Formalizing Predicate Logic (English to PL)
I'm stumped on how to formalize these two sentences properly into predicate logic.
1: There's a city to which all roads lead.
2: All roads lead to a city.
The question says nothing about defining a Universe of Discourse.
I tried using
R = is a road
C = is a city
L = leads to
1: If there exists a city and then every road leads to this city.
(∀y)(∃x)[(Cx & Ry) -> Lyx)]
I'm unsure about my use of the existential quantifier, it seems to me that what I wrote translates more into "If there exists a city and each road then each road leads to this city." I'm very confused at this point because this doesn't seem to make sense to me at all.
2: Everything that is a road leads to a city.
(∀y)(Ry -> Lyx)
With this one I started writing it and I know that x is unqualified and unquantified because I realized that if I were to add something like (∃x)(Cx) then this one would be identical to the first question. Is that right? I'm sure it is not since the first one says that "All roads lead to a specific city" whereas the second question says that "All roads lead to at least one city".