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In the last lecture. We talked about the
truth functional connective, conjunction.

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We gave the truth table for conjunction.
And we showed how we could use the truth

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table for conjunction to figure out which
inferences that use conjunction are valid

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and which inferences are not. Today, we're
going to talk about the truth functional

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connective, disjunction. We're going to
give the truth table for dis-junction, and

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we're going to show how we can use that
truth table to figure out which inferences

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that use dis-junction are valid and which
are not. Now in English, we usually

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express disjunction by using the word or
but the word or can be used in a couple

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different ways in English. For instance,
suppose that Manchester is playing

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Barcelona tonight and you ask me, who's
going to win? And I say, well, I have no

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idea who's going to win but I can tell you
this, it's going to be Manchester or

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Barcelona. Now, what I'm suggesting when I
say, it's going to be Manchester or

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Barcelona, is that it's not going to be
both. Manchester might win, Barcelona

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might win. But there's no possible way
that both of them are going to win.

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Sometimes, in English, when you want to
say that it's going to be one thing or the

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other, but not both, you say, either or
either Manchester is going to win, or

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Barcelona is going to win. But sometimes
when we use the word or, we mean it could

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be one, or the other, or both. So for
instance, suppose you ask me what we

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should have for dinner tonight and I say
well we could have chicken or fish. Well

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there's no suggestion that we couldn't
have both maybe we could have a little bit

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of chicken and a little of fish. So it has
to be chicken or fish or both. When I say

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chicken or fish, I'm not suggesting it
can't be both. Sometimes in English we use

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the phrase and, or to express that it
could be one or the other or both. I'll

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say, we could have the chicken and, or the
fish. The truth functional connective

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disjunction is expressed by the second
meaning of or. It's expressed by the

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English phrase and, or where you me an it
could be one or the other or both. That's

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what we're going to call disjunction in
this class. Now let's look at the truth

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table for disjunction. So lets look at the
truth table for dis-junction. Suppose

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you're using disjunction to combine the
propositions We eat chicken and we eat

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fish into the disjunctive proposition We
eat chicken or fish. Well when is that

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disjunctive proposition going to be true?
If it's true that we eat chicken, and it's

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true that we eat fish, then it's going to
be true that we eat chicken or fish cuz

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remember, when we use or here, we don't
mean either or, but not both. We mean and

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or. Could be one, could be the other, or
could be both. So if it's true that we eat

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chicken and it's true that we eat fish,
it's going to be true that we eat chicken

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or fish. Now supposed it's true that we
eat chicken, but its false that we eat

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fish. Well. Then, it's still going to be
true that we eat chicken or fish. Suppose

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it's false that we eat chicken, but true
that we eat fish. Then, it's still going

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to be true that we eat chicken or fish.
But suppose it's false that we eat chicken

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and it's also false that we eat fish.
Then, is it going to be true that we eat

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chicken or fish? No! Because we won't be
eating either. So then it'll be false that

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we eat chicken or fish. This is the truth
table for disjunction. And, like the truth

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table that we saw for conjunction, it's
going to work no matter what propositions

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we put into here, or here, or here. So, no
matter what proposition you have right

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here, call it P1. And, no matter what
proposition you have right here, call it

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P2. When you use the truth functional
connective disjunction. To create a new

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proposition out of those two
proposition's, so you got a new

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proposition P one or P two. That new
disjunctive proposition is going to be

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true. Whenever P1 is true, and it's also
going to be true whenever P2 is true. So

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unlike conjunction. Where you need both of
the two ingredient propositions to be true

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in order for the conjunctive proposition
to be true. In disjunc tion, you only need

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for one of the of the two ingredient
propositions to be true in order for the

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disjunctive proposition to be true. The
disjunctive proposition is false only

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when. Both of the two ingredient
propositions are false. That's the only

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time a disjunction is false. So now, let
me give you an example, of how you can use

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the truth table for disjunction. Just show
that a particular kind of argument is

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valid. We're going to discuss, a kind of
argument that is sometimes known. As

