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## Rules for Predicate Logic

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**Rules for Predicate Logic**The system for predicate logic in this course is quite simple. We will learn 3 rules which can be used for proofs and also with trees.**Rules for Predicate Logic**The system for predicate logic in this course is quite simple. We will learn 3 rules which can be used for proofs and also with trees. $O Universal Out #O Existential Out QE Quantifier Exchange There are no quantifier In rules.**Taking an Instance**$x(Px>Qx)**Taking an Instance**$x(Px>Qx) Pa>Qa**Taking an Instance**$x(Px>Qx) Pa>Qa To take an instance: 1. Remove the quantifier. Px>Qx 2. Substitute the same name (a-w) for each occurrence of the variable. Pa>Qa**Taking an Instance**$x(Px>Qx) Pa>Qa To take an instance: 1. Remove the quantifier. Px>Qx 2. Substitute the same name (a-w) for each occurrence of the variable. Pa>Qa Pa>Qx WRONG Pa>Qb WRONG Pb>Qb RIGHT**Taking an Instance**$x(Px>Qx) Pa>Qa To take an instance: 1. Remove the quantifier. Px>Qx 2. Substitute the same name (a-w) for each occurrence of the variable. Pa>Qa This should be familiar from algebra. x+7=7+x 3+7=7+3 RIGHT 3+7=7+4 WRONG**Universal Out**1) Ba A 2) Da A -$x(Dx>-Bx) GOAL**Universal Out**1) Ba A 2) Da A 3) $x(Dx>-Bx) PA ?&-? ?,? &I -$x(Dx>-Bx) 3-? -I**Universal Out**1) Ba A 2) Da A 3) $x(Dx>-Bx) PA 4) Da>-Ba 3 $O ?&-? ?,? &I -$x(Dx>-Bx) 3-? -I**Universal Out**1) Ba A 2) Da A 3) $x(Dx>-Bx) PA 4) Da>-Ba 3 $O ?&-? ?,? &I -$x(Dx>-Bx) 3-? -I $O Rule $xAx An**Universal Out**1) Ba A 2) Da A 3) $x(Dx>-Bx) PA 4) Da>-Ba 3 $O 5) -Ba 4,2 >O ?&-? ?,? &I -$x(Dx>-Bx) 3-? -I**Universal Out**1) Ba A 2) Da A 3) $x(Dx>-Bx) PA 4) Da>-Ba 3 $O 5) -Ba 4,2 >O 6) Ba&-Ba 1,5 &I 7) -$x(Dx>-Bx) 3-6 -I**Universal Out**1) Gj A 2) $x(Gx>Rx) A 3) $x(Rx>Ux) A Uj GOAL**Universal Out**1) Gj A 2) $x(Gx>Rx) A 3) $x(Rx>Ux) A 4) Gj>Rj 2 $O Uj GOAL**Universal Out**1) Gj A 2) $x(Gx>Rx) A 3) $x(Rx>Ux) A 4) Gj>Rj 2 $O 5) Rj>Uj 3 $O Uj GOAL**Universal Out**1) Gj A 2) $x(Gx>Rx) A 3) $x(Rx>Ux) A 4) Gj>Rj 2 $O 5) Rj>Uj 3 $O 6) Rj 4,1 >O Uj GOAL**Universal Out**1) Gj A 2) $x(Gx>Rx) A 3) $x(Rx>Ux) A 4) Gj>Rj 2 $O 5) Rj>Uj 3 $O 6) Rj 4,1 >O 7) Uj 5,6 >O For more click here