Ax:
Bx: Cx: d: e: f: 
x
is an aardvark x is a bullfrog x is a camel Dick Cheney Emo Phillips Francis the talking mule 
All things in the universe are bullfrogs  
At least one camel exists  
It's not true that everything in the universe is a bullfrog  
Not even one aardvark exists 


A bullfrog exists  is made true by  See the "k" (whatever "k" stands for) in the bullfrog space  
All things are aarrdvarks  is made false by  See the "b" (whatever "b" stands for) in the nonaardvark space 
A bullfrog exists  is made false by  Notice the bullfrog space is empty.  
All things are aardvarks  
is made true by  Notice the nonaardvark space is empty. 
This formula:  Is made true by:  And made false by: 

















This formula: 
Is made true by: 
And made false by: 



This formula: 
Is made true by: 
And made false by: 



This formula: 
Is made true by: 
And made false by: 



This formula: 
Is made true by: 
And made false by: 















Keith is hairy Mike is insane Ossie is hairy Noone is green 

is proved invalid by  
Jeff is green Ossie is green Everything is green 

is proved invalid by 
Jeff is not a kangaroo A kangaroo exists 

is proved invalid by 

Nigel is an elk A nonelk exists 

is proved invalid by 

Is x(Ax) true here? Is ~x(~Ax) true here? 
Is x(Ax) false here? Is ~x(~Ax) false here? 

Is x(Bx)
true here? Is ~x(~Bx) true here? 
Is x(Bx)
false here? Is ~x(~Bx) false here? 

Is ~x(Bx)
true here? Is x(~Bx) true here? 
Is ~x(Bx)
false here? Is x~(Bx) false here? 

Is ~x(Cx) true
here? Is x(~Cx) true here? 
Is ~x(Cx)
false here? Is x(~Cx) false here? 
Rule 17: Change
Quantifier (CQ) Comes in eight little rules. 1. If "xP" is an available line, then "~x~P" may be written as a new line in the derivation, and 2. If "~xP" is an available line, then "x~P" may be written as a new line in the derivation, and 3. If "x~P" is an available line, then "~xP" may be written as a new line in the derivation, and 4. If "~x~P" is an available line, then "xP" may be written as a new line in the derivation, and 5. If "xP" is an available line, then "~x~P" may be written as a new line in the derivation, and 6. If "~xP" is an available line, then "x~P" may be written as a new line in the derivation, and 7. If "x~P" is an available line, then "~xP" may be written as a new line in the derivation, and 8. If "xP" is an available line, then "~x~P" may be written as a new line in the derivation. 
Homework
12. Use your own paper or the answer sheet at logic13homework.rtf. For each of the following arguments do TWO things. First, translate it into English based on the following scheme.
Second, circle it if it's valid, cross it out if it's invalid.
