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Consider the following first order logic statements:

I: $\forall_x\forall_y p(x,y)$

II: $\forall_x\exists_y p(x,y)$

III: $\exists_y\exists_x p(x,y)$

IV: $\exists_y\forall_x p(x,y)$

Which of the following is not true about above statements?

(A) if is true then II, III, IV are true.

(B) if II is true then III, IV are true.

(C) is IV is true then II, III are true.

(D) None of these.

Ans given is (B).

Now i think that statement of (A) is also false (and can be answer too) because if domain of $x$ and $y$ is empty then statement will trivially be true. But in this case III and IV are false. So options (A) will be false.

Same for (C) if domain of $x$ is empty then IV will be true for all $y$ but in this case III will be false. So, options C is also false and hence answer too.

Is my argument correct or not?

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+1

Not sure about the argument. Let’s see what others comment. But just a little help from my side.

$\\ \forall x \forall y\leftrightarrow \forall y \forall x\\ \forall x \forall y \rightarrow \exists y \forall x\\ \exists y \forall x\rightarrow \forall x \exists y\\ \forall x \exists y\rightarrow \exists y \exists x\\ \exists y \exists x \leftrightarrow \exists x \exists y\\ \\ \forall y \forall x\rightarrow \exists x \forall y\\ \exists x \forall y\rightarrow \forall y \exists x\\ \forall y \exists x\rightarrow \exists x \exists y$

either directly observe from this or if you write all this implications and bi-implications in a systematic order you will observe a figure just like a hexagon which will indicate that B is not true

0

I'm thankful to you for your efforts.

And after I found out error in my argument that I assumed that domain of discourse can be empty. But I found on following reference link that in FOL domain of discourse must be non empty. If we take nonempty condition than my argument breaks down. (:

Here the condition of nonempty domain is mentioned.