I'm not sure what "arbitrary" means here. You don't seem to be using it in the sense that all preferences are arbitary.

a story in which a logician tries to argue me down a slippery slope of moral nihilism

If the nihilist makes a sufficiently circuitous argument, they can ensure that there's no step you can point to that's very wrong. But by doing so, they will make slight approximations in many places. Each such step loses an incremental amount of logical justification, and if you add up all the approximations, you'll find that they've approximated away any correlation with the premises. You don't need to avoid following the argument too far, if you appropriately increase your error bars at each step.

In short: "similar" is not a transitive relation.