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Let us consider a statement A that says "Statement A is false". Now is the statement A true or false? If it's false then statement that says "Statement A is false" is true therefore statement A is true which is a contradiction. If statement A is true then statement that says "Statement A is false" is false therefore statement A is false which again is a contradiction. Is statement A true or false?

edit: I was just curious if there's a known solution to it, because it seems to collide with the axiom that every statement must be either true or false and never anything else.

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    $\begingroup$ No it's not; the sentence cannot be given a truth value (or a meaning) if you want your system to be consistent. This is called the liar's paradox and is probably already discussed at length on this site and elsewhere. $\endgroup$ Commented Jun 16 at 18:53
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    $\begingroup$ This is a common logic puzzle, usually without the variable: "This statement is false." It is called the Liar Paradox..Fundamentally, the problem is the sentence can't be defined in terms of itself. Your "variable" form makes this explicit. You can't define $A$ in terms of $A.$ If I said "define $x$ to be $1+x^2,$ you'd say, "That is not a definition." $\endgroup$ Commented Jun 16 at 18:53
  • $\begingroup$ Some Self-referential constructs are used as a famous example of things which cannot be true or false. Another example is the statement " Either 2 + 2 = 5 OR this statement is false" $\endgroup$ Commented Jun 16 at 18:56
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    $\begingroup$ In general, we need to be very careful with self-referential statements. Its one of the reasons its so useful to have a very precisely defined and formal notion of recursion. $\endgroup$ Commented Jun 16 at 18:57
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    $\begingroup$ The best resource I know on this subject is the Stanford Encyclopedia of Philosophy's page on the liar paradox: plato.stanford.edu/entries/liar-paradox $\endgroup$ Commented Jun 16 at 19:12

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This is not a proper statement. A statement is a proposition that is either true or false, but neither both true and false nor neighther true nor false. As pointed out in the comments this is a very common example called the liar's paradox.

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    $\begingroup$ It is really not so easy as this to dispose with the liar's paradox. After all, in the proof of the incompleteness theorem we write down self-referential statements almost exactly like this, which instead of asserting that they are false assert that they are unprovable. So a student would be well within their rights to ask: what's the difference? Why can't we write down the liar's paradox statement using Godel's ideas? The answer is that it is not possible to say that a statement is false in this context, because of Tarski's undefinability theorem. $\endgroup$ Commented Jun 16 at 21:17

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