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So, let's go through the basics of logic.
We can think about logic as being

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statements about propositions such as
Obama is the president of the USA,

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Or Obama is the leader of the ISA,
Or Tony is the cricket captain of India,

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and things like that.
So, propositions are statements which are

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either true or false. So, if A and B are
true propositions.

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The combination A and B is true.
It means that A is true and B is true,

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And that's written in this way as A and B.
This is very basic stuff and most of you

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must have studied it.
But, this is just to get everybody back on

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the same page.
Similarly, A or B is true if either A is

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true or B is true.
So, only one of them have to be true for A

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or B to be true.
So, this is written as A or B with this

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notation.
Now comes the interesting part.

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Suppose we have a rule that says,
If A then B,

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Which is the same as saying if A is true,
then B is true.

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The kind of things that we've seen before.
If X is the president of C, then X is a

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leader of C, or the leader of C.
Well,

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The really important part about logical
reasoning or classical logic is that,

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saying that if A then B is exactly the
same thing as saying that either A is

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false or B is true.
Now, this requires a little bit of thought

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which can also be written as A implies B
is equivalent to not A or B, which is just

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the same as A is either false which is not
A being true or B is true which is just B.

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Let's go through it again.
A implies B is the same as not A or B.

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Very important critical aspect of
reasoning.

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Let's check it.
If A is true, then not A is false.

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So, the only way that not A or B can
become true is if B is true.

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In other words, by stating that not A or B
is true, we have arrived at a situation

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that whenever A is true, B has to be true
which is exactly what we meant by saying A

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implies B.
On the other hand,

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If A is false, then not A is true.
So, the statement not A or B is always

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true regardless of B being true or false.
So, the statement, rather the rule that A

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implies B remains true.
Whatever be the value of B, in the

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situation that A is false.
Which is also fine because a rule must

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remain a rule regardless of the values
that variables take, or propositions take.

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So, please go through this argument a
couple of times to convince yourselves

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that A implies B is exactly the same thing
as saying that the compound statement, not

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A or B is true.
We were dealing with statements or

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propositions which were wssential
statements of fact, and nothing more or

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less than that.
But, as we have seen before,

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We are dealing with statements which are a
little bit more complicated.

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Obama is the president of USA, is actually
stating a relationship between an entity,

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Obama, and another entity, USA.
And that relationship is that one is the

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president of the other.
Which could be written in this form, that

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Obama is the president of USA.
Now, this is rather tricky notation. But,

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all we're saying is that we have variables
which are entities like Obama and USA, and

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we have predicates which are relationships
between these variables or entities.

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Similarly, we might have a rule which says
that,

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If X is the President of C then X is the
leader of C. Which could be stated as,

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If X is president of C, then X is leader
of C.

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Now,
Unlike this particular statement on top,

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the statement about X and C applies for
all values of X and C.

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So, this rule is not a simple rule such as
Obama is the President of USA, it's a

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statement about all possible X's and C's
that says, if X and C are related by the

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predicate as president of, then they're
also related by the president, predicate

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is leader of.
So, this particular feature of predicates

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and variables is called quantification,
And that makes it significantly different

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from basic propositional logic.
Still, the process of reasoning remains

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quite similar with a few modifications.
So, let's go through that a little

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carefully.
Starting with a fact such as Obama is

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President of USA which is expressed as a
predicate, is President of Obama comma

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USA.
Together with this rule, which states that

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for all X and C, if X is the president of
C then X is leader of C.

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Using R and F,
We come to the conclusion that Obama is

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leader of USA.
In other words, this new fact, is leader

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of Obama comma USA is entailed by the rule
R applied on the fact F.

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Notice a few things have taken place here
that the variable X has been bound to a

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value Obama,
The variable C has been bound to the value

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USA.
So, the rule which applies to all values

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of X and C is combined with a fact by
binding the variables to the specific

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values that they take in a fact, like F.
Now, in response to a query which could be

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expressed as, he is leader of X comma USA
where USA has been fixed or bound, but X

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remains variable.
One could look at the bunch of rules and

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facts that one has in one's collection of
knowledge and try to answer this query by

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reasoning, which is essentially answering
by deriving new facts which match the

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query.
And we do this by a process of

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unification, which is bind, binding
different possible values of the variables

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to X, and inference, which is essentially
applying rules using the if A then B

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format to derive new facts.
The combination of unification and

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inference is called resolution, and we'll
return to resolution in a little while.

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But, notice that the process of reasoning
and predicate logic is fairly similar to

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that if you just had propositions with the
additional complication that we have

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variables which need to be bound to
different values.

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And, if you have lots of facts and lots of
rules, you have lots of different choices

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for how we bind different variables to
different specific values, and that leads

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to a degree of complexity which makes
predicate logic resolution much more

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difficult than simple logical inference
using a bunch of propositions.

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Both are quite difficult, as we shall see
during our discussions later in this

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lecture, but Y is significantly more
difficult than the other.