Lecture 4-4 - Truth Functional Connectives Disjunction (18:58)

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.