So, so much for the syntax and now we come to the semantics, i.e. we want to actually

say what propositions mean.

For that, we introduce the notion of a model.

This again is a general logic thing.

This is not just for propositional logic.

A model is a pair consisting of a universe.

In the case of propositional logic, our universe is always fixed and it's always just the set

consisting of only true and false.

But if you go over to a different logic, you might have more complicated universes,

which is why we introduce the notion of a universe here in the first place.

And then interpretation, and that interpretation tells us what our stuff actually means.

For that, we need to assign something to all of the connectives that we have.

In this case, we only use negation and conjunction as primitives, and then similar as I did here

with the exclusive disjunction, the others we can then just define as abbreviations.

So the interpretation of negation will be it maps something that has the value true

to the value false, and it maps something that has the value false to the value true.

And our interpretation of the conjunction will be it takes a pair of elements of our

universe and re-use another element of the universe such that this interpretation of

the negation applied to two values evaluates to true if and only if both alpha and beta

are true.

In all other cases, it returns false.

So basically, we interpret these symbols as functions on our domain.

This is again, this is basically something that happens in every logic.

The other connectors we can treat as abbreviations, as I mentioned, we can define the disjunction

A or B as an abbreviation for not, not A and not B. Look at this and convince yourself

that this makes sense.

And for implication, this is a little bit more contrived.

We define it as not A or B, meaning the statement if A then B is true if and only if either

A is false or B is true.

Does that make sense to you?

Really?

You just accept that?

Okay.

Notice that this in particular means that if A is false, then the statement, the implication

is always true.

This is called material implication, and that sometimes trips some people up.

Because one of the conclusions of this is that, for example, the statement if the moon

is made of cheese, then the sky is green.

That statement is true.

Just by virtue of the moon not being made of cheese.

There is logics that don't do that.

There is logics where the statement if the moon is made of cheese, then the sky is green

is false.

Simply because the statement the moon is made of cheese is false, and then those kind of

logics claim that if the premise is false, then I can't conclude anything from it.

Then you get these kind of paraconsistent logics, which are also very nice, but we're

not going to talk about them here.

I just wanted to make you aware of the fact that this is not exactly trivial.

Also, if I'm not entirely mistaken, this kind of definition implies that we're in a classical

setting where the statement IA or not A is always true.