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How do I translate the following English sentences without Uniqueness Quantifier:

  1. There is exactly one person who hates everyone

  2. All people hates exactly one person.

HoneyBee
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1 Answers1

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You can just unpack what $\exists ! x \varphi(x)$ means. There are many ways of doing it, but this is my favourite: $$\underbrace{[\exists x \varphi(x)]}_{\text{existence}} \wedge \underbrace{[\forall y \forall z(\varphi(y) \wedge \varphi(z) \to y=z)]}_{\text{uniqueness}}$$ So you can write out your two statements in terms of the $\exists!$ quantifier, and use this to write it all in terms of $\exists$ and $\forall$.

Clive Newstead
  • 64,925
  • ∃x∀y H(x,y) ∧∀z (z≠x)→¬H(z,y) can the first sentence be written like this? – HoneyBee Jun 01 '16 at 19:06
  • Not quite - be careful with those quantifiers. That means there is someone who hates everyone, and that everyone else hates no-one. Try my suggestion, using $\varphi(x) \equiv \forall y H(x,y)$ instead. It looks like your interpretation of $\exists ! x \varphi(x)$ is as $$\exists x[\varphi(x) \wedge \forall z (z \ne x \to \neg \varphi(z))]$$ which you can use instead of the form suggested in my answer. – Clive Newstead Jun 01 '16 at 19:22
  • [∃x∀yH(x,y) ]∧∀z[(z≠x→¬∀yH(z,y))]

    Basically this is what you are writing. parson..but the second part is confusing for me. :|

     – HoneyBee Jun 01 '16 at 19:30
  • The second part of the problem can be written as $$\forall x \exists ! y H(x,y)$$ so all you have to do is use exactly the same method to re-write $\exists! y H(x,y)$. (The variable now being quantified by $\exists!$ is $y$, not $x$, so you'll need to do some variable renaming...) – Clive Newstead Jun 01 '16 at 19:32
  • [∀x∃yH(x,y) ]∧∀z[(z≠y→¬∀xH(x,z))]

    So, this should be the second one's answer I hope. All people hate exactly one person

     – HoneyBee Jun 01 '16 at 19:37
  • @AnamikaAhmed: The $y$ in $z \ne y$ there is not quantified because you closed the parenthesis after $\exists y$. Ignore the $\forall x$ for now and just write out "$x$ hates exactly one person", that is $\exists! y H(x,y)$. Then "everyone hates exactly one person" is obtained by universally quantifying $x$ in the whole statement. – Clive Newstead Jun 01 '16 at 19:43