Translating English to predicate logic with units

Question

I am having difficulty translating this fragment from a larger sentence into predicate logic: three pets bathe together.

Let pets be P(x) and B(x,y) be bathe togehter.

How would I deal with the number 3 in this scenario?

Answer 1

Let us say "Two pets bathe together."

$\qquad\exists x~\exists y~\big(x\,{\neq}\,y\land P(x)\land P(y)\land B(x,y)\big)$

"There exists two distinct animals which are both pets, and they bathe together."

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Now to say it for three is just a little more busy.

$\qquad\exists x~\exists y~\exists z~\big(x\,{\neq}\,y\land x\,{\neq}\, z\land y\,{\neq}\, z\land P(x)\land P(y)\land P(z)\land B(x,y)\land B(x,z)\land B(y,z)\big)$

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Remark: If the "Bathe together" predicate is irreflexive, then it will make the inequality redundant. (Ie: When we interpret it to be that no animal may "Bathe Together" with itself, then that will ensure distinctness.)