Professional philosophers understand that many arguments rely on intuition. But they need intuition to create basic premises. Otherwise, if you have no "axioms" in your system of logic, you cannot derive any sentences.

Also, moral philosophy deals with what is right and what is wrong. These are inherently fuzzy notions and they likely require some level of intuitive reasoning. ("It is clearly wrong to kill an innocent person.") I would be extremely surprised if someone could formally define what is right and wrong in a way that captures human intuition.

It's also not worth debating philosophy with people who will argue that $10 is not clearly worth less than a finger. (And if you don't believe that, then we can consider the case with two fingers, or three, or a whole hand, etc.).