Quantified Proposition

Question

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?

Answer 1

Your statement, as it reads now, means:

There exists one such person $x$ that for every other person $y$, $x$ hates $y$ and nobody else hates $y$.

That is not what you want to state. For example, if

• $A$ hates $A$, $B$ and $C$ and
• $B$ hates $C$,

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 $C$)

Answer 2

$\exists!x\forall yH(x,y)$, where $\exists!$ is the unique existential quantifier.

Answer 3

To say "there exists a unique $x$ for which the predicate $P$ holds" : we usually say "there exists some $x$ for which the predicate holds and if the predicate holds for anything it is that $x$".

$$ \exists x \Big(P(x)\wedge \forall z\big(P(z)\to x=z\big)\Big) $$

We can use the contrapositive.

$$ \exists x \Big(P(x)\wedge \forall z\big(x\neq z \to \neg P(z)\big)\Big) $$

Or more breifly: $$ \exists x\;\forall z \Big(x=z \leftrightarrow P(z) \Big)$$

And you want to do this for the predicate "$x$ hates every one". $$P(x) \;\equiv \;\forall y\; H(x,y)\\ \neg P(x) \;\equiv\; \exists y\;\neg H(x,y)$$

So: ...