The algebraic construct of 'forming ordered pairs of the members of a set' has been used and reused with amazing success. Among these developments are Integer Fractions to better measure distances with pairs of whole numbers; Dedekind Cuts - an early way to define real numbers with ordered pairs of subsets of fractions; Vectors; Quatemians; and others like complex numbers - ordered pairs of real numbers. In the realm of Boolean Logic, ordered pairs of Boolean propositions (alternately ordered pairs of subsets of some universe set) transcend the limitations of pure Boolean Iogic to represent partially true conditional statements such as those used by conditiona1 probability. Pure Boolean logic is unable to adequately represent uncertain conditional logic, but ordered pairs of BooIean propositions can and do represent this knowledge in the logic of Boolean Fractions, also called Conditional Events. A given B can not be represented by A or not B unless P(AJB) and P(B) are both 1. Otherwise, P(A1B) < P(A or not B). But A given B can be represented by an ordered pair of propositions (AJB). These Boolean Fractions apply to the natural language and logic of A given B, they apply to classical conditional probability calculations, to expert system if-then rules, and now to the logic of quantum measurements, a logic that Boolean algebra could not represent, but which Boolean Fractions can represent without Hilbert space.
The algebraic construct of 'forming ordered pairs of the members of a set' has been used and reused with amazing success. Among these developments are Integer Fractions to better measure distances with pairs of whole numbers; Dedekind Cuts - an early way to define real numbers with ordered pairs of subsets of fractions; Vectors; Quatemians; and others like complex numbers - ordered pairs of real numbers. In the realm of Boolean Logic, ordered pairs of Boolean propositions (alternately ordered pairs of subsets of some universe set) transcend the limitations of pure Boolean Iogic to represent partially true conditional statements such as those used by conditiona1 probability. Pure Boolean logic is unable to adequately represent uncertain conditional logic, but ordered pairs of BooIean propositions can and do represent this knowledge in the logic of Boolean Fractions, also called Conditional Events. A given B can not be represented by A or not B unless P(AJB) and P(B) are both 1. Otherwise, P(A1B) < P(A or not B). But A given B can be represented by an ordered pair of propositions (AJB). These Boolean Fractions apply to the natural language and logic of A given B, they apply to classical conditional probability calculations, to expert system if-then rules, and now to the logic of quantum measurements, a logic that Boolean algebra could not represent, but which Boolean Fractions can represent without 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.
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.
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.
The logical structures studied in this paper are generalizations of the propositional calculus. The classical propositional calculus is essentially Boolean algebra or, alternatively, the theory of functions on an arbitrary set S with values in a two-element set. The generalization consists in allowing partial functions on the set S , i.e., functions defined on certain subsets of S , and defining an equivalence relation among these functions such that any two constant functions with the same constant value belong to the same equivalence class. The generalization is equally natural for functions with values in the field of real numbers and we shall consider this case first. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.