In the last lecture. We talked about the truth functional
connective, conjunction. We gave the truth table for conjunction.
And we showed how we could use the truth table for conjunction to figure out which
inferences that use conjunction are valid and which inferences are not.
Today, we're going to talk about the truth functional connective, disjunction.
We're going to give the truth table for dis-junction, and we're going to show how
we can use that truth table to figure out which inferences that use dis-junction
are valid and which are not. Now in English,
we usually express disjunction by using the word or but the word or can be used
in a couple different ways in English. For instance, suppose that Manchester is
playing Barcelona tonight and you ask me, who's going to win?
And I say, well, I have no idea who's going to win but I can tell you this,
it's going to be Manchester or Barcelona. Now, what I'm suggesting when I say, it's
going to be Manchester or Barcelona, is that it's not going to be both.
Manchester might win, Barcelona might win.
But there's no possible way that both of them are going to win.
Sometimes, in English, when you want to say that it's going to be one thing or
the other, but not both, you say, either or either Manchester is going to win, or
Barcelona is going to win. But sometimes when we use the word or, we
mean it could be one, or the other, or both.
So for instance, suppose you ask me what we should have for dinner tonight and I
say well we could have chicken or fish. Well there's no suggestion that we
couldn't have both maybe we could have a little bit of chicken and a little of
fish. So it has to be chicken or fish or both.
When I say chicken or fish, I'm not suggesting it can't be both.
Sometimes in English we use the phrase and, or to express that it could be one
or the other or both. I'll say, we could have the chicken and,
or the fish. The truth functional connective
disjunction is expressed by the second meaning of or.
It's expressed by the English phrase and, or where you mean it could be one or the
other or both. That's what we're going to call
disjunction in this class. Now let's look at the truth table for
disjunction. So lets look at the truth table for
dis-junction. Suppose you're using disjunction to
combine the propositions We eat chicken and we eat fish into the disjunctive
proposition We eat chicken or fish. Well when is that disjunctive proposition
going to be true? If it's true that we eat chicken, and
it's true that we eat fish, then it's going to be true that we eat chicken or
fish because remember, when we use or here, we don't mean either or, but not
both. We mean and or.
Could be one, could be the other, or could be both.
So if it's true that we eat chicken and it's true that we eat fish, it's going to
be true that we eat chicken or fish. Now supposed it's true that we eat
chicken, but its false that we eat fish. Well.
Then, it's still going to be true that we eat chicken or fish.
Suppose it's false that we eat chicken, but true that we eat fish.
Then, it's still going to be true that we eat chicken or fish.
But suppose it's false that we eat chicken and it's also false that we eat
fish. Then, is it going to be true that we eat
chicken or fish? No!
Because we won't be eating either. So then it'll be false that we eat
chicken or fish. This is the truth table for disjunction.
And, like the truth table that we saw for conjunction, it's going to work no matter
what propositions we put into here, or here, or here.
So, no matter what proposition you have right here, call it P1.
And, no matter what proposition you have right here, call it P2.
When you use the truth functional connective disjunction.
To create a new proposition out of those two proposition's, so you got a new
proposition P one or P two. That new disjunctive proposition is going
to be true. Whenever P1 is true, and it's also going
to be true whenever P2 is true. So unlike conjunction.
Where you need both of the two ingredient propositions to be true in order for the
conjunctive proposition to be true. In disjunction, you only need for one of
the of the two ingredient propositions to be true in order for the disjunctive
proposition to be true. The disjunctive proposition is false only
when. Both of the two ingredient propositions
are false. That's the only time a disjunction is
false. So now, let me give you an example, of
how you can use the truth table for disjunction.
Just show that a particular kind of argument is valid.
We're going to discuss, a kind of argument that is sometimes known.
As process of elimination. Here's how it goes.
Suppose, that you have to solve. A murder mystery.
Mister Jones, has been stabbed in his living room.
With a knife in the back. Now, you figured out that there were only
two people in the house at the time of his stabbing, the butler and the
accountant. You also know that the knife is
positioned in Mr. Johnson's back in such a way that he
couldn't possibly have stabbed himself. So it had to be someone else.
And whoever else it was it had to be someone who's in the house at the time of
the stabbing. So it could only have been, the butler or
the accountant, or maybe both. So you know that the butler did it, or
the accountant did it. Now you find out that the accountant is a
quadriplegic, so the accountant couldn't have stabbed Mr.
Jones in the back. So now you know that the account didn't
do it. And so, from the two premises, the butler
did it, or the accountant did it. And the accountant didn't do it.
You can conclude, the butler did it. Now, why is that argument valid?
Here's why. Think about the truth table for
disjunction again. So remember the first premise, the butler
did it or the accountant did it is a disjunction.
It's going to be true whenever one of it's disjuncts is true, one of it's
ingredient propositions is true. So it's going to be true whenever the
butler did it, and it's going to be true whenever the butler did it.
The second premise tells you that the accountant didn't do it.
So the only way for the first premise to be true, given that the accountant didn't
do it, is for the butler to have done it. And so you know, since the accountant
couldn't have done it. That the only way for the dis-junction,
the butler did it or the accountant did it to be true, is for the butler to have
done it and that's why you can conclude the butler did it and your argument is
valid. That's one example of a process of
elimination argument. Of course there are lots of others, but
with all of those others you can see why they are valid by looking at the truth
table for dis-junction. Remember how you can use the truth
functional connective conjunction to build a new proposition out of not just
two other propositions but sometimes three other propositions.
You can conjoin one proposition with a second and with a third.
Well, you can do the same thing with disjunction.
