Transcribing from predicate logic to English

Question

I am trying to transcribe the following predicate logic sentence into English,

$ ( \forall x)( \forall y)[Dx \rightarrow (Cy \rightarrow Lxy)] $

I transcribed this as 'Every dog loves every cat.' However, the book I am using says the answer is 'All dogs and cats love each other.' I would've thought that the latter answer transcribes to predicate logic as,

$ ( \forall x)( \forall y)[Dx \rightarrow (Cy \rightarrow (Lxy \land Lyx))] $

Could someone please clarify this?

Answer 1

I agree with you in the sense that your book may not have explained the manner in which these relations work sufficiently.

It seems that the book assumes that the relation $L$ is symmetrical. When a relation is symmetrical, the order doesn't matter. Thus, we can imply that both of our options are taken into account. That being said, if $L$ is symmetrical, then $Lxy$ should equivalent to $Lyx$, and it seems like the book is implying this rule by direct application.

I do, however, think your transcription is also correct.

Edit: Maybe commutativity is not the proper term to use here. Instead, let's call it symmetrical, as pointed out by @Graham Kemp.

A relation $L$ is symmetrical exactly when $\forall_x \forall_y (Lxy \to Lyx)$.