"Everyone is loved by someone or other"

I'm not sure how to translate this into predicate logic. At first, I said $\forall x\exists y,Lyx$ where $Lxy$ is x loves y. However, it is not true that there is someone who loves everyone. The sentence says that "everyone" is loved by someone or other (i.e. everyone in the domain is loved). Would it be correct to say $\exists x \exists y Lxy$ , but I don't think that 'x' in this case covers everyone in the domain.

  • $\begingroup$ Your first attempt is correct. $\endgroup$ – Brian M. Scott Oct 31 '16 at 4:52
  • $\begingroup$ Oh. Could you please break down the meaning, since I'm still confused. $\forall x \exists y Lyx$ means that there is someone who loves everyone, but the sentence says that loved by at least one person? $\endgroup$ – JC1 Oct 31 '16 at 4:55
  • $\begingroup$ No, $\forall x\,\exists y\, Lyx$ means that each person ($x$) is loved by someone ($y$). There is someone who loves everyone is $\exists x\,\forall y\,Lyx$, with the quantifiers in the other order. $\endgroup$ – Brian M. Scott Oct 31 '16 at 5:05
  • $\begingroup$ Why do not try with a more intuite example ? "for every men ($x$) there is a man $y$ that is the Father-of $x$" sounds reasonable, while "there is a man ($x$) such that : every men $y$ are Father-of $x$" sounds a little bit strange (all males are fathers of a single indivudual ?) $\endgroup$ – Mauro ALLEGRANZA Oct 31 '16 at 8:23

You seem to be unaware of the fact that the order of the arguments changes the meaning of the statement. For example, if $a$ denotes Alice, and $b$ denotes Bob, then $Lab$ means 'Alice loves Bob', but $Lba$ means 'Bob loves Alice'.

Similarly, $\forall x \exists y Lxy$ means that everyone loves someone, but when we change the arguments, we get $\forall x \exists y Lyx$, which means that everyone is loved by someone, which is a different statement.


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