[FOM] Logical Correctness

[Vaughan Pratt]

While I don't know whether Zadeh would agree with this, I would think 
that to answer his question without first reading Peter Hajek's 
excellent article at

http://plato.stanford.edu/entries/logic-fuzzy/

one would basically have to re-invent Hajek's wheel.  The two references 
Zadeh offered along with his question,

http://www2.iiia.csic.es/%7Efbou/LoMoReVI/Slides/Bou-FuzzyModalLogics.pdf

and

http://uai.sis.pitt.edu/papers/94/p278-hajek.pdf

both assume it as a prerequisite.

The former reference should be of particular interest to fans of Belnap 
and Anderson's relevant logic, Girard's linear logic, and Dunn's gaggle 
(for "generalized galois logics") theory because of the role played in 
it by Ward and Dilworth's much earlier (1939) residuated lattices in a 
nonclassical modal setting.

Zadeh's question seems to me somewhere between motivation and 
advertisement for these papers, which address a considerably wider range 
of questions about modal fuzzy logic and suitable implications.

Birkhoff and von Neumann's quantum logic based on orthocomplemented 
lattices is another logic with perfectly fine laws for conjunction, 
disjunction and (involutory) negation for which implication presents 
problems, e.g. the nice-in-some-respects Sasaki hook lacks a Deduction 
Theorem.

Vaughan Pratt