A liar is a person who tells you things that are false. Could there be a person who always lies, a super-liar who spends all day, every day saying things that are false? This probably sounds like a silly question, but at the heart of this question is a deep philosophical puzzle. Suppose that a super-liar exists. What would you think if this person said to you:
“I am lying to you right now.”
At first glance, it seems like this utterance must be true, because the super-liar always lies. If what he says is true, however, then he is actually not lying, which is impossible because he always lies. So, the super-liar must be saying something false now. If what he says is false, however, then he is actually lying; but that makes his utterance true (since he said that he is lying). How can his utterance be true and false at the same time? That is impossible.
This suggests that a super-liar cannot exist. After all, if such a person did exist, it would make contradictory things true. Maybe we have made a discovery: we have proven that it is impossible for a person to exist who always lies. That conclusion, however, is very hard to believe. Can we prove that something exists or doesn’t exist using pure philosophical reasoning? Don’t we have to look at the real world to find out if super-liars exist?
The puzzle I have just described is an example of a paradox. A paradox is a deep, and difficult puzzle that no one has solved. There are many paradoxes in the history of philosophy that teach us about different kinds of concepts. For example, the liar paradox involves the concepts of truth and falsity. The paradox shows that we should rethink some of the common assumptions we make about the concepts of truth and falsity.
My research is about these kinds of paradoxes. They are not only important to philosophy, but also linguistics, math- ematics, logic, and computer science. In the early twentieth-century there was a great interest in the concepts of inference and proof. Many brilliant people were trying to understand how we think and reason. The methods developed by people like Cantor, Frege, and Hilbert completely changed our understanding of mind and language. For example, Alan Turing created an abstract system (called “Turing Machines”) that can represent deductive reasoning in a standard algorithmic form. Because of his work, the modern computer was born. Turing even used a type of paradox to solve an important, foun- dational problem in computer science called the Halting Problem.
What about the liar paradox? Is it just a silly puzzle? Actually, I think we can learn a lot from the liar paradox. When we described the problem, we assumed that the sentence uttered by the super-liar would have to be either true or false, and it could not be true and false. These assumptions are natural, yet they lead to untenable results. The liar paradox suggests that we need to rethink the concepts of truth and falsity. This is interesting because the concepts of truth and falsity are so fundamental to how we think and communicate in general.
Paradoxes have connections to some of our deepest question in life. One of my favorite paradoxes comes from Buddhist philosophy. The goal of Buddhism is to eliminate suffering from the world. Buddha taught that people suffer because of our desires and expectations. We suffer when we want things we do not get, we suffer when we expect things to happen differently than they do, and we suffer when we realize that life itself is temporary.
Buddha claims that these harmful desires are caused by our ignorance. We believe that all of the objects we experience have “essence”, including ourselves. Buddha says that this is a mistake: objects are empty of essence. In particular, the human mind is empty of essence (e.g. we do not have an immortal soul). We cling to pleasure most of the time because we are ignorant of our own emptiness, which causes us to suffer.
The philosopher Nagarjuna realized that this has an important implication. If everything is empty, this is true of emptiness itself. Another way to express this is as follows: the real nature of things is that they are empty of essence, which means that they lack a nature -and that is their nature!
As you can see, paradoxes occur in unexpected places. Some of them are very hard to understand, but that is what makes them so interesting. I hope that this brief explanation will make you think a little more about these fascinating puzzles.