# Translating english sentences to predicate logic

Trying to practice translating english sentences into predicate logic and vice versa.

• $$E(x,y)$$: $$x$$ can eat $$y$$
• $$L(x,y)$$: $$x$$ loves eating $$y$$
• $$D$$ is the domain of all dogs
• $$S$$ is the domain of all snakes

(a) English to Predicate Logic:

A dog can eat any snake, only if they are different from some other dog who can also eat any snake:
$$\forall a \in S,\forall b \in D,\exists c \in D, E(b,a)\implies b \ \ne c \ \wedge E(c,a)$$

(b) Predicate Logic to English:

$$\forall a \in S,\forall b \in D, \forall c \in S ,\sim \ \bigg[\ a\ \ne c \ \wedge E(a,c)\bigg] \iff L(a,b)$$:

Not all snakes is the same as some other snake, or that not all snakes can eat some other snake, if and only if, all snakes loves eating all dogs. This part sounds a bit weird to me and can possibly be condensed.

Any thoughts on both (a) and (b)?