Abbreviate Universal Generalisation to UG.
The longitude of this earlier question motivates me to pose further questions separately.
Source: A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley
[p 465:] [...] 5. Py ⊃ Cy 3, 4, Hypothetical Syllogism
[ 5'. Px ⊃ Cx x and y are Variables; so I can change y to x. ]
6. (x)(Px ⊃ Cx) 5, UG [...]
Yet if we take certain liberties, we might characterize line 5 as saying “If it is a P, then it is a C, where
“it” designates any item at random in the universe.
Line 6 can then be seen as reexpressing this sense of line 5.
[p 467:] [...] 5. Qy 3, 4, Modus Ponens [...]
Line 5 states in effect that
everything in the universeis a Q.
[p 484:] 1. (x)Rx ⊃ (x)Sx [Conclusion:] (x)(Rx ⊃ Sx)
| 2. Rx Assumption for Conditional Proof
| 3. (x)Rx 2, UG (invalid)
I do not comprehend this answer: when does ℱx ≡ (x)ℱx? What if x represents the same set (or Domain/Universe of Discourse) in both?
For pp 465 and 467 above, does ℱx ≡ (x)ℱx? On p 465, the grey highlight state x as
any item at random in the universe. On p 467, th grey states y as
everything in the universe.