In this video I'm going to introduce conditional ...
derivation, which is a new kind of rule.
Conditional Derivation captures a kind of ...
hypothetical reasoning. To see why it works,
let's look at an example in English.
Let's say that I have bread. And if I have bread and ...
cheese, I can make a sandwich.
But we don't know if I have cheese.
I want to show that if I have cheese I can make a sandwich.
So I make a thought balloon, and in that thought balloon,
I think about what would happen if I had cheese.
Well, I'd have bread and cheese!
I can take the bread I actually have and use it inside my ...
thought balloon.
And it follows that I could make a sandwich.
Now we exit the thought balloon.
We don't know that, in the real world,
I can make a sandwich. But we gamed out the ...
scenario of what happens if I have cheese,
and we figured out that in that scenario I can make a ...
sandwich. So we know that IF I have ...
cheese, THEN I can make a sandwich.
Here's how that looks in our symbolic notation.
Our premises are B and (B & C) -> S.
The thing we want to show is C->S.
We're going to make a line under it and show C -> S on that line.
So we assume that C--this corresponds to entering the ...
thought balloon and saying, what if I have cheese?
Now we want to show S. If we can assume C and show ...
S is true, then we know that if C then S is true.
We make a second line--we will show S on this line.
And we start doing rules on that line.
So we can get B&C from lines 1 and 4 by &I.
Now that we have B&C, we can use MP to get S from line 2.
That's what we needed to show on this line!
So we cancel out that SHOW for
and label it Direct Derivation--since we got S,
the thing that we were trying to show,
right under that SHOW line.
Now we get to cancel out the show line for C->S!
We assumed that C was true, and we successfully showed ...
S given that assumption, so that means that if C is true ...
then S is true.
This is the rule of conditional derivation, or CD.
Here's how the rule looks written out formally.
We can only apply this rule when we have a SHOW line ...
whose major operator is an arrow.
Notice that here, having the conclusion written ...
on a special SHOW line makes a difference;
if we didn't do that, we couldn't apply conditional derivation.
So if we're showing A -> C
we get to assume A, the thing before the arrow in the SHOW line
and now we need to show C, the thing after the arrow in the SHOW line.
Now we draw a new vertical line and show C on it.
Once that's done, we get to cross out the SHOW for C,
and when we do that, we immediately get to cross ...
out the SHOW for A -> C as well.
We label A->C as having been shown through ...
conditional derivation.
So we assume what's before the arrow of the SHOW:
line, and SHOW what's after the arrow.
Here's an example.
Let's look at these premises and conclusion.
The SHOW line is an arrow statement,
so we can use conditional derivation.
Our assumption will be what's before the arrow of the SHOW line.
That's ~R, so we put that down on the ...
next line as our assumption.
Our new SHOW line will be what's after the arrow of the old SHOW line.
That's Q, so we put down Q as the new SHOW line.
Now we start a new vertical line under the new SHOW ...
line, and we do rules on it.
We can get P by vO from 2 and 4.
We can get Q by MP from 1 and 6.
Q was what we needed to show,
so we cancel that SHOW out as a direct derivation.
And when we cancel out that SHOW line,
we get to cancel out ~R->Q as a conditional derivation.
Notice that, to find the assumption and the ...
new SHOW line, we only look at the old SHOW ...
line! Whenever you need to show ...
~R->Q, you'll make that assumption ...
and new SHOW line. Don't try to get an assumption ...
out of the premises; we already know them,
we don't need to assume any part of them.
Here's another example with a more complicated SHOW line.
Given these premises and conclusion
we want to assume what's before the arrow of the SHOW line
which is P & ~S.
Our new SHOW line is what's after the arrow of the old SHOW line
which is Q & ~R
The assumption in line 4 is an &
statement, so we can do &O to it and get
and ~S.
Now we can do MP with 1 and 6 to get Q
and MT with 2 and 7 to get ~R
and then put those together to get Q&~R.
That's what we were trying to show,
so we cancel out the SHOW line for Q &
~R as a direct derivation.
And having shown that, we get to cancel out the ...
original SHOW line as a conditional derivation.
Notice that, when we're looking for inputs ...
to our rules, we can use the assumption we've made.
But we can't use the SHOW lines we've written down ...
before we've cancelled them. You can't use line 3 or line 5 ...
as inputs to rules when the SHOW lines aren't crossed ...
out, because those are things that ...
we haven't shown to be true yet.
We can even have a conditional derivation ...
inception, with CDs within CDs.
Let's look at this premise and conclusion.
The SHOW: line is an arrow,
so we set up a CD by assuming what's in front of ...
the arrow, which is
and showing what's after the arrow,
which is Q -> R. But now our new SHOW line ...
itself is an arrow statement!
So we set up a conditional derivation for it by assuming ...
what's in front of the arrow, which is
and making a new SHOW line with what's after the arrow, which is R.
Now we're trying to SHOW R. Notice that each new SHOW ...
statement gets a new vertical line under it,
so we have three vertical lines.
We do &I to put P&Q together.
We're allowed to use lines 3 and 5 as inputs,
because they're assumptions.
Then we do MP with line 1 to get R.
Now we have a crossout cascade.
We showed R directly, so we cross out that show line ...
as a direct derivation.
We assumed Q and showed R,
so we cross out the show line for Q -> R as a conditional derivation.
We assumed P and showed Q->R,
so we cross out the show line for P->(Q->R) as a conditional ...
derivation. Since that was our original ...
SHOW line, the proof is done.
To sum up how we do CD:
We do CD when our SHOW line is an arrow statement.
This only applies to SHOW lines!
If you have a premise that's an arrow statement,
that doesn't mean you can do a CD.
To set up the CD, you assume what's before the ...
arrow of the SHOW line, and your new SHOW line is ...
what's after the arrow.
You only have to look at the SHOW line to figure out how ...
to set up the proof. The premises don't play into ...
this part of the proof at all!
If the new SHOW line is an arrow statement,
set up a new CD. You keep repeating this until ...
you have a SHOW line that isn't an arrow statement.
Then you do rules until you get this last SHOW line.
And when you get it, you can cross out all the ...
SHOW lines at once.