This paper develops a general language of event configurations to discuss and compare various modes of proposition formation. It is shown that any finite orthogonality space can be numerically encoded. This encoding is applied to show that the quasimanual of all orthogonal subsets of any finite point-determining orthogonality space may be decomposed into a union of manuals and that the logic of these quasimanuals may be regarded as a composite of interlocking associative orthoalgebras.

We consider several puzzles of bounded rationality. These include the Allais- and Ellsberg paradox, the disjunction effect, and related puzzles. We argue that the present account of quantum cognition—taking quantum probabilities rather than classical probabilities—can give a more systematic description of these puzzles than the alternate treatments in the traditional frameworks of bounded rationality. Unfortunately, the quantum probabilistic treatment does not always provide a deeper understanding and a true explanation of these puzzles. One reason is that quantum approaches introduce additional (...) parameters which possibly can be fitted to empirical data but which do not necessarily explain them. Hence, the phenomenological research has to be augmented by responding to deeper foundational issues. In this article, we make the general distinction between foundational and phenomenological research programs, explaining the foundational issue of quantum cognition from the perspective of operational realism. This framework is motivated by assuming partial Boolean algebras. They are combined into a uniform system via a mechanism preventing the simultaneous realization of perspectives. Gleason’s theorem then automatically leads to a distinction between probabilities that are defined by pure states and probabilities arising from the statistical mixture of pure states. This formal distinction relates to the conceptual distinction between risk and ignorance. Another outcome identifies quantum aspects in dynamic macro-systems using the framework of symbolic dynamics. Finally, we discuss several ideas that are useful for justifying complementarity in cognitive systems. (shrink)

Using the mathematical notion of an entity to represent states in quantum and classical mechanics, we show that, in a strict sense, proper superpositions are possible in classical mechanics.

In the empirical logic approach to quantum mechanics, the physical system under consideration is given in terms of a manual of sample spaces. The resulting propositional structure has been shown to form an orthoalgebra, generalizing the structure of an orthomodular poset. An orthoalgebra satisfies the unique Mackey decomposition (UMD) property if, given two commuting propositions a and b, there is a unique jointly orthogonal triple (e, f, c) such that a=e⊕c and b=f⊕c. In a manual, E is refined by F (...) if E is logically equivalent to some partition of F, making results from F at least as informative as those from E. The main result is a characterization of the UMD property in terms of the refinement structure of an underlying manual, provided the manual is event saturated and orthogonally additive. (shrink)

This heuristic article introduces a generalization of the idea of drawing colored balls from an urn so as to allow mutually incompatible experiments to be represented, thereby providing a device for thinking about quantum logic and other non-classical statistical situations in a concrete way. Such models have proven valuable in generating examples and counterexamples and in making abstract definitions in quantum logic seem more intuitive.

Quantum Mechanics can be viewed as a linear dynamical theory having a familiar mathematical framework but a mysterious probabilistic interpretation, or as a probabilistic theory having a familiar interpretation but a mysterious formal framework. These points of view are usually taken to be somewhat in tension with one another. The first has generated a vast literature aiming at a “realistic” and “collapse-free” interpretation of quantum mechanics that will account for its statistical predictions. The second has generated an at least equally (...) large literature aiming to derive, or at any rate motivate, the formal structure of quantum theory in probabilistically intelligible terms. In this paper I explore, in a preliminary way, the possibility that these two programmes have something to offer one another. In particular, I show that a version of the measurement problem occurs in essentially any non-classical probabilistic theory, and ask to what extent various interpretations of quantum mechanics continue to make sense in such a general setting. I make a start on answering this question in the case of a rudimentary version of the Everett interpretation. (shrink)

LetA be a quasi-manual with finite operations. Associate to each E = {e 1 ,..., en} εA the set ΓE of modal formulas: □(e 1 ⋁ ··· ⋁ en), ◊ei → ∼□(e 1 ⋁ ··· ⋁ ei−1 ⋁ ei+1 ⋁ ··· ⋁ en), i=1,..., n. Set Γ A = ώ{ΓE|E εA}. We show that supports ofA are in one-to-one correspondence with certain Kripke models of Γ A where the supports are given by {x ε |A ‖ ◊ x is true}.

The notion of quantum supports introduced by Foulis, Piron, and Randall can be used to construct combinatorial versions of contextualist hidden-variable models for finite quantum logics. The original logic can be uniquely recovered from appropriate such models as a solution of a combinatorial inverse problem. One can thus set up a classical ontology for a finite quantum logics that completely specifies it. Computer studies are used to explore the ideas.

In orthodox quantum mechanics, it has virtually become the custom to identify properties of a physical system with operationally testable propositions about the system. The causes and consequences of this practice are explored mathematically in this paper. Among other things, it is found that such an identification imposes severe constraints on the admissible states of the physical system.

The connection of the structure of statistical selection procedures with measure theory is investigated. The methods of measure theory are applied in order to analyze a mathematical description of preparation and registration of physical systems that is used by G. Ludwig for a foundation of quantum mechanics.

