markh@csd4.milw.wisc.edu (Mark William Hopkins) (05/31/89)
In article <6144@cognos.UUCP> rayt@cognos.UUCP (R.) writes: >In article <14820@paris.ics.uci.edu> Wendy Sarrett writes: > ><>"induction" is the process of generalizing from lots of examples. For ><>example, suppose you see a number of examples of ducks and they are ><>all grey ( isa-duck -> grey) then you would conclude for all ducks, ><>isa-duck -> grey. ... > ><>Note that both "abduction" and "induction" are not "safe" forms of ><>inference as "deduction" is. (i.e. you can't be 100% certain your ><>inference is correct) > >Clearly the first form of induction given is not a logically valid >deduction. ... Induction is safe in the following sense. It is a process that can be broken down into 2 parts: (1) The recognition of the underlying universal pattern, and (2) its application to everything else. The first stage can be considered safe if we formulate it in the right way. For example, suppose we run across the facts: Duck #1 is grey. Duck #2 is grey. Duck #3 is grey. Duck #4 is grey. Then we can say that: for all X in [1, 2, 3, 4]: Duck #X is grey. This is step (1), recognizing the underlying pattern. Provided we keep the bounded quantifier on, it is deductively safe. And that is the sense in which induction is safe. In step (2), of course, we would then eliminate the bound and apply this to everything: for all X: Duck #X is grey. It is not "safe". But no matter, because most of the effort spent in induction is spent in completing step (1), not step (2). Anyhow, step (2) can always be rescinded and converted into the "discovery" of a new class, such as: for all X in New-Class: Duck #X is grey. At this point, our strategy would be to take all the currently observed members of this newly-inducted class and try to do some factoring with other features, such as what was done with #X -- in order to come up with independent characteristics of this new class. And this eventually leads us to the discovery of new theorems. But that's a long story ...