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

Revision as of 15:10, 28 May 2008

8,602 bytes added ,  15:10, 28 May 2008
→‎Work Area: adjust head, add working material
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<br>
 
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==Work Area==
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=Work Area 1=
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==Propositional Forms on Two Variables==
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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.
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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="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
Line 2,308: Line 2,314:  
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
 
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
 
<math>[\mathbb{D}^n]</math>
 
<math>[\mathbb{D}^n]</math>
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|}<br>
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=Work Area 2=
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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|+ '''Table 1.  Propositional Forms on Two Variables'''
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|- style="background:ghostwhite"
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| style="width:16%" | <math>\mathcal{L}_1</math>
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| style="width:16%" | <math>\mathcal{L}_2</math>
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| style="width:16%" | <math>\mathcal{L}_3</math>
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| style="width:16%" | <math>\mathcal{L}_4</math>
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| style="width:16%" | <math>\mathcal{L}_5</math>
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| style="width:16%" | <math>\mathcal{L}_6</math>
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|- style="background:ghostwhite"
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| &nbsp;
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| align="right" | <math>x\!</math> :
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| 1 1 0 0
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| &nbsp;
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| &nbsp;
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| &nbsp;
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|- style="background:ghostwhite"
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| &nbsp;
 +
| align="right" | <math>y\!</math> :
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| 1 0 1 0
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| &nbsp;
 +
| &nbsp;
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| &nbsp;
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|-
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| <math>f_{0}\!</math>
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| <math>f_{0000}\!</math>
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| 0 0 0 0
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| <math>(\!|~|\!)</math>
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| false
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| <math>0\!</math>
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|-
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| <math>f_{1}\!</math>
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| <math>f_{0001}\!</math>
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| 0 0 0 1
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| <math>(\!|x|\!)(\!|y|\!)</math>
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| neither x nor y
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| <math>\lnot x \land \lnot y</math>
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|-
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| <math>f_{2}\!</math>
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| <math>f_{0010}\!</math>
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| 0 0 1 0
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| <math>(\!|x|\!)\ y</math>
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| y and not x
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| <math>\lnot x \land y</math>
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|-
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| <math>f_{3}\!</math>
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| <math>f_{0011}\!</math>
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| 0 0 1 1
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| <math>(\!|x|\!)</math>
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| not x
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| <math>\lnot x</math>
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|-
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| <math>f_{4}\!</math>
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| <math>f_{0100}\!</math>
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| 0 1 0 0
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| <math>x\ (\!|y|\!)</math>
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| x and not y
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| <math>x \land \lnot y</math>
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|-
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| <math>f_{5}\!</math>
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| <math>f_{0101}\!</math>
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| 0 1 0 1
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| <math>(\!|y|\!)</math>
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| not y
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| <math>\lnot y</math>
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|-
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| <math>f_{6}\!</math>
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| <math>f_{0110}\!</math>
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| 0 1 1 0
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| <math>(\!|x,\ y|\!)</math>
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| x not equal to y
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| <math>x \ne y</math>
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|-
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| <math>f_{7}\!</math>
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| <math>f_{0111}\!</math>
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| 0 1 1 1
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| <math>(\!|x\ y|\!)</math>
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| not both x and y
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| <math>\lnot x \lor \lnot y</math>
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|-
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| <math>f_{8}\!</math>
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| <math>f_{1000}\!</math>
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| 1 0 0 0
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| <math>x\ y</math>
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| x and y
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| <math>x \land y</math>
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|-
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| <math>f_{9}\!</math>
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| <math>f_{1001}\!</math>
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| 1 0 0 1
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| <math>(\!|(\!|x,\ y|\!)|\!)</math>
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| x equal to y
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| <math>x = y\!</math>
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|-
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| <math>f_{10}\!</math>
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| <math>f_{1010}\!</math>
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| 1 0 1 0
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| <math>y\!</math>
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| y
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| <math>y\!</math>
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|-
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| <math>f_{11}\!</math>
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| <math>f_{1011}\!</math>
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| 1 0 1 1
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| <math>(\!|x\ (\!|y|\!)|\!)</math>
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| not x without y
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| <math>x \Rightarrow y</math>
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|-
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| <math>f_{12}\!</math>
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| <math>f_{1100}\!</math>
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| 1 1 0 0
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| <math>x\!</math>
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| x
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| <math>x\!</math>
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|-
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| <math>f_{13}\!</math>
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| <math>f_{1101}\!</math>
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| 1 1 0 1
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| <math>(\!|(\!|x|\!)\ y|\!)</math>
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| not y without x
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| <math>x \Leftarrow y</math>
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|-
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| <math>f_{14}\!</math>
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| <math>f_{1110}\!</math>
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| 1 1 1 0
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| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
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| x or y
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| <math>x \lor y</math>
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|-
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| <math>f_{15}\!</math>
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| <math>f_{1111}\!</math>
