Tip:
Highlight text to annotate it
X
Mark: Okay, we're back
at Cromwell Park here in Shoreline.
We still have water behind us, and we've got water up above.
We're sitting in this kind of nice, dry gazebo.
Paul: With a wetland behind us.
Mark: Wetland behind us.
Paul: Protected wetland.
Mark: We're gonna look
at a nested conditional proof this time,
where you might make more than one assumption.
That can be...it'll be needed for some proofs,
and...sometimes it just makes the proof easier.
It may not be needed,
but it's the easiest way to do it.
So we want to see how it can work.
Paul: Some teachers don't cover nested proofs,
and some do.
So a nested conditional proof is a conditional proof
that has within it a proof... an indented proof.
So it's an indented proof within an indented proof.
Mark: Yeah.
Paul: Some teachers don't go that far, some do.
What makes you think this... when you look at the conclusion,
what makes you think
it's gonna require a nested proof?
Mark: I don't know what to do with the premise,
so I'd be looking at the conclusion first.
I see that this is
the main operator or the main connective,
so we got a conditional,
which is making me think use conditional proof.
My goal is conditional.
I like using conditional proof.
One I would assume would be F--the antecedent.
Then, I'd be looking for the consequent.
That's the way conditional proof works.
But now that I look at this, at this point,
that would be my goal-- G horseshoe H.
That too's a conditional.
So I'm thinking, hmmm...maybe I'll make a second assumption,
assume the antecedent of my present goal,
and then try to find H.
And as I see that side,
I'd end up assuming F and assume G.
I can see right here
I've got some really easy things to do right afterwards.
So this problem's super easy,
if I'm bold enough to make some assumptions.
Paul: Ah-huh, so this horseshoe
tells you it's gonna be probably a conditional.
And then, when you see that you got to get this,
that horseshoe suggests that another conditional
within the conditional.
Mark: Yeah, so let's start off with indenting.
Paul: Okay...alright.
Mark: And I'll assume F--
the antecedent of my entire goal.
Paul: Okay, we're gonna assume F--the antecedent--
because conditional proof says if you assume...
if you wanna reach...
if you wanna prove a formula of the form P horseshoe Q,
you assume the P, reach the Q.
Then, you can assert the entire P horseshoe Q.
So we've assumed the P.
Now, we have to reach the Q in the G horseshoe H.
Mark: Which is another conditional.
Paul: It's another conditional.
Mark: At this point, what I'll do is indent a second time,
do another conditional proof, make another assumption,
and I'm gonna assume the antecedent of my goal.
And so then, afterwards, try to find H.
Paul: Good.
Mark: So let's assume G at this point.
Paul: So now, I'll indent within the indentation.
So now, I'm doing an indented proof
within an indented proof.
So I'll indent and assume G by assumed premise.
Because the rule for conditional proof says
that anywhere in a proof,
you can indent and assume an antecedent.
It doesn't restrict you from doing it again
if you've already done it.
Mark: That'll be line 3.
Paul: Okay.
Mark: At this point, now that I've assumed G--
the antecedent of my present goal--
at this point, I'm now lookin' for an H.
I really don't need to make any more assumptions,
because there's a lot of really easy things to do.
One thing I can do is a modus ponens
with lines 1 and 3.
That would get me F horseshoe H.
Paul: Okay, do you wanna do that?
Mark: So line 4 will be F horseshoe H.
Paul: Okay, so P, P horseshoe Q, Mark's bringing down the Q.
Mark: And I'm still in this sequence at this point.
Paul: And he's still in this indentation.
That was modus ponens...
Mark: Uh...1 and 3.
Paul: 1 and 3.
I'll make it like a fraction, is that alright?
Mark: Oh, yeah.
I'm lookin' for an H at this point.
At this very moment, my goal is H,
and I can get it with another modus ponens on line 2 and 4.
So have that line 5 be H.
Paul: Still in this indentation.
Mark: Yeah...and that's gonna be 2...
or modus ponens, 2 and 4.
Paul: 2 and 4...again, it's like a fraction.
It's not usually supposed to be that way,
but we don't have room.
And so, you're keeping your eye on the goal, aren't you?
That's what guides you.
Mark: I was lookin' for an H-- I got it.
When I assumed G, I was lookin' for an H.
