1. Specify the scopes of each quantifier in the following sentences:
(a) (x)(Fx - (Gx & -Hx)) & (($y)(Gy & Hy) v ($z)((-Gz & ØHz) - (w) Jwz))
(b) (x)((Fx - (Gx & -Hx)) & (y)(Gx & Hy)) v ($z)(-Gz & Hz)
2. Specify the scopes of each quantifier in the following sentences:
(a) (((x)(Fx - (Gx & -Hx)) v ($y)(Gy & Hy)) & ($z)(-Gz v -Hz)) - (w) Jww
(b) (x)((Fx - (Gx & -Hx)) v ((y)(Gy & Hx) - ($z)(-Fz & Hx)))
3. Construct fluent English readings for the PLsentences (a), (b) and (c) below, using the following symbolization key:
Universe of Discourse: A domain of people
Bxy: x is a brother of y
Sxy: x is a sister of y
Oxy: x is older than y
Lxy: x likes y
(a) - ($x)(Bxx v Sxx)
(b) (x)(y)(($z)Lyz - Lxy)
(c) ($x)(y)(Syx - Oyx)
4. Construct fluent English readings for the PLsentences (a) and (b) below, using the following symbolization key:
Universe of Discourse: A domain of people
Bxy: x is a brother of y
Sxy: x is a sister of y
Oxy: x is older than y
Lxy: x likes y
(a) ($x)(y)(Byx - Oxy)
(b) ($x)(y)(($z)Byz - Lxy) & ($x)($y)(Sxy & (z)Lzx)
5. Translate the following English argument into PL and then determine, by the Method of Truth Trees, whether it is valid or not:
No humans are mortal. Therefore all humans are immortal.
6. Translate the following English argument into PL and then determine, by the Method of Truth Trees, whether it is valid or not:
All whales are mammals. All whales have lungs. Therefore, all mammals have lungs.
7. A relation R is said to beirreflexiveiff for any x, x does not bear R to itself. It is said to be asymmetric iff for any x and y, if x bears R to y then y does not bear R to x.
Prove that if R is asymmetric it is irreflexive.
8. Determine whether the following argument is valid or not by the method of Truth Trees. If not deductively valid, explain how this can be ascertained from the tree:
(x)Px :($x)Px
9. Determine whether the following argument is valid or not by the method of Truth Trees. If not deductively valid, explain how this can be ascertained from the tree:
Pa - (x)Qx : (x)(Pa - Qx)
10. Determine whether the following argument is valid or not by the method of Truth Trees. If not deductively valid, explain how this can be ascertained from the tree:
(x) -Rxa : (y)(x)(Rxa - Rxy)
