MyWikiBiz

Changes

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

Revision as of 19:20, 28 January 2009

5,574 bytes added ,  19:20, 28 January 2009
→‎Syntactic Transformations: format table
Line 3,071: Line 3,071:  
<br>
 
<br>
   −
<pre>
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
−
Logical Translation Rule 2
+
|
−
 
+
{| align="center" cellpadding="0" cellspacing="0" width="100%"
−
If S, T are sentences
+
|- style="height:48px; text-align:right"
−
about things in the universe U
+
| width="98%" | <math>\text{Logical Translation Rule 2}\!</math>
−
 
+
| width="2%"  | &nbsp;
−
and P, Q are propositions: U -> B, such that:
+
|}
−
 
+
|-
−
L2a. [S] = P  and [T] = Q,
+
|
−
 
+
{| align="center" cellpadding="0" cellspacing="0" width="100%"
−
then the following equations hold:
+
|- style="height:48px"
−
 
+
| width="2%"  style="border-top:1px solid black" | &nbsp;
−
L2b00. [False] = () = 0 : U->B.
+
| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
−
 
+
| width="84%" style="border-top:1px solid black" |
−
L2b01. [Neither S nor T] = ([S])([T]) = (P)(Q).
+
<math>s, t ~\text{are sentences about things in the universe}~ X</math>
−
 
+
|- style="height:48px"
−
L2b02. [Not S, but T] = ([S])[T] = (P) Q.
+
| &nbsp;
−
 
+
| <math>\text{and}\!</math>
−
L2b03. [Not S] = ([S]) = (P).
+
| <math>p, q ~\text{are propositions} ~:~ X \to \underline\mathbb{B}</math>
−
 
+
|- style="height:48px"
−
L2b04. [S and not T] = [S]([T]) = P (Q).
+
| &nbsp;
−
 
+
| <math>\text{such that:}\!</math>
−
L2b05. [Not T] = ([T]) = (Q).
+
| &nbsp;
−
 
+
|- style="height:48px"
−
L2b06. [S or T, not both] = ([S], [T]) = (P, Q).
+
| &nbsp;
−
 
+
| <math>\text{L2a.}\!</math>
−
L2b07. [Not both S and T] = ([S].[T]) = (P Q).
+
| <math>\downharpoonleft s \downharpoonright ~=~ p \quad \operatorname{and} \quad \downharpoonleft t \downharpoonright ~=~ q</math>
−
 
+
|- style="height:48px"
−
L2b08. [S and T] = [S].[T] = P.Q.
+
| &nbsp;
−
 
+
| <math>\text{then}\!</math>
−
L2b09. [S <=> T] = (([S], [T])) = ((P, Q)).
+
| <math>\text{the following equations hold:}\!</math>
−
 
+
|}
−
L2b10. [T] = [T] = Q.
+
|-
−
 
+
|
−
L2b11. [S => T] = ([S]([T])) = (P (Q)).
+
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%"
−
 
+
|- style="height:52px"
−
L2b12. [S] = [S] = P.
+
| width="2%"  style="border-top:1px solid black" | &nbsp;
−
 
+
| width="14%" style="border-top:1px solid black" align="left" | <math>\text{L2b}_{0}.\!</math>
−
L2b13. [S <= T] = (([S]) [T]) = ((P) Q).
+
| width="32%" style="border-top:1px solid black" |
−
 
+
<math>\downharpoonleft \operatorname{false} \downharpoonright</math>
−
L2b14. [S or T] = (([S])([T])) = ((P)(Q)).
+
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
−
 
+
| width="28%" style="border-top:1px solid black" | <math>(~)</math>
−
L2b15. [True] = (()) = 1 : U->B.
+
| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
−
</pre>
+
| width="16%" style="border-top:1px solid black" | <math>(~)</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{1}.\!</math>
 +
| <math>\downharpoonleft \operatorname{neither}~ s ~\operatorname{nor}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p)(q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{2}.\!</math>
 +
| <math>\downharpoonleft \operatorname{not}~ s ~\operatorname{but}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(p) q\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{3}.\!</math>
 +
| <math>\downharpoonleft \operatorname{not}~ s \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{4}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{and~not}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>p (q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{5}.\!</math>
 +
| <math>\downharpoonleft \operatorname{not}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{6}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p, q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{7}.\!</math>
 +
| <math>\downharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>(p q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{8}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{and}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\downharpoonleft s \downharpoonright ~ \downharpoonleft t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>p q\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{9}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{is~equivalent~to}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((\downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright))</math>
 +
| <math>=\!</math>
 +
| <math>((p, q))\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{10}.\!</math>
 +
| <math>\downharpoonleft t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\downharpoonleft t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>q\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{11}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{implies}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>(\downharpoonleft s \downharpoonright (\downharpoonleft t \downharpoonright))</math>
 +
| <math>=\!</math>
 +
| <math>(p (q))\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{12}.\!</math>
 +
| <math>\downharpoonleft s \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>\downharpoonleft s \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>p\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{13}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{is~implied~by}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((\downharpoonleft s \downharpoonright) \downharpoonleft t \downharpoonright)</math>
 +
| <math>=\!</math>
 +
| <math>((p) q)\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{14}.\!</math>
 +
| <math>\downharpoonleft s ~\operatorname{or}~ t \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((\downharpoonleft s \downharpoonright)(\downharpoonleft t \downharpoonright))</math>
 +
| <math>=\!</math>
 +
| <math>((p)(q))\!</math>
 +
|- style="height:52px"
 +
| &nbsp;
 +
| align="left" | <math>\text{L2b}_{15}.\!</math>
 +
| <math>\downharpoonleft \operatorname{true} \downharpoonright</math>
 +
| <math>=\!</math>
 +
| <math>((~))</math>
 +
| <math>=\!</math>
 +
| <math>((~))</math>
 +
|}
 +
|}
    
<br>
 
<br>
Jon Awbrey
12,089

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/77989"