Using definite quantification in logic

Question

I am looking to translate the following sentence into formal logic:

"Haters hate the hated" Where Hxy: x hates y

I have translated it to: ∃x((Hxy ∧ ∀y(Hxy → x = y))

However I am not sure if the last part (x=y) is essential as I have already expressed any x hates y and all y is hated by x.

I would appreciate any help clarifying this, thank you.

(I am beginner in logic so please don't judge my attempt too harshly!)

Answer 1

Try and do these translations of complex statements step by step: divide and concur. And, stick to the division between subject and term (those things about which you say something) and predicate term (what you say about those things), so you can follow the standard forms for translating those.

'Haters hate the hated' is of course 'all haters hate anyone who is hated'

So, subject term: 'haters'

Predicate term: 'hate anyone who is hated'

So this becomes:

$$\forall x (\text{'x is a hater'} \rightarrow \text{'x hates anyone who is hated'})$$

Now, 'x is a hater' translates as:

$$\exists w H(x,w)$$

while 'x hates anyone who is hated' can be paraphrased as 'all those who are hated are hated by x', which has:

Subject term: 'hated'

Predicate term: 'hated by x'

So the form here is:

$$\forall y (\text{'y is hated'} \rightarrow \text{'x hates y'})$$

'y is hated' becomes:

$$\exists z H(z,v)$$

and 'x hates y' is of course:

$$H(x,y)$$

OK, so we can plug in the last two parts:

$$\forall y (\exists z H(z,y) \rightarrow H(x,y))$$

And now we can plug that into the very first form:

$$\forall x (\exists w H(x,w) \rightarrow \forall y (\exists z H(z,y) \rightarrow H(x,y)))$$

Answer 2

I think the sentence should be taken to mean that everyone who is a hater hates everyone who is hated by someone.

Formally this amounts to: $\forall x(Hx \rightarrow \forall y(\exists z Dzy \rightarrow Dxy)) $, where $H, D$ stand for the predicates 'is a hater' and 'hates' respectively.

Answer 3

$H(x,z)$ means "$x$ hates $z$".

$\exists y \ H(x,y)$ means "$x$ is an hater".

$\exists w \ H(w,z)$ means "$z$ is hated".

Thus, a possible symbolization will be:

$\forall x \forall z \ [\exists y H(x,y) \land \exists w H(w,z) \to H(x,z)]$.