Please help me to understand the following confusing reasoning:

  1. According to laws of logic either "X is false" or "X is not false". Let's call the statement1.
  2. Let's consider well known statement "this statement is false", let's call it A. So A = "A is false".
  3. Let's consider all two possible cases:
    1. A is false. Substituting A here we have "It is false that A is false", or "A is not false". We conclude that this case is impossible.
    2. A is not false. Substituting A here we have "It is not false that A is false", or "It is false that A is not false". So this case is impossible.
    3. So A neither false or not false. So statement1. is not true...

How is this possible that statement "either "X is Y" or "X is not Y"" is not true?

  • 2
    This is the liar's paradox: plato.stanford.edu/entries/liar-paradox – Dave Jun 19 '14 at 11:42
  • You are assuming that truth is an essential property, derived from nothing in particular. What would it mean, for A to be false? What states of affairs would falsify A? – Niel de Beaudrap Jun 19 '14 at 12:00

The paradox that you reference is the Liar Paradox. LP has a number of famous variants. The one you cite is the simplest. To learn more about the paradox and attempts to resolve it, go to: http://plato.stanford.edu/entries/liar-paradox/ It covers the paradox in great detail. Separately, if you need to understand bivalence, check out: http://en.wikipedia.org/wiki/Principle_of_bivalence

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