So we've been looking at the an important but somewhat slightly vague application.

An area in reality modalities, modalities as are usually of language where we

have embeddings of.

embeddings of sentences into other sentences. It is necessary that A holds, where A is the

sun is shining or something like this. A is something with sentence value, and we're embedding

it into a context which talks about the speaker's attitude towards A. This is much more widespread

than one would think. We've looked at a couple of examples, and there are various ways of embedding

these sentence level As into this mood context. It's important that it's been widely studied in

philosophy, such that we have quite a lot of different perspectives on this. All of this is

to give you context. What we're doing is we're going to look at modal logics. Modal logics as

you could say implementation is something we can actually understand. Modal logic really has

two characteristics. One is we have box and diamonds for necessity and possibility, and depending on

which necessity and possibility we have, which modal logic we have, those have different

properties. We can essentially make any logic into a modal logic by adding box and diamonds.

Any logic has some kind of a sentence, i.e. formula E that is true or false,

class, and then we can just slap on a box or a diamond and then we have modality.

That's the one thing we have box and diamond.

Yes, I can. I should. Sorry about that.

Excellent. Now you can see it, right? Yes, thanks. Excellent.

We have all these different modalities and we need to somehow distinguish them.

And for every one of them we basically have a separate modal logic with separate sets of axioms

and separate sets of entrance rules.

The important thing that all of these modal logics share is this idea of Kirke semantics,

of the semantics that's based on sets of possible worlds with an accessibility relation.

In every world, variable assignments can be different and interpretations can be different.

Except for the logical constants, they are always the same. That's the idea. For all is always,

for all and and is always in every world and otherwise we couldn't really do logic. But

whether the constant P denotes Peter or Paul might shift between worlds

and the predicates might be different and so on.

So what we do in modal logic is we essentially give the variable assignments a world argument.

For every world we have a different variable assignment. So if we fix the world here then

every variable gets in this case in a propositional modal logic true or false.

And we have a value function which essentially in every world we have an interpretation which in

every world gives us values for all the constants for the functions and predicates.

The only really important thing is that in a world W the value under an assignment phi or box A is

true if and only if the value in every related world W is true. So we have a variable assignment

which is a variable assignment. And we have a variable assignment which is a variable assignment

if and only if the value in every related world W' is actually true as well under the same assignment.

Remember that this assignment can also be world

can be world can change with the world.

Okay so this really a world structure

an accessibility relation R and an interpretation we call it.

We can do this but there's no one we can do this but very little

professional logic we would have all the cases just have them world dependent

and then we have box and diamonds. The diamond rule is essentially diamond A is true if and only if

for some accessible world we get to our future. Now one thing I would like to

rub your nose in is that just like for existentials and universals which these really are but kind of

delayed existentials and universals is that box and diamond are independent from box A we cannot follow

that diamond A is true because box A can actually be vacuously true if we are in a situation

where we have a single unrelated to anything world then