Tip:
Highlight text to annotate it
X
♪ Paul: The topic of this episode
is...translating necessary and sufficient conditions.
So let's start with some definitions.
So in logic, we say that N is necessary for X,
where N is a condition,
and X is something happening or something existing.
When it is the case that X can't exist or can't happen
without the presence of N,
or N is a requirement for X to happen.
For example, um...oxygen is necessary
for a typical gasoline motor to run.
Can you think of another example of a necessary condition?
Mark: Uh...
[sighs]
[laughing] No.
Paul: Oh...well, you just said
oxygen is necessary for a fire to burn.
Mark: For a match to light, you need oxygen.
Notice the word "need" starts with an N.
Necessary starts with an N.
A necessary condition is something that's needed
for something else to take place.
You need oxygen in order for the match to light.
Paul: Good, oxygen is necessary for a match to light.
Mark: There we go.
The rain's getting to me.
Paul: And then, we'll say S is sufficient...that was good.
Mark: Yeah, it works.
Paul: For X...and when we say that S is sufficient for X--
where X is something happening or something existing,
and S is some condition--
what we mean is S is all you need
for X to be or for X to occur.
X is...once X...once S obtains or exists, X must be.
So for example, uh...jumping in...Puget Sound behind us...
jumping in Puget Sound is sufficient for getting wet,
if you jumped in completely.
Mark: It would be enough to get wet.
Paul: It's enough to get wet.
Uh...jumping in Green Lake is sufficient for getting wet.
Mark: I'll ask you a question.
If I was to take a rock
and throw it through a window to break the window,
would throwing a rock through the window
be a sufficient or necessary condition
to break the window?
Paul: It's a sufficient condition,
but it wouldn't be a necessary condition, would it?
And why not?
Mark: You could throw me through a window,
and that would be enough.
Uh...the rock would be enough.
There's all kinds of things we might throw,
so it's not needed to throw a rock to break the window.
That would be enough to do the job.
Paul: Throwing a rock
might be sufficient to break the window,
but it's not necessary to break the window.
And to use my example, would, uh...I'll quiz you.
Mark: Okay.
Paul: Would, uh...jumping in Puget Sound...
completely jumping in... would that be suf...
would that be necessary for getting wet,
or sufficient, or both?
Mark: It would be sufficient but not necessary.
I could also jump into Green Lake
or the Pacific Ocean.
It would be enough...it would be sufficient to get wet,
uh...but I wouldn't need to jump into Puget Sound to get wet.
I could get wet other ways.
Paul: Other ways.
So...jumping in Green Lake or Puget Sound
is sufficient to get wet,
but since there's other ways to get wet besides that,
it's not necessary.
Mark: Yeah.
Paul: Okay, so now, we have our definitions.
Mark: Okay.
Paul: Now, it's typical... it's traditional in logic
to symbolize necessary and sufficient conditions
to translate them into symbols using "if then" constructions.
So we're gonna be using horseshoes and if thens.
Let's start with, uh...this.
Let O stand for oxygen is present.
And let, um...R stand for the engine is running.
And assuming it's a typical gasoline engine,
like a lawnmower engine.
The engine is running.
I'm abbreviating a little bit.
So how would we symbolize,
"Oxygen is necessary for the engine to be running,"
letting the symbol...
those letters stand for those sentences?
Mark: So oxygen is necessary for R.
Okay...if we're gonna use a horseshoe
for either a necessary or sufficient condition,
so I know I'm gonna have that.
Here's a fun, little memory device
that I think really helps a lot.
Think of the word "sun."
Paul: Like, "Here Comes the Sun,"
like The Beatles' song.
Mark: Exactly...except take the U and put it on its side.
So you actually have a condition there.
You'll notice S is the first letter for sufficient.
N is the first letter for necessary.
So the necessary condition
is gonna be the consequent of any conditional,
and the sufficient condition
is gonna be the antecedent of any conditional.
Paul: Can I translate for you?
