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Self referential statements can be paradoxical as known since ancient times. (Some of this basic paradoxes had immense impact on my own field -mathematics.) Of course, paradoxical self-referential statement plays a role also in humor. The following conversation artually took part in a department meeting two decades ago.

L: Some people in the department think that they are wiser than what they really are

J: I am really wiser than what I think I am.

J's statement seems paradoxical, and it looks different than the common ones known to me. Is it?

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    I could be wrong here but I don't see how J's statement is inherently paradoxical because I don't equate wisdom with mere thinking. Just as the statement "I am really stronger than I think I am" isn't a paradox (it could very well be true), so could J's statement be; especially if, for example, J has low self-esteem and doesn't give him or herself as much credit as s/he truly deserves. Maybe I'm missing what you're trying to get at? – stoicfury Oct 30 '11 at 17:57
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    I think that "I am really stronger than I think I am" or "I am really taller than I think I am" or "I am not as tall as I think I am" are similarlly paradoxical. Of course, once its not a self reference the paradox goes away. – Gil Kalai Oct 30 '11 at 18:04
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    Wouldn't the paradox be "I am really stronger than I am". To add "than I think I am" removes the paradoxical nature for me... but again maybe it's just completely going over my head here. XD – stoicfury Oct 30 '11 at 18:17
  • Yeah, I'm leaning towards @stoicfury 's interpretation. The I think totally changes the picture. Either that or I think I'm missing something here. – john Oct 30 '11 at 19:47
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    Is it because the "I am" implies a "I think I am"? Like, "I think I am really stronger than I think I am"? The language here is awkwardly phrased. Perhaps it is best said not in English? Or if in English, in a different way? – john Oct 30 '11 at 19:54
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I don't see how it is structurally different from any other self-referential paradox (such as the Liar), nor do I see how it offers any additional material for philosophical reflection.

The statement reduces to "I think that I am smarter than I think I am", which appears to me to be structurally the same as "I speak the truth: I am lying."

  • Maybe a better reduction is to a statement: "I think X but the truth is Y". (Like "I think that it will rain tommorow but the truth is that it will not.") So the problem is not so much that the statement is about the speaker. – Gil Kalai Oct 30 '11 at 20:21
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    The problem is still about the speaker, in that all third-person propositional statements ("S is P") are implicitly framed with a first-person statement ("I know that S is P, and am stating it to you truthfully.") So, "I think X but the truth is Y" reduces to "I think X, but I think that the truth is Y". – Michael Dorfman Oct 30 '11 at 20:29
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    @Gil Kalai: To think something is to hold a belief about something, which means you hold it to be true. The statement "I think that it will rain tommorow but the truth is that it will not" is nonsensical—you wouldn't be able to think something is true and yet not believe it simultaneously. It's like saying "I think it will rain tomorrow but I honestly believe it will not." It's a logical contradiction. – stoicfury Oct 30 '11 at 21:55
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    To @Michael: What about an anorexic who says, "I am thinner than I think I am". That would not be a paradox, would it? It is a statement about self perception (or perhaps self esteem) compared to the way that other people perceive the person. The anorexic sees fat where other people do not, so the part that says "I am thinner" is stating what others have told the anorexic. In the same way, someone could doubt their wisdom even though others think they are very wise. Thus they say they think they are wise because other people had told them they were wise. I see no paradox. – FrankH Nov 5 '11 at 0:06
  • @FrankH: If one were to phrase it accordingly ("I actually am thinner than I think I am", for example), you are correct-- the paradox disappears, because we are clearly speaking about two different things (what I think vs what others think). – Michael Dorfman Nov 5 '11 at 8:51
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I can say something without thinking it, just as I can think something without saying it. Therefore, I'd argue that "I think" is not implied in all language statements, but rather, "I say", "I state" or "I assert".

From that perspective, the paradox resolves itself.

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