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process of elimination. Here's how it
goes. Suppose, that you have to solve. A

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murder mystery. Mister Jones, has been
stabbed in his living room. With a knife

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in the back. Now, you figured out that
there were only two people in the house at

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the time of his stabbing, the butler and
the accountant. You also know that the

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knife is positioned in Mr. Johnson's back
in such a way that he couldn't possibly

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have stabbed himself. So it had to be
someone else. And whoever else it was it

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had to be someone who's in the house at
the time of the stabbing. So it could only

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have been, the butler or the accountant,
or maybe both. So you know that the butler

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did it, or the accountant did it. Now you
find out that the accountant is a

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quadriplegic, so the accountant couldn't
have stabbed Mr. Jones in the back. So now

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you know that the account didn't do it.
And so, from the two premises, the butler

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did it, or the accountant did it. And the
accountant didn't do it. You can conclude,

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the butler did it. Now, why is that
argument valid? Here's why. Think about

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the truth table for disjunction again. So
remember the first premise, the butler did

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it or the accountant did it is a
disjunction. It's going to be true

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whenever one of it's disjuncts is true,
one of it's ingredient propositions is

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true. So it's going to be true whenever
the butler did it, and it's going to be

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true whenever the butler did it. The
second premise tells you that the

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accountant didn't do it. So the only way
for the first premise to be true, given

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that the accountant didn't do it, is fo r
the butler to have done it. And so you

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know, since the accountant couldn't have
done it. That the only way for the

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dis-junction, the butler did it or the
accountant did it to be true, is for the

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butler to have done it and that's why you
can conclude the butler did it and your

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argument is valid. That's one example of a
process of elimination argument. Of course

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there are lots of others, but with all of
those others you can see why they are

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valid by looking at the truth table for
dis-junction. Remember how you can use the

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truth functional connective conjunction to
build a new proposition out of not just

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two other propositions but sometimes three
other propositions. You can conjoin one

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proposition with a second and with a
third. Well, you can do the same thing

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with disjunction. You can disjoin one
proposition with a second and a third, to

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create the proposition. Either this, or
that or the other or any combination of

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the three. What does the truth table for
that look like? Here it is. The

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disjunctive proposition, P1 or P2 or P3,
is going to be true. Whenever P1 is true,

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it's also going to be true whenever P2 is
true. And it's also going to be true

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whenever P3 is true. In fact, the only
time that P1 or P2 or P3, the only time

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that, that disjunctive proposition is
going to be false is when all these

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ingredient propositions are false. So
here's what the truth table for P1, or P2,

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or P3 looks. Now let's use the truth table
for our triple disjunction to show how a

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particular process of elimination argument
can be valid. Let's go back to our murder

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mystery in order to do that. Now suppose
that you find out contrary to what you had

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previously believed, that Butler and the
accountant were not the only people in the

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house, at the time of Mr. Jonathan's
death. In addition, the maid was in the

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house and the cook was in the house.
Alright. Well, now, you know, that the

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butler or the maid or the cook did it. We
don't yet know which of them did it, but

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we know that the butler or the maid or the
cook did it. Now suppose that yo u find

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out that the maid and the cook, at the
time of the stabbing we're off in the

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opposite corner of the house doing
something else together. Well now you

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know, that the maid didn't do it. And you
know that the cook didn't do it. So what

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can you conclude from those three
premises? Premise one, the butler or the

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maid or the cook did it. Premise two, the
maid didn't do it. And premise three: the

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cook didn't do it. Well, lets use the
truth table to figure this out. Premise

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one of the truth table tells you that the
butler or the maid or the cook did it. So

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the situation in which it falls that the
butler or the maid or the cook did it that

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situation is ruled out by premise one. So
premise one tells you at that situation is

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not the actual situation. Premise two
tells you that the maid did not do it. So

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any situation in which its true that the
maid did it is also not the actual

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situation. So this situation is one in
which its true that the maid did it so

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that's not the actual situation according
to premise two. This situation is one in

