12,080
edits
Changes
Directory:Jon Awbrey/Papers/Cactus Language (view source)
Revision as of 04:46, 27 January 2009
, 04:46, 27 January 2009→Syntactic Transformations: math markup + copy edit
| <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>
|}
|}
Going back to Rule 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 4.
<pre>
<pre>
Rule 4
Rule 4
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:
about things in the universe U
about things in the universe U
and P is a proposition : U �> B, such that:
and P is a proposition : U -> B, such that:
L1a. [S] = P,
L1a. [S] = P,
If X c U
If X c U
and P : U �> B, such that:
and P : U -> B, such that:
G1a. {X} = P,
G1a. {X} = P,
about things in the universe U
about things in the universe U
and P, Q are propositions: U �> B, such that:
and P, Q are propositions: U -> B, such that:
L2a. [S] = P and [T] = Q,
L2a. [S] = P and [T] = Q,
If X, Y c U
If X, Y c U
and P, Q U �> B, such that:
and P, Q U -> B, such that:
G2a. {X} = P and {Y} = Q,
G2a. {X} = P and {Y} = Q,
then "v = w" is a sentence about <v, w> C B2,
then "v = w" is a sentence about <v, w> C B2,
[v = w] is a proposition : B2 �> B,
[v = w] is a proposition : B2 -> B,
and the following are identical values in B:
and the following are identical values in B:
Value Rule 1
Value Rule 1
If f, g : U �> B,
If f, g : U -> B,
and u C U
and u C U
Value Rule 1
Value Rule 1
If f, g : U �> B,
If f, g : U -> B,
then the following are identical propositions on U:
then the following are identical propositions on U:
Evaluation Rule 1
Evaluation Rule 1
If f, g : U �> B
If f, g : U -> B
and u C U,
and u C U,
about things in the universe U,
about things in the universe U,
f, g are propositions: U �> B,
f, g are propositions: U -> B,
and u C U,
and u C U,
Definition 3
Definition 3
If f, g : U �> V,
If f, g : U -> V,
then the following are equivalent:
then the following are equivalent:
D5a. {X}.
D5a. {X}.
D5b. f : U �> B
D5b. f : U -> B
: f(u) = [u C X], for all u C U.
: f(u) = [u C X], for all u C U.
Rule 6
Rule 6
If f, g : U �> V,
If f, g : U -> V,
then the following are equivalent:
then the following are equivalent:
Rule 7
Rule 7
If P, Q : U �> B,
If P, Q : U -> B,
then the following are equivalent:
then the following are equivalent:
: {X} = {<u, v> C UxB : v = [S](u)}. :R
: {X} = {<u, v> C UxB : v = [S](u)}. :R
::
::
R11d. {X} : U �> B
R11d. {X} : U -> B
: {X}(u) = [S](u), for all u C U. :R
: {X}(u) = [S](u), for all u C U. :R
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".
d. R11d gives a version of the indicator function with {X} : U�>B, called its "functional form".
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.