字幕表 動画を再生する 英語字幕をプリント I have been asked to be the last speaker of today. I know, of course, that you will all be extremely tired at this moment. But the fun thing is that I'll be talking about logic and creativity. And you see the subtitle: "A False Opposition". It will mainly be about logic. But the fun thing is that you don't have to think. OK? So, just let your mind go all possible ways 'cause I will try to show you -- what I want to try to show you is this. In western philosophy for over more than 2000 years -- I'm not exaggerating here, for more than 2000 years, there was always the idea that you have logicians on one hand, and you have creative people on the other hand. And the two will never meet. So, you have this image of a logician, which does not correspond to me, namely, the kind of person with a very strict mind who follows the conclusions he or she has to follow with an inevitable force, whereas the creative mind can explore all the possibilities, etc. Well, first of all, if you do logical analysis you get problems. And, what I will do, in the time that I have at my disposal, I will show you some of these problems. And they are deep, deep, deep problems, also extremely amusing. And if you want to try to solve problems, you need creativity. So, that's me being logical, logical analysis needs creativity. It's even better. If you want to inverse the whole story, if you want to be creative, you have to explore. But, if you want to explore, then you need maps for exploration. Such maps need logical analysis, so, inevitably, creativity needs logical analysis. So, right. This is the end of the talk. (Laughter) I have made my point, so -- (Applause) But I still have 16 minutes. So, OK. Let's have some fun. OK. I'll start with a very simple problem. One of the problems that logicians think about a great deal is how do we define things. It's something that -- we don't do it on a daily basis. Very often if somebody asks you: "What is this?" Then what you do, you give it a description, you try to define what it is. OK, let's take one of the most famous examples from the British philosopher Bertrand Russell, and it goes as follows. Imagine a village, and in the village is a barber, and the barber is that person that shaves people. Now, imagine that you give the following definition. Who is the barber in the village? Well, that's that person who shaves everybody who does not shave himself. Sounds OK. People who don't shave themselves go to to the barber -- it's a very neat village, clean people, I'm not living there. (Laughter) They go to the barber to get shaved. And then you realize this definition is no good because the only question you have to ask, "What should the poor barber do?" Imagine the barber getting up in the morning. He goes into the bathroom, looks in the mirror, and he says: "Should I shave myself? Ah, no, I can't, because I'm the person who shaves everybody who does not shave himself. Now, if I would shave myself, then I can't go to my barber, but that's me. So, I can't shave myself." But then, of course, he wonders, "Should then I not shave myself?" He says: "Well, that's no good, because then I belong to those people who come to me to get shaved. So, now, I have to shave myself." So, he has to shave himself, if and only if he does not have to shave himself. Right. Can you get out of this? Sure. There's a very easy solution. Take a woman. (Laughter) (Applause) And, since I always have clever students in my college classes, one of them said, "Yes, but did you take into account the woman with a beard in the circus?" Okay, so that's why it's added a woman without a beard. Yeah. You can of course say, "Reject the definition." OK, fine, but what are you then going to do? So the question you then ask -- and this is a question that is still an open question, we have no good, decent answer to it. Namely, how can we decide, when I give you a definition, how can you decide that that definition is OK? Well, the answer is, you can't. Here's another example, one of my great favorites. OK. Watch this. Since time is staring at me at this very moment this is a perfect example. Suppose you have the following definition. If you have two watches, "one" and "two", then "one" is a better watch than "two" if "one" gives you more often correct time. That seems reasonable. No, it isn't. It's a very bad idea because -- (Laughs) -- if you have a broken watch that gives you two times a day the correct time -- Right? Whereas if you have a watch that runs ahead one minute it never gives you the right time. So the broken watch is better than the other one. OK? It's of course a bad definition because it doesn't tell you that you have to know when it gives you the correct time. Bad definition. OK. Let's forget about definitions. Let's take something more fun. Truth. Ah! (Laughs) I was very pleased that TED has as a subtitle, "Ideas Worth Spreading". Not "True Ideas Worth Spreading" because otherwise I would have not been here today. Because I will show you that I have no idea what truth is. Why? OK. Follow me for a moment. Assume the following. If I say something that is meaningful then