Okay, so good morning everybody.

We can see something.

Very good.

So last week you learned about the why and the how of propositional logic.

We were relatively specific about the logic and the Galkeli and all of those kind of things.

So one thing you should realize is that just as there are many, many natural languages for describing the world

and they can essentially do the same thing, all of them, there are lots and lots of logics.

So some logicians try to make everybody think there's essentially one logic, first-order logic.

And propositional logic is kind of a castrated little logic.

But that's wrong.

Logics are an engineering subject and if you as an agent designer see that your logic that you thought you would use

is not going to do the right thing, you just sit down and make better logic.

And people do that.

There's thousands of logics around for all different purposes.

There's first-order logics like we're going to talk about that.

They are for describing the world for infinite domains typically.

There's the propositional logic which works good with finite domains.

There's logics about believing, knowing, being able to prove, being forbidden and all of those kind of things.

Lutz Schreuder's group, there are computer scientists there sharing the same floor, 11th floor in the blue high rise.

They have a paper that has the nice little, a modal logic for counting your toes.

Which is really about counting leaves in trees.

And you could argue that your toes are leaves in your body tree.

And so there's lots of things out there.

And so what I would like to do today or start with today is kind of give you the general theory

of what we can do with logics and what are the things that are interesting and kind of cast it into a mathematical setting.

Yes.

I think you will end up with a better one.

No, that's a point I wanted to make. Thanks for reminding me.

These logics are different in expressivity.

So some logics can really do more than other logics.

Some logics are equivalent.

Some features in logics can be expressed in different ways.

But there are logics that are just much more expressive.

They can say much more, which also means that typically the inference procedures are going to be more complex.

Complex in every sense of the world, like complexity of actually searching in them.

Complexity in having many inference rules or complicated inference rules.

And there are logics that do not have sound and complete calc.

You may have heard about Gödel's incompleteness theorem, which says that certain higher order logics cannot admit sound and complete calc.

That's a bit sad, especially since the real result is that anything that can do arithmetics plus times and natural numbers.

Well, cannot admit a sound and complete calculus.

So there are contrary to natural languages and programming languages, which are essentially all created equal.

Logics have different expressivities.

Inference in propositional logic is decidable.

Undecidable in first order logic.

OK, any more questions?

But you've seen propositional logic in a classical setting, like where we use propositional variables as the atomic formula.

And you've also seen PLNQ, which is also a propositional logic.

It's actually immediately the same.

Something happened.

As you've proved or seen the proof.