8.5.x. Exercise questions

1. Use the system of derivations to establish each of the following:
a. ∃x Fx, ∀x (Fx → Gx) ⇒ ∃x Gx
b. ∃x (Fx ∧ Gx), ∀x (Gx → Hx) ⇒ ∃x (Fx ∧ Hx) [this is the syllogism Darii]
c. ∀x (Fx → Ga) ⇔ ∃x Fx → Ga
d. Fa ⇔ ∃x (x = a ∧ Fx)
e. ∃x (Fx ∧ ∀y Rxy) ⇒ ∀x ∃y (Fy ∧ Ryx)
f. ∃x (Gx ∧ Fx), ¬ Fa ⇒ ∃x (¬ x = a ∧ Gx)
g. ∀x (Fx → Ga),∀x (Ga → Fx), ∃x Fx ⇒ ∀x Fx
h. Everyone loves everyone who loves anyone, Someone loves someone ⇒ Everyone loves everyone
i. Something is such that nothing other than it is done ⇔ At most one thing is done
2. Use derivations to check each of the claims below; if a derivation indicates that a claim fails, describe a structure that divides an open gap. You need not worry about infinite derivations.
a. ∃x Fx, ∃x Gx ⇒ ∃x (Fx ∧ Gx)
b. ∃x (Fx ∧ Gx), ∃x (Fx ∧ Hx), ∀x (Fx → ∀y (Fy → x = y)) ⇒ ∃x (Gx ∧ Hx)
3. In the following, choose one of each bracketed pair of premises and one each bracketed pair of words or phrases in the conclusion so as to make a valid argument; then analyze the premises and conclusion and construct a derivation to show that the argument is valid.
a.

Some road sign was colored

[Every stop sign was a road sign | Every road sign was a traffic marker]

[If anything was red, it was colored | If anything was colored, it was painted]

Some [stop sign | traffic marker] was [red | painted]

b.

Someone who owns a snake was pleased

[Every cobra is a snake | Every snake is a reptile]

Someone who owns a [cobra | reptile] was pleased

For more exercises, use the exercise machine.

Glen Helman 28 Aug 2008