So, let's go through the basics of logic. We can think about logic as being
statements about propositions such as Obama is the president of the USA,
Or Obama is the leader of the ISA, Or Tony is the cricket captain of India,
and things like that. So, propositions are statements which are
either true or false. So, if A and B are true propositions.
The combination A and B is true. It means that A is true and B is true,
And that's written in this way as A and B. This is very basic stuff and most of you
must have studied it. But, this is just to get everybody back on
the same page. Similarly, A or B is true if either A is
true or B is true. So, only one of them have to be true for A
or B to be true. So, this is written as A or B with this
notation. Now comes the interesting part.
Suppose we have a rule that says, If A then B,
Which is the same as saying if A is true, then B is true.
The kind of things that we've seen before. If X is the president of C, then X is a
leader of C, or the leader of C. Well,
The really important part about logical reasoning or classical logic is that,
saying that if A then B is exactly the same thing as saying that either A is
false or B is true. Now, this requires a little bit of thought
which can also be written as A implies B is equivalent to not A or B, which is just
the same as A is either false which is not A being true or B is true which is just B.
Let's go through it again. A implies B is the same as not A or B.
Very important critical aspect of reasoning.
Let's check it. If A is true, then not A is false.
So, the only way that not A or B can become true is if B is true.
In other words, by stating that not A or B is true, we have arrived at a situation
that whenever A is true, B has to be true which is exactly what we meant by saying A
implies B. On the other hand,
If A is false, then not A is true. So, the statement not A or B is always
true regardless of B being true or false. So, the statement, rather the rule that A
implies B remains true. Whatever be the value of B, in the
situation that A is false. Which is also fine because a rule must
remain a rule regardless of the values that variables take, or propositions take.
So, please go through this argument a couple of times to convince yourselves
that A implies B is exactly the same thing as saying that the compound statement, not
A or B is true. We were dealing with statements or
propositions which were wssential statements of fact, and nothing more or
less than that. But, as we have seen before,
We are dealing with statements which are a little bit more complicated.
Obama is the president of USA, is actually stating a relationship between an entity,
Obama, and another entity, USA. And that relationship is that one is the
president of the other. Which could be written in this form, that
Obama is the president of USA. Now, this is rather tricky notation. But,
all we're saying is that we have variables which are entities like Obama and USA, and
we have predicates which are relationships between these variables or entities.
Similarly, we might have a rule which says that,
If X is the President of C then X is the leader of C. Which could be stated as,
If X is president of C, then X is leader of C.
Now, Unlike this particular statement on top,
the statement about X and C applies for all values of X and C.
So, this rule is not a simple rule such as Obama is the President of USA, it's a
statement about all possible X's and C's that says, if X and C are related by the
predicate as president of, then they're also related by the president, predicate
is leader of. So, this particular feature of predicates
and variables is called quantification, And that makes it significantly different
from basic propositional logic. Still, the process of reasoning remains
quite similar with a few modifications. So, let's go through that a little
carefully. Starting with a fact such as Obama is
President of USA which is expressed as a predicate, is President of Obama comma
USA. Together with this rule, which states that
for all X and C, if X is the president of C then X is leader of C.
Using R and F, We come to the conclusion that Obama is
leader of USA. In other words, this new fact, is leader
of Obama comma USA is entailed by the rule R applied on the fact F.
Notice a few things have taken place here that the variable X has been bound to a
value Obama, The variable C has been bound to the value
USA. So, the rule which applies to all values
of X and C is combined with a fact by binding the variables to the specific
values that they take in a fact, like F. Now, in response to a query which could be
expressed as, he is leader of X comma USA where USA has been fixed or bound, but X
remains variable. One could look at the bunch of rules and
facts that one has in one's collection of knowledge and try to answer this query by
reasoning, which is essentially answering by deriving new facts which match the
query. And we do this by a process of
unification, which is bind, binding different possible values of the variables
to X, and inference, which is essentially applying rules using the if A then B
format to derive new facts. The combination of unification and
inference is called resolution, and we'll return to resolution in a little while.
But, notice that the process of reasoning and predicate logic is fairly similar to
that if you just had propositions with the additional complication that we have
variables which need to be bound to different values.
And, if you have lots of facts and lots of rules, you have lots of different choices
for how we bind different variables to different specific values, and that leads
to a degree of complexity which makes predicate logic resolution much more
difficult than simple logical inference using a bunch of propositions.
Both are quite difficult, as we shall see during our discussions later in this
lecture, but Y is significantly more difficult than the other.