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Logic and math · experts disagree
The liar paradox
Eubulides of the Megarian school, fourth century BC; the modern analysis starts with Tarski (1933) and Kripke (1975).
Take the sentence: "This sentence is false." Call it L. Suppose L is true. Then what it says holds, so L is false. Suppose instead that L is false. Then what it says fails, so L is not false, which means it is true. Whichever side you start on, you are pushed to the other.
The sentence is short, grammatical and plain English. Nothing in it looks like a trick. Classical logic says every sentence is either true or false and not both, and the two steps above look valid. The puzzle goes back to the fourth century BC and is still a live research topic. What is the truth value of L?