Predicate in Java11 filters all elements

What did I do wrong?

You seem to be missing the basic De-morgan's laws which states that

!(a || b) == !a && !b

and

!(a && b) == !a || !b

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How do I correct it?

So you should change your code to use

Predicate.not(a -> a.equals("dog") || a.equals("horse")); // !(a || b)

which shall be equivalent to your existing code

Predicate<String> cats = a -> !a.equals("dog") && !a.equals("horse");

that can also be looked upon as:

Predicate<String> notDog = a -> !a.equals("dog");
Predicate<String> notHorse =  a -> !a.equals("horse");
Predicate<String> cats = notDog.and(notHorse); // !a && !b