Today's lecture is about negation. The negation that we're going to be
talking about today is a truth functional connective.
It differs from disjunction and conjunction in that instead of combining
two propositions into a new proposition, it just converts one proposition into a
new proposition. Negation is expressed in English usually
by the word not. No, no, no, not that kind of knot.
Not, not knot. Now in English, the word not can be used
in a variety of ways. For instance, you might say, hey Rom
catch. Oh, no.
Not the, Not the jacket, not, oh, no, not the pencil, not the sticky pad.
But as we're going to be using the word not, it functions as a truth functional
connective. It takes one proposition, for instance,
the proposition I am drinking coffee. And creates another proposition for
instance the proposition I am not drinking coffee..
[SOUND] Let's consider the truth table for this true functional connective.
If it's true that I'm drinking coffee, then it's going to be false that I am not
drinking coffee. And if it's false that I'm drinking
coffee, then it's going to be true that I'm not drinking coffee.
In general, for any proposition P, whenever P is true, then the negation of
P, call it not P is going to be false. And whenever P is false, then the
negation of P, not P is going to be true. We can abbreviate the negation of P by
using this symbol. That stands for negation and that's the
negation of P. Sometimes, it's not obvious how to negate
a proposition. For instance, consider the proposition,
Walter has stopped beating his dogs. What's the negation of that proposition?
Is it Walter has not stopped beating his dogs?
Well, let's see. If it's true that Walter has stopped
beating his dogs, then it's going to be false that Walter
has not stopped beating his dogs. So far, so good.
But now, what if it's false that Walter has stopped beating his dogs?
Does that mean that it's going to be true that Walter has not stopped beating his
dogs? Well, not necessarily.
Suppose the reason is false that Walter has stopped beating his dogs, is that
Walter doesn't have any dogs or suppose it's false that Walter has stopped
beating his dogs because Walter never did beat his dogs.
In either of those two cases, it would not be true that Walter has not stopped
beating his dogs. So Walter has not stopped beating his
dogs is not the negation of Walter has stopped beating his dogs.
See, if it were the negation of Walter has stopped beating his dogs then we'd
have the truth table for negation right here, but, we don't.
Since it could be false that Walter has stopped beating his dogs, but still, not
true that Walter has not stopped beating his dogs.
So this is not the negation of this. In English, the only phrase I know of
that reliably expresses negations is the phrase it is not the case that....
So if you want to negate some proposition, for instance the proposition
that Walter has stopped beating his dogs. You just apend it is not the case that,
to the beginning of it nd you can reliably negate it.
So it is not the case that Walter has stopped beating his dogs is the negation
of Walter has stopped beating his dogs, because whether Walter has not stopped
beating his dogs, or whether Walter never did beat his dogs, or whether Walter
doesn't even have any dogs, it's still going to be true that it is not the case
that Walter has stopped beating his dogs. So remember, the truth table for negation
show us what it is the negation does. Negation takes a proposition,
call it P, and creates a new proposition, the
negation of P. That's false when P is true and that's
true when P is false, but what is the negation of the negation
of P? Well, applying negation to the negation
of P, we created a new proposition. That's true whenever the negation of P is
false and that's false whenever the negation of P is true.
And notice what this means. This means that the negation of the
negation of P is going to be true when P is true and it's going to be false when P
is false. In other words, double negation cancels
itself out. I just showed you how negation can
combine with itself to cancel itself out. Let me now talk about how negation can
combine with other truth functional connectives.
Remember, this is the truth table for conjunction.
If you want to conjoin two propositions P and Q into the new proposition P & Q, we
end up with this truth table. When P is true and Q is true, then the
conjunction of P & Q is true. When P is true and Q is false, then the
conjunction of P & Q is false. When P is false and Q is true, then the
conjunction of P & Q is false. And when both P and Q are false, then the
conjunction P & Q is false. So this is the truth table for
conjunction. But now, suppose we apply negation to the
conjunction we end up with this, the negation of the conjunction of P & Q.
That's going to be false when the conjunction of P & Q is true and it's
going to be true whenever the conjunction of P & Q is false.
Remember, that's what negation does. It flips the truth values of whatever
proposition it's negating. Okay now, notice that in the truth table
that I just put on the board, I had the negation of P & Q where the conjunction P
& Q is in parentheses. Now why did I use those parentheses to
isolate the conjunction of P & Q and then apply negation to it?
