7.5.x. Exercise questions
| 1. |
Give the instances of each of the following for the terms a, b, and c (remembering that you will drop the main quantifier, and only the main one, when giving an instance): |
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| a. | ∀x Fx | |
| b. | ∀y Fy | |
| c. | ∀x Rxa | |
| d. | ∀x Saxb | |
| e. | ∀x ∀y Rxy | |
| f. | ∀x (Fx → Gx) | |
| g. | ∀x (Fx → Gd) | |
| h. | ∀x (Fx → ∀y Rxy) | |
| i. | ∀x (Fx → ∀x Rxx) | |
| 2. |
Use the system of derivations to establish each of the following. You may use detachment and attachment rules. |
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| a. | ∀x Fx, ∀x (Fx → Gx) ⊨ Ga | |
| b. | ∀x (Fx ∧ Gx) ⊨ Fa ∧ Gb | |
| c. | ∀x Rxa, ∀x (Rbx → Gx) ⊨ Ga | |
| d. | ∀x ∀y Rxy, ∀x (Rxx → Gx) ⊨ Ga | |
| e. | ∀x ∀y Rxy ⊨ (Rab ∧ Rbb) ∧ Rca | |
| f. | ∀x Fx, ∀x (Fx → Gx) ⊨ ∀x Gx | |
| g. | ∀x (Fx ∧ Gx) ≃ ∀x Fx ∧ ∀x Gx | |
| h. | Fa ≃ ∀x (x = a → Fx) | |
| i. | ∀x ∀y Rxy ⊨ ∀y Rya | |
| j. | ∀x ∀y (Rxy → ¬ Ryx) ⊨ ∀x ¬ Rxx | |
| k. | ∀x ∀y (gx = y → Fy) ⊨ ∀x F(g(hx)) | |
| 3. |
In the following, certain alternative expressions are enclosed in brackets and divided by vertical bars. Choose one of each alternative pair of premises and one of each alternative 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. You may use detachment and attachment rules. |
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| a. |
Every 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] Every [stop sign | traffic marker] was [red | painted] |
| b. |
No 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] No [stop sign | traffic marker] was [red | painted] |
| c. |
Only road signs were 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] Only [stop signs | traffic markers] were [red | painted] |
| d. |
Among road signs all except colored ones were replaced [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] Among [stop signs | traffic markers] all except [red | painted] ones were replaced |
For more exercises, use the exercise machine.