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which its true that the maid did it. So
that's not the actual situation according

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to premise two. This situation is one in
which its true that the maid did it. So

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that's not the actual situation according
to premise two, and this situation is one

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in which it is true that the maid did it.
So that's not the actual situation

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according to premise two. Premise three
tells you that the cook didn't do it. So,

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that rules out any situation in which it's
true that the cook did it. Well, here's a

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situation in which it's true that the cook
did it. So, that situation is ruled out by

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premise three. And, here's a situation in
which it's true that the cook did it. So,

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that situation is ruled out by premise
three. So, premise one rules out this

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situation. Premise two, rules out this,
this, this and this situation. And premise

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three, rules out this, this, this and this
situation. Well, whats left? The only

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situation left that could be the actual
situation is this one. See cause in this

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situation, it's t rue that the butler or
the maid or the cook did it just as

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premise one tells us. Its false that the
maid did just as premise two tells us, and

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its false that the cook did just as
premise three tells us. But, that's the

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situation in which it's true that the
butler did it. So, the conclusion that we

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can draw, based on the situations that are
ruled out by premises one, two, and three,

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is that the actual situation is this one,
and in that actual situation, it's true

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that the butler did it. So, the butler did
it That's why the process of elimination

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reasoning that we just considered is
valid. If premise one says, the butler or

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the maid or the cook did it. Premise two
says the maid didn't do it, and premise

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three says that the cook didn't do it.
Then by process of elimination we can draw

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the valid conclusion that the butler did
it and this is why. Let me give you

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another example of how you can use the
truth table for disjunction in order to

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show whether or not the process of
elimination argument is valid. Suppose we

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know that Walter is a professional
football player. Well, that means that he

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plays either American football, U.S.
Football, or European football, which

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Americans call soccer, or Australian rules
football. But now suppose we find out that

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Walter does not play American football.
And you conclude from that, that he must

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play European football. So you argue as
follows. Premise 1- Walter plays either

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American football or European football or
Australian Rules football, premise 2- he

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does not play American football and
therefore you conclude he plays European

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football. Well, that argument is invalid
and we can use the truth table for

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disjunction to show why it's invalid. Look
at this truth table. Premise one, recall,

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is that Walter plays American or European
or Australian rules football. So premise

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one rules out the situation in which it's
false that Walter plays American or

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European or Australian rules football. And
that's all it rules out. It just rules out

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the situation in which it's false that
Walter plays an y of those. Premise two,

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Walter doesn't play American Football.
That rules out the situation in which it's

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true that Walter plays American football.
So it rules out this situation. And rules

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out this situation. And it rules out this
situation. And it rules out this

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situation. So, premise one rules out the
situation represented at the bottom.

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Premise two rules out the situations
represented by these four columns at the

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top. So, can we conclude that Walter plays
European football? No. He might play

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European football but he might also play
Australian Rules football. He's looked all

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the premise one and premise two together
rule out is these five situations. But

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there is no three situations that are
possible. In one of them Walter plays both

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European football and Australian rules
football. In another one of them, Walter

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plays European football, but not
Australian rules football, and in the

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third situation, this left open by
premises one and two, it's false that

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Walter plays European football but true
that he plays Australian rules football.

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So based on the information that premises
one and two give us, we cannot conclude

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that Walter plays European football. He
might play Australian rules football

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instead. So the argument that you made is
invalid. In the next lecture, we're going

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to consider a truth functional connective
that's different from conjunction and

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disjunction in the following way. While
conjunction and disjunction are

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connectives that can be used to build
propositions out of two or more other

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propositions. Negation, the connective
that we'll talk about next time, is the

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connective that is used to build new
propositions out of just one single other

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proposition. Negation, in other words, is
a connective that you apply to one

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proposition to build a second proposition.
And that's what we'll talk about next

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time. Now, there's some exercise's for you
to do. These exercises test your

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understanding of the truth table for
disjunction and of how the truth table for

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dis-junction can be used to determine
whether a part icular argument that uses

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disjunction is a valid argument or not.