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In the Liar Paradox, someone says "I am a liar.", which we assume means "Everything I say is false." (although even that's not correct, a liar is defined as someone who says lies, not someone who only says lies).

According to the paradox, this is a contradiction because if everything he says is false, then his sentence would be false, meaning that everything he says is true, meaning that everything he says is false, and so on...

The problem with the paradox is that the negation of "Everything I say is false." is not "Everything I say is true.", but rather "Not everything I say is false.", which is equivalent to "I say some true things." (¬∀x: ¬true(x) ⇔ ∃x: true(x)), which does not necessarily mean that that particular sentence he said is true. If that sentence is false then there is no contradiction.

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You are completely correct with your analysis of the statement "everything I say is false": it must be false, but this is not a paradox in the strict logical sense.

(Some people use the word "paradox", or even more frequently the adjective "paradoxical", more loosely, meaning anything that is true, though apparently false; or false, though apparently true. You could argue that the falsity of the above statement is not immediately obvious, so there may be a paradox in this sense.)

However, the term "Liar Paradox" is more usually applied to the statement "this statement is false", and this is a genuine paradox.

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  • $\begingroup$ PS: here is an interesting list of paradoxes. Many of those in the "Mathematics" section are paradoxes in the "surprise" sense, not genuine logical paradoxes. $\endgroup$ – David Dec 5 '14 at 3:38
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The liar paradox is this statement is false (or something equivalent), not I am a liar. An alternative would be I always lie.

There are various logic puzzles, e.g. in the books by Raymond Smullyan, where the premise of the puzzle involves a society of people who either always tell the truth or always lie (or other dichotomies), but these aren't about this paradox, merely puzzles based similar ideas.

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