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?
$\endgroup$ 43 Answers
$\begingroup$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 $C$)
$\endgroup$ 7 $\begingroup$$\exists!x\forall yH(x,y)$, where $\exists!$ is the unique existential quantifier.
$\endgroup$ 3 $\begingroup$To say "there exists a unique $x$ for which the predicate $P$ holds" : we usually say "there exists some $x$ for which the predicate holds and if the predicate holds for anything it is that $x$".
$$ \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: ...
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