Lecture 4-2 - Propositions & Propositional Connectives (24:18)

In the last lecture, I said that, we were
going to spend this unit learning the

rules for evaluating deductive arguments.
Now this week, we're gonna learn the rules

for evaluating deductive arguments that
involve, what I'm gonna call propositional

connectives. Now, what are propositional
connectives? Well, in order to explain

what propositional connectives are. I
first have to tell you what propositions

are and then I can talk about connectives.
So first, propositions. What's a

proposition? A proposition, is the kind of
thing that can be true, or false and that

can serve as the premise, or the
conclusion of an argument. Here let me

give you an example. See this book this
book is not a proposition. It can't be

true or false, and it can't serve as the
premise or the conclusion of an argument.

See this hand, this hand is not a
proposition it can be true or false, and

it can serve as the premise or the
conclusion of a argument. But now suppose

I say, the book is in my hand. Now what I
just said, that the book is in my hand is

a proposition, it can be true in fact, it
is true, or it can be false and right now

it is false. It can also serve as the
premise of an argument. I could say, the

book is in my hand, therefore, my hand is
not free to shake yours. And it could

serve as the conclusion of an argument. I
could say, you just gave me the book and I

haven't let go of it, therefore, the book
is in my hand. I just told you what

propositions are. But what's a
propositional connective? A propositional

connective is something that takes
propositions and makes new propositions

out of it. Let me give you an example to
illustrate. Consider a proposition, the

book is under my hand. Now consider the
proposition, my foot is under the book. We

can combine those two propositions using
the propositional connective end to make

the new proposition. The book is under my
hand, and by foot is under the book. Now

what I've just said that the book is under
my hand and my foot is under the book,

that's a proposition. It's the kind of
thing that could be true or false. For

instance, right now it's true and right
now it's false. It's also the kind of

thing that can be premise or the
conclusion of the argument. For instance,

I could say, what you're seeing right now
is really happening, therefore, the book

is under my hand and my foot is under the
book. Or I could say, the book is under my

hand, and my foot is under the book.
Therefore, my foot is under my hand. So

you see that the book is under my hand,
and my foot is under the book. That's the

kinda thing that can be true or false and
it's the kind of thing that can be the

premise, or the conclusion of an argument.
So it's a proposition, but it's a

proposition that we made by combining two
other propositions. See how propositional

connectives work? They're beautiful,
aren't they? I just gave you an example of

a propositional connective. I called it
the proposition connective and. But in

English, the word and can be used in
different ways. It's not always used as a

proposition connective. Let me give you
different examples of how and can be used.

Think about the sentence, Jack and Jill
finally talked. Okay now, there are three

different ways to understand what that
sentence is saying. Jack and Jill could be

the name of a fast food company that
serves a special stew that's very popular

with its patrons. Now, maybe lawyers have
been wondering what the ingredients are in

Jack and Jill's special stew because many
of Jack and Jill's patrons have been

coming down with an unusual disease. So,
lawyers have been asking Jack and Jill to

disclose what's in their stew. Jack and
Jill has been refusing to do so. But

finally, the spokesman for Jack and Jill
discloses what's in their stew. I might

tell you about that situation by saying
Jack and Jill finally talked. There, I'm

telling you there's a particular company,
a company called Jack and Jill, and that

company finally talked through their
spokesperson. Here's a second way to

understand the sentence, Jack and Jill
finally talked. Suppose that Jack and Jill

ar e a couple, and recently they've been
going through a tough time. They've been

angry and resentful towards each other.
And they haven't talked about what's been

bothering them. Now, I might tell you,
Jack and Jill finally talked. And what I

mean by that is, that they finally talked
to each other about what's bothering them.

Now that's different, from the first
example in which I said, Jack and Jill

finally talked. There, I was saying that a
particular thing the company, Jack and

Jill. Finally talked about the ingredients
in its special stew. But in this second

example, I'm saying that two things, Jack
and Jill finally talked to each other. Let

me give you a third way to understand the
sentence Jack and Jill finally talked.

Imagine that Jack, Jill, and Roger are
having a silence contest. They're having a

contest to see who can go for the longest
period of time without talking. And I'm

watching them to see who wins. Well, you
call be periodically and ask me as anyone

of them talked? And for the first few
hours I might say to you nope, none of

them has talked yet. And then at one point
you call me and you ask me, has anyone of

them talked? And I say well Jack and Jill
finally talked. Now here I'm not saying

that Jack or Jill talked to each other.
I'm saying that Jack finally talked, and

Jill finally talked. So, when I say Jack
and Jill finally talked. I'm expressing a

proposition that's made up of two other
propositions. The proposition that Jack

finally talked, and the proposition that
Jill finally talked. When I say Jack and

