Next thing you have in every logic is a semantics, a mapping into the world.

Here since it's such an easy logic we're going to have a very, very simple world which

is just big enough that we can talk about the things that we're interested in.

Is this thing true or is it false?

So we have a logic that has a semantics that just says true or false.

Other logics?

Bigger universes, more interesting.

Right now we keep it simple.

And the minimal thing we're interested in is to distinguish true and false so that's

what we're going to do.

And then what we do is what we do in computer science.

In other words, we have a recursively set up set of formulae which they are as terms

or trees or whatever you want them to think about.

You kind of follow the shape of the tree of your formula and divide and conquer if you

know what the meaning of not is, in this case a function that flips truth values, and if

you know what the meaning of and is, in this case a kind of a maximum function where true

is bigger than false, then you can kind of ripple all the way down to the propositional

variables.

If you have a value for that you can actually compute the value of the whole formula.

Very simple.

Okay, we use the compositionality principle here to define what we call a value function

which can take any syntactically well formed formula and give it a value.

And the value only depends on the value of the propositional variables.

Why?

Because the meaning of the connectives is fixed.

Remember we only talk about two connectives here because we can define the others from

them which is very convenient.

Otherwise if I had all these, what is it, eight or so connectives I would need to write

down eight cases here and not have any space left, which is something I don't need to do

so I don't do it.

Once we have a value function we can think of formulae whether they're true or false

in a situation and we're going to call them true under some variable assignment or false

under some variable assignment in the obvious ways or we say something like that, the variable

assignment which we think of as a situation satisfies A or falsifies A.

The next thing is if you have multiple assignments you can call a formula satisfiable if there

is one that satisfies it and falsifiable if there's one that falsifies it.

If there's none that falsify it or in other words all that satisfy it then we call it

valid and otherwise we call it unsatisfiable.

These words are important.

Very importantly we're not talking about true or false.

Any formula with variables can be true in some situation and false in another.

What we're really interested in is the ones that are valid, that are always true or true

in my situation but usually I don't know the situation completely.

My sensing is incomplete and all of those kind of things.

We're really interested in those things that are true in all possible situations, i.e. valid.

Valid is the most important word here and it has a dual which is unsatisfiable as we

will see and they're basically just one knot apart so we kind of treat them the same with

a slight preference for valid.

That's the semantics for this particular logic.

It's just a mapping into a world and where you can say true or false and that is something