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We had a lot of stuff yesterday as exercises and this is not healthy.

We should actually do some of these things in the classroom and apart from those which

I actually leave as proper house of government, still probably I will still want to prove

that I would like to do at least some of the things that we have not done last time.

So let's say as an example, let us prove the theorem that we had last time.

Suppose I could write k as a substitute to stress that we are talking about this minimal

model logic.

No, not the one.

I wanted to have the one with the conjunction.

So let's try to do it.

It was done in previous tutorial sessions.

So tell me what to do or alternatively somebody can simply come forward and solve it for everybody.

How am I going to prove something like this?

We have not proved any theorem of this kind, right?

We have not proved that some formulas are valid universally.

I only stated several such things and left all the proofs of exercises.

So we have to do at least one proof here in the classroom.

What does this mean?

So maybe let's agree that we are allowed implicitly to use something that we should have proved

actually, namely that all the Boolean connectives work the way they are expected to.

So maybe it would be better to start at least with the clause for implication.

I said that this is easy to derive what it is and I left it as exercise, but maybe let's

do it first to make sure that we can actually use it in further proofs.

So for any model and any five side model formulas, we have that implication is satisfied at the

given point if and only if what happens.

Anybody can tell me?

This is actually perhaps the most important connective and in future when we are going

to introduce other non-classical logic like intuitionistic, bunch implications, substructure

or whatever, you will see how differently one can read the implication connective.

But here our underlying logic is propositional logic.

So first of all, this thing is the same as what?

Equivalent to what?

Not phi or psi.

Not phi or psi.

Ah, not phi or psi, okay, yes.

If we were very pedantic, we would observe that actually this junction is not a primitive

in our setup either, so we would have to reformulate in terms of conjunction what that would be

in terms of conjunction.

Because we have not stated explicitly the clause for conjunction, disjunction either,

right?

It was also left as exercise.

So yes?

Not phi and not psi.

Not phi, so, yes, okay.

And not psi.

Yes, exactly.

So it cannot happen that this thing becomes true without this being true.

Okay, so that should suggest what is happening here.

So for this we can actually use directly the clauses that we had last time.