Let us.

So if we want to deal with probabilities.

One good technique is to kind of use linear algebra

and think of vectors of probabilities.

Thinking of all the four outcomes

and what their probabilities are as a forevector.

And we'll be able to write the formula down

with much less sums, and so on.

So that will be very convenient.

So we have the probability distribution

for a random variable.

Or even the joint probability distribution

of a set of random variables.

And then we introduce a very convenient trick.

If you go back here and think of the set of random variables,

then really in this full joint probability distribution,

what you're really thinking of is this and that and that

and that.

So it's very convenient to be able to build up complex events

by conjoining simple events or having

events over sets of random variables

and bringing them down to atomic events.

So it's very convenient to have a language for talking

about events that not only has the atomic events,

this random variable has that value,

but also combinations with and of those.

And that is something that we kind of implicitly

did when we did constraint propagation

or constraint-based reasoning.

Had a language that had ands in it.

Turned out that sometimes you really want to have this or that.

So why not go all the way and have

a language that allows us just all the propositional

connectives and or not?

And if you have two of those, then you

can define all the others.

The typical way of doing things is

that you give yourself, when you're doing probability theory,

a language that can not only talk about atomic events,

but also of Boolean propositional combinations of any events.

And that regularizes a lot of things,

gives you a lot of mileage.

So we almost always do that.

You can take any language on top of this,

first order logic, modal logic, all kinds of logics

as languages for talking about events,

but that gets interesting.

So we're not going to do it here.

We're only going to essentially talk

about propositional logic here as the language