Yeah, welcome to another week of AI.

We're talking about ontologies as world description languages and the idea there is that we kind

of try to chart the middle ground between propositional logic, which is too inexpressive

because we want to talk about concepts, sets of individuals, things like dogs and cats

and tigers and elephants and maybe numbers and all of those kind of things and their

members and it's important to understand that propositional logic cannot talk about individuals.

Even though we can kind of...

Do I have chalk?

Okay, somebody...

Somebody sold the chalk so I can't write to the blackboard.

So I'll wave my hands even more than I normally do.

So even though we can write something like Peter loves Mary in PLNQ, which is the same

thing, we cannot say...

We cannot really make any relations between.

We can't say if Peter loves Mary then John hates Mary or something like that, but we

can't have systematic relationships.

First of all, logic can do that.

We can say for all x's that Peter loves, John hates.

And so we want to talk about individuals like the elephant Clyde or the number one

or something like this.

We have to have more than propositional logic and nothing in the world can actually change

that.

And on the other hand, we have first of all logic, which is perfectly fine about talking

about individuals.

That's what it's made for.

But of course, it's undecidable.

Satisfiability is undecidable.

And so we want to have something in between and the logics that can do these things that

we want, namely describe kind of the world without becoming undecidable.

Undecidable means that the agent might actually not get an answer to in the deliberations

of what should I do.

So we want to talk about concepts, we want to talk about individuals, and we want to

talk about relations.

And the logics for that which are still decidable but better than propositional logic we call

description logics.

We've looked at propositional logic as a description logic, but what you want to understand here

is that propositional logic, and you can see it in the examples we looked at, is a purely

T-box only language.

It can talk about children being sons and daughters.

But it can't talk about Peter being a child of Mary or something like this.

So we have to do a little bit more, but before we actually do, and this is the picture to

Before we do something more, I would like to look at the role of inference.

So essentially when we use and build ontologies, we have three kinds of things we want from

inference.

Consistency subsumption and instance tests.

Let me go over them.

So consistency essentially is something we understand in logic, namely a set of formulae

is inconsistent if I can derive A and at the same time not A from it.

So obviously there is some contradiction in there.

That's something we don't want.