Minds and Computers : An Introduction to the Philosophy of Artificial Intelligence

Exercise 13.3
Before reading on, attempt to apply the first three steps of
this procedure to the initial state of our kinship system.
The first conditional in our resident information is:
grandparent_of (x , y) & male (x) 
→ grandfather (x)
We don’t have any statements in our initial state of the form grand-
parent_of (x , y), so we are unable to satisfy the antecedent of this
conditional. We can satisfy one of the conjuncts as we do have state-
ments of the form male (x), but we need to satisfy both conjuncts to
satisfy the conjunctive antecedent.
The next two conditionals in our resident information are:
parent (x) & male (x) 
→ father (x)
parent (x) & female (x) 
→ mother (x)
Once again, we don’t have any statements of the form parent (x), so
we can’t satisfy the antecedents of either of these conditionals.
Do be careful here, for while we do know that parent_of (j , m),
and we know – as a matter of common sense – that if someone is a
parent of someone else they are ipso facto a parent, we can’t just
assume ‘parent (j)’. The symbols ‘parent’ and ‘parent_of ’ are just
that – symbols. We have used symbols that are meaningful to us but
as far as the operations of the system are concerned, any relations
between the predicates that symbols represent need to be encoded
explicitly in the rules. This is precisely what the next conditional
does.
The fourth conditional in our resident information is:
parent_of (x , y) 
→ parent (x)
We do have a statement in the initial state which satisfies the
antecedent of this conditional – parent_of (j , m) – so we can add the
consequent (being careful to substitute the correct name for the vari-
able) to our list of deduced statements – namely parent (j).
 
139

---

We also have another statement in the initial state which satisfies
the antecedent of the same conditional – parent_of (m , h) – so we can
also add parent (m) to our list of deduced statements.
The fifth conditional in our resident information is
parent_of (x , y) 
→ child_of (y , x)
We have two statements in the initial state which satisfy the ante-
cedent of this conditional – parent_of (j , m) and parent_of (m, h) –
so we can add child_of (m , j) and child_of (h , m) to our list of
deduced statements.
The final conditional in our resident information is:
parent_of (x , y) & parent_of (y , z) 
→ grandparent_of (x , z)
We need to be careful here – the name which we substitute for y in the
first conjunct of the antecedent must be the same name which we sub-
stitute for y in the second conjunct. As it turns out, we do have state-
ments which satisfy the antecedent of this conditional – parent_of
(j, m) and parent_of (m , h) – which allows us to deduce grandpar-
ent_of (j , h).
We have now considered each of the conditionals in the resident
information and have deduced five new statements:
parent ( j )
parent (m)
child_of (m , j)
child_of (h , m)
grandparent_of (j , h)
So we add the list of deduced statements to our initial state (checking
to make sure none of them are redundant) to get the following state:
parent_of (j , m)
parent_of (m , h)
male (j)
male (h)
female (m)
parent (j)
parent (m)
child_of (m , j)
child_of (h , m)
grandparent_of (j , h)
The next thing to do is to check each of the conditionals again in turn
to see if the new statements in our derived state allow us to deduce
any further novel statements.
140
  

---