بسط یک جمله در منطق محمول ها - ویکی‌پدیا، دانشنامهٔ آزاد

(${\displaystyle \forall \;x}$) ${\displaystyle \mathrm {F} x}$

(${\displaystyle \mathrm {F} a}$ ${\displaystyle \And }$ ${\displaystyle \mathrm {F} b}$ ) ${\displaystyle \And }$ (${\displaystyle \mathrm {F} c}$ ${\displaystyle \And }$ ${\displaystyle \mathrm {F} d}$ )

(${\displaystyle \exists \;x}$) ${\displaystyle \mathrm {F} x}$

(${\displaystyle \mathrm {F} a}$ ${\displaystyle \vee \;}$ ${\displaystyle \mathrm {F} b}$ ) ${\displaystyle \vee \;}$ (${\displaystyle \mathrm {F} c}$ ${\displaystyle \vee \;}$ ${\displaystyle \mathrm {F} d}$ )

${\displaystyle \sim \;}$ [( ${\displaystyle \sim \;}$ ${\displaystyle \mathrm {A} a}$ ${\displaystyle \And }$ ${\displaystyle \mathrm {B} a}$ ) ${\displaystyle \vee }$ ( ${\displaystyle \sim \;}$ ${\displaystyle \mathrm {A} b}$ ${\displaystyle \And }$ ${\displaystyle \mathrm {B} b}$ ) ]

( ${\displaystyle \mathrm {A} a}$ ${\displaystyle \And }$ ${\displaystyle \mathrm {C} a}$ ) ${\displaystyle \vee }$ ( ${\displaystyle \mathrm {A} b}$ ${\displaystyle \And }$ ${\displaystyle \mathrm {C} b}$ )

${\displaystyle \sim \;}$ [( ${\displaystyle \mathrm {C} a}$ ${\displaystyle \supset }$ ${\displaystyle \sim \;}$ ${\displaystyle \mathrm {B} a}$ ) ${\displaystyle \And }$ ( ${\displaystyle \mathrm {C} b}$ ${\displaystyle \supset }$ ${\displaystyle \sim \;}$ ${\displaystyle \mathrm {B} b}$ ) ]

${\displaystyle {\underset {\mathrm {T} }{\sim \;}}}$ [( ${\displaystyle {\underset {\mathrm {F} }{\sim \;}}}$ ${\displaystyle {\underset {\mathrm {T} }{\mathrm {A} a}}}$ ${\displaystyle {\underset {\mathrm {F} }{\And }}}$ ${\displaystyle {\underset {\mathrm {F} }{\mathrm {B} a}}}$ ) ${\displaystyle {\underset {\mathrm {F} }{\vee }}}$ ( ${\displaystyle {\underset {\mathrm {F} }{\sim \;}}}$ ${\displaystyle {\underset {\mathrm {T} }{\mathrm {A} b}}}$ ${\displaystyle {\underset {\mathrm {F} }{\And }}}$ ${\displaystyle {\underset {\mathrm {F} }{\mathrm {B} b}}}$ ) ]

( ${\displaystyle {\underset {\mathrm {T} }{\mathrm {A} a}}}$ ${\displaystyle {\underset {\mathrm {T} }{\And }}}$ ${\displaystyle {\underset {\mathrm {T} }{\mathrm {C} a}}}$ ) ${\displaystyle {\underset {\mathrm {T} }{\vee }}}$ ( ${\displaystyle {\underset {\mathrm {T} }{\mathrm {A} b}}}$ ${\displaystyle {\underset {\mathrm {T} }{\And }}}$ ${\displaystyle {\underset {\mathrm {T} }{\mathrm {C} b}}}$ )

${\displaystyle \therefore }$ ${\displaystyle {\underset {\mathrm {F} }{\sim \;}}}$ [( ${\displaystyle {\underset {\mathrm {T} }{\mathrm {C} a}}}$ ${\displaystyle {\underset {\mathrm {T} }{\supset }}}$ ${\displaystyle {\underset {\mathrm {T} }{\sim \;}}}$ ${\displaystyle {\underset {\mathrm {F} }{\mathrm {B} a}}}$ ) ${\displaystyle {\underset {\mathrm {T} }{\And }}}$ ( ${\displaystyle {\underset {\mathrm {T} }{\mathrm {C} b}}}$ ${\displaystyle {\underset {\mathrm {T} }{\supset }}}$ ${\displaystyle {\underset {\mathrm {T} }{\sim \;}}}$ ${\displaystyle {\underset {\mathrm {F} }{\mathrm {B} b}}}$ ) ]