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Directory:Jon Awbrey/Papers/Dynamics And Logic (view source)

Revision as of 01:40, 14 June 2009

3,959 bytes added ,  01:40, 14 June 2009
→‎Note 7: + tables
Line 862: Line 862:  
==Note 7==
 
==Note 7==
   −
<pre>
+
To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>P \times Q \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> A few Tables are set here that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.
−
To broaden our experience with simple examples, let us
  −
now contemplate the sixteen functions of concrete type
  −
!P! x !Q! -> B and abstract type B x B -> B.  For ease
  −
of future reference, I will set here a few tables that
  −
specify the actions of E and D on the 16 functions and
  −
allow us to view the results in several different ways.
     −
By way of initial orientation, Table 7 lists equivalent expressions
+
Tables&nbsp;A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.
−
for the sixteen functions in several different formalisms, indexing
  −
systems, or languages for the propositional calculus, also known as
  −
"zeroth order logic" (ZOL).
     −
Table 7. Propositional Forms on Two Variables
+
<br>
−
o---------o---------o---------o----------o------------------o----------o
+
 
−
| L_1    | L_2    | L_3    | L_4      | L_5              | L_6      |
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
−
|         |         |         |         |                 |          |
+
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
−
| Decimal | Binary  | Vector  | Cactus  | English          | Ordinary |
+
|- style="background:#f0f0ff"
−
o---------o---------o---------o----------o------------------o----------o
+
| width="15%" |
−
|         |       p : 1 1 0 0 |         |                 |         |
+
<p><math>\mathcal{L}_1</math></p>
−
|         |       q : 1 0 1 0 |         |                 |         |
+
<p><math>\text{Decimal}</math></p>
−
o---------o---------o---------o----------o------------------o----------o
+
| width="15%" |
−
|         |        |        |          |                  |          |
+
<p><math>\mathcal{L}_2</math></p>
−
| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
+
<p><math>\text{Binary}</math></p>
−
|        |        |        |          |                  |          |
+
| width="15%" |
−
| f_1    | f_0001  | 0 0 0 1 | (p)(q) | neither p nor q | ~p & ~q |
+
<p><math>\mathcal{L}_3</math></p>
−
|        |        |        |          |                  |          |
+
<p><math>\text{Vector}</math></p>
−
| f_2    | f_0010  | 0 0 1 0 |  (p) q   | q and not p     | ~p &  q |
+
| width="15%" |
−
|        |        |        |          |                  |          |
+
<p><math>\mathcal{L}_4</math></p>
−
| f_3    | f_0011  | 0 0 1 1 |  (p)    | not p           | ~p       |
+
<p><math>\text{Cactus}</math></p>
−
|        |        |        |          |                  |          |
+
| width="25%" |
−
| f_4    | f_0100  | 0 1 0 0 |  p (q)  | p and not q     |  p & ~q |
+
<p><math>\mathcal{L}_5</math></p>
−
|         |         |        |          |                  |          |
+
<p><math>\text{English}</math></p>
−
| f_5    | f_0101  | 0 1 0 1 |    (q)  | not q            |      ~q  |
+
| width="15%" |
−
|        |        |        |          |                  |          |
+
<p><math>\mathcal{L}_6</math></p>
−
| f_6    | f_0110  | 0 1 1 0 | (p, q) | p not equal to q |  p +  q |
+
<p><math>\text{Ordinary}</math></p>
−
|        |        |        |          |                  |          |
+
|- style="background:#f0f0ff"
−
| f_7    | f_0111  | 0 1 1 1 |  (p q) | not both p and q | ~p v ~q |
+
| &nbsp;
−
|        |        |        |          |                  |          |
+
| align="right" | <math>p\colon\!</math>
−
| f_8    | f_1000  | 1 0 0 0 |  p q   | p and q         |  p &  q |
+
| <math>1~1~0~0\!</math>
−
|         |         |         |         |                 |         |
+
| &nbsp;
−
| f_9    | f_1001  | 1 0 0 1 | ((p, q)) | p equal to q    | p = q  |
+
| &nbsp;
−
|         |         |         |         |                 |         |
+
| &nbsp;
−
| f_10    | f_1010  | 1 0 1 0 |     q  | q                |       q  |
+
|- style="background:#f0f0ff"
−
|         |         |         |         |                 |         |
+
| &nbsp;
−
| f_11    | f_1011  | 1 0 1 1 | (p (q)) | not p without q | p => q  |
+
| align="right" | <math>q\colon\!</math>
−
|         |         |        |          |                  |          |
+
| <math>1~0~1~0\!</math>
−
| f_12    | f_1100  | 1 1 0 0 |   p     | p               | p       |
+
| &nbsp;
−
|         |         |        |          |                  |          |
+
| &nbsp;
−
| f_13    | f_1101  | 1 1 0 1 | ((p) q) | not q without p | p <= q |
+
| &nbsp;
−
|         |         |         |         |                 |         |
+
|-
−
| f_14    | f_1110  | 1 1 1 0 | ((p)(q)) | p or q           | p v  q |
+
|
−
|         |        |        |          |                  |          |
+
<math>\begin{matrix}
−
| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |   1     |
+
f_0
−
|         |        |        |          |                  |          |
+
\\[4pt]
−
o---------o---------o---------o----------o------------------o----------o
