Quantified Propositional Logics and Quantifier Elimination
(joint work with Norbert Preining and Viennese collegues)
In this lecture we discuss quantified propositional logics with linearly
ordered truth values and condidtions for the elimination of propositional
quantifiers.
Propositional quantifiers are defined in a natural way by suprema and infima.
We concentrate on Gödel logics on closed subsets of [0,1] containing 0,1 and
on Lukasiwicz logic.
For the quantified propositional Gödel logic on [0,1] we demonstrate
eliminability of quantifiers and exhibit a finite aximatization based on an
axiomatic variant of the Takeuti-Titani rule. For the quantified propositional
Gödel logics Garrowdown (on {1/n : n in N}u{0}) and
Garrowup (on {1-1/n : n in N}u{1}) we calculate the minmial propositional
extension where elimination of quantifiers is possible and provide a finite
axiomatization.
As a corrollary we prove, that Garrowup is the intersection of quantified
propositional finitely valued Gödel logics.
For quantified propositional Lukasiewicz logic we calculate the minimal
propositional extension admitting quantifier elimination. As a corrolary we
obtain, that quantified propositional Lukasiewicz logic in the original
language is decidable, but not finitely axiomatizable.