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, present a counterexample that lurks in 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] |
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| 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 |
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