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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?

user932571
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  • More direct way: $\exists d: P(d)$ and it is not the case that there exists $d_1, d_2$ such that $[P(d_1), P(d_2), d_1 \neq d_2].$ – user2661923 May 27 '21 at 18:21
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    hi user. You should use our standard typesetting system, MathJax. I paste the link for you here: https://math.meta.stackexchange.com/a/10164/ – 311411 May 27 '21 at 18:29
  • Your final line says something like "There is exactly one object with property $P$, and it is a dog. And there exists exactly one object with property $P$ and it is a cat, and the dog chaces the cat." Note that by the stated uniqueness of the object with property $P$, it follows that your cat is identical to your dog and it chases itself! – Hagen von Eitzen May 27 '21 at 18:56
  • Also, what do you want to express: "There exist exactly one dog and one cat, and this dog chases this cat"? "There exists exactly one dog that chases exactly one cat" (i.e., other dogs may chase more cats or no cats)? "There exists exactly one dog that chases a single target, and that target is a cat" (i.e., now we also exclude the possibility of a dog chasing a zebra or his own tail)? "There exists exactly one chaser-chasee pair, and the chaser is a dog and the chasee is a cat"? – Hagen von Eitzen May 27 '21 at 19:01

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