>> Okay, this is the first video in the Introduction
to Symbolic or Propositional Logic Series.
So, the first question you might have is why bother studying this?
Why bother studying all these Ps and Qs, and here are some reasons.
First, it'll help you better understand how computers work at a deep level.
It'll lay the foundations for that.
It'll help you better understand and do math.
It'll create more logical circuitry in your brain, so to speak, neural networks.
It'll help you learn habits of thinking that are just more logical, and they become ingrained.
It'll help you symbolize arguments and thereby,
quickly and more accurately identify their validity or invalidity.
So, a lot of arguments you don't even that too think about the content.
You can just put them in symbols in your mind and quickly see they're valid or invalid.
And finally, I think it's fun and challenging.
So, hopefully, that will keep you motivated through all of this.
The categorical versus modern.
So, we're leaving the categorical Aristotelian traditional logic and moving to the Modern,
and we're going to start with this propositional logic.
The difference is that categorical logic dealt with actual words and categories.
Like if I wanted to say all dinosaurs are funny creatures,
I was symbolize that with Aristotle's method as all S are P. But now,
with modern symbolic logic, the letters will represent entire statements.
So, all dinosaurs are funny, which just be represented
as D. It's the whole proposition that's making a truth claim.
It's either true or false, right?
So, you're dealing with propositions not classes,
and we can do much more with modern logic.
So, but that in mind, let's look at an example here.
S, let's say that represents a proposition that I'm wearing a blue shirt.
So, if S is true, then it's true that I'm wearing a blue shirt.
And if S is false, then it's false that I'm wearing a blue shirt, right?
Now if not S is true, that it's true that it is not the case
that I'm wearing a blue shirt, right?
And you want to, instead of just saying I'm not wearing a blue shirt, it's helpful in logic
to say it's not the case that I'm wearing a blue shirt for reasons we'll see later.
Okay? So, also, when you write statements,
you don't want to have S represent I'm not wearing a blue shirt.
Rather, you want to make a positive assertion and then put not S
if you're not wearing a blue shirt.
We'll see why later.
Don't worry about right now, okay?
So, the letters represent true or false statements, but there are some sentences
that there that don't make truth claims, like who are you or close that door.
And we won't be able to translate those sentences into our modern logic.
So, keep that in mind.
It doesn't translate everything well.
Here's a compound statement, because it's making two different truth claims.
So, I'm wearing a blue shirt, and I'm wearing clogging shoes, right?
Or I'm wearing a blue shirt and clogging shoes.
That would be represented as S and P or B and C. It doesn't matter what letters you use, okay?
But that's a compound statement.
So, here's some practice.
See if you can just put these into symbolic form using some letter.
So, for example, for number one, you might write J or just S, you know?
For number two, you might write S and P, right?
Number three might be -- I'm either clogging or singing might be C or S. Okay?
I'm going to show you the answer slide in just a minute, and there you go.
Notice, number four, I put dogs like cats.
And then I put a not in front of that.
It's not the case the dogs like cats, all right?
Okay, now instead of using these little words here like and and or
and not, we want to symbolize those, too.
We want to put everything into symbols, okay?
So, we introduce these five operators.
You can see the first operator here, right here, is called the tilde.
It's a nice name, and it represents negation.
So, instead of saying not or it's not the case that I'm going to use a little tilde,
and the dot will represent and also, and it's what we call a conjunction.
It can joins two propositions.
The VA will represent the wedge.
It's a disjunction.
Usually or captures that.
By the way, the and is sometimes represented with an upside down V in some books, okay?
The horseshoe represents if-then sentences, and that represents implication.
We'll go over that later, and in some books, it'll be an arrow pointing
to the right instead of the horseshoe.
But we'll use the horseshoe, and the last one is the triple bar, which represents equivalence.
And that's if and only if statements.
And in some books, that's represented with arrows going both ways.
Okay, so, here's those same sentences again, statements I should say,
and they're represented now with letters and operators.
So, look at number two.
"John Denver is a great singer and pilot."
J.P, right?
Number three, "I'm either clogging or singing."
C, whoop, I capitalized the V. I meant to lowercase it.
But C wedge S, right?
"Dog's don't like cats," not D, and so on.
Okay, all right, so let's talk about each one of these in a little more detail
and do some -- the truth tables for them,okay?
And if you look at the tilde, which represents negation, this is --
let me make sure I'm going the right -- yes, okay.
So, the tilde represents negation.
The tilde is the only operator that can occur right after another operator.
So, if I say I'm eating green beans or I'm not healthy,
that would be G. I'm eating green beans, right?
G wedge not H. H represents I'm healthy.
Not H, I'm not healthy.
Okay, so, it can occur right after the wedge operator.
It's kind of neat.
Anyway, P, let's say a P represents I'm wearing a blue shirt.
That's either true or false.
A statement can be true or false.
Based on that, we can figure out it's not the case that P. So,
if it's true that I'm wearing a blue shirt, then it's not the case that P,
it's not the case in wearing a blue shirt must be false.
If it's false that I'm wearing a blue shirt, then it's not the case
that I'm wearing a blue shirt must be true.
Okay? So, that's what this truth table means.
You just reverse the values.
So, sit translate these real quick.
Monkeys don't fly, and it's not the case if I turn in my homework,
I will ace this course, and so on.
And here's the answers.
Not M. Notice number two, it's not the case that the whole statement, if H then A,
because they're not saying if I don't turn my homework, I will ace this course.
That would be not H then A with no parentheses.
However, they're saying it's not the case that if I turn
in my homework, I will ace this course, right?
Look at at the last one, number three.
Neither Clemson or Virginia will win the championship.
So it's not the case that Clemson will win or Virginia will win.