Why didn't I just write it this way, the negation of P & Q?
Well, the reason I didn't do this is because these two expressions mean
different things. This truth table shows you how those two
expressions mean different things. Alright. Here is P and Q and here is the
truth table for their conjunction, P & Q. Okay? Now, the negation of the
conjunction of P & Q is going to be false whenever P is true and Q is true and it's
going to be true in every other situation.
In contrast, the negation of P & Q is going to be false when P and Q are both
true, it's going to be false when P is true and Q is false, it's going to be
true when P is false and Q is true, and it's going to be false when P and Q are
both false. The only situation where the negation of
P & Q is going to be true is the situation in which P is false and Q is
true. Since the truth table for not P & Q is
different from the truth table for not P & Q.
It follows that not P & Q must mean something different from not P & Q.
Let me give you an example. Suppose the proposition P is Walter is
busy working and the proposition Q is Rom is busy working.
Well then, the conjunction of those two is going to be Walter and Ram are both
busy working. Now, the negation of that conjunction is
going to say it's not the case that Walter and Rahm are both busy working.
Now, that statement could be true if Walter's not busy working, and it could
be true if Rom is not busy working, and it could be true if both of them are
not busy working. Now, that's a very different statement
from the statement it's not the case that Walter is busy working and Rom is busy
working. And that statement could not be true if
Ram is not busy working, because part of what you're saying there
is Ram is busy working and Walter isn't. That's a very different statement from
saying it's not the case that Walter and Ram are both busy working.
So, there's an example to illustrate the difference between negating P and then
conjoining that with Q on the one hand or negating the conjunction of P & Q on the
other hand. Now, just as you can apply negation to a
conjunction of propositions, you can also apply negation to a disjunction of
propositions. Now remember how disjunction works,
if you have a proposition P and a proposition Q,
the disjunction of those two propositions P or Q is going to be true whenever P and
Q are both true. Its going to be true when P is true and Q
is false. It's going to be true when P is false and
Q is true and its going to be false when P and Q are both false.
So that's the truth table for disjunction.
Now, suppose you apply negation to this disjunction,
well then, you get this truth table. The negation of P or Q is going to be
false when P and Q are both true. It's going to be false when P is true and
Q is false. It's going to be false when P is false
and Q is true. And the only situation in which it's
going to be true is the situation in which P and Q are both false.
Now, you may have noticed, that on that last truth table, I wrote the negation of
P or Q where the P or Q was in parentheses.
And you might wonder, well, why didn't I just write the negation of P or Q?
Well again, it's because these two expression mean two different things.
Here's a truth table that shows how those two expressions differ in meaning.
So, the negation of the disjunction P or Q,
that's going to be false in the situation when P and Q are both true.
It's going to be false in the situation where P is true and Q is false.
It's going to be false in the situation where P is false and Q is true.
And the only situation where that proposition is going to be true is the
situation where P and Q are both false. In contrast, the proposition the negation
of P or Q is going to be true when P and Q are both true.
It's going to be false when P is true and Q is false.
It's going to be true when P is false and Q is true and it's going to be true when
P and Q are both false. So, this proposition and this proposition
are true in different situations and so they must mean different things.
Here's an example to illustrate the point.
Again, suppose that the proposition P is Walter is busy working and the
proposition Q is Rom is busy working. Well, then, disjunction P or Q is
going to be Walter or Ram is busy working.
And the negation of that disjunction is going to be it's not the case that Walter
or Rom is busy working or in other words, neither Walter nor Rom is busy working.
That's what this says. This says neither Walter nor Rom is busy
working, but, the disjunction of not P and Q says,
Walter is not busy working or Rom is busy working and that means something very
different from neither Walter nor Rom is busy working.
So that's an example to illustrate how this proposition and this proposition are
going to mean two very different things. A moment ago, I showed you the truth
table for the disjunction of the negation of P and Q.
Alright? That disjunction is going to be true whenever P and Q are both true and
it's also going to be true whenever P is false.
The only situation in which that disjunction is going to be false is the
situation in which P is true and Q is false.
Now, I want you to remember this truth table, because it's going to be relevant
in our next lecture on conditionals.