quantifier exchange what to do when a - precedes a quantifier
TRANSCRIPT
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Quantifier Exchange
What to do when a - precedes a quantifier.
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Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A
$x(Hx>-Ix) GOAL
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Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
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Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) -(Ha>-Ia) 3 $O
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
ILLEGAL!Rules apply to
main connectivesonly!
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Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 6: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/6.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x-
Not all people are tall =Some people are not tall
![Page 7: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/7.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-#x = $x-
Not some people are tall =all people are not tall
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Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 9: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/9.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE
The part after the quantifier stays the same.
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 10: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/10.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE ??? TILT ??? #x without & next!
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
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Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE
??? TILT ??? #x without & next!
Not to worry! We have -(Hx>-Ix) here.
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 12: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/12.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 13: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/13.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
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Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 15: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/15.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O
What to do with this step?
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 16: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/16.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O8) Ha&Ia 5 AR
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 17: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/17.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O8) Ha&Ia 5 AR9) Ha 8 &O10) Ia 8 &O
?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 18: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/18.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O8) Ha&Ia 5 AR9) Ha 8 &O10) Ia 8 &O11) Ga 6, 9 >O12) -Ia 7,11 >O ?&-? ?,? &I $x(Hx>-Ix) 3-? -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 19: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/19.jpg)
Quantifier Exchange1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O8) Ha&Ia 5 AR9) Ha 8 &O10) Ia 8 &O11) Ga 6, 9 >O12) -Ia 7,11 >O13) Ia&-Ia 10,12 &I14) $x(Hx>-Ix) 3-13 -O
QuantifierExchange:
-$x = #x--#x = $x-
![Page 20: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/20.jpg)
A Fancy Trick1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O How to use -(Ha>-Ia)? ?&-? ?,? &I $x(Hx>-Ix) 3-? -O
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A Fancy Trick1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O8) Ha>-Ia 6,7 CH ?&-? ?,? &I $x(Hx>-Ix) 3-? -O
A>BB>CA>C CHain
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A Fancy Trick1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O8) Ha>-Ia 6,7 CH ?&-? ?,? &I $x(Hx>-Ix) 3-? -O
![Page 23: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/23.jpg)
A Fancy Trick1) $x(Hx>Gx) A2) $x(Gx>-Ix) A3) -$x(Hx>-Ix) PA4) #x-(Hx>-Ix) 3 QE5) -(Ha>-Ia) 4 #O6) Ha>Ga 1 $O7) Ga>-Ia 2 $O8) Ha>-Ia 6,7 CH9) (Ha>-Ia)&-(Ha>-Ia) 8,5 &I10) $x(Hx>-Ix) 3-9 -O
![Page 24: Quantifier Exchange What to do when a - precedes a quantifier](https://reader036.vdocument.in/reader036/viewer/2022062309/56649cae5503460f94972309/html5/thumbnails/24.jpg)
QE Strategy
It is a good idea to apply rules in the following order:
QE#O$O
For more click here