Symbolic Logic by teachphilosophy

>> 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.