| 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 | |
| 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. | |||
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