Masked man fallacy

The masked man

“I know who my father is. I have no idea who the masked man at the party is. So the masked man isn’t my father.”

If the masked man is in fact your father, both premises are still true. Your not recognizing him is a fact about you and the description “the masked man”, not a fact about him that your father lacks. Taking off the mask changes what you know; it doesn’t change who he is.

Two names for one café

“Jordan says the little café next to the library has the best coffee in town. Jordan has never heard of Bean There. So Bean There can’t be the café next to the library.”

If Bean There is the café’s name, the conclusion is false while both premises hold: Jordan knows the café by its location and not by its name. This version runs on belief rather than knowledge, and on a place rather than a person. The error is the same: someone’s knowing it under one description and not another says nothing about whether there are one or two things.

A certain thing and a doubtful thing

“I can be completely certain that I’m in pain right now. I can’t be completely certain that any nerve cells in my body are firing. So being in pain isn’t the same thing as nerve cells firing.”

This one sounds more like philosophy than a mix-up, but “I’m certain of it” and “I can doubt it” are facts about the person thinking, and about the two ways the thing is described. They can differ even if the two descriptions pick out one and the same event. That doesn’t show pain is nerve cells firing either; the argument just can’t settle the question. Philosophers disagree about whether more careful arguments against identifying the mind with the body avoid this problem.

The two columns look alike: a difference, then “so they’re different things”. The valid form uses a property that belongs to the things themselves. If P and Q were one person, that person couldn’t be both heights, so the difference really does show there are two.

In the fallacy, the “difference” sits inside a clause about what someone knows, believes or doubts. Those clauses don’t behave like ordinary properties. The Stanford Encyclopedia’s entry on identity separates Leibniz’s law from a different principle: that two names for the same thing can always be swapped without changing whether a sentence is true. That second principle is false. Inside “Jordan has never heard of ___”, swapping “the café next to the library” for “Bean There” turns a false sentence into a true one, even though both refer to the same café. Such contexts are called referentially opaque (or intensional). As that entry puts it, failures of swapping like these are not counterexamples to Leibniz’s law, which says nothing about substituting names.

It is treated as a formal fallacy because the flaw shows in the shape alone: an argument that infers “not the same” from a difference in what someone knows, believes or doubts about them can have true premises and a false conclusion, whoever the people are and whatever is believed. (One family of philosophical views locates the problem in a premise instead; see “Why it happens”.)

• Knowledge (“who is it?”): the classic masked man. “I know who P is; I don’t know who Q is.”
• Belief: “Lois believes Superman is strong; she doesn’t believe Clark Kent is strong; so Clark Kent isn’t Superman.” The Stanford Encyclopedia’s entry on attitude reports uses this pair of beliefs to set out the underlying puzzle.
• Doubt or certainty: “I can’t doubt P; I can doubt Q; so P isn’t Q”, as in the pain example.

• When the property belongs to the thing itself: its height, location, age, what it did or where it was at a given time. A real difference there does show two things.
• When the knowledge or belief is the thing’s own. “My father knows my middle name” is a fact about your father, the person doing the knowing. If the masked man doesn’t know it, that is a genuine difference between them.
• As evidence rather than proof. If you’d normally recognize someone you know and you get a clear look at a person, not recognizing them is some evidence they’re a stranger. That’s fine as long as the conclusion stays “probably” and rests on how reliable your recognition is, not on treating your ignorance as a property of the person.

The test: is the difference in the thing itself, or only in what someone knows or believes about it?

A difference in height

“The masked man is well over six feet tall. My father is five foot six. So the masked man isn’t my father.”

Height is a property of the man, not of anyone’s knowledge about him. One person can’t be two heights at once, so this is Leibniz’s law used correctly and the argument is valid. Whether the premises are true (platform shoes, say) is a separate question; see Valid vs. true.

What the masked man himself knows

“My father would recognize my voice at once. I spoke to the masked man for five minutes and he clearly had no idea who I was. So he’s probably not my father.”

This has “recognize” and “no idea” in it, which makes it look like the fallacy. But here the knowing is done by the masked man and by your father: recognizing your voice is a fact about each of them, so it’s a real difference in the thing itself. The “probably” allows for the premises being wrong, for example if he was pretending not to know you.

Leibniz’s law is correct, and we rely on it constantly. The Stanford Encyclopedia’s entry on identity notes that we show our commitment to it whenever we infer from something being true of one thing and false of another that they are not the same. In grammar, “is known by me” looks just like “is six feet tall”, so the rule gets applied to both.

The difficulty is an old one: the Oxford Companion to Philosophy calls the masked man fallacy ancient. A closely related puzzle is named after Gottlob Frege: swapping two names for the same thing inside a belief report can seem to change whether the report is true. Philosophers still disagree about why. On Frege’s view, names inside those clauses stand not for the thing but for the way it is presented (what he called their sense). Some later philosophers argue instead that “Lois believes Superman is strong” and “Lois believes Clark Kent is strong” are true or false together, however strange that sounds. On that view, one of the premises in an argument like the masked man’s only seems true. Either way, the argument doesn’t show that the masked man isn’t your father.