ \lambda (x, \in, Q).(x \in Q).</math>
 
| <math>\downharpoonleft x \in Q \downharpoonright ~=~ \lambda (x, \in, Q).(x \in Q).</math>
 
|}
 
|}
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Going back to Rule&nbsp;1, we see that it lists a pair of concrete sentences and authorizes exchanges in either direction between the syntactic structures that have these two forms.  But a sentence is any sign that denotes a proposition, and so there are any number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged.  For example, a larger collection of equivalent sentences is recorded in Rule&nbsp;4.
    
<pre>
 
<pre>
−
As it is presently stated, Rule 1 lists a couple of manifest sentences, and it authorizes one to make exchanges in either direction between the syntactic items that have these two forms.  But a sentence is any sign that denotes a proposition, and thus there are a number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged.  Consider the sense of equivalence among sentences that is recorded in Rule 4.
  −
   
Rule 4
 
Rule 4
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R4e. {X}(u) = 1.
 
R4e. {X}(u) = 1.
   −
The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U �> B.  At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence.  On reflection, and taken in context, these problems are not as serious as they initially seem.  For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence.  As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one.
+
The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U -> B.  At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence.  On reflection, and taken in context, these problems are not as serious as they initially seem.  For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence.  As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one.
    
The use of the basic connectives can be expressed in the form of a STR as follows:
 
The use of the basic connectives can be expressed in the form of a STR as follows:
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about things in the universe U
 
about things in the universe U
   −
and P is a proposition : U �> B, such that:
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and P is a proposition : U -> B, such that:
    
L1a. [S]  =  P,
 
L1a. [S]  =  P,
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If X c U
 
If X c U
   −
and P : U �> B, such that:
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and P : U -> B, such that:
    
G1a. {X}  =  P,
 
G1a. {X}  =  P,
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about things in the universe U
 
about things in the universe U
   −
and P, Q are propositions: U �> B, such that:
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and P, Q are propositions: U -> B, such that:
    
L2a. [S] = P  and  [T] = Q,
 
L2a. [S] = P  and  [T] = Q,
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If X, Y c U
 
If X, Y c U
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and P, Q U �> B, such that:
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and P, Q U -> B, such that:
    
G2a. {X} = P  and  {Y} = Q,
 
G2a. {X} = P  and  {Y} = Q,
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then "v = w" is a sentence about <v, w> C B2,
 
then "v = w" is a sentence about <v, w> C B2,
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[v = w] is a proposition : B2 �> B,
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[v = w] is a proposition : B2 -> B,
    
and the following are identical values in B:
 
and the following are identical values in B:
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Value Rule 1
 
Value Rule 1
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If f, g : U �> B,
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If f, g : U -> B,
    
and u C U
 
and u C U
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Value Rule 1
 
Value Rule 1
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If f, g : U �> B,
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If f, g : U -> B,
    
then the following are identical propositions on U:
 
then the following are identical propositions on U:
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Evaluation Rule 1
 
Evaluation Rule 1
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If f, g : U �> B
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If f, g : U -> B
    
and u C U,
 
and u C U,
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about things in the universe U,
 
about things in the universe U,
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f, g are propositions: U �> B,
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f, g are propositions: U -> B,
    
and u C U,
 
and u C U,
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Definition 3
 
Definition 3
   −
If f, g : U �> V,
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If f, g : U -> V,
    
then the following are equivalent:
 
then the following are equivalent:
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D5a. {X}.
 
D5a. {X}.
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D5b. f : U �> B
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D5b. f : U -> B
    
: f(u) = [u C X], for all u C U.
 
: f(u) = [u C X], for all u C U.
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Rule 6
 
Rule 6
   −
If f, g : U �> V,
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If f, g : U -> V,
    
then the following are equivalent:
 
then the following are equivalent:
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Rule 7
 
Rule 7
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If P, Q : U �> B,
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If P, Q : U -> B,
    
then the following are equivalent:
 
then the following are equivalent:
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: {X} = {<u, v> C UxB : v = [S](u)}. :R
 
: {X} = {<u, v> C UxB : v = [S](u)}. :R
 
::
 
::
−
R11d. {X} : U �> B
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R11d. {X} : U -> B
    
: {X}(u) = [S](u), for all u C U. :R
 
: {X}(u) = [S](u), for all u C U. :R
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c. R11c gives a version of the indicator function with {X} c UxB, called its "extensional form".
 
c. R11c gives a version of the indicator function with {X} c UxB, called its "extensional form".
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d. R11d gives a version of the indicator function with {X} : U�>B, called its "functional form".
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d. R11d gives a version of the indicator function with {X} : U->B, called its "functional form".
    
Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.
 
Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.
Jon Awbrey
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