I've been trying to translate the following sentences into quantified propositions by making sure I state all propositional functions that I use and any assumptions that I make.
There is exactly one person who hates everyone.
Let $H(x, y)$ be '$x$ hates $y$,'
where the domain of $x$ is all people in the world.
Then, $\exists x \forall y\ (\ H(x,y)\ \land \forall z\ (z \neq x) \rightarrow \neg H(z,y)\ ) $.
Can you see if I'm on the right track here?
Best Answer
Your statement, as it reads now, means:
That is not what you want to state. For example, if
then the original statement (there exists a person who hates everybody) is true, but your statement is false (because it is not true that nobody else hates $C$)