🚀 go-pugleaf

In article <287@kubix.UUCP> flach@kubix.UUCP (Peter Flach) writes:
|In article <526@thor.wright.EDU> vdasigi@silver.UUCP (Venu Dasigi) recalls
|Peirce's beautiful characterisation of deduction, induction and abduction:
|
|>Starting with
|>
|>1. A --> B (if a sample is from this bag, the sample is white.)
|>2. A (this sample s is from this bag.)
|>3. B (the sample s is white.)
|>
|>Deduction amounts to concluding 3 from 1 and 2.
|>Induction amounts to concluding 1 from 2 and 3.
|>Abduction amounts to concluding 2 from 1 and 3.
|
|The formulation of induction given here, brings into mind a problem that
|has been bothering me for some time. Given premises A and B, why should
|I prefer the inductive conclusion A --> B over B --> A (any white sample
|is from this bag)? The same problem arises with the prototypical
|crows-argument: seeing a number of black crows might amount to the
|inductive conclusion, that everything that is black is a crow. Of
|course, this hypothesis would be falsified by the observation of a black
|non-crow, rendering the set of premises non-symmetrical. But it seems to
|me that any set of premises of the form
|	{A(a)&B(a), A(b)&B(b), ...}
|give equal evidence for two possible inductive hypotheses:
|	forall(x) A(x) --> B(x), and
|	forall(x) B(x) --> A(x).
|
|Any comments on this?

In principle you are correct.  Both are "valid" inductions.  This is why
programs that do inductive inference need to have good criteria for creating
inductions, rather than creating them haphazardly.  In a way, this is what AI 
and heuristic search are all about.  Suppose, for example, that you use 
conditional probabilities to guide your search for "good" inductions.  Going
back to the crow example, if all the crows you have seen are black and all
the black things you have seen are crows then P(black|crow) and P(crow|black)
will both be high and you will decide that black(X) --> crow(X) and
crow(X) --> black(X).  I believe that this is a good induction given that
particular data.  

What is more likely, though, is that you will have seen
many objects that are black but not crows, making P(black|crow) relatively
low.  In this case, the statement black(X) --> crow(X) will be a very poor
predictor (i.e., it will usually be wrong) and is therefore a ``bad''
induction.  Remember that you can only base inductive inferences on what
you have seen and know about the world.  Then, you can test them by 
measuring their predictive ability when you see new facts and objects.

--
Quagmire Jones