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arXiv:2101.07498

Paraconsistent Foundations for Quantum Probability

Ben Goertzel

Published Jan 19, 2021• 0 citations
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Abstract

It is argued that a fuzzy version of 4-truth-valued paraconsistent logic (with truth values corresponding to True, False, Both and Neither) can be approximately isomorphically mapped into the complex-number algebra of quantum probabilities. I.e., p-bits (paraconsistent bits) can be transformed into close approximations of qubits. The approximation error can be made arbitrarily small, at least in a formal sense, and can be related to the degree of irreducible "evidential error" assumed to plague an observer's observations. This logical correspondence manifests itself in program space via an approximate mapping between probabilistic and quantum types in programming languages.

DOI
10.48550/arxiv.2101.07498

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It is argued that a fuzzy version of 4-truth-valued paraconsistent logic (with truth values corresponding to True, False, Both and Neither) can be approximately isomorphically mapped into the complex-number algebra of quantum probabilities. I.e., p-bits (paraconsistent bits) can be transformed into close approximations of qubits. The approximation error can be made arbitrarily small, at least in a formal sense, and can be related to the degree of irreducible "evidential error" assumed to plague an observer's observations. This logical correspondence manifests itself in program space via an approximate mapping between probabilistic and quantum types in programming languages.
In quantum logic, i.e., within the structure of the Hilbert lattice imposed on all closed linear subspaces of a Hilbert space, the assignment of truth values to quantum propositions (i.e., experimentally verifiable propositions relating to a quantum system) is unambiguously determined by the state of the system. So, if only pure states of the system are considered, can a probability measure mapping the probability space for truth values to the unit interval be assigned to quantum propositions? In other words, is a probability concept contingent or emergent in the logic of quantum propositions? Until this question is answered, the cause of probabilities in quantum theory cannot be completely understood. In the present paper it is shown that the interaction of the quantum system with its environment causes the irreducible randomness in the relation between quantum propositions and truth values.
In this paper, we propose a new interpretation of quantum mechanics according to which Born probabilities are interpreted as many-valued truth values of statements concerning results of not-yet-performed experiments on quantum objects, or equivalently [0,1]-valued degrees to which quantum objects possess all their properties prior to measurements. The interpretation is based on mathematical results already published by the author, according to which any Birkhoff–von Neumann quantum logic can be isomorphically represented as a family of fuzzy sets or, equivalently, many-valued propositional functions.
We derive an analogue of the quantum total probability rule by constructing a probability theory based on paraconsistent logic. Bayesian probability theory is constructed upon classical logic and a desiderata, that is, a set of desired properties that the theory must obey. We construct a new probability theory following the desiderata of Bayesian probability theory but replacing the classical logic by paraconsistent logic. This class of logic has been conceived to handle eventual inconsistencies or contradictions among logical propositions without leading to the trivialisation of the theory. Within this Paraconsistent bayesian probability theory it is possible to deduce a new total probability rule which depends on the probabilities assigned to the inconsistencies. Certain assignments of values for these probabilities lead to expressions identical to those of Quantum mechanics, in particular to the quantum total probability rule obtained via symmetric informationally complete positive- operator valued measure.
The logic of approximation gives a precise mathematical formulation to the notion of approximate truth and to deductions based on it. We study properties of quantum propositions expressed in this logic, in connection to a modal accessibility predicate which induces a metric on the set of states of a mechanical system. We then relate the logic to some quantum logical calculi, and discuss possible applications to probability theory and to the Hilbert space formulation of quantum mechanics. Natural deductions in mathematical logic are concerned with expressions of the form Ft-$ (read: ‘I’ yields @‘), where @ is a formula of a certain formal language L and I is a set of formulae of L . Semantically, the expression Ft@ means: ‘if all formulae in the set F are true then $J is also true’. In this paper we deal with natural deductions applied to the natural sciences, in particular to physics. The problem is that in the context of scientific theories we cannot always speak about (absolute) truth: we have to take into account the uncertainties and errors arising in experiments and
Rei-AIOS D-FUMT8 logic distinguishes eight truth-like values. Paper 69 anchored the first four to Schnorr's randomness hierarchy L1-L4. Paper 75 adds a second rigorous anchor: five of the eight values are placed in numerically verified correspondence with specific quantum objects on a single qubit or its Bell-reduced extension via the QuTiP 5.2.3 Python library. RIGOROUS (5): TRUE/FALSE = computational basis eigenstates; BOTH = superposition |+> with |<0|+>|^2 = 1/2; NEITHER = maximally mixed state I/2 with S = 1 bit; FLOWING = Bell-reduced state with S = 1 bit AND concurrence = 1. STRUCTURAL (3): INFINITY / ZERO / SELF require multi-mode extension. 25/25 tests pass. BOTH-as-superposition resolves the long-standing paraconsistent-logic worry by exhibiting BOTH as a well-defined physical state. Open source CC-BY-4.0.
According to quantum mechanics, statements about the future made by sentient beings like us are, in general, neither true nor false; they must satisfy a many-valued logic. I propose that the truth value of such a statement should be identified with the probability that the event it describes will occur. After reviewing the history of related ideas in logic, I argue that it gives an understanding of probability which is particularly satisfactory for use in quantum mechanics. I construct a lattice of future-tense propositions, with truth values in the interval [0, 1], and derive logical properties of these truth values given by the usual quantum-mechanical formula for the probability of a history.