In this video, we're going to go over logic arguments and how to prove them using the
method of derivations.
I wrote the definition here.
It says, an argument consists of a set of propositions called the premises and a proposition
called the conclusion.
An argument is valid if and only if the conclusion is true whenever all the premises are true.
Here is an example of a logic argument written out in text form.
It says, If math et al is a youtube channel, it has videos.
Math et al is a youtube channel.
And then it has a line.
This is called the line of inference.
And then it has this symbol, which stands for therefore.
Then it says, math et al has videos.
This top part above the line of inferences, these are called the premises.
Each line is a premise and it's just a proposition.
And then you have the line of inference and this is your conclusion.
If you translate this into a logic statement, you have if P then Q, P, line of inference,
therefore Q. When you have problems like this, you're going to be asked to prove that the
argument is valid.
You want to prove that you can draw this conclusion form the proposition or from the premises
that you're given.
One way of doing this is using the method of derivations.
The method of derivations is where you write out line by line how you get from the premises
to the conclusion.
This is a really simple example and this is actually a law in logic called Modus Pones.
If we were going to write this out as a derivation, what you do is you number each line and you
start by writing the premises.
And then you write over here on the right side what the justification is for each line
or each formula.
The first lines, you'll always just be repeating what the premises are.
You know this is true because it's the premise.
You know P is true because it's the premise.
Then you want to write out your justifications and the final line, which will be some number
line, will be Q and your justification for it will be here.
In the example here, because this is actually a law that you can use, your justification
would just be the next line.
And it would be, you know this from 1, 2, and Modus Ponens.
But we'll go over that later.
I'll go over the logic laws that we're going to use in this video and then do some examples.
We already know that this is our first law of inference.
The second law of inference that we can just know is Modus Tollens.
That is a manner of denying.
That is P implies Q, not Q, and then the inference you can draw from that is therefore not P.
The third law of inference that we can use is called Modus Tollendo Ponens.
This is sometimes called disjunctive syllogism.
This is similar to Modus Ponens in that, if you have P or Q is true and you know that
not P or not Q is true, but in this case not P is true, then you know that you have Q because
one of these has to be true for this premise to be true.
Using these three laws in different ways, you can actually prove that a lot of arguments
are valid.
And we'll go over some examples.
Okay, let's do a simple derivation with this logic argument.
We're going to write out the premises first.
Our first premise is P. Then we have two more lines.
P then Q. And if Q then R. Since we're given in the premises that P is true and if P is
true then Q is true, then the first derivation we can make is that we have Q is true.
And this is from Modus Ponens and lines 1 and 2.
You write here on the side your justification.
We know that this is true.
Next we also know that if Q is true, then R is true.
Again, that's going to be R and we're using Modus Ponens again, this time with lines 3
and 4.
And that's actually the conclusion that we wanted.
Again, it wasn't a very difficult argument to prove that it's valid but this is just
the format for how you would do a derivation.
The next problem we can do will be using some other rules as well.
Here's a second example of an argument that we can prove.
It says, if P is true, then Q or R is true.
If P is true, then not R. Then you have P, so the conclusion is Q. Again, we're just
going to write out the premises in our derivations first.
If P, then Q or R. These are just justified premises.
Modus Ponens told us, if P then Q and you have P, then you can conclude Q.
We have, if
P then, let's call this Q, then
you have P. So we know that Q or R true, given Modus Ponens, lines 1 and 3.
Then we know that Modus Tollendo Ponens says that if you have P or Q and not P, then you
can conclude Q.
We have Q or R and we want to have not R so that we can conclude Q.
We can get not R from P and from P implies not R.
We have P implies not R and we have P. Again from Modus Ponens, have can conclude that
we have not R.
We can conclude not R, which is what we wanted, from Modus Ponens and then that's going to
be lines 2 and 3.
Now we have what's necessary to conclude Q from Modus Tollendo Ponens for lines 4 and
5.
And this is our conclusion for the derivation of this problem.
That problem was a little bit more involved.
Let's just do one more that uses all three inferences.
Writing out the premises for this problem, we have not P implies Q or R, P implies Q,
and we have not Q. The first thing I see is that we have P implies Q and we have not Q.
From Modus Tollens, that means we have not P. Have not P, Modus Tollens, and that was
line 2 and 3.
We have not P implies Q or R, and we have not P. From Modus Ponens, that implies Q or
R. That's the next thing we'll derive.
We have Q or R, and that's from Modus Ponens, lines 1 and 4.
Then if we have Q or R and we have not Q from 3, then we can conclude R. That's Modus Tollendo
Ponens.
We can conclude R from Q or R, which was line 5 and from not Q, which is line 3, and Modus
Tollendo Ponens.
So we have come to the conclusion that this is a valid argument as well and that was using
the three laws of inferences.
These problems were all pretty basic.
There are actually a lot more laws of inference and I might make a video with more problems
that explain some of the additional laws, but this is just a quick introduction.