After a long meeting, I went to the cafeteria with a friend to have a cup of cappuccino, my favourite drink at office. In the dialogue below, P is my friend and colleague, and S is me.
S: Hey, they have a 'Silence Please' board here. We can't talk.
P: Rules are made to be broken.
S: So, your rule is to break rules?
S: This implies that you should break your rule of breaking rules and thus not break rules and be silent.
P: Umm… My rule is to break rules made by others.
S: Ah! You avoided a self-reference there. It makes sense now.
I could have still trapped her by making a rule that P breaks rules made by others but this didn't occur to me in the cafeteria.
Such self references often set up paradoxes in logic. One of the simplest ones is the liar paradox.
Now, is this statement true? Is it false? Is it both true and false, or is it neither? This is discussed pretty nicely in the Wikipedia page for this paradox.
How do we avoid such paradoxes? Yes, one way is to avoid self-references by restricting a statement from talking about itself. But does it really solve the problem? Let us have a look at this.
Is the first statement true or false? Yes, we have set up a paradox again despite avoiding self-reference here. The correct way to prevent these paradoxes from happening is to categorize the statements into various levels and allow a statement to talk about statements belonging to a lower level only.
So, if we categorize the first statement "The next statement is true" as a level 1 statement, then the next statement automatically becomes a level 2 statement as it is talking about level 1 statement. But wait! This implies that the level 1 statement is talking about level 2 statement which is illegal as per our new restriction of not allowing a statement to talk about another statement at the same or higher level. So, we see that we can avoid such paradoxes with this additional restriction.
A friend of mine, shown as W below, often claims something strange.
W: I only ask questions in a debate. I don't take sides.
S: Ok. Let us debate this. Side 1: You take sides. Side 2: You don't take sides. Which side do you take?
Comic by Randall Munroe: http://xkcd.com/468/
Let me list down some of my favourite logical paradoxes.
- Russel's paradox: A set containing exactly the sets that are not member of themselves.
- Epimenides paradox: All Cretans are liars.
- Interesting number paradox: All natural numbers are interesting.
- Grelling–Nelson paradox: An adjective is heterological if and only if it does not describe itself.
- Barber's paradox: A barber shaves all and only those men in town who do not shave themselves.