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 want to 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 going to 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
going to be Walter [SOUND] 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 going to 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 going to be true to say Walter is poor but happy?
Well, it's not going to 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 going to be false that Walter
is poor but happy. It's also not going to 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 going to 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 want to 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
going to 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 going to 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 going to 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 going to 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
going to to be false. [SOUND] 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, because he's not going to be happy.
So, in these situations right here where it's false that Walter is happy, it's
also going to be false [SOUND] that Walter is poor but happy and popular.
And if it's false that Walter is popular, then of course, it's also going to 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 going to 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
going to 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 Washington 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 going to 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 lecture.