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 going to learn the
rules for evaluating deductive arguments that involve, what I'm going to 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 kind of 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 are 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 connective 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 way, 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, because 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 finally 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 talked. 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 going to 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 and 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.