Logical Liars, Paradoxical Politicians

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Lies, lies, lies. Prone to stretching logic, alleging deceit, and passing on gossip, politicians and the media that report on them are a natural setting for a few classic puzzles involving lying and self-reference.

Superficially political scenarios are also a bit easier to relate to than are the original ones, so I've dressed up some of these conundrums in this more modern garb.

Proceeding from simple to more difficult examples, I'll start with the very well-known liar paradox.

It can result, for example, if a news anchor were simply to announce, "This very statement I'm making is false." If his statement is true, then it's false, and if it's false, then it's true.

Less obvious and more realistic occurrences involving two or more people can also easily arise.

If Senator S says that Senator T's comment about the health care bill is false, there is nothing paradoxical about her statement. If Senator T says that Senator S's remark about the issue is true, there is nothing paradoxical about this statement either. But if we combine these two statements, we have a paradox.

It's not too hard to imagine a larger collection of such comments from a variety of people, each individually plausible, yet leading to an equally potent paradox.

Who's Telling the Truth?

Another old puzzle, again in a slightly different setting, concerns the reporter who has two very knowledgeable and sources, A and B.

In crucial political situations, A always tells the truth, B always lies, but the reporter has forgotten who is who. The reporter wants to know if Senator S is involved in a certain scandal and for whatever reason can ask only one of his sources, say by email, a single Yes or No question. What should it be?

Answer: One solution (there are others) is to ask either source the following question: Are the two statements -- 1) you are a truth-teller, and 2) the Senator is involved in this scandal -- either both true or both false? The remarkable thing about this question is that both truth-teller and liar will answer Yes if the Senator is involved.

If the source is a truth-teller, the source will answer Yes since both statements are true, and if the source is a liar, the source will answer Yes since only one of the two statements is true. A similar argument shows that both sources will answer No if the Senator is not involved.

Note that a completely useless question in this situation is, "Are you telling the truth about the Senator?" since both liars and truth-tellers would answer Yes.

Complicating the Situation

The answer above gives rise to a general principle. If you want to know if any proposition P is true and your source is a liar or a truth-teller, ask him if the two statements -- you are a truth-teller, and proposition P -- are either both true or both false. You can trust the answer even if you don't know whether it was given by a truth-teller or a liar.

The reporter might confront an intriguing, but more difficult problem, originally formulated in a slightly different scenario by logician Ray Smullyan. Assume the reporter again wants to know if Senator S is implicated in the scandal, but this time he has three knowledgeable informants, A, B, and C.

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