Lecture 4-3 - Truth Functional Connectives Conjuction (16:11)

In the last lecture I explained what
propositional connectives are, I described

a particular category of propositional
connectives that we called truth

functional connectives, and I gave you an
example of one truth functional

connective. And, another word, and in
English isn't always used to mean a truth

functional connective, but sometimes it
is. And one thing I'd like to point out

right now, is that there are other words
in English that can be used to indicate

the very same truth functional connective
that the word and is used to indicate. For

instance, think about the English words
also, moreover, furthermore, and but. Now,

you might think the word and, and the word
but" mean two very different things. If I

say Walter is poor and happy, that seems
to mean something very different from

Walter is poor but happy. In particular,
when I say Walter is poor but happy, I'm

suggesting that there is contrast between
his poverty and his happiness. But when I

say Walter is poor and happy, I'm not
suggesting any such contrast. Still,

whatever contrast there might be between
his poverty and his happiness doesn't

effect the truth table for the truth
functional connective but. Let's consider

when it would be true to say Walter is
poor but happy. To show you what I mean,

about the words" but and, and," let's go
back to the truth table for the truth

functional connective and. So remember, if
you have two propositions, p1 and p2, and

you use the truth functional connective
and to put them together to make another

proposition, the proposition p1 and p2.
And now you wanna know when is that new

proposition, the proposition p1 and p2,
when is that going to be true? Well, the

answer is it's going to be true only when
p1 is true and p2 is true. In any other

scenario, the proposition p1 and p2 is
gonna be false. Let's take an example so I

can illustrate. Let's suppose, for p1 we
use the proposition Walter is poor, and

for p2 we use the proposition Walter is
happy, then we use the truth functional

connective, and, to put those put two
propositions together into a new

proposition and the new proposition is
gonna be Walter is poor and happy. Okay.

Now, when it going to be true that Walter
is poor and happy? Well, if it's true that

Walter is poor and it's also true that
Walter is happy then its going to be true

that Walter is poor and happy. But, if
it's false that Walter is poor, then it is

not going to be true that Walter is poor
and happy. And if it's false that Walter

is happy, then it's not going to be true
that Walter is poor and happy. So the

proposition Walter is poor and happy is
gonna be true, only when Walter is poor is

true and Walter is happy is true. In any
other possible scenario, the proposition

Walter is poor and happy will end up being
false. So, lets compare that to the

proposition that we get by combining
Walter is poor and Walter is happy with

the connective, but, Walter is poor but
happy. Now, when is it gonna be true to

say Walter is poor but happy? Well, it's
not gonna be true to say Walter is poor

but happy in any situation where it's
false that Walter is poor. Right? If it's

false that Walter is poor, then it's also
gonna be false that Walter is poor but

happy. It's also not gonna be true to say
Walter is poor but happy in any situation

where it's false that Walter is happy. If
it's false that Walter is happy, then it's

gonna be false that Walter is poor but
happy. So when is it going to be true that

Walter is poor but happy? The only
possible situation where it could be true

is the situation where it's true that
Walter is poor and it's also true that

Walter is happy. Now, you might think,
wait a second. When I say Walter is poor

but happy, I'm saying more than just that
Walter is poor and that Walter is happy.

I'm also suggesting a contrast between his
poverty and his happines. And maybe that

suggestion is misleading, maybe poor
people are often happy. But notice, what

you say can be misleading even if it's
true. For example, suppose someone comes

up to me with a car that's sputtering.
They might say, do you know where there's

a gas station around here? I need to fill
up this car with gas. And I might say,

there's a gas station just around the
corner. Now, what I say might be true,

there might really be a gas station just
around the corner even if I know that,

that gas station has been closed for three
years and has no gas. So, what I say is

misleading because I've lead them to
believe falsely, that they can get gas if

they can just get their car around the
corner. But even though what I've said is

misleading, it's still true because there
is a gas station around the corner, only a

closed one. So what you say can be true
but misleading and I suggest that when you

say Walter is poor but happy. That can be
true even if it's misleading to suggest

that poverty and happiness are somehow at
odds with each other. I've just said that

the word but in English can be used to
indicate the same truth functional

connective that the word and is sometimes
used to indicate. And there are other

words in English that can be used to
indicate that same truth functional

connective, also, furthermore, moreover,
and sometimes we even use the word too,

too. But now I wanna introduce a term
that's going to describe that truth

functional connective no matter what word
in ordinary language we use to indicate

that connective. The term is conjunction.
And the term conjunction, as I'm using it

here and as philosophers use it, is not
the same term that grammarians use when

they talk about conjunctive terms like
but, or, and, therefore. Here's something

that could help you understand what
conjunctions in the grammarian sense are

like. All of those terms are conjunctions
in the grammarian sense, but they're not

conjunctions in the philosopher's sense. A
conjunction in the philosopher's sense is

just the truth functional connective that
has this particular truth table. You can

use the conjunction to create a new
proposition out of joining two other

propositions and that new proposition that
you create using conjunction is gonna be

true, only when the other two propositions
are true. In any other case, the new

proposition is going to be false, that's
what a conjunction is. And we can use the

