Minds and Computers : An Introduction to the Philosophy of Artificial Intelligence
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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 133 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 134 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) 135 |
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