Fuzzy Sets pp 49-57 | Cite as

Fuzzy Logic and Non-Distributive Truth Valuations

  • David McGoveran


In 1936, Birkhoff and von Neumann were able to show that a non-distributive, orthocomplemented, modular lattice was equivalent to the traditional mathematical representation of quantum mechanics. In such a lattice of propositions, it is demonstrable that there can be no coherent truth valuation set of cardinality greater than two inasmuch as a finite probability measure is not homomorphic with a non-distributive valuation on the real interval [0,1]. In the present paper, it is empirically demonstrated that the logic of natural languages is non-distributive. Thus, the lattice theoretic representation reduces from the traditional Boolean one to that proposed by Birkoff and von Neumann for quantum mechanics, commonly known as quantum logic. This result implies that fuzzy logics, probability logics, and multi-valued logics are inappropriate representations of either natural linguistic or quantum mechanical propositions. At best, such representations are valid only under limited conditions in which the lattice is locally Boolean (known as the isles of Boole). A criteria is presented for the determination of these conditions from empirical data.


Membership Function Fuzzy Logic Distributive Lattice Fuzzy Variable Modular Lattice 
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Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • David McGoveran
    • 1
  1. 1.Alternative TechnologiesBoulder CreekUSA

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