Introductory Logic
Quiz #11
December 9, 2005
 

Name______________________
R. Hammack
Score _________
1. Use conditional or indirect proof to derive the conclusion.

1.     (∃x)(Ax ∨ Bx) ⊃ ~ (∃x)Ax /     (x)~Ax
  |2.     ~(x)~Ax AIP
  |3.     (∃x)~~Ax 2, CQ
  |4.     (∃x)Ax 3, DN
  |5.     ~~(∃x)Ax 4, DN
  |6.     ~(∃x)(Ax ∨ Bx) 1, 5, MT
  |7.     (x)~(Ax ∨ Bx) 6, CQ
  |8.     Am 4, EI
  |9.     ~(Am ∨ Bm) 8, UI
  |10.     ~Am • Bm 9, DM
  |11.     ~Am 10, Simmp
  |12.     ~Am • Am 8, 11 Conj
13     (x)~Ax 2-12, IP





2. Use any applicable method to prove the following argument is invalid.

1.     (∃x)(Ax • Bx)  
2.     (x)(Cx ⊃ Ax) /    (∃x)(Cx • Bx)

Consider a Universe with just one element, denoted a. Then the argument becomes:

Aa
•
 Ba
/
Ca
⊃
 Aa
//
Ca
•
Ba
T
T
 T
F
T
 T
F
F
 T

The indirect truth table can be filled out without contradiction, so the argument is invalid in the small universe.
Hence, it is also INVALID in our universe.