Jill finally talked, I'm using the word
and, as a propositional connective. It

takes two propositions. First, that Jack
finally talked, adn second that Jill

finally talked. And it combines those two
propositions into a third proposition. The

proposition that, Jack and Jill. Finally
talked. Now that's a third way to

understand the sentence Jack and Jill
finally talked and it's different from the

first two. But that's the only way of
understanding the word and, so that it's a

propositional conn ective in that
sentence. So I've just explained to you

what propositional connectives are. I gave
you an example of a propositional

connective and, and I showed how the
English word and, can sometimes be used as

a propositional connective but sometimes
not. Now the English language like every

other natural language, contains lots and
lots of different phrases that can be used

as propositional connectives. For
instance, consider the phrase, I believe

that. You tell me Jack and Jill finally
talked. Well, I might say I believe that

Jack and Jill finally talked. There, I'm
taking one proposition Jack and Jill

finally talked, and attaching the phrase
to it, I believe that to make another

proposition namely, I believe that Jack
and Jill finally talked. Or consider the

phrase, I hate it when. You might tell me,
it's raining, and I might say, I hate it

when it's raining. See there, I take the
proposition it's raining, and I attach a

phrase to it, I hate it when, to make
another proposition, I hate it when it's

raining. So these are other examples of
how phrases in English, I believe that, or

I hate it when, can be used as
propositional connectives. They can take

other propositions and make new
propositions out of them. But and is a

very different kind of propositional
connective from I believe that or I hate

it when. And it's different in two ways.
One way it's different is that it makes a

new proposition out of two other
propositions, not just one other

proposition. So when you make a new
proposition by attaching I hate it when to

another proposition, what you're doing is
turning one proposition into another

proposition. But when you make a new
proposition by combining two other

propositions using the word and, you're
making a new proposition out of two other

propositions. Jack finally talked, and
Jill finally talked. You make the new

proposition, Jack and Jill finally talked.
That's one way in which the propositional

connective and is different from the
propositional connective, I hate it when.

But there's a second wa y, in which the
propositional connective, and, differs

from the propositional connective, I hate
it when. And that is, that when you use

the propositional connective and, to
combine two other propositions into a new

proposition. Whether that proposition is
true or false doesn't depend upon anything

other than whether the two original
propositions that you used to build it

were true or false, that's all it depends
upon. Let me illustrate this point using

something we call a truth table. A truth
table is a way of representing various

possible situations, and how the truth of
a proposition depends upon which of those

various possible situations is real. For
instance, consider the two possibilities

there are with Jack finally talked. The
proposition Jack finally taught, could be

true. Or it could be false. The
proposition Jill finally talked could be

true or it could be false. So, there are
four possibilities we have to consider.

Either Jack finally talked is true and
Jill finally talked is true. Or Jack

finally talked is true and Jill finally
talked is false. Or Jack finally talked is

false and Jill finally talked is true. Or
finally, Jack finally talked is false and

Jill finally talked is false. Those are
the four possible situations. Now, if the

first of those four situations is the real
situation, so it's true that Jack finally

talked and it's true that Jill finally
talked, then is it going to be true or

false that Jack and Jill finally talked?
It's going to be true. All right, cuz Jack

finally talked, and Jill finally talked,
so Jack and Jill finally talked. Now

suppose it's true that Jack finally
talked, but it's false that Jill finally

talked. Then is it going to be true or
false that Jack and Jill finally talked?

It'll be false, and suppose it's false
that Jack finally talked, but it's true

that Jill finally talked. Then, is it
going to be true or false that Jack and

Jill finally talked, again it's going to
be false. And finally, suppose it's false

that Jack finally talked and it's also
false that Jill final ly talked. Then is

it going to be true or false that Jack and
Jill finally talked? Well obviously in

that situation, it's going to be false
that Jack and Jill finally talked. So what

this truth table demonstrates is that the
truth of the proposition Jack and Jill

finally talked. Just depends, it depends
on nothing other than the truth of the

proposition Jack finally talked, and the
truth of the proposition Jill finally

talked. In other words, the truth of the
proposition that we've use the

propositional connective and to build.
Depends on nothing other than the truth of

the two ingredient propositions that we
connected by means of the propositional

connective and. Because the propositional
connective and, works that way because it

builds new propositions whose truth
depends on nothing other than the truth of

the ingredient propositions that go into
building them. That kind of propositional

connective is one that we're going to call
a truth functional connective. Now and,

the propositional connective and, is a
truth-functional connective. But not all

propositional connectives are
truth-functional connectives. For

instance, suppose we try to construct a
truth table for I hate it when, the

propositional connective I hate it when.
Well, so consider the proposition, it's

raining. Now that proposition could be
true or it could be false. Sometimes its

true, sometimes its false. So lets
consider these two possible situations. So

suppose the proposition Its raining is
true. In that situation is it going to be

true or false that I hated when its
raining. Could be either one, it could be

raining even though I enjoyed the rain or
it could be raining even though I hate the

rain. Or it could be raining even though
I'm indifferent to the rain. So the truth,

of I hate it when it's raining isn't
determined by its raining. So, if it's

true that it's raining. It's unclear
whether, I hate it when it's raining.