You can disjoin one proposition with a second and a third, to create the
proposition. Either this, or that or the other or any
combination of the three. What does the truth table for that look
like? Here it is.
The disjunctive proposition, P1 or P2 or P3, is going to be true.
Whenever P1 is true, it's also going to be true whenever P2 is true.
And it's also going to be true whenever P3 is true.
In fact, the only time that P1 or P2 or P3, the only time that, that disjunctive
proposition is going to be false is when all these ingredient propositions are
false. So here's what the truth table for P1, or
P2, or P3 looks. Now let's use the truth table for our
triple disjunction to show how a particular process of elimination
argument can be valid. Let's go back to our murder mystery in
order to do that. Now suppose that you find out contrary to
what you had previously believed, that Butler and the accountant were not the
only people in the house, at the time of Mr.
Jonathan's death. In addition, the maid was in the house
and the cook was in the house. Alright.
Well, now, you know, that the butler or the maid or the cook did it.
We don't yet know which of them did it, but we know that the butler or the maid
or the cook did it. Now suppose that you find out that the
maid and the cook, at the time of the stabbing we're off in the opposite corner
of the house doing something else together.
Well now you know, that the maid didn't do it.
And you know that the cook didn't do it. So what can you conclude from those three
premises? Premise one, the butler or the maid or
the cook did it. Premise two, the maid didn't do it.
And premise three: the cook didn't do it. Well, lets use the truth table to figure
this out. Premise one of the truth table tells you
that the butler or the maid or the cook did it.
So the situation in which it falls that the butler or the maid or the cook did it
that situation is ruled out by premise one.
So premise one tells you at that situation is not the actual situation.
Premise two tells you that the maid did not do it.
So any situation in which its true that the maid did it is also not the actual
situation. So this situation is one in which its
true that the maid did it so that's not the actual situation according to premise
two. This situation is one in which its true
that the maid did it. So that's not the actual situation
according to premise two. This situation is one in which its true
that the maid did it. So that's not the actual situation
according to premise two, and this situation is one in which it is true that
the maid did it. So that's not the actual situation
according to premise two. Premise three tells you that the cook
didn't do it. So, that rules out any situation in which
it's true that the cook did it. Well, here's a situation in which it's
true that the cook did it. So, that situation is ruled out by
premise three. And, here's a situation in which it's
true that the cook did it. So, that situation is ruled out by
premise three. So, premise one rules out this situation.
Premise two, rules out this, this, this and this situation.
And premise three, rules out this, this, this and this situation.
Well, whats left? The only situation left that could be the
actual situation is this one. See cause in this situation, it's true
that the butler or the maid or the cook did it just as premise one tells us.
Its false that the maid did just as premise two tells us, and its false that
the cook did just as premise three tells us.
But, that's the situation in which it's true that the butler did it.
So, the conclusion that we can draw, based on the situations that are ruled
out by premises one, two, and three, is that the actual situation is this one,
and in that actual situation, it's true that the butler did it.
So, the butler did it That's why the process of elimination reasoning that we
just considered is valid. If premise one says, the butler or the
maid or the cook did it. Premise two says the maid didn't do it,
and premise three says that the cook didn't do it.
Then by process of elimination we can draw the valid conclusion that the butler
did it and this is why. Let me give you another example of how
you can use the truth table for disjunction in order to show whether or
not the process of elimination argument is valid.
Suppose we know that Walter is a professional football player.
Well, that means that he plays either American football, U.S.
Football, or European football, which Americans call soccer, or Australian
rules football. But now suppose we find out that Walter
does not play American football. And you conclude from that, that he must
play European football. So you argue as follows.
Premise 1- Walter plays either American football or European football or
Australian Rules football, premise 2- he does not play American football and
therefore you conclude he plays European football.
Well, that argument is invalid and we can use the truth table for disjunction to
show why it's invalid. Look at this truth table.
Premise one, recall, is that Walter plays American or European or Australian rules
football. So premise one rules out the situation in
which it's false that Walter plays American or European or Australian rules
football. And that's all it rules out.
It just rules out the situation in which it's false that Walter plays any of
those. Premise two, Walter doesn't play American
Football. That rules out the situation in which
it's true that Walter plays American football.
So it rules out this situation. And rules out this situation.
And it rules out this situation. And it rules out this situation.
So, premise one rules out the situation represented at the bottom.
Premise two rules out the situations represented by these four columns at the
top. So, can we conclude that Walter plays
European football? No.
He might play European football but he might also play Australian Rules
football. He's looked all the premise one and
premise two together rule out is these five situations.
But there is no three situations that are possible.
In one of them Walter plays both European football and Australian rules football.
In another one of them, Walter plays European football, but not Australian
rules football, and in the third situation, this left open by premises one
and two, it's false that Walter plays European football but true that he plays
Australian rules football. So based on the information that premises
one and two give us, we cannot conclude that Walter plays European football.
He might play Australian rules football instead.
So the argument that you made is invalid. In the next lecture, we're going to
consider a truth functional connective that's different from conjunction and
disjunction in the following way. While conjunction and disjunction are
connectives that can be used to build propositions out of two or more other
propositions. Negation, the connective that we'll talk
about next time, is the connective that is used to build new propositions out of
just one single other proposition. Negation, in other words, is a connective
that you apply to one proposition to build a second proposition.
And that's what we'll talk about next time.
Now, there's some exercise's for you to do.
These exercises test your understanding of the truth table for disjunction and of
how the truth table for dis-junction can be used to determine whether a particular
argument that uses disjunction is a valid argument or not.