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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

Revision as of 19:06, 28 May 2008

88 bytes added ,  19:06, 28 May 2008
→‎Version 2: try \circ\!(...)\!\circ
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===Version 2===
 
===Version 2===
−
  −
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&nbsp;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%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
Line 2,784: Line 2,780:  
| <math>f_{0000}\!</math>
 
| <math>f_{0000}\!</math>
 
| 0 0 0 0
 
| 0 0 0 0
−
| <math>(\!|~|\!)</math>
+
| <math>\circ\!(~)\!\circ</math>
 
| false
 
| false
 
| <math>0\!</math>
 
| <math>0\!</math>
Line 2,791: Line 2,787:  
| <math>f_{0001}\!</math>
 
| <math>f_{0001}\!</math>
 
| 0 0 0 1
 
| 0 0 0 1
−
| <math>(\!|x|\!)(\!|y|\!)</math>
+
| <math>\circ\!(x)\!\circ\circ\!(y)\!\circ</math>
 
| neither x nor y
 
| neither x nor y
 
| <math>\lnot x \land \lnot y</math>
 
| <math>\lnot x \land \lnot y</math>
Line 2,798: Line 2,794:  
| <math>f_{0010}\!</math>
 
| <math>f_{0010}\!</math>
 
| 0 0 1 0
 
| 0 0 1 0
−
| <math>(\!|x|\!)\ y</math>
+
| <math>\circ\!(x)\!\circ\ y</math>
 
| y and not x
 
| y and not x
 
| <math>\lnot x \land y</math>
 
| <math>\lnot x \land y</math>
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| <math>f_{0011}\!</math>
 
| <math>f_{0011}\!</math>
 
| 0 0 1 1
 
| 0 0 1 1
−
| <math>(\!|x|\!)</math>
+
| <math>\circ\!(x)\!\circ</math>
 
| not x
 
| not x
 
| <math>\lnot x</math>
 
| <math>\lnot x</math>
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| <math>f_{0100}\!</math>
 
| <math>f_{0100}\!</math>
 
| 0 1 0 0
 
| 0 1 0 0
−
| <math>x\ (\!|y|\!)</math>
+
| <math>x\ \circ\!(y)\!\circ</math>
 
| x and not y
 
| x and not y
 
| <math>x \land \lnot y</math>
 
| <math>x \land \lnot y</math>
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| <math>f_{0101}\!</math>
 
| <math>f_{0101}\!</math>
 
| 0 1 0 1
 
| 0 1 0 1
−
| <math>(\!|y|\!)</math>
+
| <math>\circ\!(y)\!\circ</math>
 
| not y
 
| not y
 
| <math>\lnot y</math>
 
| <math>\lnot y</math>
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| <math>f_{0110}\!</math>
 
| <math>f_{0110}\!</math>
 
| 0 1 1 0
 
| 0 1 1 0
−
| <math>(\!|x,\ y|\!)</math>
+
| <math>\circ\!(x,\ y)\!\circ</math>
 
| x not equal to y
 
| x not equal to y
 
| <math>x \ne y</math>
 
| <math>x \ne y</math>
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| <math>f_{0111}\!</math>
 
| <math>f_{0111}\!</math>
 
| 0 1 1 1
 
| 0 1 1 1
−
| <math>(\!|x\ y|\!)</math>
+
| <math>\circ\!(x\ y)\!\circ</math>
 
| not both x and y
 
| not both x and y
 
| <math>\lnot x \lor \lnot y</math>
 
| <math>\lnot x \lor \lnot y</math>
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| <math>f_{1001}\!</math>
 
| <math>f_{1001}\!</math>
 
| 1 0 0 1
 
| 1 0 0 1
−
| <math>(\!|(\!|x,\ y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(x,\ y)\!\circ)\!\circ</math>
 
| x equal to y
 
| x equal to y
 
| <math>x = y\!</math>
 
| <math>x = y\!</math>
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| <math>f_{1011}\!</math>
 
| <math>f_{1011}\!</math>
 
| 1 0 1 1
 
| 1 0 1 1
−
| <math>(\!|x\ (\!|y|\!)|\!)</math>
+
| <math>\circ\!(x\ \circ\!(y)\!\circ)\!\circ</math>
 
| not x without y
 
| not x without y
 
| <math>x \Rightarrow y</math>
 
| <math>x \Rightarrow y</math>
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| <math>f_{1101}\!</math>
 
| <math>f_{1101}\!</math>
 
| 1 1 0 1
 
| 1 1 0 1
−
| <math>(\!|(\!|x|\!)\ y|\!)</math>
+
| <math>\circ\!(\circ\!(x)\!\circ\ y)\!\circ</math>
 
