As the logic of propositions of quantum measurements,G.Birkhoff and J.von Neumann's quantum logic has been regarded as including Orthogonal complement that is weaker than classic negation,quantum disjunction that is neither classical logic nor distributive,and poset that is a non-Boolean algebra.Based on the unitary operator functional analyses,this paper points out that those apparent nonclassic features of quantum logic are only classic functional nature,and the non-distribution ∩(x ∪ x_) = y ∩1 = y,but(y ∩ x) ∪(y ∩ x_) = ∪ 0 = x at the benzene ring can not be correct.As the logic of propositions of quantum measurements,G.Birkhoff and J.von Neumann's quantum logic has been regarded as including Orthogonal complement that is weaker than classic negation,quantum disjunction that is neither classical logic nor distributive,and poset that is a non-Boolean algebra.Based on the unitary operator functional analyses,this paper points out that those apparent nonclassic features of quantum logic are only classic functional nature,and the non-distribution ∩(x ∪ x_) = y ∩1 = y,but(y ∩ x) ∪(y ∩ x_) = ∪ 0 = x at the benzene ring can not be correct.

Kexue jishu zhexue yanjiuJan 1, 20130 citations

Logic and Quantum Logic

Peter Mittelstaedt

Boston studies in the philosophy of scienceJan 1, 19760 citations

Recovering quantum logic within an extended classical framework

Claudio Garola, Sandro Sozzo

We present a procedure which allows us to recover classical and nonclassical logical structures as \emph{concrete logics} associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory $\mathcal T$ and classical language $\mathcal L$ expressing $\mathcal T$ , an observative sublanguage $L$ of $\mathcal L$ with a notion of truth as correspondence, introducing in $L$ a derived and theory-dependent notion of \emph{C-truth} (\emph{true with certainty}), defining a \emph{physical preorder} induced by C-truth, and finally selecting a set of sentences that are \emph{verifiable} (or \emph{testable}) according to $\mathcal T$ , on which a \emph{weak complementation} is induced by $\mathcal T$ . The triple consisting of the set of verifiable sentences, physical order and weak complementation is then the desired concrete logic. By applying our procedure we recover a classical logic as the concrete logic associated with classical mechanics and standard quantum logic as the concrete logic associated with quantum mechanics. We also show that our alternative view of standard quantum logic, which can be constructed in a purely formal way, can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Our results then show that some nonstandard logics can be obtained as mathematical structures formalizing the properties of different notions of verifiability in different physical theories. More generally, they strongly support the idea that many nonclassical logics can coexist without conflicting with classical logic (\emph{global pluralism}), for they formalize metalinguistic notions that do not coincide with the notion of truth (described by Tarski's truth theory).We present a procedure which allows us to recover classical and nonclassical logical structures as \emph{concrete logics} associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory $\mathcal T$ and classical language $\mathcal L$ expressing $\mathcal T$, an observative sublanguage $L$ of $\mathcal L$ with a notion of truth as correspondence, introducing in $L$ a derived and theory-dependent notion of \emph{C-truth} (\emph{true with certainty}), defining a \emph{physical preorder} induced by C-truth, and finally selecting a set of sentences that are \emph{verifiable} (or \emph{testable}) according to $\mathcal T$, on which a \emph{weak complementation} is induced by $\mathcal T$. The triple consisting of the set of verifiable sentences, physical order and weak complementation is then the desired concrete logic. By applying our procedure we recover a classical logic as the concrete logic associated with classical mechanics and standard quantum logic as the concrete logic associated with quantum mechanics. We also show that our alternative view of standard quantum logic, which can be constructed in a purely formal way, can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Our results then show that some nonstandard logics can be obtained as mathematical structures formalizing the properties of different notions of verifiability in different physical theories. More generally, they strongly support the idea that many nonclassical logics can coexist without conflicting with classical logic (\emph{global pluralism}), for they formalize metalinguistic notions that do not coincide with the notion of truth (described by Tarski's truth theory).

Jan 1, 20139 citations

Nonclassical features in higher-dimensional systems through logical qudits

Sooryansh Asthana, V. Ravishankar

In a recent work [S. Asthana. New Journal of Physics 24.5 (2022): 053026], we have shown the interrelation of different nonclassical correlations in multiqubit systems with quantum coherence in a single logical qubit. In this work, we generalize it to higher-dimensional systems. For this, we take different choices of logical qudits and logical continuous-variable (cv) systems in terms of their constituent physical qudits and physical cv systems. Thereafter, we show reciprocity between conditions for coherence (in logical qudits and logical cv systems) and conditions for nonlocality and entanglement (in their underlying constituent qudits and cv systems). This shows that a single nonclassicality condition detects different types of nonclassicalities in different physical systems. Thereby, it reflects the interrelations of different nonclassical features of states belonging to Hilbert spaces of nonidentical dimensions.In a recent work [S. Asthana. New Journal of Physics 24.5 (2022): 053026], we have shown the interrelation of different nonclassical correlations in multiqubit systems with quantum coherence in a single logical qubit. In this work, we generalize it to higher-dimensional systems. For this, we take different choices of logical qudits and logical continuous-variable (cv) systems in terms of their constituent physical qudits and physical cv systems. Thereafter, we show reciprocity between conditions for coherence (in logical qudits and logical cv systems) and conditions for nonlocality and entanglement (in their underlying constituent qudits and cv systems). This shows that a single nonclassicality condition detects different types of nonclassicalities in different physical systems. Thereby, it reflects the interrelations of different nonclassical features of states belonging to Hilbert spaces of nonidentical dimensions.

arXiv:2203.06635Mar 13, 20220 citations

\mathcal T

and classical language

\mathcal L

expressing

\mathcal T

, an observative sublanguage

L

\mathcal L

with a notion of truth as correspondence, introducing in

L

a derived and theory-dependent notion of \emph{C-truth} (\emph{true with certainty}), defining a \emph{physical preorder} induced by C-truth, and finally selecting a set of sentences that are \emph{verifiable} (or \emph{testable}) according to

\mathcal T

, on which a \emph{weak complementation} is induced by

\mathcal T

. The triple consisting of the set of verifiable sentences, physical order and weak complementation is then the desired concrete logic. By applying our procedure we recover a classical logic as the concrete logic associated with classical mechanics and standard quantum logic as the concrete logic associated with quantum mechanics. We also show that our alternative view of standard quantum logic, which can be constructed in a purely formal way, can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Our results then show that some nonstandard logics can be obtained as mathematical structures formalizing the properties of different notions of verifiability in different physical theories. More generally, they strongly support the idea that many nonclassical logics can coexist without conflicting with classical logic (\emph{global pluralism}), for they formalize metalinguistic notions that do not coincide with the notion of truth (described by Tarski's truth theory).