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From: OsherD on 26 May 2010 21:06
From Osher Doctorow

Since my wife Marleen and I first pointed out in 1980 that Probable
Causation/Influence (PI), which we then called the Probability of
Material Implication) is analogous to Lukaciewicz/Rational Multivalued
Logics, and Jan Lukaciewicz lived from 1878 to 1956, it seems strange
that India's Sumit Mohinta and T. K. Samantha of Uluberia College
India are claiming today in arXiv: 1005.4140 v1 [math.GM] 22 May 2010,
20 pages, that the line of fuzzy logics in effect developed from Zadeh
in 1965 to Katsaras in 1984 to Felbin in 1992, and reading their
definitions of "fuzzy norm" and "fuzzy norm on a linear space" yields
exactly the Multivalued Logical fuzzy t-norms (see also Pavel Hajek,
"Metamathematics of Fuzzy Logics", Kluwer: Dordrecht 1998, which is
actually mostly on Multivalued Logics).

I will continue this later hopefully - I have to leave to do some
outside tasks.
Osher Doctorow
From: OsherD on 26 May 2010 21:43
From Osher Doctorow

The title of Mohinta et al's article is:

1) "A note on generalized intuitionistic fuzzy phi normed linear
space,".

It is 20 pages long.

Their "continuous t-norm" and "continuous t-conorm" are extremely
similar to Multivalued Logic t-norms. Moreover, it turns out in
Multivalued Logics that Multivalued Logical Conditionals/Implications
are far more important in practice than the t-norms which in fact
obscure the relationship with Probability. Even Hajek in his
Probability chapter didn't notice the analogy which Marleen and I
noticed in 1980 and 1983 in U.C. Berkeley USA philosophy seminars
which were later published.

The Multivalued Logical Conditions or Implications are:

2) (x-->y) = 1 + y - x for y < = x (Lukaciewicz/Rational Pavelka)
3) (x-->y) = y/x for x not 0 (Product/Goguen)
4) (x-->y) = y (Godel)

which Marleen and I found to correspond to Probability - respectively
Probable Causation/Influence (PI), Conditional Probability, and
Independent Probability/Statistics. Here x, y assume any values
between 0 and 1 except for x not being 0 in (3).

Osher Doctorow
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