In predicate Logic, I'm trying to work out when both the subject and object are exactly one how is this translated?
I've seen that to work out how to say there exists only one dog such that P(d) is
∃d: (P(d) & ∀x: P(x) ⇒ d=x
but where do we add the rest of the predicates, like that it is a dog or that it chased exactly one cat? And what if I want to say there is more than one dog or cat? do we change the d=x to d>x or something?
So so far I've got this to say one dog chased one cat, is this a mess?
∃d ∃c [(DOG(d)) & (P(d) & ∀x: P(x) ⇒ d=x) & (CAT(c)) & (P(c) & ∀y: P(y) ⇒ c=y) & (CHASE(d,c))]
And if I wanted to say one dog chased multiple cats, how would I do that?