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| 1 1 1 1
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| <math>(\!|(\!|~|\!)|\!)</math>
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| true
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| <math>1\!</math>
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|}<br>
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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.
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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|+ '''Table 2.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
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|- style="background:ghostwhite"
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| style="width:16%" | &nbsp;
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| style="width:16%" | <math>f\!</math>
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| style="width:16%" | <math>\operatorname{E}f|_{xy}</math>
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| style="width:16%" | <math>\operatorname{E}f|_{x(\!|y|\!)}</math>
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| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)y}</math>
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| style="width:16%" | <math>\operatorname{E}f|_{(\!|x|\!)(\!|y|\!)}</math>
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|-
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| <math>f_{0}\!</math>
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| <math>(\!|~|\!)</math>
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| <math>(\!|~|\!)</math>
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| <math>(\!|~|\!)</math>
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| <math>(\!|~|\!)</math>
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| <math>(\!|~|\!)</math>
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|-
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| <math>f_{1}\!</math>
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| <math>(\!|x|\!)(\!|y|\!)</math>
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| <math>\operatorname{d}x\ \operatorname{d}y</math>
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| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
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| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
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| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
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|-
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| <math>f_{2}\!</math>
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| <math>(\!|x|\!) y</math>
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| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
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| <math>\operatorname{d}x\ \operatorname{d}y</math>
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| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
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| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
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|-
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| <math>f_{4}\!</math>
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| <math>x (\!|y|\!)</math>
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| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
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| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
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| <math>\operatorname{d}x\ \operatorname{d}y</math>
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| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
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|-
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| <math>f_{8}\!</math>
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| <math>x y\!</math>
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| <math>(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)</math>
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| <math>(\!|\operatorname{d}x|\!) \operatorname{d}y</math>
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| <math>\operatorname{d}x (\!|\operatorname{d}y|\!)</math>
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| <math>\operatorname{d}x\ \operatorname{d}y</math>
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|-
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| <math>f_{3}\!</math>
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| <math>(\!|x|\!)</math>
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| <math>\operatorname{d}x</math>
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| <math>\operatorname{d}x</math>
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| <math>(\!|\operatorname{d}x|\!)</math>
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| <math>(\!|\operatorname{d}x|\!)</math>
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|-
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| <math>f_{12}\!</math>
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| <math>x\!</math>
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| <math>(\!|\operatorname{d}x|\!)</math>
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| <math>(\!|\operatorname{d}x|\!)</math>
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| <math>\operatorname{d}x</math>
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| <math>\operatorname{d}x</math>
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|-
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| <math>f_{6}\!</math>
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| <math>(\!|x, y|\!)</math>
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| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
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| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
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|-
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| <math>f_{9}\!</math>
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| <math>(\!|(\!|x, y|\!)|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
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| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
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| <math>(\!|\operatorname{d}x, \operatorname{d}y|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x, \operatorname{d}y|\!)|\!)</math>
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|-
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| <math>f_{5}\!</math>
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| <math>(\!|y|\!)</math>
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| <math>\operatorname{d}y</math>
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| <math>(\!|\operatorname{d}y|\!)</math>
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| <math>\operatorname{d}y</math>
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| <math>(\!|\operatorname{d}y|\!)</math>
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|-
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| <math>f_{10}\!</math>
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| <math>y\!</math>
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| <math>(\!|\operatorname{d}y|\!)</math>
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| <math>\operatorname{d}y</math>
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| <math>(\!|\operatorname{d}y|\!)</math>
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| <math>\operatorname{d}y</math>
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|-
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| <math>f_{7}\!</math>
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| <math>(\!|x y|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
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| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
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| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
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|-
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| <math>f_{11}\!</math>
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| <math>(\!|x (\!|y|\!)|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
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| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
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| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
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|-
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| <math>f_{13}\!</math>
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| <math>(\!|(\!|x|\!) y|\!)</math>
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| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
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| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
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|-
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| <math>f_{14}\!</math>
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| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
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| <math>(\!|\operatorname{d}x\ \operatorname{d}y|\!)</math>
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| <math>(\!|\operatorname{d}x (\!|\operatorname{d}y|\!)|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x|\!) \operatorname{d}y|\!)</math>
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| <math>(\!|(\!|\operatorname{d}x|\!)(\!|\operatorname{d}y|\!)|\!)</math>
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|-
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| <math>f_{15}\!</math>
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| <math>(\!|(\!|~|\!)|\!)</math>
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| <math>(\!|(\!|~|\!)|\!)</math>
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| <math>(\!|(\!|~|\!)|\!)</math>
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| <math>(\!|(\!|~|\!)|\!)</math>
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| <math>(\!|(\!|~|\!)|\!)</math>
 
|}<br>
 
|}<br>
Jon Awbrey
12,080

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