The following content has been provided by the University of Erlangen-Nürnberg.

We are talking about logics. Logics as ways that agents can build world models

and describe world models, that's what logics are for,

describe the world in terms of a world model.

And after propositional logic, which is a very simple logic,

has good properties, it's decidable and so on,

and we know exactly what to do.

We've kind of graduated to first order logic,

which is a much more expressive language,

which means we can describe the world in much smaller formulae.

And we can talk about infinite structures.

Now normally you would say why would one want to do that as an agent, right?

There's only finitely many atoms on the Earth,

and who cares about the rest of the universe, which also only has finitely many.

But if you think about, just thinking about time,

you can think about tomorrow and the day after tomorrow,

and then you can keep on going.

And sometimes you do not want to kind of commit

on your expected lifetime or something like this.

You want to have infinite structures.

And that allows us to do reasoning and world modelling

with arbitrarily large structures.

So it makes sense to have a language like that.

First of all, logic is such a language.

We essentially get that language by opening up propositions.

So instead of having a proposition like P,

which is something you naturally can't look into,

we now have propositions like love, Peter, marry.

We have discernible parts and we can look into them.

That's the one thing we're doing.

We add individual variables and we allow to quantify over them.

And this is the real big innovation of first law logic.

We have quantification.

And with universal quantification for all X, A holds,

we also get there exists an X, A holds.

And those are the things we want to say about the world.

And first law logic is in a way the strongest language

that still has good properties.

There are stronger languages and sometimes you would like to have them,

for instance, to do math.

But there we lose the property that we have.

Sound and complete calc.

Which causes all kinds of interesting things to happen,

which we do not want to go into here.

Okay, first law logic is one of the languages we want to describe the world in

because we can have universal and existential sentences.

And we've been kind of dealing with the consequences of that much.

One of the consequences of that move is that we have individual variables.

And if we want to understand quantification, we need substitutions.

So we talked about substitutions last week.