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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 17:40, 30 May 2008
, 17:40, 30 May 2008archive old work
| <math>(\!|(\!|~|\!)|\!)</math>
| <math>(\!|(\!|~|\!)|\!)</math>
| <math>(\!|(\!|~|\!)|\!)</math>
| <math>(\!|(\!|~|\!)|\!)</math>
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<br>
=Archive 3=
{| 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"
| <math>\mathcal{L}_1</math>
| <math>\mathcal{L}_2</math>
| <math>\mathcal{L}_3</math>
| <math>\mathcal{L}_4</math>
| <math>\mathcal{L}_5</math>
| <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>
| <math>\operatorname{false}</math>
| <math>0\!</math>
|-
| <math>f_{1}\!</math>
| <math>f_{0001}\!</math>
| 0 0 0 1
| <math>(x)(y)\!</math>
| <math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math>
| <math>\lnot x \land \lnot y\!</math>
|-
| <math>f_{2}\!</math>
| <math>f_{0010}\!</math>
| 0 0 1 0
| <math>(x)\ y\!</math>
| <math>y\ \operatorname{without}\ x</math>
| <math>\lnot x \land y\!</math>
|-
| <math>f_{3}\!</math>
| <math>f_{0011}\!</math>
| 0 0 1 1
| <math>(x)\!</math>
| <math>\operatorname{not}\ x</math>
| <math>\lnot x\!</math>
|-
| <math>f_{4}\!</math>
| <math>f_{0100}\!</math>
| 0 1 0 0
| <math>x\ (y)\!</math>
| <math>x\ \operatorname{without}\ y</math>
| <math>x \land \lnot y\!</math>
|-
| <math>f_{5}\!</math>
| <math>f_{0101}\!</math>
| 0 1 0 1
| <math>(y)\!</math>
| <math>\operatorname{not}\ y</math>
| <math>\lnot y\!</math>
|-
| <math>f_{6}\!</math>
| <math>f_{0110}\!</math>
| 0 1 1 0
| <math>(x,\ y)\!</math>
| <math>x\ \operatorname{not~equal~to}\ y</math>
| <math>x \ne y\!</math>
|-
| <math>f_{7}\!</math>
| <math>f_{0111}\!</math>
| 0 1 1 1
| <math>(x\ y)\!</math>
| <math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math>
| <math>\lnot x \lor \lnot y\!</math>
|-
| <math>f_{8}\!</math>
| <math>f_{1000}\!</math>
| 1 0 0 0
| <math>x\ y\!</math>
| <math>x\ \operatorname{and}\ y</math>
| <math>x \land y\!</math>
|-
| <math>f_{9}\!</math>
| <math>f_{1001}\!</math>
| 1 0 0 1
| <math>((x,\ y))\!</math>
| <math>x\ \operatorname{equal~to}\ y</math>
| <math>x = y\!</math>
|-
| <math>f_{10}\!</math>
| <math>f_{1010}\!</math>
| 1 0 1 0
| <math>y\!</math>
| <math>y\!</math>
| <math>y\!</math>
|-
| <math>f_{11}\!</math>
| <math>f_{1011}\!</math>
| 1 0 1 1
| <math>(x\ (y))\!</math>
| <math>\operatorname{not}\ x\ \operatorname{without}\ y</math>
| <math>x \Rightarrow y\!</math>
|-
| <math>f_{12}\!</math>
| <math>f_{1100}\!</math>
| 1 1 0 0
| <math>x\!</math>
| <math>x\!</math>
| <math>x\!</math>
|-
| <math>f_{13}\!</math>
| <math>f_{1101}\!</math>
| 1 1 0 1
| <math>((x)\ y)\!</math>
| <math>\operatorname{not}\ y\ \operatorname{without}\ x</math>
| <math>x \Leftarrow y\!</math>
|-
| <math>f_{14}\!</math>
| <math>f_{1110}\!</math>
| 1 1 1 0
| <math>((x)(y))\!</math>
| <math>x\ \operatorname{or}\ y</math>
| <math>x \lor y\!</math>
|-
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| 1 1 1 1
| <math>((~))\!</math>
| <math>\operatorname{true}</math>
| <math>1\!</math>
|}
|}
<br>
<br>
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.
===Table 1 : Variant 1===
===Table 1===
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
| height="36px" | <p><math>1\!</math></p>
| height="36px" | <p><math>1\!</math></p>
|}
|}
|}
|}
<br>
<br>