There's a couple of things we want to say about this. I have taken and and not as

the only connectives and I've defined A or B as not not A and not B via the

De Morgan formula. Okay. Which is good because that allows me to make logic

even simpler than it is. Instead of having ten or so connectives I pick out

two namely and and not in my case and I can define all the others from that via

the usual De Morgan things. Right. I have an or, A implies B is not A or B, A

if and only if B is A implies B implies and B implies A. What else do we need?

True is A or not A and false is A and not A. Now I have all of them. The choice doesn't

matter. I could have just as well done or and not. I could have done implies and

and and all of those kind of things. Any two of them give you the rest. It's even

the case that if we have slightly weirder connectives which are sometimes

denoted that way we can which is a NAND and an or we can define all the others

from that as well. Okay. We have a lot of choice how to engineer our logic and

depending on the application we will use that choice. If you want to use

propositional logic as a model for chips then using NAND or nor is a good idea

because those are easy to make in silicon and if you only have nor's and

you can make it in silicon you can do everything so that's a good choice. You

want to take humans you do not want to talk NAND and nor. You want to talk and

and not and so on. You should not go out in the rain without an umbrella. It's

something about not a nor nor b nor nor nor a and so on right. So there's a good

and the not is good and for humans but not for machines and so we're going to

tailor our languages to the application and it's almost always the case that we

want to use a larger language and or not if or only if NAND nor everything you

can think of but when we want to define the language we want to have a small

language because as you're going to see every new connective is going to cause

us work. So we actually want to have both lots of them so I can express myself as

I want to and few of them so I can so when I'm talking about the logic it

stays small and I can put it into my pocket. Good and I'm going to skip over

all of this math stuff. I'm going to show you an example of a real-life logic

with an inference system and it all fits onto half a slide and I can even

do a nice proof example on the other half slide. Okay remember a logic is a

language plus a semantics plus a calculus. Okay so those things we want to do now.

The first thing we define is the language and it's kind of a sub language

of what we've seen before and this doesn't want to collaborate anymore. Okay

I hate this thing. So our language consists of propositional variables and

one connective that's the implication. Okay I can write which means I can write

down things like P implies Q implies P and this actually gives me the chance to

show you one of the things that makes logic difficult. Brackets. Brackets are

very important in logic because you need them to make a language formal but as

humans you don't want to see them. Okay what do we do? Well we leave them out

with the understanding that when you read them you kind of can put them in if

you want to. So what we do is if we have a P implies Q implies R with brackets

and the brackets are on the right we leave them out. What for you that means

when I read P implies Q implies R which is not in my language really you kind of

silently put the brackets back in to understand them and it's very important

that you understand that we're only leaving out right brackets. These two

things are different. Well obviously they have different brackets so they must be

different but now if I erase brackets this is an allowed bracket erasing and

this is a non-allowed bracket erasing so now they look the same. So never erase

these otherwise things look the same that are not the same. Okay good so