I have to some translations. Here is what I need to translate and what I have so far.
- Every philosopher respects some self-respecting logician.
Let "x" denote philosophers and "y" denote logicians. And let "R" be the respecting relation.
$\forall x \exists y (Rxy \rightarrow Ryy)$
- There is someone that loves everyone who respects themselves.
Let "R" be the respecting relation and let "L" be the loving relation.
$\exists x \forall y (Lxy \land Ryy)$
I also think that the English sentence is tantamount to saying that there is some x s.t. x loves everyone and if x loves everyone then everyone loves themselves, so I translated that as...
$\exists x ((\forall y)Lxy \land (\forall yLxy \rightarrow Ryy))$
- Everyone who loves everyone else also loves everyone who is loved by someone else.
$\forall x (\exists y Lxy \land \forall y Ly \rightarrow \exists z(Lzx \land y \neq z))$
Does anyone see where I've gone wrong? Are there any tips about how to proceed? Any help would be greatly appreciated.