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 gonna learn the rules
for evaluating deductive arguments that involve, what I'm gonna 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
kinda 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
ar e 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 conn ective 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 wa y, 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, cuz 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 final ly 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 tal ked. 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 gonna 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 an
d 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.