In their paper “A note on Misunderstandings of Piron's Axioms for Quantum Mechanics,” Foulis and Randall undertake a reply to our critique of Piron's “question-proposition system” (qp-s) which appeared in previous issues of this journal. In the present paper, we want briefly to refute the points of criticism raised by Foulis and Randall (FR). We argue that the “misunderstandings” are not ours, and we prove it.

Quantum stochastic models are developed within the framework of a measure entity. An entity is a structure that describes the tests and states of a physical system. A measure entity endows each test with a measure and equips certain sets of states as measurable spaces. A stochastic model consists of measurable realvalued function on the set of states, called a generalized action, together with measures on the measurable state spaces. This structure is then employed to compute quantum probabilities of test (...) outcomes. We characterize those measure entities that are isomorphic to a quantum probability space. We also show that stochastic models provide a phase space description of quantum mechanics and a realistic model of spin. (shrink)

The standard interpretation of quantum physics (QP) and some recent generalizations of this theory rest on the adoption of a rerificationist theory of truth and meaning, while most proposals for modifying and interpreting QP in a “realistic” way attribute an ontological status to theoretical physical entities (ontological realism). Both terms of this dichotomy are criticizable, and many quantum paradoxes can be attributed to it. We discuss a new viewpoint in this paper (semantic realism, or briefly SR), which applies both to (...) classical physics (CP) and to QP. and is characterized by the attempt of giving up verificationism without adopting ontological realism. As a first step, we construct a formalized observative language L endowed with a correspondence truth theory. Then, we state a set of axioms by means of L which hold both in CP and in QP. and construct a further language Lv which can express bothtestable andtheoretical properties of a given physical system. The concepts ofmeaning andtestability do not collapse in L and Le hence we can distinguish between semantic and pragmatic compatibility of physical properties and define the concepts of testability and conjoint testability of statements of L and Le. In this context a new metatheoretical principle (MGP) is stated, which limits the validity of empirical physical laws. By applying SR (in particular. MGP) to QP, one can interpret quantum logic as a theory of testability in QP, show that QP is semantically incomplete, and invalidate the widespread claim that contextuality is unavoidable in QP. Furthermore. SR introduces some changes in the conventional interpretation of ideal measurements and Heisenberg’s uncertainty principle. (shrink)

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.

Piron's axioms for a realistically interpreted quantum mechanics are analyzed in detail within the context of a formal mathematical structure expressed in the conventional set-theoretic idiom of mathematics. As a result, some of the serious misconceptions that have encouraged recent criticisms of Piron's axioms are exposed.

We implement a recent characterization of metaphysical indeterminacy in the context of orthodox quantum theory, developing the syntax and semantics of two propositional logics equipped with determinacy and indeterminacy operators. These logics, which extend a novel semantics for standard quantum logic that accounts for Hilbert spaces with superselection sectors, preserve different desirable features of quantum logic and logics of indeterminacy. In addition to comparing the relative advantages of the two, we also explain how each logic answers Williamson’s challenge to any (...) substantive account of determinacy: For any proposition p, what could the difference between “p” and “it’s determinate that p” ever amount to? (shrink)

from non-conscious components by positing that consciousness is a universal primitive. For example, the double aspect theory of information holds that infor- mation has a phenomenal aspect. How then do you get from phenomenal infor- mation to human consciousness? This paper proposes that an entity is conscious to the extent it amplifies information, first by trapping and integrating it through closure, and second by maintaining dynamics at the edge of chaos through simul- taneous processes of divergence and convergence. The origin (...) of life through autocatalytic closure, and the origin of an interconnected worldview through conceptual closure, induced phase transitions in the degree to which informa- tion, and thus consciousness, is locally amplified. Divergence and convergence of cognitive information may involve phenomena observed in light e.g. focusing, interference, and resonance. By making information flow inward- biased, clo- sure shields us from external consciousness; thus the paucity of consciousness may be an illusion. (shrink)

We outline the rationale and preliminary results of using the State Context Property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. The quantum formalism was developed to cope with problems arising in the description of (1) the measurement process, and (2) the generation of new states with new properties when particles become entangled. Similar problems arising with concepts motivated the formal treatment introduced (...) here. Concepts are viewed not as fixed representations, but entities existing in states of potentiality that require interaction with a context---a stimulus or another concept---to `collapse' to observable form as an exemplar, prototype, or other (possibly imaginary) instance. The stimulus situation plays the role of the measurement in physics, acting as context that induces a change of the cognitive state from superposition state to collapsed state. The collapsed state is more likely to consist of a conjunction of concepts for associative than analytic thought because more stimulus or concept properties take part in the collapse. We provide two contextual measures of conceptual distance---one using collapse probabilities and the other weighted properties---and show how they can be applied to conjunctions using the pet fish problem. (shrink)

We propose a theory for modeling concepts that uses the state-context-property theory (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties. This theory enables us to incorporate context into the mathematical structure used to describe a concept, and thereby model how context influences the typicality of a single exemplar and the applicability of a single property of a concept. We introduce the notion `state of a concept' to account for this contextual influence, and show (...) that the structure of the set of contexts and of the set of properties of a concept is a complete orthocomplemented lattice. The structural study in this article is a preparation for a numerical mathematical theory of concepts in the Hilbert space of quantum mechanics that allows the description of the combination of concepts. (shrink)