This paper begins with an overview of quantum mechanics, and then recounts a relatively recent algebraic extension of the Boolean algebra of probabilistic events to “conditional events” (order pairs of events). The main point is to show that a so-called “superposition” of two (or more) quantum events (usually with mutually inconsistent initial conditions) can be represented in this algebra of conditional events and assigned a consistent conditional probability. There is no need to imagine that a quantum particle can simultaneously straddle two inconsistent possibilities.
This paper begins with an overview of quantum mechanics, and then recounts a relatively recent algebraic extension of the Boolean algebra of probabilistic events to “conditional events” (order pairs of events). The main point is to show that a so-called “superposition” of two (or more) quantum events (usually with mutually inconsistent initial conditions) can be represented in this algebra of conditional events and assigned a consistent conditional probability. There is no need to imagine that a quantum particle can simultaneously straddle two inconsistent possibilities.
This paper shows that the non-Boolean logic of quantum measurements is more naturally represented by a relatively new 4-operation system of Boolean fractions—conditional events—than by the standard representation using Hilbert Space. After the requirements of quantum mechanics and the properties of conditional event algebra are introduced, the quantum concepts of orthogonality, completeness, simultaneous verifiability, logical operations, and deductions are expressed in terms of conditional events thereby demonstrating the adequacy and efficacy of this formulation. Since conditional event algebra is nearly Boolean and consists merely of ordered pairs of standard events or propositions, quantum events and the so-called “superpositions” of states need not be mysterious, and are here fully explicated. Conditional event algebra nicely explains these non-standard “superpositions” of quantum states as conjunctions or disjunctions of conditional events, Boolean fractions, but does not address the so-called “entanglement phenomena” of quantum mechanics, which remain physically mysterious. Nevertheless, separating the latter phenomena from superposition issues adds clarity to the interpretation of quantum entanglement, the phenomenon of influence propagated at faster than light speeds. With such treacherous possibilities present in all quantum situations, an observer has every reason to be completely explicit about the environmental–instrumental configuration, the conditions present when attempting quantum measurements. Conditional event algebra allows such explication without the physical and algebraic remoteness of Hilbert space.
This paper uses a non-distributive system of Boolean fractions (a ib), 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 con- cepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a lb) 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 (cid) 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 quan- tum 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 entangle- ment 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.
Quantum probability and quantum mechanics The formalism of quantum probability theory Discussions of quantum mechanics are often confused by a lack of clarity about what exactly constitutes ‘quantum mechanics’. It is therefore useful to try at the start to isolate a consistent mathematical core of quantum mechanics, and consider anything that goes beyond this core to be ‘interpretation’. For us, this core is quantum probability theory. Quantum probability is a generalization of classical probability, and therefore I begin with a brief review of the latter. I assume that the reader has some familiarity with the ideas of probability theory. What follows is just to provide a quick review, and to establish some notation and terminology. In modern classical probability theory, probabilities are defined over algebras of events. The motivation is straightforward: we begin with a set of ‘primitive’, or ‘simple’, events (the ‘sample space’), and form an algebra of events by taking all logical combinations of the simple events. For example, let us take the simple events to be the possible results of rolling a six-sided die one time, so that the sample space is the set {1,2,3,4,5,6}. We then form an algebra of events from the sample space by taking all possible logical combinations of the simple events. Logical combinations include, for example, ‘either 3 or 5’ and ‘not 3 and not 2’.
Abstract With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s∈S s ∈ S . A function p:S→[0,1] p : S → [ 0 , 1 ] is called a numerical event or alternatively, an S-probability. If a set P of S-probabilities is ordered by the order of real functions, it becomes a poset which can be considered as a quantum logic. In case the logic P is a Boolean algebra, this will indicate that the underlying physical system is a classical one. The goal of this paper is to study sets of S-probabilities which are not far from being Boolean algebras by means of the addition and comparison of functions that occur in these sets. In particular, certain classes of so-called Boolean posets of S-probabilities are characterized and related to each other and descriptions based on sets of states are derived.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
The nature of quantum mechanical probability has often seemed mysterious. To shed some light on this topic, the present paper analyzes the logical form of probability assignment in quantum mechanics. To begin the paper, I set out and criticize several attempts to analyze the form. I go on to propose a new form which utilizes a novel, probabilistic conditional and argue that this proposal is, overall, the best rendering of the quantum mechanical probability assignments. Finally, quantum mechanics aside, the discussion here has consequences for counterfactual logic, conditional probability, and epistemic probability.