Negating a quantified predicate

Question

Let H(x,y) defined as x hates y. Then translate "No one hates Susan and Harry."

An answer could be $ \neg \exists x(H(x,Susan) \land H(x,Harry).$ But if we use De morgan for Quantifiers we get $ \forall x (\neg H(x,Susan) \lor \neg H(x,Harry).$ However this statement is not true because this evaluates to true for example when someone hates Susan but not hates Harry.

Also $ \forall x (\neg H(x,Susan) \land \neg H(x,Harry)$ seems to be a correct answer. What is the contradiction in these answers. Thanks in advance .

Answer 1

The natural language sentence

No one hates Susan and Harry

is ambiguous. It could mean either

(1) No one hates both Susan and Harry

or

(2) No one hates Susan and no one hates Harry.

These are two different statements which have different translations in predicate logic. I find it a bit hard to see (2) where the quantifier distributes over the two predicates and would claim that (1) is the more natural interpretation of the English sentence, but you seem to find (2) more in line with the intended truth conditions.

(1) (non-distributive reading) translates as

¬∃((,)∧(,))

which is logically equivalent to

∀(¬(,)∨¬(,))

which may be true if someone hates Susan but not Harry or the other way round, as you said. These truth conditions are in line with the meaning of interpretation (1).

(2) (distributive reading) translates as

¬∃((,)) ∧ ¬∃((,))

which is logically equivalent to

∀(¬(,)∧¬(,))

and is only true if noone hates either of Susan or Harry.

TL;DR: The apparent contradiction arises from the fact that the English sentence can mean two different statements, however the first one is the more likely intended interpretation.