English Sentence To First Order Logic

Question

I have to convert the following sentence to First Order Logic :

Ken doesn't like what Goerge likes, and likes what Goerge doesn't like.

I tried the following convertion , but I am not sure it is right :

∀x(¬Likes(x,George) ^ Likes(x,Ken)) ^ (Likes(x,George) ^ ¬Likes(x,Ken))

Is the above convertion right , and even if it is , are there any better ways to express it ?

Answer 1

The first half of your answer,

$\forall x(\neg\text{Likes}(x,\text{George}\wedge\text{Likes}(x,\text{Ken}))$,

says that it holds of anything that George doesn't like it and Ken does, while the second half,

$\text{Likes}(x,\text{George}\wedge\neg\text{Likes}(x,\text{Ken})$

is not well-formed, as the $x$ is not in the scope of a quantifier.

Another way to put the English sentence is "given any thing, Ken likes it if and only if George doesn't like it". So it can be formalized like this: $\forall x(\text{Likes}(x,\text{Ken})\leftrightarrow\neg\text{Likes}(x,\text{George}))$.

Answer 2

Change the second $\land$ in your sentence into $\lor$, and it's correct!

... Though personally I would also change the order of the arguments in the $Likes$ predicate, e.g. instead of $Likes(x,George)$ I would use $Likes(George,x)$ ... While you can of course interpret these predicates any way you want, typically in a 2-place predicate the first argument is the subject, and the second the object.