Could be true, could be false. Suppose
it's false, that it's raining. Then, is it

going to be true or false that I hate it
when it's raining. Again, could be either

one. The truth of I hate it when it's
raining isn't determined by the falsehood

of it's raining. So, even if it's not
raining, that doesn't mean anything one

way or the other for whether I hate it
when it's raining. So once again, if it's

false that it's raining, I hate it when
it's raining, could be true or could be

false. So the proposition, I hated when
it's raining. Whether that proposition is

true or false doesn't just depend on the
truth or the falsehood of the proposition

it's raining, that you built this
proposition out of using the propositional

connective, I hate it when. Because of
that, the propositional connective I hate

it when is not a truth-functional
connective. It's different from the

propositional connective and which is a
truth-functional connective. A moment ago,

we built a truth table for a proposition
that was built using a truth-functional

connective, specifically the
truth-functional connective and. But I'd

like us to notice something about that
truth table. Notice that if we replace the

particular propositions that we are
putting together using the

truth-functional connective and, to make a
different resultant proposition. Even if

we change the ingredient propositions the
truth table looks the same. Let me show

you what I mean. Suppose instead of having
Jack and Jill finally talked. We have Jack

finally walked and Jill finally talked. So
now, we're connecting two different

propositions using the truth-functional
connective and. There's Jack finally

walked, there's Jill finally talked. And
then we connect them up into Jack finally

walked and Jill finally talked. Okay, now
the truth of that resultant proposition,

Jack finally walked and Jill finally
talked. How does that depend on the truth

or false sort of the ingredient
propositions?Jack Jack finally walked and

Jill finally talked. Well it's the same
patent we saw earlier. If it's true that

Jack finally walked, and it's also true
that Jill finally talked. and it's also

true that Jill finally tal ked.
Then it's going to be true that Jack

finally walked and Jill finally talked. If
it's true that Jack finally walked, but

it's false that Jill finally talked, then
it's going to be false that Jack finally

walked and Jill finally talked. If it's
false that Jack finally walked but it's

true that Jill finally talked, then it's
going to be false that Jack finally walked

and Jill finally talked. And if it's false
that Jack finally walked and it's false

that Jill finally talked, then it's going
to be false that Jack finally walked and

Jill finally talked. So even if we change
one of the ingredient propositions as long

as we're combining propositions using the
truth functional connective and, the

overall truth table looks the same. We
could change them some more to illustrate

this point. I suppose if we changed Jill
finally talked to the zebra escaped.

Change it here, so notice what we have
here. We take two propositions, the

proposition Jack finally walked, and the
proposition the zebra escaped. And we put

them together with the truth functional
connective, and to create a resultant

proposition Jack finally walked and the
zebra escaped. Now, when is that

proposition gonna be true? Well, again it
depends just on, when these propositions

are true. So if it's true, that Jack
finally walked, and it's also true that

the zebra escaped, then it's going to be
true that Jack finally walked and the

zebra escaped. If it's true that Jack
finally walked, but it's false that the

zebra escaped, then it's going to be false
that Jack finally walked and the zebra

escaped. If it's false that Jack finally
walked and it's true that the zebra

escaped, then it's going to be false that
Jack finally walked and the zebra escaped.

And finally, if it's false that Jack
finally walked and it's false that the

zebra escaped, then of course it's going
to be false that Jack finally walked and

the zebra escaped. So once again, same
truth table even if we change the

ingredient propositions that we're putting
together with the functional connective an

d to make the resultive proposition. Now,
since the truth table stays the same even

when we change these propositions up on
top. We could represent that fact, by

replacing these propositions altogether
with variables that can range over any

proposition. So, for instance, instead of
saying Jack finally walked, we could just

have a variable here call it P1 Our first
proposition. Instead of saying the zebra

escaped we can have a variable there, call
it P2 our second proposition. And finally,

when we put those two propositions
together using the truth functional

connective and. We'll have P1 one and. P2.
So that's going to be our resultant

proposition, P1 and P2. And whatever
exactly that is, is going to depend of

course on what P1 is and what P2 is. But
whether this third proposition is true or

false again is only going to depend on the
truth or falsehood of P1 and of P2, when

P1 whatever exactly that is, is true and
P2 is true then. The proposition p1 and p2

is going to be true. Whatever proposition
that is, is going to be true. And in every

other possible situation, that proposition
is going to be false. So no matter what

proposition we have for P1 and P2 their
conjunction P1 and P2 is going to be true,

just in those situations when P1 and P2
are both true. That's the truth table for

the truth functional connective and, which
we'll also call conjunction. In the next

lecture, we're going to see how we can use
the truth table for the truth functional

connective and. To figure out the rules
for evaluating deductive arguments that

rely on the truth functional connected
and. And in the following three lectures,

we'll see how we can use the truth tables
for other truth functional connectives, to

figure out the rules for evaluating
deductive arguments that use those other

connectives. See you in next lecture.