# Transcribing from predicate logic to English

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,

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

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.
A relation $$L$$ is symmetrical exactly when $$\forall_x \forall_y (Lxy \to Lyx)$$.
• The word you are after is "symmetrical". A relation, $L$, is symmetrical exactly when $\forall x\forall y~(Lxy\to Lyx)$ . – Graham Kemp Jul 18 '19 at 4:44