Okay, let's start the last lecture of AI in 2018.

We've talked about logic yesterday and we've mainly, I tried to convince you of a couple

of things. One was that logic is easy. Logics can be very small. And the other thing I wanted

to convince you of is there's more than one logic. In fact, logics make very good pets.

You might have more than one logic as a pet. Okay, and you develop logic as descriptions

of particular worlds. Remember, in the agent, the logic is actually the language in which

to describe the world model. The world model is just a set of sentences in that particular

logic that's implemented in the agent. There are lots of different agents in different

environments. And so it's not a surprise that they need different tools to survive. I.e.,

they need different logics. Logics that are kind of tailored to the world they live in.

And yesterday I mainly tried to kind of teach you logic, independent logic. What are the

things we'll see every time? We'll see a formal language. And no matter how we do it, there

will be a formal language which we can decide well-deformedness in. There will be a semantics

which is just essentially a mapping from that language into the world. That's what we need.

That's kind of the reverse of the sensory mapping. In the sensory mapping, you see something

in the world and map it into your language, whereas the semantics should be the inverse

or at least partial inverse to that. And the last thing is we need a calculus. A way of

taking world models and deriving better world models out of that. So the first thing you

should realize is this is a description layer process, just like we learned in constraint

propagation, where we took constraint descriptions and made tighter equivalent constraint descriptions

out of it. This is exactly what we're doing with the calculus. We take a world model and

make a quote unquote tighter world model, something where we can see directly what the consequences

are. Without changing the meaning, that's the important thing, which is why we're studying

meaning, but we're really interested in these calculi here. Okay, I could directly stop

now. That's all you really have to know about logic. Except, of course, there are many calculi

and some of them are better for some things and other are better for other things. So

we're going to learn a couple of calculi and we're going to learn a couple of calculi,

I hope we can get to them, which are very suited for implementing on the machine. We're

doing not philosophy, but AI after all. Okay, so it's always also the question is, can we

engineer this so that we can actually build it? Good. Okay, so let's start. And we've

looked at two logics, actually. One is called propositional logic is very important logic.

We're going to concentrate on that. And the other one was this little Hilbert calculus

example I gave you, which was kind of a subset of this. So the logic we were talking about

has essentially propositions. We're going to call them propositional variables for weird

and wonderful reasons. But if you see a proposition which we're going to write as P or Q or P

17 or so, think of things that can be true or not. sentences of languages that can be

true or not. It is currently 10 o'clock is such a proposition, it happens to be false.

Okay. And there are lots of those infinitely many of those. And we'll just abbreviate them

by single letters because that fits on the slides nicely. It also and that's important

to realize is they're essentially black boxes. You will see that we cannot look into them.

Okay, this is something we're going to change that we really have a logic here that has

black box propositions. And we're going to in our language only be able to talk about

P and Q P implies Q P but not Q and all of those kind of things. But we're not going

to be able to look into the propositions, which will change after Christmas, where we

learn another logic which is called first order logic, which allows us to look into

those. Okay. So that's really what we have. We have propositional variables. Think of

Peter loves Mary, we have connectives, things like and or implies if and only if and so

on. And using those connectives, we can actually build up complex formulae out of the propositions

and the connectors in the obvious way. We're going to call propositional formulae without

connective atomic and propositions with connective complex. Okay, and the atomic ones boring