🚀 go-pugleaf

In article <6144@cognos.UUCP> rayt@cognos.UUCP (R.) writes:
>In article <14820@paris.ics.uci.edu> Wendy Sarrett writes:
><>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 there is also induction in mathematics...
> 
><>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. I am surprised to here that the SECOND isn't, since MANY
>mathematical proofs rest upon it. 

What exactly do you mean by "logically valid"?  The definition in my
math logic class gives a couple of really basic axioms about how
logical (truth-functional) connectives and quantifiers relate to one
another, and says something is "logically valid" iff it holds in all
models of those axioms.  Those axioms do not include mathematical
induction, and in fact there are lots of models of those minimal
axioms (indeed, there are lots of models of the more powerful
Robinson arithmetic, which looks as though it's talking about natural
numbers) in which induction does not hold.  Induction happens to be
true in the Platonic object we call the natural numbers, so you can
draw valid conclusions about the natural numbers using it, but it's
certainly not logically valid.
By the way, there are different flavors of mathematical induction,
which are equivalent in the unbounded logic of the natural numbers but
not equivalent in computational systems with bounded resources.

"The above opinions are my own.  But that's just my opinion."
Stephen Bloch