Waclaw
thanks for this
I was hoping to get the mathematical representation of this reasoning so that I can learn somthing from my 'to do' list
while I reason about nature -
the problem is that y (fred, the bat) is not a bird
y is a mammal -
therefore, if y is not a bird, and it flies,
not only birds fly
(tell me what I am missing)
P
On 7/17/07, Waclaw Kusnierczyk
<Waclaw.Marcin.Kusnierczyk@xxxxxxxxxxx> wrote:
Re induction.
The pattern John shows as induction is (correct me if I am wrong):
From:
P(x1), P(x2), ..., P(xn)
Q(x1), Q(x2), ..., Q(xn)
induce:
forall x, P(x) => Q(x)
e.g., x1, x2, ..., xn are birds and they fly, therefore if y is a bird,
it flies.
(Note that this is symmetrical wrt. P and Q; but if you add Q(y) and
~P(y), you can still induce the above but not the inverse.)
The pattern above is a case of generalization: from a number of
examples, you infer (here induce) a general rule.
John, would you agree that the following is also a case of induction (an
inductive specialization)?
exists x, P(x) and Q(x)
P(y)
induce (?): Q(y)
e.g., some birds fly, y is a bird, therefore y flies.
vQ
paola.dimaio@xxxxxxxxx wrote:
> Thanks John
>
>
> I do not mean to insist on this lilliputian quibble but...
>
>
> > Given: Tweety, Polly, and Hooty are birds. Fred is bat.
> > Tweety, Polly, and Hooty fly. Fred flies.
> > Assume: Every bird flies.
>
>
>
> could you explain how does the bat part of your statement end up in the
> assumption that every bird flies? or does the assumption completely
> ignores the fact that Fred is a bat. yet it flies?
> if the assumption is based on taking into account both parts of the
> statement, then the assumption as stated above seems incomplete -
>
> this logic right?
>
> not everything that flies is a bird, and not every bird flies
>
> I ll make something up for the class
>
> (feel free to ignore )
>
> P
>
> On 7/14/07, * John F. Sowa* <sowa@xxxxxxxxxxx <mailto:sowa@xxxxxxxxxxx
>>
> wrote:
>
> Paola,
>
> In my previous note, I forgot to answer the following question:
>
> PDM> My assumption, following your example 2 would be:
> > not only birds fly - would I be right?
>
> JFS> 2. Induction. Assume a general principle that subsumes many facts.
> >
> > Given: Tweety, Polly, and Hooty are birds. Fred is bat.
> > Tweety, Polly, and Hooty fly. Fred flies.
> > Assume: Every bird flies.
>
> PDM> NOT ONLY BIRDS FLY.
>
> Yes, that is true. But that is a separate observation.
>
> The assumption made by induction is "Every bird flies."
>
> The additional statement "Not only birds fly" follows
> from two facts plus some background knowledge plus an
> inference. Following are the facts as given:
>
> Fred is a bat. Fred flies.
>
> The additional background knowledge, which was not stated
> in the slide, is
>
> No bat is a bird.
>
> From that statement and the preceding facts, one can infer
> by deduction:
>
> Some things fly that are not birds.
>
> Then it is possible to rephrase that conclusion as a qualifier
> to the preceding:
>
> Every bird flies, but not only birds fly.
>
> In short, you could add that statement, but it is derived
> by a more complex series of steps. For teaching purposes,
> it might be better to show that in a separate slide.
>
> (And, of course, the statement "Every bird flies" has to be
> qualified when you consider penguins and kiwis. You also have
> to exclude injured birds, baby birds, sleeping birds, and
> dead birds -- remember the dead parrot from Monty Python.)
>
> John
>
>
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