it is either true or false. At least in first order we can accept it, I know we have plenty of occasions where we have doubts. Is it raining, or not raining? It's drizzling. OK. But in that case, is it drizzling, or not drizzling? OK, fine. Keep the world simple for a moment. Either true or false. Right? Either the one or the other. That seams, OK -- that's trivial. And this one too. Not both of them. You can't say of something that is both true and false. That's excluded. Right? Doesn't it sound perfectly reasonable? It doesn't. (Laughs) And this is the reason why. The famous liar paradox. OK. This sentence says, "This sentence is not true." So this sentence says of itself that it is not true. It is meaningful. I assume that everybody here present knows what this statement says. It says about itself that is not true. So, we understand it. So that means it must have a truth value: either true or false. But now what happens? If it is true, then it turns out that is not true. Of course. Assume that a sentence is true. Then what the sentence says must be the case. What does it say? That it is not true. So if it is true, it is not true. That's OK. That's fine. That's OK. You can conclude from that that is not true. So then assume that is not true. What then? Well, if it is not true, then that is exactly what the sentence says. Now, if something says exactly what is the case, then it is true. So, if it's not true, false, then it is true. It is true, if and only if it is false. So, there you have it. Ah! Then you say: "How can we get out of this?" First a warning. I have to confess I'm a professor, so I'm a teacher and I can't resist teaching. So, here's a short moment of teaching. You will learn something you can embarrass logicians with. OK? So I'm now working against my own Trade Union, the United Force of Logicians Worldwide. If you now meet the logicians you can say, "Tell me, how about that problem," -- namely this problem. The paradox that I've just shown you is also known as the Epimenides paradox. And it goes as follows. All Cretans -- you are on the island of Creta, and a Cretan says to you, "I have to warn you, all Cretans are liars." Now, what are you supposed to do here? Well, very funny, first of all a bit of theology. If you say logic, you say theology. What would theologians be without logicians to prove the existence of God -- which, of course, doesn't work. Neither does the opposite, but that's another problem. And that's a different talk, by the way, also. Which I have, so, OK, worth spreading. I told you, I'm doing the thinking for you. OK? So you don't have to think. Actually, the Epimenides paradox, the first reference you get is in the bible. It's in the Epistle of Paul to Titus. Titus, being sent off with his family, yes, his family, to Crete, to convert people there. And Paul gives him a warning. And that's chapter 1, 1-12, -- you may remain seated -- "One of themselves, a prophet of their own said, 'Cretans are always liars, evil beasts, idle gluttons.' " And then Paul makes a horrible mistake. He says, "This testimony is true." (Laughs) Which proves that in the early Roman-Catholic church there weren't that many logicians around. Because they would have said, "Paul, don't write this, I mean, it's silly." Because what you have it's the following situation. There is no paradox. So any logician will tell you, "Oh, this is definitely a paradox." It isn't. Why? Because it is not so that all Cretans are liars. Because you know it has been said by a Cretan, so if what he says it would be the case, then they are all liars, so he must be a liar. OK. Now what is the meaning of "It is not so that all Cretans are liars"? That is, that some of them are liars and some of them speak the truth. Now if it so happens that the Cretan that is telling you this, is a liar, everything is fine. It's basically a liar who has told you a lie. If he had been a truth teller, then you would have had a problem. And that's exactly what Paul wrote. I'm not going into a theological discussion here, Hence this must be -- OK, what? How to solve it? I'll be very brief. We don't know. One of the brute -- yes, well, I mean -- (Applause) This is no exaggeration. There are plenty of logicians who would say, "Don't say such thing." "I'm now saying a lie." "Shut up." Or you could say, well, there's more than true and false. You have true, you have false, and you have stuff in between. That's a possibility. It's not a good possibility. Or, why not -- and this is something that logicians have been working on since the 1950's-60's -- why can't we reason with sentences, statements that are both true and false? If you say, "What's typical for a sentence like, 'It's Saturday today?' " Well, in this case it is true. Tomorrow it will be false. Fine. What is typical for a sentence such as, "I'm rambling." Well, you have to decide. And if you then ask, what's typical for "This sentence is not true." Answer, that is is both true and false. That's the characteristic of it. Ah, lovely. OK. Let's get closer to the world. I'm sure you are all familiar with Zeno's paradoxes. And we have a solution, that's nice.