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