# Equivalences in Sentence Translation with Quantificational Logic

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?

1. (x)( ( (Bx & Hx) → Px) → Pc)
2. ((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).

• They are not equivalent. To move quantifiers out of implications correctly see prenex normal form: (∀x ϕ) → ψ is equivalent to ∃x(ϕ → ψ), not to ∀x(ϕ → ψ). Commented Nov 13, 2019 at 18:53
• a priori that Charlie is a bird. Commented Nov 14, 2019 at 4:49
• Well 2 isn't a well-formed formula. Did you mean to add another closed parenthesis? Moreover there is an effective procedure for adding parenthesis such that these questions shouldn't arise. Commented Nov 15, 2019 at 19:18