I have to some translations. Here is what I need to translate and what I have so far.

  1. Every philosopher respects some self-respecting logician.
    Let "x" denote philosophers and "y" denote logicians. And let "R" be the respecting relation.

$\forall x \exists y (Rxy \rightarrow Ryy)$

  1. 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...

$\exists x ((\forall y)Lxy \land (\forall yLxy \rightarrow Ryy))$

  1. 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 ..

  • $\begingroup$ I've changed 1 to something like this: $\forall x (\exists y Ly \land Ryy \rightarrow Rxy)$. My thought that the sentence was tantamount to saying that for all x if x respects some y that is a logician that also respects itself then x respects that y. $\endgroup$ – Rusty Nov 3 '17 at 17:50
  • $\begingroup$ @Rusty Make sure to say that x is a philosopher $\endgroup$ – Bram28 Nov 3 '17 at 17:51
  • $\begingroup$ Oh god, so my conditional's antecedent now looks like a conjunction: $\forall x (Px \land \exists y(Ly \land Ryy) \rightarrow Rxy)$ $\endgroup$ – Rusty Nov 3 '17 at 17:52
  • $\begingroup$ And as per your suggestion for 2, I'm getting $\exists x (\forall yLxy \land (Ryy \rightarrow Lxy)$ which says, I think, "there is some x s.t. x is in love with every y and if y respects itself then y then x loves y. $\endgroup$ – Rusty Nov 3 '17 at 18:11
  • $\begingroup$ OK, for 1: The claim is about philosophers, so it will be of the form $\forall x (P(x) \rightarrow ...)$. What you say about those philosophers goes in the consequent, so that is the part where you say that they respect some self-respecting logician .. try again. $\endgroup$ – Bram28 Nov 3 '17 at 18:26

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