8

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.

  • See http://math.stackexchange.com/a/1888389 for a more precise version and one possible resolution. – user21820 Jan 08 '17 at 14:46
  • If "This sentence is not true" was true that would make it false so we can know for sure that it is not true. –  Apr 22 '20 at 21:35

3 Answers3

10

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.

David
  • 82,662
2

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.

aes
  • 7,653
0

The Liar is more usually stated as "This sentence is false."

The problem with "This sentence is false" is that it would seem to be a true sentence if and only if it is a false sentence. From this, we can infer that it is not a true sentence AND not a false sentence. Hence, its seemingly paradoxical nature.

There are, however, many such sentences in daily discourse, e.g. "What time is it?" and "Wash your hands." Using only some basic set theory and ordinary true-or-false logic, "This sentence is false" can be shown to be one such sentence, i.e. one of indeterminate truth-value. (Hint: Consider the trichotomy: A sentence is either (1) a true sentence, (2) a false sentence, or (3) a sentence of indeterminate truth value. Not to be confused with three-valued logic.)

Dan Christensen
  • 14,529