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"Is quantum logic really logic?" This book argues for a positive answer to this question once and for all. There are many quantum logics and their structures are delightfully varied. The most radical aspect of quantum reasoning is reflected in unsharp quantum logics, a special heterodox branch of fuzzy thinking.
For the first time, the whole story of Quantum Logic is told; from its beginnings to the most recent logical investigations of various types of quantum phenomena, including quantum computation. Reasoning in Quantum Theory is designed for logicians, yet amenable to advanced graduate students and researchers of other disciplines.

Inhaltsverzeichnis

Frontmatter

Mathematical and Physical Background

Frontmatter

Chapter 1. The mathematical scenario of quantum theory and von Neumann’s axiomatization

Abstract
In his celebrated book Mathematische Grundlagen der Quantenmechanik (von Neumann, 1932),1 John von Neumann proposed an axiomatic version of sharp QT. This theory is often referred to as orthodox quantum theory.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 2. Abstract axiomatic foundations of sharp QT

Abstract
We will now develop an abstract analysis of the basic conceptual structures of QT in a framework that is relatively independent of the Hilbert space formalism. On this basis, orthodox QT will be reconstructed as a particular model of such a general formal approach.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 3. Back to Hilbert space

Abstract
We will now return to orthodox QT. We will compare the abstract event-state systems, studied in the previous chapter, with the concrete examples that emerge in the framework of Hilbert space structures.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 4. The emergence of fuzzy events in Hilbert space quantum theory

Abstract
The event-structures we have studied so far (both in the abstract and in the Hilbert space situation) are typically sharp: any event always satisfies the noncontradiction principle.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 5. Effect algebras and quantum MV algebras

Abstract
In Chapter 4 we have seen how the system of all effects in a Hilbert space gives rise to a “genuine” Brouwer Zadeh poset (which is neither an orthoposet nor a lattice). We will now study other interesting ways of structuring the set of all concrete effects. In particular, we will see how effect algebras and quantum MV algebras naturally emerge from effect-systems.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 6. Abstract axiomatic foundations of unsharp quantum theory

Abstract
In the previous chapters we have studied the emergence of fuzzy events in the framework of Hilbert space QT. We now abstract from the Hilbert space formalism, paralleling the case of sharp QT, in order to obtain an abstract axiomatic foundation of unsharp QT.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 7. To what extent is quantum ambiguity ambiguous?

Abstract
A natural question arises in the framework of the unsharp approach to QT: to what extent is “sharp/unsharp” an unsharp, ambiguous distinction? So far, we have generally called sharp any quantum effect (or property) that satisfies the noncontradiction principle. However, different definitions of “sharp effect” can be proposed. One can distinguish two basic kinds of characterizations (Cattaneo, Dalla Chiara and Giuntini, 1999): (i) purely algebraic definitions, that only refer to the algebraic structure of the quantum effects; and (ii) probabilistic definitions, that also refer to the relationships between effects and states.
M. Dalla Chiara, R. Giuntini, R. Greechie

Quantum Logics as Logic

Frontmatter

Chapter 8. Sharp quantum logics

Abstract
We will first study two interesting examples of logics that represent a natural logical abstraction from the class of all Hilbert lattices. These are represented respectively by orthomodular quantum logic (OQL) and by the weaker orthologic (OL), which for a long time has been also termed minimal quantum logic. In fact, the name “minimal quantum logic” appears today quite inappropriate for two reasons. First, a number of, in a sense, weaker forms of quantum logic have recently attracted much attention; and second, the models for the “minimal quantum logic” do not provide for the possibility of an adequate modeling of the generalized probabilities that are induced by states of QT. However these probabilities do not usually play a fundamental role in the logical developments that follow. And the “minimal quantum logic” provides a “floor” for the other logics, so we include it. In the following we will use QL as an abbreviation for either OL or OQL.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 9. Metalogical properties and anomalies of quantum logic

Abstract
Some metalogical distinctions that are not interesting in the case of a number of familiar logics weaker than classical logic turn out to be significant for QL (and for nondistributive logics in general).
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 10. An axiomatization of OL and OQL

Abstract
QL is an axiomatizable logic. Many axiomatizations are known: both in the Hilbert-Bernays style and in the Gentzen-style (natural deduction and sequent-calculi).1
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 11. The metalogical intractability of orthomodularity

Abstract
As we have seen, the proposition-ortholattice in a Kripkean realization k = (I, R, Pr, V) does not generally coincide with the (algebraically) complete ortholattice of all propositions of the orthoframe 〈I, R〉. When Pr is the set of all propositions, k will be called standard. