12,089
edits
Changes
Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 19:34, 27 May 2008
, 19:34, 27 May 2008→Differential Propositions: HTML --> TeX
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
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="background:lightcyan; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 1. Propositional Forms on Two Variables'''
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:paleturquoise"
|-
| 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:paleturquoise"
|-
|
|
| align="right" | <math>x\!</math> :
| align="right" | <math>x\!</math> :
|
|
|
|
|- style="background:paleturquoise"
|-
|
|
| align="right" | <math>y\!</math> :
| align="right" | <math>y\!</math> :
| <math>f_{0000}\!</math>
| <math>f_{0000}\!</math>
| 0 0 0 0
| 0 0 0 0
| ( )
| <math>(\!|~|\!)</math>
| false
| false
| 0
| 0
| <math>f_{0001}\!</math>
| <math>f_{0001}\!</math>
| 0 0 0 1
| 0 0 0 1
| (x)(y)
| <math>(\!|x|\!)(\!|y|\!)</math>
| neither x nor y
| neither x nor y
| ¬x ∧ ¬y
| ¬x ∧ ¬y
| <math>f_{0010}\!</math>
| <math>f_{0010}\!</math>
| 0 0 1 0
| 0 0 1 0
| (x) y
| <math>(\!|x|\!)\ y</math>
| y and not x
| y and not x
| ¬x ∧ y
| ¬x ∧ y
| <math>f_{0011}\!</math>
| <math>f_{0011}\!</math>
| 0 0 1 1
| 0 0 1 1
| (x)
| <math>(\!|x|\!)</math>
| not x
| not x
| ¬x
| ¬x
| <math>f_{0100}\!</math>
| <math>f_{0100}\!</math>
| 0 1 0 0
| 0 1 0 0
| x (y)
| <math>x\ (\!|y|\!)</math>
| x and not y
| x and not y
| x ∧ ¬y
| x ∧ ¬y
| <math>f_{0101}\!</math>
| <math>f_{0101}\!</math>
| 0 1 0 1
| 0 1 0 1
| (y)
| <math>(\!|y|\!)</math>
| not y
| not y
| ¬y
| ¬y
| <math>f_{0110}\!</math>
| <math>f_{0110}\!</math>
| 0 1 1 0
| 0 1 1 0
| (x, y)
| <math>(\!|x,\ y|\!)</math>
| x not equal to y
| x not equal to y
| x ≠ y
| x ≠ y
| <math>f_{0111}\!</math>
| <math>f_{0111}\!</math>
| 0 1 1 1
| 0 1 1 1
| (x y)
| <math>(\!|x\ y|\!)</math>
| not both x and y
| not both x and y
| ¬x ∨ ¬y
| ¬x ∨ ¬y
| <math>f_{1000}\!</math>
| <math>f_{1000}\!</math>
| 1 0 0 0
| 1 0 0 0
| x y
| <math>x\ y</math>
| x and y
| x and y
| x ∧ y
| x ∧ y
| <math>f_{1001}\!</math>
| <math>f_{1001}\!</math>
| 1 0 0 1
| 1 0 0 1
| ((x, y))
| <math>(\!|(\!|x,\ y|\!)|\!)</math>
| x equal to y
| x equal to y
| x = y
| x = y
| <math>f_{1010}\!</math>
| <math>f_{1010}\!</math>
| 1 0 1 0
| 1 0 1 0
| y
| <math>y\!</math>
| y
| y
| y
| y
| <math>f_{1011}\!</math>
| <math>f_{1011}\!</math>
| 1 0 1 1
| 1 0 1 1
| (x (y))
| <math>(\!|x\ (\!|y|\!)|\!)</math>
| not x without y
| not x without y
| x → y
| x → y
| <math>f_{1100}\!</math>
| <math>f_{1100}\!</math>
| 1 1 0 0
| 1 1 0 0
| x
| <math>x\!</math>
| x
| x
| x
| x
| <math>f_{1101}\!</math>
| <math>f_{1101}\!</math>
| 1 1 0 1
| 1 1 0 1
| ((x) y)
| <math>(\!|(\!|x|\!)\ y|\!)</math>
| not y without x
| not y without x
| x ← y
| x ← y
| <math>f_{1110}\!</math>
| <math>f_{1110}\!</math>
| 1 1 1 0
| 1 1 1 0
| ((x)(y))
| <math>(\!|(\!|x|\!)(\!|y|\!)|\!)</math>
| x or y
| x or y
| x ∨ y
| x ∨ y
| <math>f_{1111}\!</math>
| <math>f_{1111}\!</math>
| 1 1 1 1
| 1 1 1 1
| (( ))
| <math>(\!|(\!|~|\!)|\!)</math>
| true
| true
| 1
| 1