# Directory talk:Jon Awbrey/Papers/Syntactic Transformations

## Alternate Version : Needs To Be Reconciled

#### 1.3.12. Syntactic Transformations ✔

##### 1.3.12.1. Syntactic Transformation Rules
```Value Rule 1

If	v, w	C	B

then	"v = w" is a sentence about <v, w> C B2,

[v = w] is a proposition : B2 -> B,

and the following are identical values in B:

V1a.	[ v = w ](v, w)

V1b.	[ v <=> w ](v, w)

V1c.	((v , w))
```
```Value Rule 1

If	v, w	C	B,

then the following are equivalent:

V1a.	v = w.

V1b.	v <=> w.

V1c.	(( v , w )).
```

A rule that allows one to turn equivalent sentences into identical propositions:

(S <=> T) <=> ([S] = [T])

Consider [ v = w ](v, w) and [ v(u) = w(u) ](u)

```Value Rule 1

If	v, w	C	B,

then the following are identical values in B:

V1a.	[ v = w ]

V1b.	[ v <=> w ]

V1c.	(( v , w ))
```
```Value Rule 1

If	f, g	:	U -> B,

and	u	C	U

then the following are identical values in B:

V1a.	[ f(u) = g(u) ]

V1b.	[ f(u) <=> g(u) ]

V1c.	(( f(u) , g(u) ))
```
```Value Rule 1

If	f, g	:	U -> B,

then the following are identical propositions on U:

V1a.	[ f = g ]

V1b.	[ f <=> g ]

V1c.	(( f , g ))\$
```
```Evaluation Rule 1

If	f, g	:	U -> B

and	u	C	U,

then the following are equivalent:

E1a.	f(u) = g(u).	:V1a

::

E1b.	f(u) <=> g(u).	:V1b

::

E1c.	(( f(u) , g(u) )).	:V1c

:\$1a

::

E1d.	(( f , g ))\$(u).	:\$1b
```
```Evaluation Rule 1

If	S, T	are sentences

about things in the universe U,

f, g	are propositions: U -> B,

and	u	C	U,

then the following are equivalent:

E1a.	f(u) = g(u).	:V1a

::

E1b.	f(u) <=> g(u).	:V1b

::

E1c.	(( f(u) , g(u) )).	:V1c

:\$1a

::

E1d.	(( f , g ))\$(u).	:\$1b
```