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=Work Area 3=
=Work Area 3=
===Propositional Forms on Two Variables===
To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> For future reference, I will set here a few Tables 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.
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:ghostwhite"
| style="width:16%" | <math>\mathcal{L}_1</math>
| style="width:16%" | <math>\mathcal{L}_2</math>
| style="width:16%" | <math>\mathcal{L}_3</math>
| style="width:16%" | <math>\mathcal{L}_4</math>
| style="width:16%" | <math>\mathcal{L}_5</math>
| style="width:16%" | <math>\mathcal{L}_6</math>
|- style="background:ghostwhite"
|
| align="right" | <math>x\!</math> :
| 1 1 0 0
|
|
|
|- style="background:ghostwhite"
|
| align="right" | <math>y\!</math> :
| 1 0 1 0
|
|
|
|-
| <math>f_{0}\!</math>
| <math>f_{0000}\!</math>
| 0 0 0 0
| <math>(~)\!</math>
| false
| <math>0\!</math>
|-
| <math>f_{1}\!</math>
| <math>f_{0001}\!</math>
| 0 0 0 1
| <math>(x)(y)\!</math>
| neither x nor y
| <math>\lnot x \land \lnot y\!</math>
|-
| <math>f_{2}\!</math>
| <math>f_{0010}\!</math>
| 0 0 1 0
| <math>(x)\ y\!</math>
| y and not x
| <math>\lnot x \land y\!</math>
|-
| <math>f_{3}\!</math>
| <math>f_{0011}\!</math>
| 0 0 1 1
| <math>(x)\!</math>
| not x
| <math>\lnot x\!</math>
|-
| <math>f_{4}\!</math>
| <math>f_{0100}\!</math>
| 0 1 0 0
| <math>x\ (y)\!</math>
| x and not y
| <math>x \land \lnot y\!</math>
|-
| <math>f_{5}\!</math>
| <math>f_{0101}\!</math>
| 0 1 0 1
| <math>(y)\!</math>
| not y
| <math>\lnot y\!</math>
|-
| <math>f_{6}\!</math>
| <math>f_{0110}\!</math>
| 0 1 1 0
| <math>(x,\ y)\!</math>
| x not equal to y
| <math>x \ne y\!</math>
|-
| <math>f_{7}\!</math>
| <math>f_{0111}\!</math>
| 0 1 1 1
| <math>(x\ y)\!</math>
| not both x and y
| <math>\lnot x \lor \lnot y\!</math>
|-
| <math>f_{8}\!</math>
| <math>f_{1000}\!</math>
| 1 0 0 0
| <math>x\ y\!</math>
| x and y
| <math>x \land y\!</math>
|-
| <math>f_{9}\!</math>
| <math>f_{1001}\!</math>
| 1 0 0 1
| <math>((x,\ y))\!</math>
| x equal to y
| <math>x = y\!</math>
|-
| <math>f_{10}\!</math>
| <math>f_{1010}\!</math>
| 1 0 1 0
| <math>y\!</math>
| y
| <math>y\!</math>
|-
| <math>f_{11}\!</math>
| <math>f_{1011}\!</math>
| 1 0 1 1
| <math>(x\ (y))\!</math>
| not x without y
| <math>x \Rightarrow y\!</math>
|-
| <math>f_{12}\!</math>
| <math>f_{1100}\!</math>
| 1 1 0 0
| <math>x\!</math>
| x
| <math>x\!</math>
|-
| <math>f_{13}\!</math>
| <math>f_{1101}\!</math>
| 1 1 0 1
| <math>((x)\ y)\!</math>
| not y without x
| <math>x \Leftarrow y\!</math>
|-
| <math>f_{14}\!</math>
| <math>f_{1110}\!</math>
| 1 1 1 0
| <math>((x)(y))\!</math>
| x or y
| <math>x \lor y\!</math>
|-
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| 1 1 1 1
| <math>((~))\!</math>
| true
| <math>1\!</math>
|}<br>
The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 2. <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
|- style="background:ghostwhite"
| style="width:16%" |
| style="width:16%" | <math>f\!</math>
| style="width:16%" | <math>\operatorname{E}f|_{xy}\!</math>
| style="width:16%" | <math>\operatorname{E}f|_{x(y)}\!</math>
| style="width:16%" | <math>\operatorname{E}f|_{(x)y}\!</math>
| style="width:16%" | <math>\operatorname{E}f|_{(x)(y)}\!</math>
|-
| <math>f_{0}\!</math>
| <math>(~)\!</math>
| <math>(~)\!</math>
| <math>(~)\!</math>
| <math>(~)\!</math>
| <math>(~)\!</math>
|-
| <math>f_{1}\!</math>
| <math>(x)(y)\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
|-
| <math>f_{2}\!</math>
| <math>(x) y\!</math>
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
|-
| <math>f_{4}\!</math>
| <math>x (y)\!</math>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
|-
| <math>f_{8}\!</math>
