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Quantifier Exchange

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.

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Quantifier Exchange

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

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

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

  4. 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!

  5. 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-

  6. 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

  7. 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

  8. 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-

  9. 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-

  10. 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-

  11. 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-

  12. 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-

  13. 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-

  14. 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>Ga 1 $O 7) Ga>-Ia 2 $O ?&-? ?,? &I $x(Hx>-Ix) 3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

  15. 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>Ga 1 $O 7) Ga>-Ia 2 $O What to do with this step? ?&-? ?,? &I $x(Hx>-Ix) 3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

  16. 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>Ga 1 $O 7) Ga>-Ia 2 $O 8) Ha&Ia 5 AR ?&-? ?,? &I $x(Hx>-Ix) 3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

  17. 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>Ga 1 $O 7) Ga>-Ia 2 $O 8) Ha&Ia 5 AR 9) Ha 8 &O 10) Ia 8 &O ?&-? ?,? &I $x(Hx>-Ix) 3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

  18. 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>Ga 1 $O 7) Ga>-Ia 2 $O 8) Ha&Ia 5 AR 9) Ha 8 &O 10) Ia 8 &O 11) Ga 6, 9 >O 12) -Ia 7,11 >O ?&-? ?,? &I $x(Hx>-Ix) 3-? -O Quantifier Exchange: -$x = #x- -#x = $x-

  19. 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>Ga 1 $O 7) Ga>-Ia 2 $O 8) Ha&Ia 5 AR 9) Ha 8 &O 10) Ia 8 &O 11) Ga 6, 9 >O 12) -Ia 7,11 >O 13) Ia&-Ia 10,12 &I 14) $x(Hx>-Ix) 3-13 -O Quantifier Exchange: -$x = #x- -#x = $x-

  20. 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>Ga 1 $O 7) Ga>-Ia 2 $O How to use -(Ha>-Ia)? ?&-? ?,? &I $x(Hx>-Ix) 3-? -O

  21. 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>Ga 1 $O 7) Ga>-Ia 2 $O 8) Ha>-Ia 6,7 CH ?&-? ?,? &I $x(Hx>-Ix) 3-? -O A>B B>C A>C CHain

  22. 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>Ga 1 $O 7) Ga>-Ia 2 $O 8) Ha>-Ia 6,7 CH ?&-? ?,? &I $x(Hx>-Ix) 3-? -O

  23. 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>Ga 1 $O 7) Ga>-Ia 2 $O 8) Ha>-Ia 6,7 CH 9) (Ha>-Ia)&-(Ha>-Ia) 8,5 &I 10) $x(Hx>-Ix) 3-9 -O

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

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