Is the following a logical truth?
$(\exists x)(\forall y) S(x,y)\Rightarrow (\forall x)(\exists y)S(y,x)$
Intuitively, the sentence seems for me to say: If there is some thing that causes every thing, then for every thing there is something that is the cause of it.
But I'm still not sure if this is a logical truth or not.
Thank!
Yes it is!
Your reasoning was very close to a proof already:
Suppose there is something x that stands in some relation R to y. OK, that x is a very special something then: let's call it 'Bob'! Then for every y, is there something x that stands in the relation R to y? Yes, of course, for any y, we can pick Bob! So yes, for every y, there is something x that stands in the relation R to y.
In general then:
$(\exists x)( \forall y) \phi(x,y) \Rightarrow (\forall y)(\exists x) \phi(x,y)$
And of course you can swap variables, so:
$(\forall y)(\exists x) \phi(x,y) \Leftrightarrow (\forall x)(\exists y) \phi(y,x)$
And thus:
$(\exists x)( \forall y) \phi(x,y) \Rightarrow (\forall x)(\exists y) \phi(y,x)$
Here is a formal proof:
All check marks, so it must be right! :)