In this video I want to talk about an advanced technique ...
using new SHOW lines that you don't get from a particular ...
derivation rule. I want to start by emphasizing ...
that I'm not going to test you on this material;
if you find it confusing, concentrate on the earlier material.
I call this spontaneous SHOW lines.
Usually, when we write down a new ...
SHOW line, it's because we're setting up a ...
particular derivation strategy--showing what's after ...
the arrow for a conditional derivation,
showing a contradiction for an indirect derivation.
But you can actually make a new SHOW line anytime you ...
want, if you need a certain ...
statement for your proof.
Then you draw a vertical line under it where you show the statement
and when you've shown the statement,
you cross out the SHOW line and end the vertical proof line.
Now that the spontaneous SHOW line is cancelled,
you can use the statement on it as an input to other statements!
Crossing out one of these spontaneous SHOW lines ...
does *not* let you cross out any previous ...
SHOW lines you've written down--that's only for SHOW ...
lines that come from particular derivation rules.
And once you finish a proof line,
you're not allowed to use any of the statements within that ...
proof line. The point of writing SHOW ...
and drawing the vertical proof line is to get a statement to ...
use in the proof.
One reason to do spontaneous SHOWS is if ...
we're trying to show an & statement and we need to use ID.
It's usually difficult to get an & statement directly by ID;
you'd be assuming a negated &
statement, which turns into an arrow ...
statement, which doesn't give you that ...
much to work with.
A better technique is to put one side of the &
statement on a SHOW line.
Then you should be able to use that SHOW line to set up an ID.
Once you've finished that, you might have to repeat the ...
technique with the other side of the SHOW line,
or you might be able to use the statement you just got to ...
get the other side.
This can also work if you're showing an &
statement where one half of the & is an arrow
You write down the arrow statement as a SHOW line,
and then you can use CD to get it.
Here's an example of how to use ID to get half of an &
statement that you need to show.
Here's our premises, and we need to SHOW P & Q.
So we start by SHOWING P.
Now we set up an indirect derivation to SHOW P; assume ~P
and SHOW a contradiction. Note that we started a new ...
vertical line right under the SHOW statement for P.
Now we start another vertical line right under the SHOW ...
statement for the contradiction;
we're going to show the contradiction on this line.
We start by getting Q by vO from 1 and 5.
Now we need to do something with the double arrow.
Since we have Q, which is one side of the ...
double arrow, we make sure that Q is in ...
front of the arrow statement
so we can do MP to get P.
This gives us a contradiction with P and ~P.
So we've shown the contradiction
and since we've assumed ~P and shown a contradiction,
we've shown P by indirect derivation.
But we haven't shown P&Q yet!
We don't get to cross that out; there's no derivation rule that ...
says that if you show one half of an &
statement, we get to cross out the whole thing.
Now we go ahead and finish the proof,
using the P that we've shown. Notice that we've ended the ...
proof line under SHOW: contradiction,
and the proof line under SHOW:
P, because we did SHOW those.
So those lines have served their purposes.
And every statement on those lines is something that we got ...
using an assumption that ~P so we can't use those lines 5 ...
through 10 as inputs anymore.
We can only use our premises 1 and 2,
and the thing we showed in 4. So we start by doing double ...
arrow out on 2 to get P -> Q. Since we have P in line 4,
we want to get P in front of the arrow.
So now we can do MP with 4--which we can use now,
since we cancelled the SHOW line--to get Q.
Since we have P in 4, and Q in 12,
we can use &I to put them together and get P&Q
and we've shown P&Q through direct derivation.
So, again,
once we cancel the SHOW line in 4,
we can use it as an input to more rules.
But once we've ended a vertical proof line,
we can't use the statements on it anymore,
since they may depend on assumptions that aren't in ...
effect any longer. So we can't use lines 5 ...
through 10 outside the proof lines that they're on.
I call this the Vegas rule: what happens in a proof line stays there.
And we couldn't cancel the SHOW in line 3 right away ...
when we cancel the SHOW line in line 4,
because we didn't get 4 from a derivation rule for showing ...
3. We had to wait until we get ...
P&Q on the proof line under 3;
then we could cancel it out using Direct Derivation.
We can use the spontaneous SHOW technique to show double arrows.
You need two single arrows to get a double arrow.
So you write down each of the single arrows you need on a SHOW line.
And then you can use CD to SHOW each of these single arrows.
Here's an example of that.
We have P double arrow Q, and we want to SHOW ~P ...
double arrow ~Q.
We know we're going to need ~P -> ~Q
and ~Q -> ~P, so we write those down on ...
separate SHOW lines.
Now we set up CD to show line 3 by assuming what's ...
before the arrow, ~P
and SHOWing what's after the arrow, ~Q.
Since we have ~P, the negation of one side of the ...
double arrow, we put P after the arrow
and do modus tollens to get ~Q.
We've shown ~Q
so we've finished the conditional derivation and we ...
can cancel the SHOW line in 3.
Now we set up the CD for 8; we assume what's before the ...
arrow, ~Q
and SHOW what's after the arrow, ~P.
We have ~Q, so we make sure Q is after ...
the arrow when we do double arrow out
and we can get ~P by MT.
Now we've shown ~P
and we can cancel out the SHOW for ~Q->~P using ...
conditional derivation. Now we can use lines 3 and 8 as inputs
so we put them together and get ~P double arrow ~Q,
by the double arrow in rule.
So we've shown ~P double arrow ~Q.
There are more things we can do with spontaneous SHOW lines.
For some proofs, we need to show the negation ...
of one side of a wedge in order to do vO.
Those particular proofs are extra tricky to deal with.
As you can see from the examples we worked with,
all the proof involving spontaneous SHOWs tend to ...
be long and involved. For instance,
when you're getting a double arrow this way,
you have to show two single arrows,
so the proof is twice as long. That's why I'm not going to ...
test you on these; I just wanted to make sure ...
that you had the chance to see them.