When I talked about type theory, I thought some people asked about this and I thought, It's a good bit more to go a bit more to detail.

Explain things a bit.

Yeah.

So what does this mean?

The nature you could say the relation between proofs and propositions, A straight cop for short is the same, in some sense, as programs.

Basically commission, which I would like to explain.

So first of all, let me explain a little bit by proposition.

I mean, a mathematical statement.

Like the injury.

Many prime numbers are this program swords, numbers, stuff like this.

Yeah, and you'll even explain things.

I sometimes use examples from the view bird like trains.

Today we go to the zoo or something like this.

But really, it's not about the propositions I'm interested in here are not like everyday propositions, but they're so precise.

Statements about programs are mathematical objects or whatever.

What do I mean by by types?

Because they are the types.

We haven't a computer.

Pull them and a computer.

You know something?

Pool warrant, ages but I'm going to use maybe some more primitive types, which maybe not s as coming.

Well, that explains it.

Uh, it's a proof want survey to justify a proposition, I'd and what's the problem?

I should be watching computer file if you know what this is.

The thing about about logic and reasoning.

So traditionally there's another creation I would liketo I don't This is the idea that top equals booth.

What's Boo?

It's a type.

It's just two elements to enforce.

This idea is very widespread on mathematics.

It's related to what we call classical logic.

And initially, if you first learn a bit of a logical looks quite well, there's these operations we like and conjunction or disjunction implication navigation.

We can explain them as operations on Duel, and maybe you've seen these tooth tables which defying all this operations.

But that's cool.

But then you learn about predicated logic.

So Critical Lodging has some other symbols, like this one, which means for all and this one which means exist.

So, for example, we can work something like for all natural numbers for all ex in N.

And then something holds.

So something holds for financial numbers you know nothing.

F off x the F It's a function which assigns to every national number two civilians.

That's a predicated for every natural number.

We have a truce value.

Now, if I want to define the meaning of this statement, I have to compute a 1,000,000,000 this bullion it's okay to if this function returns to for every input.

But I don't know how to pull them such a function because I have to check infinitely many inputs to say that I can't return to what it has to be to everywhere, a kind of guy to pull them.

Which justice?

I mean, I can't start tying f zero f one f or two and if it's always to it if it ever falls, because it must be for you.

But if it stays too, I never know when to return to.

To me, that means that this idea of Popsicles school doesn't really work for me.

As a computer scientist, maybe may say OK and mathematics, that's okay, but I like to use a deal's from computer science toe, explain mathematics and here let me turn over.

There is the other idea, which is pop.

It's tight, but that's just me.

You have a proposition.

Instead of trying to explain minutes tool, which is tricky, it may explain what it's evidence for this proposition.

I say a proposition and then explain what sort off, except as evidence evidence to accept this proposition and the evidence of a proposition.

It's a type off its pools.

So the idea is we assigned to every proposition a type off its proofs or evidence.

And whenever you find can deliver an element off this type, you have very fine this proposition, so it's really, really a fundamental, different point of view to logic.

Instead of talking about tooth, we talk about evidence and the proposition as types.

Translation translates propositions as types is a heartless is work.

So I want to do an example.

I want just to tell you what a proof, a simple proposition and proposition a logic.

I want to prove what's called a tautology, p and Q or our implies P and Q or peace sent.

I find obviously useful to think off complete association so that C P means it rains today.

Q.

Is we go to the zoo and are we have chicken for dinner.

So I say, Okay, it rains and we go to the zoo where we have chicken.

Then either it drains and we go to the zoo over it rains and we have chicken for dinner.

This implication hasn't got anything to do this to concrete instance, she ations off P Q and R.

If ever replace it with something else, it would still remain to.

So it doesn't tell you anything about whether it rains.

But what zoo or chicken dinner?

It is what's called a tautology, which means it's, too for any peak.

You are being propositions.

Okay, so everyone to check that this is a tautology and we use the previous idea.

Proposition equals bull populace poor.

We have just took a ball to stable and shake.

That's two everywhere, but I won't know to use this proposition.

Is types principle to explain viruses a tautology?

What I'm going to do is I'm going to consummate this into a type and the town station.

It's actually quite straightforward.

I have to give a translation for every operation here for every logical operation for the end of the hall for authentication.

Okay, let's do this.

So what's end What is evidence for P and Q evidence for P and Q is a pair off evidence for P and evidence or Q and the D notice as the product?

P Times Q.

So this is what mathematics called cottage in Poland or product off types.

It corresponds to constructing records.

And the basic rule is if you have an element X off P and an element of eye off, you can put them together like two pools.

Thanks, Come.

All right, That's element off P times killed.

I mean, he knows us from coordinates.

Real times, Lou have a peer off numbers.

That's the product was quite widely used, so we translate conjunction.

That's the product, because evidence would pee in, few disappear off evidence would be and evidence work.

You know, the next thing is P or kick you out of it.

Complete peor que.

