Epimenides Paradox

Going to Crete made me think of the famous Epimenides paradox. Epimenides was a philosopher who lived in Knossos around 6th century B.C. His most well known statement was "All Cretans are liars." The reasons it is a paradox is because Epimenides was a Cretan. So if all Cretans are liars that would include Epimenides and so his statement must be a lie. That means that all Cretans tell the truth. However, we derived the statement "all Cretans tell the truth" by positing that Epimenides was a liar. See the paradox?

The statement, "all Cretans are liars" may sound familiar because Paul quoted it in his letter to Titus. Titus was put in charge of the church in Crete. Paul's letter mainly gives advice about how Paul should lead the church in Crete. In verse 12, Paul tells Titus, "One of the Cretans, a prophet of their own said, 'Cretans are liars, evil beasts, lazy gluttons." Clearly, Paul did not take Epimenides statement as a paradox nor did anyone else back then. It was understood to mean that a liar is someone who lies most of the time or some time but not all the time.

In modern times, people began to view the statement as a paradox. However it was treated as kind of fun thing to think about and really no more than that. No one saw it as particularly useful.

Traditionally, it was Aristotle who wrote the last word on logic. At the beginning of the 20th century, a new group arose working on a different type of logical called formal logic. Formal logic breaks the world down into propositions which are proven to be true according to the rules of the formal system. This is called the verification principle.

This was especially influential in the field of mathematics. Attempts were made to build formal systems of mathematics. Bertrand Russell attempted to build an entire system of mathematics based on formal logic in his work Principia Mathematica. The outcome of this project is that it means that mathematics is entirely a human construct. Mathematics, is merely a logical game where we completely define the rules. There is no independent existence. In effect it makes "man the measure of all things," which is a quote from another Greek called Protagoras who lived in the 5th century.

Protagoras' statement is quite radical and typical of a school of thought called Sophism. According to the Sophist there was no objective truth. The Sophist taught the sons of rich Greek nobles the art of rhetoric or oratory for high fees. Since they did not believe in truth they mostly taught their pupils how to argue. It did not really matter which side was true since there was no truth - so it was the force of your argument that was really important.

They were opposed by Socrates and several of Plato's dialogues recount the arguments Socrates used against them. Mostly, they were a variation of an argument that goes like this - so you say that there is no truth and therefore we cannot know anything objectively? However, the statement "there is no truth" is an objective statement. So in refuting truth and objectivity you are using truth and objectivity. It is a paradox kind of like Epimenides' paradox.

Interestingly, the same line of argument was used against the verification principle. Here is it how it worked - so you say the only things we know to be true are by what we can prove through the verification principle? Well have you bothered to verify the verification principle? See same problem.

Kurt Godel, was a 20th century Czech logician and mathematician, and hated formal logic. He did not believe that mathematics was something constructed by human beings, rather it was something that had an independent existence and that man discovers. In 1931, he effectively destroyed the formal mathematical project by publishing his "Incompleteness Theorem" The Incompleteness Theorem proved any mathematical system complex enough to account for arithmetic cannot prove itself. He did that by trying to prove the statement "this statement is false." Essentially, this is the same as Epimenides' paradox. Turns out it was useful.

The reason Godel so passionately hated formal logic and believed so strongly in the independent existence of mathematic was because of a class he took in college. That class was a philosophy class in which he learned about Plato. Plato wrote about the independent existence of the world of the forms. The world of the forms is the perfect, unchanging world of which our world is a crude copy. The world of the forms contains truth. The world we experience by our senses is only a shadow of the world of the forms. Mathematics fits perfectly in this conception since it is an idealized version of our own world like the world of the forms.

So Betrand Russell using the arguments of Protagoras and the Sophists is defeated again by Plato using a statement of Epimenides by Kurt Godel. There is nothing new under the sun.