1. | instance for a | instance for b | instance for c | |||||
a. | ∀x Fx | Fa | Fb | Fc | ||||
b. | ∀y Fy | Fa | Fb | Fc | ||||
c. | ∀x Rxa | Raa | Rba | Rca | ||||
d. | ∀x Saxb | Saab | Sabb | Sacb | ||||
e. | ∀x ∀y Rxy | ∀y Ray | ∀y Rby | ∀y Rcy | ||||
f. | ∀x (Fx → Gx) | Fa → Ga | Fb → Gb | Fc → Gc | ||||
g. | ∀x (Fx → Gd) | Fa → Gd | Fb → Gd | Fc → Gd | ||||
h. | ∀x (Fx → ∀y Rxy) | Fa → ∀y Ray | Fb → ∀y Rby | Fc → ∀y Rcy | ||||
i. | ∀x (Fx → ∀x Rxx) | Fa → ∀x Rxx | Fb → ∀x Rxx | Fc → ∀x Rxx |
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The term a could have been used again as the parameter of the second general argument of the derivation on the left since we require only that a parameter not appear outside its scope line in the gap which is developed by introducing the general argument, and the first general argument is boxed off from the gap that is developed by setting up the second one. But we will not be short of letters to use as parameters, so it will be easier to see that the requirement is satisfied if we use a new parameter for each general argument in a derivation. |
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