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Understanding Quantifier Exchange: Rules and Strategies for Logical Statements

This guide explores the concept of Quantifier Exchange in logical statements, detailing how to handle situations where a quantifier precedes an expression. It outlines relevant rules, illegal transitions, and strategies to effectively manipulate logical formulas. Key transformations are illustrated, such as how to correctly interpret statements involving "some" and "not all." Ideal for students and educators in fields like mathematics and philosophy, this resource emphasizes clarity in logical reasoning and the application of formal rules in various contexts.

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Understanding Quantifier Exchange: Rules and Strategies for Logical Statements

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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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