Examples of Logical Paradoxes

Date: 03/14/2006 at 22:45:20
From: Andrea
Subject: logic statements

What kind of logic statement is "I am telling you the truth when I say
I am a liar"?

Date: 03/16/2006 at 18:45:20
From: Doctor Achilles
Subject: Re: logic statements

Hi Andrea,

Thanks for writing to Dr. Math.  The sentence "This sentence is false"
is a logical paradox.

A tautology is a sentence which is always true.  For example:

  "This sentence is either true or not true."

  "My front door is either white or it is not white."

  "If it is raining, then it is raining."

A contradiction is a statement that is always false.  For example:

  "This object is red and it is not red."

Notice that for all of these you don't even have to know what the
words "red", "white", "raining", or "true" mean.  The very structure
of the sentences dictates that they are true or false.

There is nothing in the *structure* of the sentence "This sentence is
false" which creates a problem.  The problem comes for two reasons: 

  1) the sentence refers to itself

  2) once we interpret the meaning of the sentence, then we are left
     to conclude that if it is true, it must be false and if it is
     false, it must be true.  We end up like a dog chasing our tail.  

That is what is called a paradox.  Once you understand the meaning,
you are compelled to make a conclusion which contradicts your original
interpretation, and no stable interpretation can be reached.

Hope this helps.  If you have other questions or you'd like to talk
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/