Quantifier Exchange What to do when a - precedes a quantifier.

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Presentation transcript:

Quantifier Exchange What to do when a - precedes a quantifier.

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A $x(Hx>-Ix)GOAL

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA ?&-??,? &I $x(Hx>-Ix)3-? -O

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) -(Ha>-Ia)3 $O ?&-??,? &I $x(Hx>-Ix)3-? -O ILLEGAL! Rules apply to main connectives only!

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- Not all people are tall = Some people are not tall

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -#x = $x- Not some people are tall = all people are not tall

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE The part after the quantifier stays the same. ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE ??? TILT ??? #x without & next! ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE ??? TILT ??? #x without & next! Not to worry! We have -(Hx>-Ix) here. ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O What to do with this step? ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O 8) Ha&Ia5 AR ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O 8) Ha&Ia5 AR 9) Ha8 &O 10) Ia8 &O ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O 8) Ha&Ia5 AR 9) Ha8 &O 10) Ia8 &O 11) Ga6, 9 >O 12) -Ia7,11 >O ?&-??,? &I $x(Hx>-Ix)3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

Quantifier Exchange 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O 8) Ha&Ia5 AR 9) Ha8 &O 10) Ia8 &O 11) Ga6, 9 >O 12) -Ia7,11 >O 13) Ia&-Ia10,12 &I 14) $x(Hx>-Ix)3-13 -O Quantifier Exchange: -$x = #x- -#x = $x-

A Fancy Trick 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O How to use -(Ha>-Ia)? ?&-? ?,? &I $x(Hx>-Ix)3-? -O

A Fancy Trick 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O 8) Ha>-Ia6,7 CH ?&-??,? &I $x(Hx>-Ix)3-? -O A>B B>C A>C CHain

A Fancy Trick 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O 8) Ha>-Ia6,7 CH ?&-??,? &I $x(Hx>-Ix)3-? -O

A Fancy Trick 1) $x(Hx>Gx)A 2) $x(Gx>-Ix)A 3) -$x(Hx>-Ix)PA 4) #x-(Hx>-Ix)3 QE 5) -(Ha>-Ia)4 #O 6) Ha>Ga1 $O 7) Ga>-Ia2 $O 8) Ha>-Ia6,7 CH 9) (Ha>-Ia)&-(Ha>-Ia)8,5 &I 10) $x(Hx>-Ix)3-9 -O

QE Strategy It is a good idea to apply rules in the following order: QE #O $O For more click here