There's another operation, which is less familiar in mathematics.

It's wouldn't plus, it's a son, and this coast wants to either.

Your element of P.

We have an element of Cuba, but we're labeling which one of this so it's not.

It's different to the union operator on, Dhe said.

Clearly, we just put things together.

But here we put them together.

But we label what they are on an example.

If you have some database and maybe members off the university can be either stuff.

Okay, So members of the university can be as a dog's over humans.

Yeah.

So then you may have different data for dogs and for humans.

So you say that that type of a member of the university is Doc plus human.

But you're basically member who was a dog and he was human.

You don't mix him up to be either, or so let me write on the world.

If you have an element X in P.

So if you have a dog, then you have also element left.

Thanks off P plus que And if you have an element of eye off you, then you come there right by and P plus Que Okay, here's the words left and right.

The orbital.

You could use other symbols.

Something they will in the I don't know whatever.

Okay.

The one thing I have to explain as well is what means p implies Q.

There used the same symbol.

P employees Q.

It's the type of functions frumpy took, You know, if I want to convince you that if peace and Q the evidence was this, it's a function which, whenever you feed it, evidence for P spits out evidence.

Work you?

Yeah, And instead of inventing lots of notations for functions, I just drew a picture.

So function f off Pierrot Q is a box.

Whenever I feed it Elements off P on the other end elements of Q come out.

Okay, so this is my explanation.

Translation off oppositional symbols into operations on types.

And now let's go back So proposition.

And I think I have to go to a different page Snow Oh, boy.

Translators as p and cute.

Plus, uh, ho p times cules class peace times are so I'm just applying the translation which are just explain.

So this proposition and now to convince you that a tautology I have to write a function whenever it gets this input to reverse this output, how do we find such a function?

Cf off some input, X equals some output.

So ex CNN element was in court.

My question.

Look, it's an element of the output.

How can I write such a function now?

I kind of really given answer.

I first have to analyze with my input, men put its first of all a product.

So I know it must be appear X comma and then something.

And there's something is an element off you plus r.

There are two possibilities that as a left off lives or right off that define dysfunction have to give to your creations.

I have to give two definitions.

Two lines.

One for the case to hear the first element X is in peace.

And the 2nd 1 element of Coop Lazar was avoid being a pew.

And here I also have another P and this that is an element off.

What I've done is I've analyzed the input.

What could be the possible elements off the input for each element of the input?

I have to give an output to define dysfunction.

So in this case Ah, I haven't p and Q.

And if you look at this, I can could use an output by going left left.

Thanks.

Come over.

And the whole thing to x comma by is an element of P.

Times two and left X comma virus.

Another mint off this type here.

Same right?

Thanks.

Common is it for what I've done here is having written a simple program in the functional style which inhabits this type.

Anderson's evidence for this proposition.

Whenever I feel defeated an element off this type and the output an element of system.

Okay, This ah, very to proof propositions.

Just using programming is instead off instead off looking up to the stables or something is just about the program.

And, um so what's what?

What follows from this virus is good or how this is compare.

That's a previous idea of using boo.

I mean, I've always say that the idea just using bull for propositions doesn't really scale.

So there, there, other differences.

So the logic we get here is different from the logic from the classical logic were used to.

So there are statements because they're not provable using the understand station, the famous fondness to be or not to be with e p or not peace.

But what does this mean?

I haven't really explained What notice?

Let me do this.

What is not not off p means that if p holds them false to say not you see, if P then you're in trouble, okay?

And how do we consider it?

Falls it translate Force see Empty type.

The empty type has no elements.

So in this case, they cannot be evidence for P.

Because if there would be evidence for P, then we won't have evidence for the empty type.

And there's no evidence for the empty type.

There's no problem in the end to top.

OK, Is that like no.

Ah, I was finally is like picks can fly, huh?

So if if if a girl wants to see she doesn't want to marry you, she could say If I marry, use and ticks can fly, which is a bit off, a complicated way to seem normal.

So here, if if you translate this, this means p class pea gravel empty.

So now what does ask you is for every type to either return element off p or not, Pee pee implies empty.

And you can't do this because you can't even look at it.

The type, so you have to decide either or propositions to old positions affords.

That's not the case because there's some tools and some of forts.

So this statement here, which is ah, classical tautology.

You can even your check that this horse and toast tables because two stables just realized this idea that every proposition is either to a fault is not provable.

This is evidence.

Semantics.

Let me That sounds like it.

Disadvantage.

Something we would maybe want to use in your proof.

It doesn't work on Dhe.

Some people, some mathematicians actually think it's a disadvantage.

Uh, miss most famously, Hibbert, a famous German mathematician.

He didn't like this on and had a big argument with Dutch Guy called pro.

But I think especially from a computer science perspective, it's actually quite useful, not effort, because in computer science, I'm often interested in something that's decided well, so we may have a predicated, as called Q, which gives me, for every natural number a proposition.