| <math>x y\!</math>
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
|-
| <math>f_{3}\!</math>
| <math>(x)\!</math>
| <math>\operatorname{d}x\!</math>
| <math>\operatorname{d}x\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>(\operatorname{d}x)\!</math>
|-
| <math>f_{12}\!</math>
| <math>x\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>(\operatorname{d}x)\!</math>
| <math>\operatorname{d}x\!</math>
| <math>\operatorname{d}x\!</math>
|-
| <math>f_{6}\!</math>
| <math>(x, y)\!</math>
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
|-
| <math>f_{9}\!</math>
| <math>((x, y))\!</math>
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
|-
| <math>f_{5}\!</math>
| <math>(y)\!</math>
| <math>\operatorname{d}y\!</math>
| <math>(\operatorname{d}y)\!</math>
| <math>\operatorname{d}y\!</math>
| <math>(\operatorname{d}y)\!</math>
|-
| <math>f_{10}\!</math>
| <math>y\!</math>
| <math>(\operatorname{d}y)\!</math>
| <math>\operatorname{d}y\!</math>
| <math>(\operatorname{d}y)\!</math>
| <math>\operatorname{d}y\!</math>
|-
| <math>f_{7}\!</math>
| <math>(x y)\!</math>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
|-
| <math>f_{11}\!</math>
| <math>(x (y))\!</math>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
|-
| <math>f_{13}\!</math>
| <math>((x) y)\!</math>
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
|-
| <math>f_{14}\!</math>
| <math>((x)(y))\!</math>
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
|-
| <math>f_{15}\!</math>
| <math>((~))\!</math>
| <math>((~))\!</math>
| <math>((~))\!</math>
| <math>((~))\!</math>
| <math>((~))\!</math>
|}<br>
<pre>
Table 3. Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | dx | dx |
| | | | | | |
| f_12 | x | dx | dx | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | dy | dy | dy |
| | | | | | |
| f_10 | y | dy | dy | dy | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
| f_11 | (x (y)) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_13 | ((x) y) | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_14 | ((x)(y)) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
</pre>
<pre>
Table 4. Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| f_12 | x | (x) | (x) | x | x |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
| | | | | | |
| f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | y | (y) | y | (y) |
| | | | | | |
| f_10 | y | (y) | y | (y) | y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) |
| | | | | | |
| f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| | | | | | |
| f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| | | | | | |
| f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | |
| Fixed Point Total | 4 | 4 | 4 | 16 |
| | | | | |
o-------------------o------------o------------o------------o------------o
</pre>
<pre>
Table 5. Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | ((x, y)) | (y) | (x) | () |
| | | | | | |
| f_2 | (x) y | (x, y) | y | (x) | () |
| | | | | | |
| f_4 | x (y) | (x, y) | (y) | x | () |
| | | | | | |
| f_8 | x y | ((x, y)) | y | x | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | (()) | (()) | () | () |
| | | | | | |
| f_12 | x | (()) | (()) | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | () | (()) | (()) | () |
| | | | | | |
| f_9 | ((x, y)) | () | (()) | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | (()) | () | (()) | () |
| | | | | | |
| f_10 | y | (()) | () | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x, y)) | y | x | () |
| | | | | | |
| f_11 | (x (y)) | (x, y) | (y) | x | () |
| | | | | | |
| f_13 | ((x) y) | (x, y) | y | (x) | () |
| | | | | | |
| f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
</pre>
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