| not y without x
 
| not y without x
 
| <math>x \Leftarrow y</math>
 
| <math>x \Leftarrow y</math>
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| <math>f_{1110}\!</math>
 
| <math>f_{1110}\!</math>
 
| 1 1 1 0
 
| 1 1 1 0
−
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(x)\!\circ\circ\!(y)\!\circ)\!\circ</math>
 
| x or y
 
| x or y
 
| <math>x \lor y</math>
 
| <math>x \lor y</math>
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| <math>f_{1111}\!</math>
 
| <math>f_{1111}\!</math>
 
| 1 1 1 1
 
| 1 1 1 1
−
| <math>(\!|(\!|~|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(~)\!\circ)\!\circ</math>
 
| true
 
| true
 
| <math>1\!</math>
 
| <math>1\!</math>
 
|}<br>
 
|}<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%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
Line 2,902: Line 2,896:  
| style="width:16%" | <math>f\!</math>
 
| style="width:16%" | <math>f\!</math>
 
| style="width:16%" | <math>\operatorname{E}f|_{xy}</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\circ\!(y)\!\circ}</math>
−
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)y}</math>
+
| style="width:16%" | <math>\operatorname{E}f|_{\circ\!(x)!\circ y}</math>
−
| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)(\!|y|\!)}</math>
+
| style="width:16%" | <math>\operatorname{E}f|_{\circ\!(x)\!\circ\circ\!(y)\!\circ}</math>
 
|-
 
|-
 
| <math>f_{0}\!</math>
 
| <math>f_{0}\!</math>
−
| <math>(\!|~|\!)</math>
+
| <math>\circ\!(~)\!\circ</math>
−
| <math>(\!|~|\!)</math>
+
| <math>\circ\!(~)\!\circ</math>
−
| <math>(\!|~|\!)</math>
+
| <math>\circ\!(~)\!\circ</math>
−
| <math>(\!|~|\!)</math>
+
| <math>\circ\!(~)\!\circ</math>
−
| <math>(\!|~|\!)</math>
+
| <math>\circ\!(~)\!\circ</math>
 
|-
 
|-
 
| <math>f_{1}\!</math>
 
| <math>f_{1}\!</math>
−
| <math>(\!|x|\!)(\!|y|\!)</math>
+
| <math>\circ\!(x)\!\circ\circ\!(y)\!\circ</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 \circ\!(\operatorname{d}y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ \operatorname{d}y</math>
−
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ\circ\!(\operatorname{d}y)\!\circ</math>
 
|-
 
|-
 
| <math>f_{2}\!</math>
 
| <math>f_{2}\!</math>
−
| <math>(\!|x|\!) y</math>
+
| <math>\circ\!(x)\!\circ y</math>
−
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
+
| <math>\operatorname{d}x \circ\!(\operatorname{d}y)\!\circ</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>\circ\!(\operatorname{d}x)\!\circ\circ\!(\operatorname{d}y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ \operatorname{d}y</math>
 
|-
 
|-
 
| <math>f_{4}\!</math>
 
| <math>f_{4}\!</math>
−
| <math>x (\!|y|\!)</math>
+
| <math>x \circ\!(y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ \operatorname{d}y</math>
−
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ\circ\!(\operatorname{d}y)\!\circ</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 \circ\!(\operatorname{d}y)\!\circ</math>
 
|-
 
|-
 
| <math>f_{8}\!</math>
 
| <math>f_{8}\!</math>
 
| <math>x y\!</math>
 
| <math>x y\!</math>
−
| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ\circ\!(\operatorname{d}y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ \operatorname{d}y</math>
−
| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
+
| <math>\operatorname{d}x \circ\!(\operatorname{d}y)\!\circ</math>
 
| <math>\operatorname{d}x\ \operatorname{d}y</math>
 
| <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
 
| <math>f_{3}\!</math>
 
| <math>f_{3}\!</math>
−
| <math>(\!|x|\!)</math>
+
| <math>\circ\!(x)\!\circ</math>
 
| <math>\operatorname{d}x</math>
 
| <math>\operatorname{d}x</math>
 
| <math>\operatorname{d}x</math>
 
| <math>\operatorname{d}x</math>
−
| <math>(\!|\operatorname{d}x|\!)</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ</math>
−
| <math>(\!|\operatorname{d}x|\!)</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ</math>
 