Now, for number three, you might have representatives
like this down here on the bottom.
It's not the case that Clemson will win, and it's not the case that Virginia will win,
and these two are equivalent, the same thing.
So, you would be correct if you did it that way, and later,
we'll use this logical equivalence in proofs, and so on.
Okay, the main operator is very important.
So, we want to understand this concept, because we'll later use it, the main operator,
to determine whether the entire statement is true or false, okay?
It's the one operator that covers the entire statement.
So, for example, my socks are not a main operator metaphorically,
and because they only cover my feet.
My shirt only covers my waist and chest and arms.
So, it's not a main operator.
But if I got, you know, well, I guess.
Well, let's say I'm inside a tent.
The tent would be a main operator, because it covers all of me, okay?
So, it looks like the main operators here, and this will help later.
If I say S and P are the main operators, the only operator and, right?
If I say, let's say, number six, not A or B, the main operators or because it applies
to the whole sentence or most of A or B. The tilde is not the main operator,
because it only applies to A, not to B. Look at number two, if S and P, then Q or R. The little
and dot here only applies to S and P not to Q and R. Therefore, it's not the main operator.
The little wedge, Q or R, only applies -- between Q and R, only applies to Q and R not
to S and P. So, it's not the main operator.
Here is the main operator, the little horseshoe, because it connects the whole.
So, once I know the value of the horseshoe, whether true or false,
that I know the value of this whole statement.
Once I know the value of this and, the number one, this dot,
I know the value of this whole statement.
Okay, so, on this slide, I'll show you the answers, and there they are.
Here's the main operators, and you can kind of tell the main operator will always be outside
of the parentheses, if you have parentheses,
more than one parentheses, then this one is complex, right?
All these little brackets and stuff.
But if you think it through, I think you'll see
that the tilde is what applies to the whole statement.
None of these apply to the whole statement.
The and -- because of the tilde here only applies to P or Q,
not to M or B. So, the and is the main operator.
Okay, the next one is the conjunction.
The conjunction is P and Q. so, I'm wearing a blue shirt and I'm clogging.
P represents blue shirt.
Q is I'm clogging.
So, if both are true, I'm wearing a blue shirt and I'm clogging, then the conjunctive sentence,
the main operator of P and Q, is true.
But if anyone of those is false, either P or Q is false, then it's false.
So, if I'm not clogging but I'm wearing a blue shirt, then this P and Q is false, right?
So, they both have to be true in order for the dot to be true.
And this is something that you've got to memorize.
You'll be using it over and over again, and it's helpful to talk through it,
like with my example with blue shirts and clogging.
Let's do the wedge now.
The wedge, over here, the main operator for P or Q, this is true unless both disjunct or false,
and one disjunct is P and one is Q. Okay, now before I even get started,
I'm going to explain truth tables in the next lesson, but notice when you have two letters,
you're going to have four rows, because you're giving all possible combinations.
It's like flipping two coins, twice, right?
You're going to have four possible combinations, heads/heads,
head/tails, tails/heads, and tails/tails.
So, that's what we're capturing here, all possible combinations.
We'll get to that later, but P. I'm wearing a blue shirt or I'm clogging.
If they're both true, it's true.
It's what we call an inclusive or, and then, as long as one is true,
either I'm wearing a blue shirt or I'm clogging, then this or, the wedge, is true.
But if I'm not wearing a blue shirt and I'm not clogging, then the wedge,
of course, combining them, is false.
Okay? Now, sometimes when you have an or sentence in English
like I'm in Austin or Orlando, Florida.
I'm in Austin Texas Orlando Florida.
You can't represent that with A or O, because you can't be in both, right?
So, this first line wouldn't apply.
I can't be in both right now.
So, we'll learn later that you can still represent that sentence by saying A or O,
but it's not the case that A and O, and we'll get to that later, okay.
The next one is a conditional.
All right, so this is called material implication.
When you see if/then sentences, they will probably be expressed as if P then Q,
and the best way to remember this is
that it's always true unless you have a true antecedent and a false consequence.
And I did a little video on conditionals that you can check out later, but, so,
if it's raining then the roads are wet, okay?
So, P represents raining, Q the roads are wet.
Both are true, it's true.
But if it's raining and the roads aren't wet, then my statement must be false.
Now if it's not raining and the roads are wet, that's true, right?
If the antecedent is false, then the conditional is going to be true.
It's kind of counterintuitive.
So, Hurley [assumed spelling] suggest you use the example of if I make an A
on the final, then I'll ace the course.
Okay? Now, if it's false that you make in a on the final but still true
that you ace the course, well, the teacher didn't lie to you.
So, it could very well be true.
Or if it's false that you made an A on the final and false that you made an A in the course,
then again, the teacher didn't lie to you.
So, then if P then Q is true here now again, there's some English statements
that involve causation with if-then P then Q that we just need to capture a different way.
So, you have to be careful, but for now, just memorize the truth table,
and we'll get to those more complicated ones later.
Finally, there's a biconditional, and this is P, and then you see three lines and Q. Okay,
so this is P if and only if Q, and when I see this symbol, I think of if P horseshoe Q
and if Q horseshoe P. That's basically what it means, but the bottom line is
that this is true whenever P and Q have the same truth value.
So, I'll jump over a cliff if, and only if, you do, or off a cliff, right?
So, if you jump over, then I will, right, and vice versa.
So, now if you don't jump over, then I won't so it's true if they're both false,
P or Q are both false, and it's true if P and Q are both true.
But if you jump over and I don't, then it's false.
Or if I jump over and you don't then it's false.
So, it's a pretty easy one to remember, right?
Okay, so in the next video, I will go over well-formed formulas
and gives some more practice on determining the truths of compound statements.
Thanks.