View Full Version : The Liar Paradox

The Void
July 24th, 2011, 01:44 AM
Think of this sentence:

This sentence is false.

If we were to say that this were true, then the sentence would be false, which would make it true, but would then make it false, ad infinitum.

If we were to say it were false, then the sentence would be true, making it false, which would make it true, but would also make it false, and so on and so forth.

This is the liar paradox. I was amazed by this and thought of sharing this to the community.

Anyway, moving on.

St. Jerome also speaks of one version of the liar paradox when he quoted about David in the Bible:

I said in my alarm, 'Every man is a liar!' (Psalm. 116:11) Is David telling the truth or is he lying? If it is true that every man is a liar, and David's statement, "Every man is a liar" is true, then David also is lying; he, too, is a man. But if he, too, is lying, his statement: "Every man is a liar," consequently is not true. Whatever way you turn the proposition, the conclusion is a contradiction. Since David himself is a man, it follows that he also is lying; but if he is lying because every man is a liar, his lying is of a different sort.

So what to do you think? Is the statement, "This sentence is false" true or false?

Blue Nocturne
July 24th, 2011, 02:23 AM
"Err... I'll go with true! Phew, that was easy" - Wheatley, Portal 2.

Best not to think to hard about these things!

July 24th, 2011, 02:27 AM
The second one is easy to answer. How many times do you have to lie to become a liar? Not every sentence you say. Therefore, a man can be a liar and still say a true sentence.

July 24th, 2011, 05:30 AM
By breaking it down into logic notation: A = !A, it's truth value is unsolvable. So it's neither true nor false. :P

July 24th, 2011, 09:57 AM
The problem is the language used to describe the situation. It's self-modifying, which creates a number of logical problems.

First of all, the modification isn't even a real liar's paradox. "All men are liars" could mean men lie some of the time but not all of it. This makes the situation easy to explain. Even if we assume he meant "all men are liars all of the time," the situation is resolvable. If we describe the situation in logical terms, it's easy to solve.

Let us try to determine the truth value of the statement. First, we assume the statement "all men (including David) are liars all of the time". Then, the following:
- David lies all of the time
- All of David's claims must be untrue
- David's claim was "all men are liars all of the time"
The statement "all men are liars all of the time" must be false. Q.E.D.

The reverse is even simpler. I'll take some liberties with the phrasing to make my job easier.
All men must lie when speaking --> All men are capable of telling the truth while speaking (definition of contrapositive)
This means we can take "all men are capable of telling the truth" to be true.
- David has the ability to tell the truth
- David claims "all men lie when speaking"
- this does not invalidate the initial claim (David still has the capacity to tell the truth, he has just lied in this case).

Again. Q.E.D. David's claim is false.

As for the original truth paradox, like I said, it's a problem with language. The language is self-modifying.
"This statement is false."
The predicate of the sentence (a statement is false) modifies the subject (this). This is what allows the paradox to exist. There's a better version which eliminates one potential answer (that the statement is neither false nor true): "this statement is not true." If we assume that the statement cannot be both false and true at the same time, then essentially we are saying (as someone said earlier) A == !A when A and !A are mutually exclusive, or "'this statement is not true' is both true and not true when it cannot be both at the same time." This would be how the sentence would be said in a language that wasn't flawed in the way most (including English) are, and it is obviously not logically possible by definition. In other words, it is a paradox of language, and with proper modeling is not a problem at all.