Help with predicate logic translation

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  1. Translate into predicate form. No one hates Bob. [H = “is hate”] ∼∃x∀y H_b

  2. Translate into predicate form. Dogs are not reptiles. [D = “is a dog”, R = “is a reptile”] ∀x(Dx⟶ ∼Rx)

  3. Translate into predicate form. Nothing is blue. [B = “is blue”] ∼∃xBx or ∀x∼Bx

  4. Translate into predicate form. [C = “is cat”, M = “Mars”, A = “mammals”] No cat lives on Mars. Unless all cats are mammals, Bob, the cat, lives on Mars. Therefore, only mammals are cats. ∀x(Cx⟶∼M), ∀x(∼Cx⟶Ax) ⟶(C B ⟶Mx) ⊢ Mx⟶∃xCx

What I need help is are these the correct predicate translations for these sentences so I need some feedback if they are translated correctly.

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  1. Translate into predicate form. No one hates Bob. [H = “is hate”] ∼∃x∀y H_b

You quantify an $x$ and a $y$ ... but don't use them in the actual formulae that is being quantified. So, the quantifiers end up doing nothing.

Also, what are the $x$ and $y$ supposed to be? The $x$ could be the person (not) doing the hating of Bob .. but who or what is $y$? .... The $y$ is unnecessary.

So ... what we need to say is 'there is not a $x$ such that $x$ hates Bob'

This translates as $\neg \exists x \ Hxb$

  1. Translate into predicate form. Dogs are not reptiles. [D = “is a dog”, R = “is a reptile”] ∀x(Dx⟶ ∼Rx)

Good. You correctly surmised that this English statement is most likely meant to be about all dogs.

  1. Translate into predicate form. Nothing is blue. [B = “is blue”] ∼∃xBx or ∀x∼Bx

Also good! Note that likewise for number 4. you could have done $\forall x \ \neg Hxb$

  1. Translate into predicate form. [C = “is cat”, M = “Mars”, A = “mammals”] No cat lives on Mars. Unless all cats are mammals, Bob, the cat, lives on Mars. Therefore, only mammals are cats. ∀x(Cx⟶∼M), ∀x(∼Cx⟶Ax) ⟶(C B ⟶Mx) ⊢ Mx⟶∃xCx

First premise is: $\forall x (Cx \to \neg Mx)$ (you forgot the $x$ in $Mx$). But we can also ranslate this as $\neg \exists x (Cx \land Mx)$ ('there is no cat that lives on Mars')

Second premise: "Unless [P], [Q]$ translates to: 'If not [P}, then [Q]

Now, your [P] here is: all cats are mammals, which is $\forall x (Cx \to Ax)$

While the [Q] is: 'Bob is a cat and lives on Mars', which is $Cb \land Mb$

Putting these together, we thus get as the translation for premise 2:

$\neg \forall x (Cx \to Ax) \to (Cb \land Mb)$

Conclusion: 'only mammals are cats' This can be understood as: 'For anything: if it is a cat, then is is a mammal'. And so this translates as: $\forall x (Cx \to Ax)$.