Consider the sentence "If all the birds in my house are peacocks, then Charlie is a peacock," and let the provided symbols be denoted as B for bird, H for house, P for peacock, and c for Charlie.
Are the following two translations exactly equivalent?
- (x)( ( (Bx & Hx) → Px) → Pc)
- ((x)( ( (Bx & Hx) → Px)) → Pc
If they are different, what is the exact difference? In particular, what is an example that works for (1) and not (2) (and vice versa). If they are equivalent, how can I generalize the situations for which the correctness of the translation is not changed by moving the position of the parentheses?
As a note, the textbook that I am using denotes (x) as a universal qualifier instead of using (∀x).