I can now pull out one column to the left,
and I can write if G's true, then H must be.
Now, I'm in this column.
Paul: So now, we're gonna drop out to this column,
and discharge our assumption, and write, if G, then H, by CP.
Mark: CP, uh...3 through 5.
Paul: 3...okay, 3...3, 4, 5.
Mark: And now, we're at line 6.
Paul: And that was line 6.
Now, it's important to remember when you do a nested proof
that you have to close off the inner proof
before you do the outer proof.
Mark: And I would not really be able to use
lines 3, 4, 5 individually after this,
'cause they're locked off.
Paul: That's a good point to remember.
Mark: I can use 1, 2, or 6 at this point.
Paul: That's right.
Once you've discharged an assumption, and disindented,
and drawn the inference, you cannot appeal to these lines
if you go further.
So these are out of the proof now.
They're out of play.
And just remember, you have...
if you start an indented proof within an indented proof,
you have to close off the inner proof
before you close off the outer proof.
So, uh...
Mark: Okay, when I assumed F--
the antecedent of my big goal--
I was lookin' for G horseshoe H, and I got it.
So if this is true, it looks like this must be.
Paul: Given that these... given that this is true.
Mark: Yeah, given that.
So for line 7, I'm gonna discharge once again and say,
if F is true, then G horseshoe H is true.
Over there.
Paul: Okay.
Mark: And that will be CP, 2 through 6.
Paul: Including all of this, CP 2 through 6.
So in other words, given 2 through 6--
assuming this is true-- this must be true.
So the justification appeals to all the lines
that were indented to get to this.
And so, very good, Mark.
Mark: Sweet.
Paul: A nice, nested CP.
And I wanna, uh...make one quick point.
I meant to make it at the end of the last video.
There's a common error that people sometimes make.
It's easy to get mixed up
when you're working with lots of symbols.
And you'll remember this, Mark.
A lot of people, when they're doing a conditional proof,
they assume...I'm gonna make a little model here.
They're, um...
Thank you.
They're doing a conditional proof...
blah, blah, blah.
They've got to get, you know, A horseshoe B,
and so, they assume A as their assumed premise.
And they're trying to reach what?
Mark: B.
Paul: Trying to reach B.
And they get goin'.
And then, they get mixed up, and they forget.
And they try to derive A horseshoe B,
down here at the bottom of their indentation,
instead of just B, and that's a mistake.
What's the mistake?
Mark: Well, what they're lookin' for...
what their goal is actually B.
If they end up getting A horseshoe B,
they don't know that A horseshoe B's true.
It was all based on this assumption.
What they need to do is get the B,
so that they can use CP to pull back out
and with confidence say A horseshoe B.
Paul: Right, so when you're doing a conditional proof
for P horseshoe Q, and you reach,
uh...your goal at the bottom of the indentation
is not to reach the conditional.
It's only to reach the consequent
of the conditional-- the B alone.
Mark: I tell my students, you got to chant like a mantra.
Go home and do some yoga.
Assume the antecedent.
Try to find the consequent.
Assume the antecedent.
Try to find the consequent.
Paul: Then, assert the conditional.
Mark: Yeah, you're tryin' to find the...
assume the antecedent and try to find the consequent.
Then, you'll get the conditional after that.
Paul: The chant could be assume the antecedent.
Try to get the consequent.
Assert the conditional if you do.
Would that be a good chant?
Mark: Could work.
It's almost a ditty at that point,
but you could chant a ditty.
Paul: You could make it like a sea chantey.
Mark: I could give you that.
Paul: But in the Middle Ages...
Mark: A CP chantey.
Paul: A CP chantey, good...good one.
In the Middle Ages, the monks would have
a little rhyming, uh...mean... mnemonic devices.
That they would chant
to remember their little logic rules.
So there'd be a rule put into a chant,
and then, they'd chant it.
Mark: Early rap...or...maybe.
Paul: Monk rap.
Mark: Monk rap.
Paul: Like an early form of rap.
Medieval monk rap.
Mark: Ah-huh.
Paul: So, um...just remember
that when you're trying to reach B,
uh...when you assume A and try to reach B,
only a...only reach B.
Don't try to get the whole conditional at the bottom.
So that's the lesson.
We hope this is helpful.
Mark: We do.