David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Philosophical Logic 34 (4):363 - 401 (2005)
This paper uses a non-distributive system of Boolean fractions (a|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, 'a if b' or 'a given b', ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due to the mutual inconsistency of their experimental conditions, this algebra of conditionals can express indeterminacy. In fact, this system is able to express the crucial quantum concepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a|b) is said to be "inapplicable" (or "undefined") in those instances or models for which b is false. Otherwise the conditional takes the truth-value of proposition a. Thus the system is technically 3-valued, but the 3rd value has nothing to do with a state of ignorance, nor to some half-truth. People already routinely put statements into three categories: true, false, or inapplicable. As such, this system applies to macroscopic as well as microscopic events. Two conditional propositions turn out to be simultaneously verifiable just in case the truth of one implies the applicability of the other. Furthermore, two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b = d, their conditions are equivalent. Since all aspects of quantum mechanics can be represented with this near classical logic, there is no need to adopt Hilbert space logic as ordinary logic, just a need perhaps to adopt propositional fractions to do logic, just as we long ago adopted integer fractions to do arithmetic. The algebra of Boolean fractions is a natural, near-Boolean extension of Boolean algebra adequate to express quantum logic. While this paper explains one group of quantum anomalies, it nevertheless leaves no less mysterious the 'influence-at-a-distance', quantum entanglement phenomena. A quantum realist must still embrace non-local influences to hold that "hidden variables" are the measured properties of particles. But that seems easier than imaging wave-particle duality and instant collapse, as offered by proponents of the standard interpretation of quantum mechanics
|Keywords||compatible propositions conditional events conditional logic conditional probability orthoalgebra orthogonal simultaneously measurable simultaneously observable simultaneously verifiable superposition|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Othman Qasim Malhas (1994). Abacus Logic: The Lattice of Quantum Propositions as the Poset of a Theory. Journal of Symbolic Logic 59 (2):501-515.
Geoffrey Hellman (1981). Quantum Logic and the Projection Postulate. Philosophy of Science 48 (3):469-486.
J. L. Bell (1986). A New Approach to Quantum Logic. British Journal for the Philosophy of Science 37 (1):83-99.
Slawomir Bugajski (1980). Only If 'Acrobatic Logic' is Non-Boolean. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1980:264 - 271.
Peter Milne (2004). Algebras of Intervals and a Logic of Conditional Assertions. Journal of Philosophical Logic 33 (5):497-548.
Peter Mittelstaedt (1978). The Metalogic of Quantum Logic. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1978:249 - 256.
Fernando Guzmán (1994). A Gentzen System for Conditional Logic. Studia Logica 53 (2):243 - 257.
Jeffrey Bub (1982). Quantum Logic, Conditional Probability, and Interference. Philosophy of Science 49 (3):402-421.
Patrick Suppes (1966). The Probabilistic Argument for a Non-Classical Logic of Quantum Mechanics. Philosophy of Science 33 (1/2):14-21.
Jeffrey Bub (1991). The Problem of Properties in Quantum Mechanics. Topoi 10 (1):27-34.
Added to index2009-01-28
Total downloads22 ( #76,845 of 1,098,973 )
Recent downloads (6 months)8 ( #27,099 of 1,098,973 )
How can I increase my downloads?