|-
 
|-
 
| <math>f_{12}\!</math>
 
| <math>f_{12}\!</math>
 
| <math>x\!</math>
 
| <math>x\!</math>
−
| <math>(\!|\operatorname{d}x|\!)</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ</math>
−
| <math>(\!|\operatorname{d}x|\!)</math>
+
| <math>\circ\!(\operatorname{d}x)\!\circ</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>f_{6}\!</math>
−
| <math>(\!|x, y|\!)</math>
+
| <math>\circ\!(x, y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x, \operatorname{d}y)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x, \operatorname{d}y)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x, \operatorname{d}y)\!\circ)\!\circ</math>
−
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x, \operatorname{d}y)\!\circ</math>
 
|-
 
|-
 
| <math>f_{9}\!</math>
 
| <math>f_{9}\!</math>
−
| <math>(\!|(\!|x, y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(x, y)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x, \operatorname{d}y)\!\circ)\!\circ</math>
−
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x, \operatorname{d}y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x, \operatorname{d}y)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x, \operatorname{d}y)\!\circ)\!\circ</math>
 
|-
 
|-
 
| <math>f_{5}\!</math>
 
| <math>f_{5}\!</math>
−
| <math>(\!|y|\!)</math>
+
| <math>\circ\!(y)\!\circ</math>
 
| <math>\operatorname{d}y</math>
 
| <math>\operatorname{d}y</math>
−
| <math>(\!|\operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}y)\!\circ</math>
 
| <math>\operatorname{d}y</math>
 
| <math>\operatorname{d}y</math>
−
| <math>(\!|\operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}y)\!\circ</math>
 
|-
 
|-
 
| <math>f_{10}\!</math>
 
| <math>f_{10}\!</math>
 
| <math>y\!</math>
 
| <math>y\!</math>
−
| <math>(\!|\operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}y)\!\circ</math>
 
| <math>\operatorname{d}y</math>
 
| <math>\operatorname{d}y</math>
−
| <math>(\!|\operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}y)\!\circ</math>
 
| <math>\operatorname{d}y</math>
 
| <math>\operatorname{d}y</math>
 
|-
 
|-
 
| <math>f_{7}\!</math>
 
| <math>f_{7}\!</math>
−
| <math>(\!|x y|\!)</math>
+
| <math>\circ\!(x y)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x)\!\circ\circ\!(\operatorname{d}y)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x)\!\circ \operatorname{d}y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\operatorname{d}x \circ\!(\operatorname{d}y)\!\circ)\!\circ</math>
−
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x\ \operatorname{d}y)\!\circ</math>
 
|-
 
|-
 
| <math>f_{11}\!</math>
 
| <math>f_{11}\!</math>
−
| <math>(\!|x (\!|y|\!)|\!)</math>
+
| <math>\circ\!(x \circ\!(y)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x)\!\circ \operatorname{d}y)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x)\!\circ\circ\!(\operatorname{d}y)\!\circ)\!\circ</math>
−
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x\ \operatorname{d}y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\operatorname{d}x \circ\!(\operatorname{d}y)\!\circ)\!\circ</math>
 
|-
 
|-
 
| <math>f_{13}\!</math>
 
| <math>f_{13}\!</math>
−
| <math>(\!|(\!|x|\!) y|\!)</math>
+
| <math>\circ\!(\circ\!(x)\!\circ y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\operatorname{d}x \circ\!(\operatorname{d}y)\!\circ)\!\circ</math>
−
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x\ \operatorname{d}y)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x)\!\circ\circ\!(\operatorname{d}y)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x)\!\circ \operatorname{d}y)\!\circ</math>
 
|-
 
|-
 
| <math>f_{14}\!</math>
 
| <math>f_{14}\!</math>
−
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(x)\!\circ\circ\!(y)\!\circ)\!\circ</math>
−
| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\operatorname{d}x\ \operatorname{d}y)\!\circ</math>
−
| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\operatorname{d}x \circ\!(\operatorname{d}y)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x)\!\circ \operatorname{d}y)\!\circ</math>
−
| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(\operatorname{d}x)\!\circ\circ\!(\operatorname{d}y)\!\circ)\!\circ</math>
 
|-
 
|-
 
| <math>f_{15}\!</math>
 
| <math>f_{15}\!</math>
−
| <math>(\!|(\!|~|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(~)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|~|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(~)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|~|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(~)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|~|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(~)\!\circ)\!\circ</math>
−
| <math>(\!|(\!|~|\!)|\!)</math>
+
| <math>\circ\!(\circ\!(~)\!\circ)\!\circ</math>
 
|}<br>
 
|}<br>
Jon Awbrey
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