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Which of the following first order logic statement is equivalent to below statement?

If anyone cheats, everyone suffers.

• $S_1 \forall x (\text{cheat}(x) \to \forall y \text{ suffer}(y))$
• $S_2: \forall x\forall y (\text{cheat}(x) \to \text{ suffer}(y))$
1. Only $S1$
2. Only $S2$
3. Both $S1$ and $S2$
4. None
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is option C ?

C. Both S1 and S2 are true

$\text{Any}$ can mean "for all" or "there exists" depending on the sentence.

$\exists x \ \text{cheat}(x) \rightarrow \forall y \ \text{suffers} (y)$, this is also True.

but $\forall x \ (\text{cheat}(x) \rightarrow \ \text{suffers} (x))$ is incorrect

because that means "If anyone cheats he suffers".

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Hey can you tell me how "everyone suffers " is done here?

like there exists someone who suffers is what i'm thinking.
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Got it. thanks. :)
+1

Both are correct.

There can be many representations for this statement.

https://math.stackexchange.com/questions/3033559/english-statement-to-logic/3033618#3033618