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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 18:15, 29 January 2009

1,648 bytes added , 18:15, 29 January 2009

→‎Variant 3

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====Variant 3====

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~~<pre>~~

A rule that allows one to turn equivalent sentences into identical propositions:

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(S <=> T) ~~<=>~~ (~~[S]~~ = ~~[T]~~)

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{| align="center" cellpadding="8" width="90%"

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| <math>(s ~\Leftrightarrow~ t) \quad \Leftrightarrow \quad (\downharpoonleft s \downharpoonright ~=~ \downharpoonleft t \downharpoonright)</math>

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|}

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Consider ~~[ v = w ](v, w) and [ v(u) = w(u) ](u)~~

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Consider the following pair of expressions:

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~~Value Rule 1~~

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{| align="center" cellpadding="8" width="90%"

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| <math>\downharpoonleft v ~=~ w \downharpoonright (v, w)</math>

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|-

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| <math>\downharpoonleft v(x) ~=~ w(x) \downharpoonright (x)</math>

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|}

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~~If v, w C B,~~

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<br>

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then the following are identical values in B:

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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"

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|

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{| align="center" cellpadding="0" cellspacing="0" width="100%"

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|- style="height:48px; text-align:right"

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| width="98%" | <math>\text{Value Rule 1}\!</math>

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| width="2%" |  

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|}

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|-

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|

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{| align="center" cellpadding="0" cellspacing="0" width="100%"

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|- style="height:48px"

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| width="2%" style="border-top:1px solid black" |  

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| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>

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| width="84%" style="border-top:1px solid black" | <math>v, w ~\in~ \underline\mathbb{B}</math>

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|- style="height:48px"

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|  

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| <math>\text{then}\!</math>

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| <math>\text{the following are identical values in}~ \underline\mathbb{B}:</math>

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|}

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|-

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|

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{| align="center" cellpadding="0" cellspacing="0" width="100%"

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|- style="height:56px"

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| width="2%" style="border-top:1px solid black" |  

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| width="14%" style="border-top:1px solid black" | <math>\text{V1a.}\!</math>

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| width="84%" style="border-top:1px solid black" | <math>\downharpoonleft v = w \downharpoonright</math>

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|- style="height:56px"

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|  

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| <math>\text{V1b.}\!</math>

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| <math>\downharpoonleft v \Leftrightarrow w \downharpoonright</math>

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|- style="height:56px"

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|  

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| <math>\text{V1c.}\!</math>

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| <math>\underline{((}~ v ~,~ w ~\underline{))}</math>

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|}

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|}

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~~V1a. [ v = w ]~~

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<br>

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~~V1b. [ v~~ <~~=> w ]~~

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~~V1c. (( v , w ))~~

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~~</pre~~>

====Variant 4====

Jon Awbrey

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edits