Okay

so I was a little bit slow last time because I wanted to teach you the general

principles of logic a bit more gently.

I want to try to be faster today.

So today's goal is to look at lots of different calculi for propositional logic.

You've already seen the Hilbert and the natural reduction calculus.

I explained to you that these calculi are like algorithms that we can use to try to

derive our theorems.

Consequently

there's all kinds of different algorithms you could use.

Some work better than others.

There's a question of efficiency or ease of implementation.

This is very different from the semantics.

The semantics is fixed.

There could be variance, but usually we would call it a whole different logic.

Whereas

generally for one logic

we have one semantics

and then we can try many different

trying to find one that helps us derive theorems the best in practice.

The notes have another chapter about formal systems in general.

That's kind of already what I told you using the example of propositional logic.

Here it is done now in full generality.

So the question is what kind of logics are there?

So already we can see we can already get a couple of different logics by just looking

at variance of propositional logic.

Just by tweaking this primitive connectives

we can get all kinds of different variance

of propositional logic.

They're kind of trivial variants because it's intuitively still the same logic, but for

technical reasons, since we haven't seen that many logics so far, we don't have that many

examples yet.

This is a way to throw, even with the little thing that we've seen already, we can construct

lots of different logics.

And we like to define things like model and being true in a model and these kind of things

generally for our logics so that we don't always have to repeat the definitions every

time we introduce a new logic.

So we need one general definition of what a logic is

and one thing to be aware of is

that this definition isn't the one official standard definition.

There is no such official definition of what a logic is because it's such a fundamental

concept that whatever definition you might come up with

someone will come and find something

that doesn't

that has

someone will come up with a logic that doesn't cover your

sorry

whatever definition of logic you might propose

someone else might come up with some logic

that is not covered by your definition.