While reviewing my AI textbook, I came across a paragraph that baffled me. It attempts to explain why the truth table for implication turns out to be perfect, as put by Russel and Norvig in their textbook.
Logically speaking, the sentence ∀x King(x)→ Person(x) seems quite reasonable, since if an object turns out to be a king, then this king must indeed be a person. What I find strange, however, is that such a sentence will be asserted for all and every x, even if x is not a king, since, the implication is always true incase the premise is true. Russel and Norvig comment on this by the following:
the other assertions are true in the model, but make no claim whatsoever about the personhood qualifications of legs, crowns, or indeed Richard.
and
by asserting the universally quantified sentence, which is equivalent to asserting a whole list of individual implications, we end up asserting the conclusion of the rule just for those objects for whom the premise is true and saying nothing at all about those individuals for whom the premise is false.
Could someone please clarify what they mean? and, in this case, how is the truth table for implication perfect as they stated?
I guess they are making the following general point (to tinker with an answer I gave to another question here):
As background, start with the trite remark that mathematicians are very, very often concerned with statements of generality and especially statements of multiple generality – you know the kind of thing, e.g. the definition of continuity that starts for any $\epsilon$ ... there is a $\delta$ ... And the formalized quantifier-variable notation which we all learn serves mathematicians brilliantly to regiment such statements of multiple generality and make them utterly unambiguous and transparent (as contrasted with some of their ordinary-language counterparts).
OK, so now think about restricted quantifiers that talk about only some of a domain (e.g. talk not about all numbers but just about all the even ones). How might we render Goldbach's Conjecture, say? As a first step, we might try
Here restrict the quantifier using a conditional, and this seems at first sight to give us what we want. But now think about the embedded conditional here. In particular, ask: what if $n$ is odd, so the antecedent of the conditional is false??? If we say this instance of the conditional lacks a truth-value, or may be false, then the quantification would have non-true instances and so would not be true! But of course we, we all agree that we can't refute Goldbach's Conjecture by looking at odd numbers!! So if the quantified conditional is indeed to come out true when Goldbach is right, then we'll have to say that the irrelevant instances of the conditional with a false antecedent have to be treated as non-false by default. And in classical logic, non-falsity implies truth. Which means that the embedded conditional will have to be treated as a material (truth-functional) conditional.
So the moral seems to be: if mathematicians are to deal nicely with expressions of generality using the quantifier-variable notation they will have to get used to using material conditionals too.