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

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a particular category of propositional
connectives that we called truth

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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then also combine the resulting
proposition with another proposition using

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

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proposition uses two conjunctions to
combine three other propositions into a

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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number of propositions that can be
connected using the truth functional

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

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

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

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

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

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

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

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

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

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