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. |
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| b. |
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