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 me an 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 cuz
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 disjunc tion, 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 fo r 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 yo u 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 t rue 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 an y 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 part icular argument that uses
disjunction is a valid argument or not.