symbol ampersand, like that, in order to
signify conjunction. Now that we know the

truth table for the conjunction, let's
consider how we can use that truth table

to figure out when an argument that uses
conjunction is valid. Consider the

argument Walter is poor but happy,
therefore, Walter is happy. Is that

argument valid or invalid? Well, pretty
obviously, that argument is valid. There's

no possible way for the premise to be true
while the conclusion is false. But, can

you see why the argument is valid using
the truth table for conjunction? You

should be able to in a situation in which
the premise is true, Walter is poor but

happy, there are gonna have to be two
other propositions that are true, namely

Walter is poor and Walter is happy. So, if
its true that Walter is true but happy,

then its gonna have to be true that Walter
is happy and that's why the argument is

valid. That's why there is no possible way
for the premise to be true while the

conclusion is false. Let's consider some
other arguments that involve conjunction.

Consider the argument Walter is poor,
walter is happy, therefore, Walter is poor

and happy. Is that argument valid? Clearly
it is. And again, you can use the truth

table for conjunctions to see why it's
valid. In a situation where the first

premise Walter is poor is true, and in
which the second premise Walter is happy

is true. In that situation, the conclusion
Walter is poor and happy, is gonna have to

be true. So there's no possible way for
the premises of that argument to both be

true while the conclusion is false and so
that argument is also valid. Now notice,

just as we can combine two propositions
with each other using conjunction, we can

then also combine the resulting
proposition with another proposition using

conjunction. So, consider the proposition
Walter is poor but happy and popular. That

proposition uses two conjunctions to
combine three other propositions into a

single conjunctive proposition. To
understand how that works, let's look at

the truth table for that. So when is it
going to be true that Walter is poor but

happy and popular? When is that going to
be true? Well, if it's false that Walter

is poor, then it's definitely not going to
be true that Walter is poor but happy and

popular. So in all of these situations
right down here, walter is poor but happy

and popular, is gonna to be false. If it's
false that Walter is happy, then it's

definitely not going to be true that
Walter is poor but happy and popular, cuz

he's not gonna be happy. So, in these
situations right here where it's false

that Walter is happy, it's also gonna be
false that Walter is poor but happy and

popular. And if it's false that Walter is
popular, then of course, it's also gonna

be false that Walter is poor but happy and
popular. So, in this situation right here,

it'll be false that Walter is poor but
happy and popular. So, is it ever gonna be

true that Walter is poor but happy and
popular? Yes. It'll be true just when it's

true that Walter is poor, it's true that
Walter is happy, and it's true that Walter

is popular. That's the only situation when
it's gonna be true that Walter is poor but

happy and popular. In general, this is the
kind of truth table that we get when we

combine three propositions using
conjunction. So now, considered how we can

use the truth table for conjunctions of
three propositions to figure out whether

certain deductive arguments are valid or
not. So consider the following deductive

argument. From the premises Paris is the
capital of France, Jakarta is the capital

of Indonesia, and Washington DC is the
capital of the United States. Let's

conclude Paris is the capital of France,
and Jakarta is the capital of Indonesia,

and Washington D.C. is the capitol of the
United States. Valid or not? Well,

clearly, that argument is valid and the
truth table shows us why. The conclusion

Paris is the capital of France, and
Jakarta is the capital of Indonesia, and

Wash ington D.C. is the capital of the
United States is true just when it's true

that Paris is the capital of France, and
it's also true that Jakarta is the capital

of Indonesia, and it's also true that
Washington D.C. is the capital of the

United States. So whenever the premises
are true, the conclusion is also true, and

that's why that argument is valid. The
truth table explains why the argument is

valid. Now consider a different one. From
the premise Mick Jagger is a singer, a

man, and a septuagenarian. We could draw
the conclusion Mick Jagger is a

septuagenarian. Now is that argument
valid? Yes, it is and the truth table for

conjunction explains why it's valid. Think
about the situation in which it's true

that Mick Jagger is a man, a singer, and a
septuagenarian. The only situation in

which that's true is the situation in
which it's true that Mick Jagger is a man,

it's true that Mick Jagger is a singer,
and it's true that Mick Jagger is a

septuagenarian. But that means that if the
premises is true, then the conclusion has

got to be true. The premise is only true
in a situation in which the conclusion is

true and so that argument has got to be
valid and the truth table for conjunction

explains why. I have said that conjunction
can be used to connect two other

propositions into a new proposition. And
conjunction can also be used to connect

three other propositions into a new
proposition. But there's no limit to the

number of propositions that can be
connected using the truth functional

connective conjunction, or as we could
say, there's no limit to the number of

propositions that can be conjoined. You
can conjoin four propositions, five

propositions, or however many you like,
and notice that there's a pattern to the

truth tables for all of these
conjunctions. In every case, the conjoined

proposition is gonna be true only when all
of the propositions that are conjoined in

it are true. Now, I'ld like you to take
several minutes and look at the following

truth tables, and identify which of these
truth tables are truth tables for

conjunction and which of them are not.
Well, that's it for our discussion of

conjunction and reasoning with
conjunctions. In the next lecture we'll

introduce the topic of disjunction and
reasoning with disjunctions. See you next