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

inductive proof, on the other hand, is a display of amassed observa-
tions, made under certain conditions, in support of the claim that
future such conditions will yield the same observations.
It is a feature of deductive proofs that they are unrevisable. There
is nothing that can be added to a deductive proof such that it will, in
light of the addition, fail to yield its original conclusion. Inductive
proof, on the other hand, is essentially revisable. Inductive proofs
only establish their conclusions with a certain degree of probability
and are only ever one countervailing observation away from failing to
deliver their purported conclusions.
Although I’ve only given a rough sketch of inductive reasoning
here, I’ve illustrated the distinction with deductive reasoning for two
reasons: firstly, in order that you appreciate that there are legitimate
and established kinds of reasoning which are distinct from deduction;
secondly, so that you realise that the word ‘proof ’ means something
quite di
fferent in the mouths of scientists to what it does in the
mouths of logicians or mathematicians. When you hear that scientists
have ‘proven’ something or that ‘studies have shown’ something, it
pays to realise that the very thing proved or shown may be revised and
disproved in light of subsequent investigation.
There is a lot more to be said about scientific reasoning and I cer-
tainly wouldn’t want to be charged with having given only a caricature
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of scientific process so, once again I refer you to the suggestions for
further reading.
Before we proceed to examine expert systems, we will need to develop
a little bit of terminology concerning conditionals and predicates.
13.2 CONDITIONALITY AND PREDICATION
Natural language conditionals are statements of the form ‘if . . .
then . . .’. The study of conditionals, and the determination of an
adequate formal account thereof, is of central importance to logic.
Many logics are distinguished solely by virtue of their treatment of
the conditional.
We can represent conditionals by using an arrow. The statement ‘if
today is Monday then tomorrow is Tuesday’ can be represented as
follows:
today is Monday 
→ tomorrow is Tuesday
The left-hand side of a conditional – which represents the bit
between ‘if ’ and ‘then’ – is the antecedent of the conditional. The
right-hand side – the bit which comes after ‘then’ – is the consequent
of the conditional.
If the antecedent of a conditional is satisfied then we can derive the
consequent according to a simple logical principle. So, if it is actually
the case that today is Monday, we can – given the above conditional –
deduce that tomorrow is Tuesday. This logical principle is known as
modus ponens and can be symbolised as follows:


→ 


_______

 
The logical principle of modus ponens – which tells us that given a con-
ditional with a satisfied antecedent we can deduce its consequent – is
the only logical principle we will be appealing to in our examination
of expert systems.
The last thing to do before looking at an example expert system is
to discuss predicates and logical forms.
Consider the following two statements. If something is a dog then
it is a mammal. If something is a mammal then it has a heart. One
way to represent these statements would be as follows:
something is a dog 
→ that thing is a mammal
something is a mammal 
→ that thing has a heart
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However, we can do better than that. Notice that in each antecedent
and consequent, a property is applied to – or predicated of – a thing.
Note also that in each conditional, it is the same thing referred to in
both the antecedent and the consequent.
If we take ‘dog’ to represent the property of being a dog, ‘mammal’
to represent the property of being a mammal, and ‘heart’ to represent
the property of having a heart, we can recast the above conditionals
to capture the fact they are applying properties to the same thing in
their antecedents and consequents:
dog (x) 
→ mammal (x)
mammal (x) 
→ heart (x)
These conditionals are as close to logical form as we require for the
purposes of this chapter. The symbols ‘dog’, ‘mammal’ and ‘heart’ –
which could, of course, be substituted uniformly for any other symbol
we choose – represent predicates. For our purposes, predicates can be
understood as encoding properties and relations.
The symbol x in the above conditionals is a variable – as you have
no doubt discerned. We will say that the antecedent of one of these
conditionals is satisfied if we have a statement which has the same
logical form.
Statements, for our purposes, apply predicates to names (not vari-
ables). So if, for instance, we know that Mia is a dog, we can represent
this by using the symbol m as a name for Mia, as follows:
dog (m).
We can now use the two conditionals we have symbolically repre-
sented to do some simple deduction. The statement – dog (m) –
is of the same logical form as the antecedent of our first condi-
tional. This means that the antecedent of the conditional is satisfied
so we can deduce the consequent, namely mammal (m). We now
have a statement which satisfies the antecedent of the second con-
ditional, so we can deduce its consequent and derive heart (m).
Given that we know that ‘heart’ represents the property of having
a heart and that m is a name for Mia, we have just deduced that
Mia has a heart.
We can represent this deduction symbolically, as follows:
dog (m)
dog (x) 
→ mammal (x)
____________________

 mammal (m)
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mammal (x) 
→ heart (x)
____________________

 heart (m)