Fuzzy Sets and Infinite-Valued Łukasiewicz Logic in Foundations of Quantum Mechanics

Abstract We show that measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability theory. The transition from Boolean logic to Łukasiewicz logic has a categorical background and the resulting divisible probability theory possesses both fuzzy and quantum qualities. Observables of the divisible probability theory play an analogous role as classical random variables: to convey stochastic information from one system to another one. Observables preserving the Łukasiewicz logic are called conservative and characterize the “classical core” of divisible probability theory. They send crisp random events to crisp random events and Dirac probability measures to Dirac probability measures. The nonconservative observables send some crisp random events to genuine fuzzy events and some Dirac probability measures to nondegenerated probability measures. They constitute the added value of transition from classical to divisible probability theory.

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Mathematical foundations of quantum mechanics. By John von Neumann, translated from the German by R. T. Beyer. Pp. xii, 445. 48s. 1955, (Princeton University Press; Geoffrey Cumberlege, Oxford University Press)

The Mathematical Gazette ◽

10.1017/s0025557200034999 ◽

1956 ◽

Vol 40 (332) ◽

pp. 150-150

Author(s):

R. G. Cooke

Keyword(s):

Quantum Mechanics ◽

Foundations Of Quantum Mechanics ◽

John Von Neumann ◽

Von Neumann ◽

Oxford University ◽

Princeton University ◽

Mathematical Foundations ◽

Oxford University Press

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Dynamic Łukasiewicz logic and its application to immune system

Soft Computing ◽

10.1007/s00500-021-05955-3 ◽

2021 ◽

Author(s):

Antonio Di Nola ◽

Revaz Grigolia ◽

Nunu Mitskevich ◽

Gaetano Vitale

Keyword(s):

Immune System ◽

Scalar Multiplication ◽

Kripke Semantics ◽

Łukasiewicz Logic ◽

Lukasiewicz Logic ◽

Regular Algebras

AbstractIt is introduced an immune dynamic n-valued Łukasiewicz logic $$ID{\L }_n$$ I D Ł n on the base of n-valued Łukasiewicz logic $${\L }_n$$ Ł n and corresponding to it immune dynamic $$MV_n$$ M V n -algebra ($$IDL_n$$ I D L n -algebra), $$1< n < \omega $$ 1 < n < ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras $$(\mathcal {M}, \mathcal {R}, \Diamond )$$ ( M , R , ◊ ) that combine the varieties of $$MV_n$$ M V n -algebras $$\mathcal {M} = (M, \oplus , \odot , \sim , 0,1)$$ M = ( M , ⊕ , ⊙ , ∼ , 0 , 1 ) and regular algebras $$\mathcal {R} = (R,\cup , ;, ^*)$$ R = ( R , ∪ , ; , ∗ ) into a single finitely axiomatized variety resembling R-module with “scalar” multiplication $$\Diamond $$ ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic $$ID{\L }_n$$ I D Ł n with application in immune system.

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