Which path to take

Can we ever ascertain the truth of a single statement? This question has been pondered since the days of Socrates, who purportedly said “The one thing that I know is that I know nothing” – a paradox, for if Socrates knows nothing, he cannot know that this statement is true.

People tend to make certain logical assumptions about their thoughts. First, we assume that some statements are true (in philosophy this is called the law of identity). Second, we often assume that no statement can be both true and false (the law of contradiction). Third, we often assume, mistakenly, that every statement must be either true or false (the law of the excluded middle). If a friend tells you that it is raining, he is either lying or telling the truth. Or is he? After all, it is probably raining somewhere in the world at that moment.

But what happens if a man comes up to you and says, “I am a liar”? if the statement is true, that he’s not a liar, so the statement is false. But the statement can’t be false, because if that were the case, he would be telling the truth and would not be a liar. The statement is an example of an unresolvable contradiction, and thus is neither true nor false. “This painting is beautiful” is another example of a statement that could be true for one person and not another. Neurologically, truth and fiction are subjective values created by the brain. As far as human survival is concerned, it isn’t necessary to know what is absolutely true; you simply have to use logic, as the following riddle demonstrates. It is called the Liar’s Paradox, and Godel used a version of it to illustrate his incompleteness theorem.

Imagine that you are journeying down a path and you come to a ford in the road. There you meet two people and sign that reads:

“One of these paths will lead you to safety, but the other will lead to death. The two men in front of you both know which path is which. However, one personal always lies and the other always tells the truth. You may ask only one question in order to decide which path to take.

If you could ask two questions, the problem would be imple. You could ask one person how many eyes you have. You would immediately know who was telling the truth and could then ask that person which path to take. Unfortunately, you get only one question.

Here’s the solution. Ask either man which path the other person would tell you to take to reach safety, and then take the opposite path. Assume, for a moment, that path A leads to safety and path B to death. The liar will tell you that the truth teller would say to take path B. The truth teller will tell you that the liar would say to take path B. obviously, you should take path A. Godel tried to use similar logic to prove the existence of God, but he overlooked his own theorem, which implies that every equation contains assumptions that might be false. In the example above, you probably assumed that the sign was stating the truth. What if it had been written by a liar?