numbers; x=y and y=z, therefore x=z. Here, the first layer of content is “x”, “y” and “z”, which are described as numbers. The second level is made of things like “x=z” and “y=z”, which are described as formulae. A third level is made of the full utterance, that we’ll describe as a statement. Again, we’re simplifying here, but bear with us. We are formally recognizing the layering that we talked about earlier when we were speaking of Madonna. Now, we see that the logic we learned at school allows us to reason on content. If we can reason on it, certainly we need to consider that content is knowledge. As a bonus, we now understand how to build more complex content out of elementary content. We know how to increase knowledge. In short, we know how to learn. Logic, of course, doesn’t apply only to mathematics. John, Mary and Virginia are children, John is the brother of Mary, Mary is the sister of Virginia, therefore John is the brother of Virginia. As we see, logic is the way we can organize content and build with it more content, whether we talk about numbers, people or anything else that we might be interested in.

In the 1970’s, when Artificial Intelligence started to be deeply interested in knowledge and its use, the community of scientists was very innocent, or naïve, if you wish. The pundits were announcing that now that we are masters both of computers and of logic, computers are soon going to perform human feats like law and medicine. Just give us big salaries, and you’ll see. Big salaries they got, but nothing was seen. Artificial Intelligence was derided and forgotten, and it took twenty years for the field to recover credibility. What happened? Well in fact, something was forgotten. Incredibly, as we can see in retrospect, because the Greeks had seen the problem and an Austrian called Kurt Gödel had seen it too. More to the point, everyone knew that. So, what happened? What the Greeks had identified is the following paradox: when I say, “This sentence is false.” is this sentence true? Well, if it’s true, then it is false, isn’t it? And if it is false, then it is false that this sentence is false; therefore this sentence is true, isn’t it? But then … What we see is that there are expressions of logic for which it is impossible to say whether they are true or false, no matter how long we try. Does that apply to computers? That’s what Gödel went out to investigate. Of course he was not thinking about computers at the time, but if you read his famous proof, not an easy reading by the way, you’ll see that what he was describing would be called today a computer program. Anyway, Gödel showed that in any form of rich enough mathematics, there are statements that are similar to “This sentence is false.” and that’s unavoidable. Therefore, if a computer uses logic, which it does because that’s how computers work, it is bound to have trouble, because some of the situations it will encounter are, as we say in computer science, well, undecidable. So, this is how Artificial Intelligence failed at the time. Applying logic in the most general way, computers just couldn’t do the job they were asked to.

Well then, “What about humans?” you will say. “They use logic, don’t they?” Moreover, “They function, don’t they? So, what’s the deal with computers, really? In one sentence you just said that all computers use logic, and in the next you said that they can’t do the job they’re asked to? Isn’t that a contradiction?” Well, those are good questions, and it took 20 years to sort them out. Concerning humans, we’ll get back to the question later. Concerning computers, the fact is that they use logic and they work most of the time. They work because we design them and then we test them extensively to make sure that they will perform the tasks that they are asked to do. As long as what they do was understood beforehand in the tests, we’re in good shape. If not, then we just hope they’ll work. That looks strange to the uninitiated, but really, that’s how the computer world works. In fact, things are not too bad, because for a long time computers have been asked to do what they know well how to do, that is manipulating numbers, filing data and exchanging messages. These are all operations that are well understood, well tested, and that we therefore are confident in. The problem comes when the computers try to do what humans do, because humans do precisely the introspective things like “This sentence is false.” For example, “Did I say what I

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