Mark: Go ahead.
Paul: What Mark just said was that
what this little device helps us remember
is that when you're symbolizing a necessary condition,
it'll go in the consequent of a conditional.
When you're symbolizing a sufficient condition,
the sufficient condition will go
in the antecedent of a conditional.
That's what he just said.
Mark: Okay...so Paul asked me to translate,
"O is a necessary condition for R."
So O is the necessary condition,
so O goes in the N spot, if you will.
I now got a horseshoe,
and if O's a necessary condition,
it goes in this spot.
R's gotta go somewhere.
There's only one place left.
So that's gonna be the correct translation
if O is a necessary condition for R.
If we have something like A is a sufficient condition...for B.
Paul: For instance, A would be jumping in the lake.
B would be getting wet.
Mark: Or yet, or anything like that.
Here, we're having A is a sufficient condition.
Again, if I think sun, sufficient starts with S.
A's gonna go as the antecedent.
It's in a sufficient spot.
The B's gotta go somewhere.
There's only one place left, so we put it there.
That will be the translation
of A is a sufficient condition for B.
If we had A is a necessary condition for B,
it would be the opposite,
'cause then A would be in the N spot.
So it's not bad.
Paul: Let me add something, then, to this.
Let's take this off.
[loud squeaking sound]
There's a little train action
goin' on in the background there.
[squeaking]
So...so we...we're saying that oxygen
is necessary for the engine to run,
and so, Mark says we symbolize it this way.
If oxygen is necessary for the engine to run,
then we put the oxygen in the consequent,
and the engine running as the antecedent.
And this says, if the engine's running,
then oxygen is present.
Now, let's...that's the correct way to symbolize
oxygen is necessary for an engine to run.
Let's take a look and see what it sounds like
if we had done it in the reverse.
Suppose someone symbolizes,
oxygen is necessary for an engine to run,
and they symbolize it that way.
What does this say?
Mark: Now, one way of thinking about it is if there's oxygen,
then the engine would be running.
But that clearly isn't always the case.
Right now, there's oxygen in the air,
and my engine isn't running.
So it's not saying the same thing.
Paul: Right, you could have oxygen,
but your lawnmower engine's not running.
It's broken.
Or it doesn't have a spark plug.
It needs electricity too.
So...this is correct.
This is not correct.
And suppose I say, jumping in Green Lake
is sufficient for getting wet.
So we've...Mark said the way to symbolize this is...
thank you... uh...X...
Mark: [laughing]
Paul: Anyways, so...Mark says we symbolize it this way.
We put the sufficient condition in the antecedent.
If...so, what does this read?
Mark: If you jump in the lake, you're gonna get wet.
Paul: Yeah, and that just says,
jumping in the lake is sufficient for getting wet.
Now, let's see, suppose someone symbolizes it this way.
If you jump in Green Lake, then you're gonna get...
or...jumping in Green Lake is sufficient for getting wet,
and they symbolize it that way.
What's wrong with doing it that way?
Mark: This would be saying that, uh...getting wet
is a sufficient condition for being in Green Lake.
Paul: Yeah, it would, but how would you read that...
Mark: If you're wet, then you jump in Green Lake,
which is not necessarily the case.
I can get wet in any number of other ways.
Paul: So if you saw someone, and they're soaking wet,
you wouldn't necessarily suppose
they must've jumped in Green Lake.
Like, you could get wet standing in the rain.
Mark: Yeah, which is about to happen.
♪ Paul: Which is happening.
Mark: Yeah.
Paul: And so... so Mark's reminding us
that we have the sun memory device.
The sufficient condition goes to the left
when we symbolize it.
When we symbolize a necessary condition,
we put it to the right of a horseshoe.
And so, that's symbolizing
necessary and sufficient conditions.
Is that sufficient?
Mark: I think it's sufficient... indeed.
Necessary, perhaps not, but sufficient.
Paul: Thank you.
Mark: Alright.
Paul: Thank you... that's it.