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Disclaimer: I am not affiliated with the following two users in any way:

User33948, Arbitrary Kangaroo


What I would like to know is:

  • Does the following paradox have a name?
  • What is the fallacy of the argument?
  • If possible, where did it originate?

Two logicians, Alice and Bob, are having a conversation.

A: If I am not mistaken, magicians exist.
B: If you are not mistaken, magicians exist.
A: So I was right.
B: Yes.
A: So I am not mistaken.
B: Correct.
A: So magicians exist!


I heard this from a friend in a version pertaining to Santa Claus.

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    $\begingroup$ Why the disclaimer? $\endgroup$ – Deusovi Feb 5 '17 at 20:52
  • $\begingroup$ Also who is user33948? And Arbitrary Kangaroo currently has 1 rep and apparently hasn't done anything here (except possibly chat) for almost 2 months..? $\endgroup$ – tilper Feb 5 '17 at 21:02
  • $\begingroup$ Anyway, all that aside, I think I know what this is. $\endgroup$ – tilper Feb 5 '17 at 21:04
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    $\begingroup$ I take it the disclaimer is because a question with a similar title was posted here recently by one of the two users named there, who might be suspected of being a sockpuppet of the other. $\endgroup$ – Gareth McCaughan Feb 5 '17 at 23:31
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This sounds like

Curry's paradox.

Short explanation: The "paradox" comes from the difference in the truth value of a statement of the form $\text{If $P$ then $Q$}$ as opposed to the truth value of just $P$ itself. A paradox arises if $P$ is phrased in a way that makes its truth value equivalent to that of $\text{If $P$ then $Q$}$ in natural language, because those statements do not have the same truth value in first-order logic.

Longer explanation:

A: If I am not mistaken, magicians exist.

This is a statement of the form $\text{If $P$ then $Q$}$, where $P$ represents "I am not mistaken" and $Q$ represents "magicians exist." The following truth table lays out all the possible cases for the truth value of an $\text{If $P$ then $Q$}$ statement (in general, not just in this case). In the table, $T$ means true and $F$ means false. \begin{array}{c|c|c} P & Q & \text{If $P$ then $Q$}\\ \hline T & T & T\\ T & F & F\\ F & T & T\\ F & F & T \end{array} The only surprising one here may be the last line (or possibly the last two lines). If $P$ is a false statement, then $\text{If $P$ then $Q$}$ is true. This is sometimes called vacuous truth.

Here's where things get paradoxical. "I am not mistaken" can be equivalently phrased as "I am correct" or, perhaps more clearly for this explanation, "This sentence is true." So we can rephrase Alice's statement as:

A: If this sentence is true, magicians exist.

But in this specific case, $P$ being true means the entire $\text{If $P$ then $Q$}$ is true (because $P$ is the statement "This sentence is true"). But we've already seen from the table above that this is not the case. In fact, $P$ and $\text{If $P$ then $Q$}$ only have the same truth value in that first row. Hence the paradox.

As for the fallacy...

A: So I was right.
B: Yes.
A: So I am not mistaken.
B: Correct.

"So I was right" is interpreted as being right about the entire $\text{If $P$ then $Q$}$ statement. And "So I am not mistaken" is interpreted by Bob as the exact same thing, so he says "correct." But then in the next line:

A: So magicians exist!

Alice interprets Bob's affirmation ("Correct") as saying $P$ is true, not the entire $\text{If $P$ then $Q$}$. And then we know from our table above that if $P$ is true, then $Q$ must also be true, i.e., magicians exist.

Re: the history, I'm sure the link in my spoiler tag above has some info but I haven't checked it thoroughly for that.

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  • $\begingroup$ Would the downvoter care to explain? $\endgroup$ – tilper Feb 6 '17 at 1:37
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(I am not sure if this question is on-topic, but I will try to answer it anyway. It may be better suited to Philosophy SE.)

I believe the fallacy is based on an equivocation of "mistaken". It is thus a fallacy of equivocation. In fact, it is a fallacy of equivocation based on a certain kind of ellipsis.

The first thing to notice is that one can be mistaken with respect to some of one's beliefs (or statements) and correct with respect to others.

When we say that someone is mistaken, we usually make implicit or elided reference to the belief that they're mistaken about. This implicit reference can usually be make explicit.

For example, if A declares "Pigs fly," B can offer either of the following responses:

B: You are mistaken.
B: You are mistaken with respect to your belief that pigs fly.

If we make explicit which beliefs are being talked about in the puzzling conversation between logicians, we can paraphrase the argument this way:

A: If I am not mistaken [with respect to my belief that magicians exist], then magicians exist.
B: I agree. If you are not mistaken [with respect to your belief that magicians exist], then magicians exist.
A: So I was right.
B: Yes.
A: So I am not mistaken [with respect to my belief that magicians exist].
B: Correct.
A: So magicians exist!

But B is wrong to say "correct". In doing so, she allows equivocation on "mistaken".

What B should say is:

B: That doesn't follow. You were correct about your conditional belief that [if you are not mistaken with respect to your belief that magician's exist, then magician's exist]. But you may still be mistaken with respect to your belief that magician's exist.

I don't think this paradox has a name and I am not sure when it originated.

That said, it is similar to the types of problems and paradoxes explored by philosophers like Frege, Meinong, and Russell at the beginning of the 20th century. These puzzles involved various non-denoting expressions like "Santa Clause", "the present King of France", and so on.

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The fallacy (the only "puzzling" element here) is

that A's statement is an accurate logical statement - the first part, "If I am not mistaken", correctly implies the second, "magicians exist". B agrees that the logical statement is accurate. A's conclusion that the accuracy of the logical statement has any bearing on whether or not the first part is true is the fallacy; an if/then statement can be accurate:

  If humans did not need to breathe, they could stay underwater for hours without aid

without meaning the conditions are actually met for both halves to be true.

The other questions posed here I'm not sure belong on PSE. I don't know the answers in any case.

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