arXiv:2506.10045

Eigenlogic and Probabilistic Inference; when Bayes meets Born

François Dubois, Zeno Toffano

Published Jun 11, 2025• 0 citations

Abstract

This paper shows how inference is treated within the context of Eigenlogic projection operators in linear algebra. In Eigenlogic operators represent logical connectives, their eigenvalues the truth-values and the associated eigenvectors the logical models. By extension, a probabilistic interpretation is proposed using vectors outside the eigensystem of the Eigenlogic operators. The probability is calculated by the quantum mean value (Born rule) of the logical projection operators. We look here for possible connections between the Born rule in quantum mechanics and Bayes' theorem from probability theory and show that Eigenlogic offers an innovative approach to address the probabilistic version of logical inference (material implication) in a quantum context.

DOI: 10.48550/arxiv.2506.10045

Related papers

Eigenlogic and probabilistic inference: when Bayes meets Born

Zeno Toffano, François Dubois

This article shows how inference is treated within the context of Eigenlogic projection operators in linear algebra. In Eigenlogic, operators represent logical connectives, their eigenvalues the truth-values and the associated eigenvectors the logical models. By extension, a probabilistic interpretation is proposed using vectors outside the eigensystem of the Eigenlogic operators. The probability is calculated by the quantum mean value (Born rule) of the logical projection operators. We look here for possible connections between the Born rule in quantum mechanics and Bayes' theorem from probability theory and show that Eigenlogic offers an innovative approach to address the probabilistic version of logical inference (material implication) in a quantum context. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering SciencesDec 11, 20250 citations

When Bayes meets Born; Quantum-like Probabilistic and Eigenlogic Inference

Zeno Toffano, François Dubois

We show how inference is treated within the context of Eigenlogic projection operators in linearalgebra. In Eigenlogic, operators represent logical connectives, their eigenvalues the truth‑values andthe associated eigenvectors the logical models. By extension, a probabilistic interpretation is proposedusing vectors outside the eigensystem of the Eigenlogic operators. The probability is calculated by the quantummean value (Born rule) of the logical projection operators. We look here for possible connections between the Born rule in quantum mechanics and Bayes’ theorem from probability theory and show that Eigenlogic offers an innovative approach to address the probabilistic version of logical inference (material implication) in a quantum context.

Jun 10, 20250 citations

Quantum Inspired Eigenlogic and Truth Table Semantics

Zeno Toffano

Eigenlogic proposes a new method in logic inspired from quantum theory using operators. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet {0,1} using projection operators to the binary alphabet {+1,-1} employing reversible involution operators. Also, many-valued logical operators are synthesized, for whatever alphabet, using operator methods based on Lagrange interpolation and on the Cayley-Hamilton theorem. Considering a superposition of logical input states one gets a fuzzy logic representation where the fuzzy membership function is the quantum probability given by the Born rule. Eigenlogic is essentially a logic of operators and its truth-table logical semantics is provided by the eigenvalue structure which is shown to be related to the universality of logical quantum gates, a fundamental role being played by non-commutativity and entanglement.

Jun 2, 20200 citations

Adapting Logic to Physics: The Quantum-Like Eigenlogic Program

Zeno Toffano, François Dubois

Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet { 0 , 1 } using projection operators to the binary alphabet { + 1 , - 1 } employing reversible involution operators. Also, many-valued logical operators are synthesized, for whatever alphabet, using operator methods based on Lagrange interpolation and on the Cayley-Hamilton theorem. Considering a superposition of logical input states one gets a fuzzy logic representation where the fuzzy membership function is the quantum probability given by the Born rule. Historical parallels from Boole, Post, Poincaré and Combinatory Logic are presented in relation to probability theory, non-commutative quaternion algebra and Turing machines. An extension to first order logic is proposed inspired by Grover's algorithm. Eigenlogic is essentially a logic of operators and its truth-table logical semantics is provided by the eigenvalue structure which is shown to be related to the universality of logical quantum gates, a fundamental role being played by non-commutativity and entanglement.

EntropyJan 24, 202018 citations

The Born rule and logical inference in quantum mechanics

Tsubasa Ichikawa

Abstract Logical inference leads to one of the major interpretations of probability theory called logical interpretation, in which the probability is seen as a measure of the plausibility of a logical statement under incomplete information. Assuming that our usual inference procedure is working rationally for every set of logical propositions represented in terms of commuting projectors on a given Hilbert space, we extend the logical interpretation to quantum mechanics and derive the Born rule. Our result implies that, from epistemological viewpoints, we can regard quantum mechanics as a natural extension of classical probability theory.

arXiv:1804.10067Progress of Theoretical and Experimental PhysicsOct 19, 20213 citations

Eigenlogic: Interpretable Quantum Observables with applications to Fuzzy Behavior of Vehicular Robots

Zeno Toffano, François Dubois

This work proposes a formulation of propositional logic, named Eigenlogic, using quantum observables as propositions. The eigenvalues of these operators are the truth-values and the associated eigenvectors the interpretations of the propositional system. Fuzzy logic arises naturally when considering vectors outside the eigensystem, the fuzzy membership function is obtained by the Born rule of the logical observable.This approach is then applied in the context of quantum robots using simple behavioral agents represented by Braitenberg vehicles. Processing with non-classical logic such as multivalued logic, fuzzy logic and the quantum Eigenlogic permits to enlarge the behavior possibilities and the associated decisions of these simple agents.

arXiv:1707.05654Jul 17, 20173 citations