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| </pre> | | </pre> |
| | | |
| + | <br> |
| + | |
| + | {| 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%" |
| + | | |
| + | {| align="center" cellpadding="0" cellspacing="0" width="100%" |
| + | |- style="height:40px; text-align:center" |
| + | | width="80%" | |
| + | | width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~8}</math> |
| + | |} |
| + | |- |
| + | | |
| + | {| align="center" cellpadding="0" cellspacing="0" width="100%" |
| + | |- style="height:40px" |
| + | | width="2%" style="border-top:1px solid black" | |
| + | | width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math> |
| + | | width="60%" style="border-top:1px solid black" | <math>s, t ~\text{are sentences about things in}~ X</math> |
| + | | width="20%" style="border-top:1px solid black; border-left:1px solid black" | |
| + | |- style="height:40px" |
| + | | |
| + | | <math>\text{then}\!</math> |
| + | | <math>\text{the following are equivalent:}\!</math> |
| + | | style="border-left:1px solid black" | |
| + | |} |
| + | |- |
| + | | |
| + | {| align="center" cellpadding="0" cellspacing="0" width="100%" |
| + | |- style="height:40px" |
| + | | width="2%" style="border-top:1px solid black" | |
| + | | width="18%" style="border-top:1px solid black" | <math>\operatorname{R8a.}</math> |
| + | | width="60%" style="border-top:1px solid black" | <math>s ~\Leftrightarrow~ t</math> |
| + | | width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | |
| + | <math>\operatorname{R8a~:~D7a}</math> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:60px" |
| + | | |
| + | | <math>\operatorname{R8b.}</math> |
| + | | <math>\downharpoonleft s \downharpoonright ~=~ \downharpoonleft t \downharpoonright</math> |
| + | | style="border-left:1px solid black; text-align:center" | |
| + | <p><math>\operatorname{R8b~:~D7b}</math></p> |
| + | <p><math>\operatorname{R8b~:~R7a}</math></p> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:60px" |
| + | | |
| + | | <math>\operatorname{R8c.}</math> |
| + | | <math>\overset{X}{\underset{x}{\forall}}~ (\downharpoonleft s \downharpoonright (x) ~=~ \downharpoonleft t \downharpoonright (x))</math> |
| + | | style="border-left:1px solid black; text-align:center" | <math>\operatorname{R8c~:~R7b}</math> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:40px" |
| + | | |
| + | | <math>\operatorname{R8d.}</math> |
| + | | <math>\operatorname{Conj_x^X}~ (\downharpoonleft s \downharpoonright (x) ~=~ \downharpoonleft t \downharpoonright (x))</math> |
| + | | style="border-left:1px solid black; text-align:center" | <math>\operatorname{R8d~:~R7c}</math> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:40px" |
| + | | |
| + | | <math>\operatorname{R8e.}</math> |
| + | | <math>\operatorname{Conj_x^X}~ (\downharpoonleft s \downharpoonright (x) ~\Leftrightarrow~ \downharpoonleft t \downharpoonright (x))</math> |
| + | | style="border-left:1px solid black; text-align:center" | <math>\operatorname{R8e~:~R7d}</math> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:60px" |
| + | | |
| + | | <math>\operatorname{R8f.}</math> |
| + | | <math>\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft s \downharpoonright (x) ~,~ \downharpoonleft t \downharpoonright (x) ~\underline{))}</math> |
| + | | style="border-left:1px solid black; text-align:center" | <math>\operatorname{R8f~:~R7e}</math> |
| + | |- style="height:20px" |
| + | | |
| + | | |
| + | | |
| + | | style="border-left:1px solid black; text-align:center" | <math>::\!</math> |
| + | |- style="height:40px" |
| + | | |
| + | | <math>\operatorname{R8g.}</math> |
| + | | <math>\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft s \downharpoonright ~,~ \downharpoonleft t \downharpoonright ~\underline{))}^\$ (x)</math> |
| + | | style="border-left:1px solid black; text-align:center" | <math>\operatorname{R8g~:~R7f}</math> |
| + | |} |
| + | |} |
| + | |
| + | <br> |
| | | |
| ===Rule 10=== | | ===Rule 10=== |