OK.

So let's tackle this very daunting slide.

So let's define the syntax that we want to use for propositional logic.

We start with...

Ah, there we go.

We start with a set of propositional variables, where propositional variable really just means

some symbol.

Again, the thing has no meaning, it's literally just a symbol.

In this case, we just pick out the symbols P, Q, R, P1, P2.

It's kind of a matter of taste which ones you use.

What I've often seen is that people use A, B, C, D, E, or A0, A1, A2, A3, all of that

stuff.

We assume that our set of propositional variables is infinite, but at least countably infinite,

but you might as well just stick to a finite one.

For most examples, it really doesn't matter that much.

We have some kind of propositional variables, and then we have the following connectives.

T standing for true, F standing for false, this bar here standing for negation.

So if you ever see something like this, think not A. That's all it means.

We have this V thingy here which stands for R, so this is A or B. We have this wedge thingy

which stands for A and B. Then we have the implication which we've already seen, which

just means if A, then B, and the by implication which says A if and only if B.

Of course, you can add several ones.

The truth is in practice, basically all you need is negation and conjunction, and then

you can define all the others.

Or you just take negation and implication and then define all the others.

Or you just take negation and disjunction, i.e. or, and define all the others.

Or you just take the Schaeffer stroke and define all the others with that one alone,

or NAND or NOR or yeah.

Yes.

Sorry, can you throw the cube over there?

There are symbols for if or or.

You mean exclusive or?

In this instance, not, usually you don't need it.

You can of course introduce one.

Like nobody's stopping you.

Once you have those, you can define all the others.

I can define exclusive or as A or B and not A and B. There you go.

And then of course you can introduce a new symbol for that.

That would be fine.

But to be honest, I don't know if there is a canonical symbol for that.

There probably is.

It's just that logicians usually only ever use inclusive or.

And now, given these connectives, we now define the full syntax of propositional logic and

we do that recursively.

So we basically say a well-formed propositional formula is either a propositional variable

or in the negation of a well-formed propositional formula, that's where the recursive aspect

comes in, or the conjunction between two well-formed propositional formulas or the disjunction

of two well-formed propositional formulas or the implication between or the equivalence

of etc.

The convention we will use here is that normal letters like this are propositional variables