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.