Fuzzy quantum logics and infinite-valued ?ukasiewicz logic

The Problem of Conjunction and Disjunction in Quantum Logics

“…The results of investigations performed by one of the authors through years [12,13,15,16] (all results are collected in [17]) show that every OML (even orthomodular poset) with an ordering set of states S can be isomorphically represented as a family L(S) of infinite-valued propositional functions defined on S endowed with Łukasiewicz conjunction:…”

Modelling of Uncertainty and Bi–Variable Maps

“…However, various authors use this name to denote different objects, such as just orthomodular lattices (without mentioning probability measures at all) [32], or orthomodular posets with ordering sets of probability measures [25]. Therefore, we decided not to use this name in the present paper.…”

“…In [25] the following representation theorem was proved: Theorem 2. Every orthomodular poset L with an ordering set of probability measures M can be isomorphically represented by a family Λ of fuzzy subsets of M endowed with the inclusion of fuzzy sets as partial order, the standard fuzzy set complementation as orthocomplementation, and such that (i) The empty set belongs to Λ , (ii) Λ is closed with respect to the standard fuzzy set complementastion, (iii) Λ is closed with respect to Lukasiewicz unions of sequences of pairwisely weakly disjoint sets, (iv) the empty set is the only set in Λ that is weakly disjoint with itself.…”

Bell-Type Inequalities for Bivariate Maps on Orthomodular Lattices

“…However, one of the authors in a series of papers (see, e.g., [26][27][28]) promoted an idea that 'quantum logic' can be equivalently regarded as a specific ∞-valued Łukasiewicz logic, which opens the possibility of working out a new interpretation of quantum mechanics [29]. In this approach conjunctions and disjunctions of propositions about quantum systems are modelled by a pair of partially defined operations used in a specific version of Łukasiewicz many-valued logic.…”

“…The notion of an s-map opens a new possibility: if propositions a and b are non-compatible, the value p(a, b) can be thought of as representing probability of simultaneous verification of a and b in a 'counterfactual measurement': 'what would be the probability of simultaneous verification of propositions a and b if we were able to perform it' or, according to the approach propounded in [26][27][28] Let us note that…”