logicial equivalence in nested quantifiers

Asked Sep 19 '20 at 18:46

Active Sep 19 '20 at 18:50

Viewed 83 times

I know the question might be repeated but I am not able to understand this particular question regarding nested quantifier. So I need some clarification and guidance on this.

If: $$A(x,y) = \text{x attends y's office hour} \\ \text{where the domain of discourse is all students } \times \text{all teachers}$$

For this sentence: "All teachers have at least one student attend their office hours"

Is $$\forall y \exists x A(x,y) \equiv \exists x \forall y A(x,y)$$

The left part is the right answer from the book, but I don't know why the one on the right doesn't work. And what is the difference?

Thanks!

edited Sep 19 '20 at 18:50

asked Sep 19 '20 at 18:46

Yan Zhuang

Presumably the domain is actually Students $\times$ Teachers, since $x$ is always a Student and $y$ is always a Teacher. Accordingly, your statement (the right hand) would translate as "there is some student who attends the office hours of every teacher". – lulu Sep 19 '20 at 18:49
So when we have a nested quantifier we cannot change the order of the quantifier? Or it depends? – Yan Zhuang Sep 19 '20 at 18:53
1

Indeed, you can not freely change the order of quantifiers. – lulu Sep 19 '20 at 18:56
Thanks a lot !! – Yan Zhuang Sep 19 '20 at 18:58
1

See this nice 1-page document by S. Marc Cohen. Also, Intuitive Reason that Quantifier Order Matters. – Dave L. Renfro Sep 19 '20 at 19:01
These are really good notes. Thanks a lot! – Yan Zhuang Sep 19 '20 at 19:05

logicial equivalence in nested quantifiers

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