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.
For 1: You can't just say $x$ are philosophers and $y$ are logicians: every variable you use will be quantified over the whole domain ... which will have to include both philosophers and logicians.
So, what you need to do is to use predicates for the philosophers and logicians, e.g. $P(x):$ '$x$ is a philosopher' and $L(x):$ '$x$ is a logician'
OK, give that one another try.
For 2., there is a conditional missing. Right now, it ends up saying: 'there is someone x for which it is true that everyone y respects themselves and that first person x will love y' ... which implies, among other things, that everyone definitely respects themselves! That is a far stronger statement than the original English sentence. Paraphrase it as: 'there is some person x such that for every person y: if y respects themselves, then x will love y'
OK, give that one another try as well.
For 3, you'll first need to fix your parentheses before I'll comment ..