Question 8

Given premises:

1. "All hill-stations have a lake." In predicate logic:
   $$\forall x (\text{HillStation}(x) \implies \text{HasLake}(x))$$
   Here, having "a lake" means having at least one lake $(\ge 1)$. It does not restrict the number of lakes to exactly one, nor does it state that only hill-stations have lakes.
2. "Ooty has two lakes." That is, $\text{Lakes}(\text{Ooty}) = 2$.

Evaluate statement (i): "Ooty is not a hill-station."
Since Ooty has 2 lakes, it satisfies the condition of having at least one lake. It is entirely possible that Ooty is a hill-station, or that it is not. The premises do not entail that Ooty is not a hill-station. Thus, (i) cannot be inferred.

Evaluate statement (ii): "No hill-station can have more than one lake."
The premise states that every hill-station has at least one lake; it sets a lower bound, not an upper bound. A hill-station can certainly have multiple lakes without violating the premise. Thus, (ii) cannot be inferred.

Therefore, neither (i) nor (ii) is logically valid.