Tl;tr: Can we logically refute a charlatan? No, we can’t.
A typical charlatan says with conviction that the Earth is flat, or that water can cure illnesses, showing an unusual ability to reason and push his convictions. This latter fact is very curious, how can someone argue that water has an inside mechanism that cure things? how do we explain ships that, far on the horizon, looks like sailing over a curved crest of water? Well, a charlatan laugh at these doubts, dismissing them with one hand and smoking a cigarette with the other hand, blowing the smoke on our face.
Now I was wondering, what logic charlatans use for reasoning? They must follow some rules that allow them to appear so self-confident. Actually, logic is their only instrument, because they do not have experiments to show (which would immediately refute them).
Let’s start with Aristotle syllogisms. For example
“All dragons are animals” (premise)
This is a given fact that we can use to conclude
“Some animal is a dragon” (tesi)
The astute reader may be startled, “Impossible!”.But actually this is not a deduction, rather in the Prior Analytics (part 2) Aristotle himself explains us that premises necessarily have this semantic property; if the contrary thesis “No animal is a dragon” were true, we would falsify the premise. The problem here is when someone states “homeopathy cures illnesses”, implying that “this homepathic cure will cure your illness”. It’s rather nasty that from a false premise we get the existence of something, albeit quite false too.
With the development of First Order Logic there had been some step forward and the language of syllogisms is translated conveniently as:
“If something is a dragon, then it is an animal” (premise)
and now we aren’t forced to claim the existence of a dragon, because the opposite thesis “No animal is a dragon” doesn’t contradict the premise, but it implies, naturally, that there aren’t any dragons! The premise is still valid, and we can repeat it like a mantra, logically sound, and if one day, walking down the street, we’ll spot a dragon, we’ll know it’s an animal. Wait and see.
Instead of dragons, that I greatly respect, charlatans use “Not Even Wrong” terms. He’ll never precisely define “homepathic water”, and he will use this at its own advantage when someone will try to refute him. “I took your homepathic water and I didn’t get any better” “OK, but what was the shape of your glass?”, and so on. A charlatan can play with generic terms in an indirect way too, like “Homepathic water is full of positive energy” and “positive energy cure illnesses”. How dare you to attack positive energy, you piece of contrarian waste! Something really similar happens with Flat Earth, whose real shape is unclear, because “the flat earth community doesn’t have consensus on a specific map”, but surely “it has consensus on Earth not a Globe”. Don’t ask me where I did take these sentences, but they’re real.
So we reach the bitter conclusion that if someone talks about curing water or flat earths, he will develop his logic as he wishes and he will counter any observation, because of nothing we can talk for hours and hours.