Re: [ontolog-forum] Toward the logic of everyday reasoning

[Paola Di Maio <paola.dimaio@xxxxxxxxx>]

Thanks a lot John, Pavitra, Adrian and all (Zadeh!)

For the comments and Interesting references

I learned about fuzzy logic together with other logic, when learning about NL systems, so for me it was not a problem accepting it, but I was told that it was a hard one to pass through the thight net of certain black and white academic circles back in its days (and probably still is)

I will study FLu in more detail when I get the chance

Mathematics has rarely been intuitive for me, I wonder if quasi-mathematics would come easier?
(LOL)


Only found one reference to Lacan

http://books.google.co.uk/books?id=PvfpyjM0PBYC&lpg=PA109&ots=e5wTsRTfV8&dq=QUASI%20MATHEMATICS&pg=PA109#v=onepage&q=QUASI%20MATHEMATICS&f=false


Best

PDM

On Sun, Nov 22, 2009 at 3:30 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:

Adrian,

That is just a notational method for burying the phrase
"very likely" in the name of the predicate:

AW> some-person is a Swedish national

 > it is very likely that that-person is tall
 > --------------------------------------------
 > it is likely that that-person is blond

That is equivalent to writing

(Ax)(is_a_Swedish_national(x) & it_is_very_likely_that_is_tall(x)
    -> it_is_very_likely_that_is_blond(x))

I have said that your method of writing rules with very readable
predicate names is useful.  But it's not a general solution to
the question of how to deal with probability, fuzzy qualifiers,
etc.

A more general method, which Bob Kowalski proposed for Prolog
many years ago, is to use a metalevel approach.  For example,
consider Lotfi's original statement:

LZ>  Most Swedes are tall.

 >    Most tall Swedes are blond.
 >    Magnus is a Swede (picked at random)
 >    What is the probability that Magnus is blond?

I'll use the IKL notation, which uses the operator 'that'
for making statements about propositions.  Following is
a translation of the three premises to IKL:

  (forall (x)
    (if (Swedish x)
        (gt (ProbabilityOf (that (tall x)) 0.5)) ))

  (forall (x)
    (if (and (tall x) (Swedish x))
        (gt (ProbabilityOf (that (blond x)) 0.5)) ))

  (Swedish Magnus)

 From the first premise and the third premise, we derive

   (gt (ProbabilityOf (that (tall Magnus)) 0.5))

To go further we have to add some axioms about probability.
The first one is that if a proposition p is true, then its
probability is 1.0:

   (forall (p)
      (if p (Probability p 1.0)) )

That allows us to deduce

   (Probability (that (Swedish Magnus)) 1.0)

Then we need more axioms for computing the probabilities
of statements combined with various Boolean operators:

   (forall (p q x y))
      (if (and (Probability p x) (Probability p y)
          (Probability (that (and p q)) (* x y))) ))

 From this we can conclude that

   (gt (Probability (that (and (tall Magnus) (Swedish Magnus)))) 0.5)

Then we need to add another axiom for 'if'.

With this approach, Kowalski showed how to use Prolog to reason
about probabilities, fuzzy logic, and many other kinds of reasoning.
I am not recommending it or opposition it.  I'm just showing how
it can be done by using any version of FOL that supports metalevel
statements.

John


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