David Bernstein’s Paradox

You know that the negation of a true statement is a false statement, and the negation of a false statement is a true statement. You also know that you can negate a sentence by preceding it with “It is not true that ….”

Now look at the following statement and its negation, invented by David Bernstein. Which one is true?

  • This sentence contains five words.
  • It is not true that this sentence contains five words.

How about this pair?

  • This sentence contains ten words.
  • It is not true that this sentence contains ten words.
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6 Comments

  1. Danny Cook:

    It’s not necessarily possible to negate a self-referential statement by preceding it with “It is not true that”. To find the true negation for such a statement, you must explicitly mention the original statement in its negation.

    This sentence contains five words.
    It is not true that the sentence “This sentence contains five words” contains five words.

    The former, of course, is true, making the latter false.

    This sentence contains ten words.
    It is not true that the sentence “This sentence contains ten words” contains ten words.

    The former is false, the latter true.

  2. Bill:

    Also, the word “this” has a different referent in each case. I could make statements like “this is a banana” or “he is an attorney” that you wouldn’t exactly call paradoxical, yet would have different truth values depending on context.

    Quine overcame this problem nicely with his invention “Yields falsehood when preceded by its quotation.” Then, if we subject it to Danny’s “explicit mention” requirement, we arrive at the following:

    “Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.

    It manages to be paradoxical without being self-referential.

  3. Emmanuel Vantieghem:

    It is not right that every sentence is either true or not true.
    To say something about the truth or untruth of a sentence requires that the sentences would be members of a consistent system of sentences. But to say that a system of sentences is consistent or not is of course a sentence that is not in that system …
    Should I reread “Gödel Escher Bach” ?

  4. Thirteenth Linkfest:

    […] Tanya Khovanova: Star Trek TNG Science Quiz, David Bernstein’s Paradox […]

  5. Felipe Pait:

    Also, a double affirmation is a negation. Sometimes. An example: in the 1950’s, the British philosopher J.L. Austin came to Columbia to present a paper about the close analysis of language. He pointed out that although two negatives make a positive, nowhere is it the case that two positives make a negative. ”Yeah, yeah,” Dr. Morgenbesser said.

    http://www.nytimes.com/2004/08/04/us/sidney-morgenbesser-82-kibitzing-philosopher-dies.html#

  6. Ahaan Rungta:

    This reminds me of the two very similar paradoxes:

    1. Is the answer to this question “No”?
    2. This statement is False.

    Do they have names, though?

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