+
f_1
−
</pre>
+
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_3
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_5
 +
\\[4pt]
 +
f_6
 +
\\[4pt]
 +
f_7
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0000}
 +
\\[4pt]
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0011}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{0101}
 +
\\[4pt]
 +
f_{0110}
 +
\\[4pt]
 +
f_{0111}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~0
 +
\\[4pt]
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~0~1~1
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
0~1~0~1
 +
\\[4pt]
 +
0~1~1~0
 +
\\[4pt]
 +
0~1~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)~~~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~~~(q)
 +
\\[4pt]
 +
(p,~q)
 +
\\[4pt]
 +
(p~~q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{false}
 +
\\[4pt]
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
\text{not}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
\text{not}~ q
 +
\\[4pt]
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
\text{not both}~ p ~\text{and}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0
 +
\\[4pt]
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
\lnot p
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
\lnot q
 +
\\[4pt]
 +
p \ne q
 +
\\[4pt]
 +
\lnot p \lor \lnot q
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_8
 +
\\[4pt]
 +
f_9
 +
\\[4pt]
 +
f_{10}
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{12}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\\[4pt]
 +
f_{15}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{1000}
 +
\\[4pt]
 +
f_{1001}
 +
\\[4pt]
 +
f_{1010}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1100}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\\[4pt]
 +
f_{1111}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
1~0~0~0
 +
\\[4pt]
 +
1~0~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\\[4pt]
 +
1~1~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~p~~q~~
 +
\\[4pt]
 +
((p,~q))
 +
\\[4pt]
 +
~~~~~q~~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
~~p~~~~~
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{and}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
\text{not}~ p ~\text{without}~ q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
\text{not}~ q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{or}~ q
 +
\\[4pt]
 +
\text{true}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p \land q
 +
\\[4pt]
 +
p = q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
p \Rightarrow q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
p \Leftarrow q
 +
\\[4pt]
 +
p \lor q
 +
\\[4pt]
 +
1
 +
\end{matrix}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
 +
|- style="background:#f0f0ff"
 +
| width="15%" |
 +
<p><math>\mathcal{L}_1</math></p>
 +
<p><math>\text{Decimal}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_2</math></p>
 +
<p><math>\text{Binary}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3</math></p>
 +
<p><math>\text{Vector}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4</math></p>
 +
<p><math>\text{Cactus}</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5</math></p>
 +
<p><math>\text{English}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6</math></p>
 +
<p><math>\text{Ordinary}</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>f_{0000}\!</math>
 +
| <math>0~0~0~0</math>
 +
| <math>(~)</math>
 +
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{1000}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
1~0~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
p ~\text{and}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
p \land q
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0011}
 +
\\[4pt]
 +
f_{1100}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ p
 +
\\[4pt]
 +
p
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p
 +
\\[4pt]
 +
p
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0110}
 +
\\[4pt]
 +
f_{1001}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~0
 +
\\[4pt]
 +
1~0~0~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p \ne q
 +
\\[4pt]
 +
p = q
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0101}
 +
\\[4pt]
 +
f_{1010}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ q
 +
\\[4pt]
 +
q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot q
 +
\\[4pt]
 +
q
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0111}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~1
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p~~q)~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not both}~ p ~\text{and}~ q
 +
\\[4pt]
 +
\text{not}~ p ~\text{without}~ q
 +
\\[4pt]
 +
\text{not}~ q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{or}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p \lor \lnot q
 +
\\[4pt]
 +
p \Rightarrow q
 +
\\[4pt]
 +
p \Leftarrow q
 +
\\[4pt]
 +
p \lor q
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>f_{1111}\!</math>
 +
| <math>1~1~1~1</math>
 +
| <math>((~))</math>
 +
| <math>\text{true}\!</math>
 +
| <math>1\!</math>
 +
|}
 +
 
 +
<br>
    
==Note 8==
 
==Note 8==
Jon Awbrey
12,080

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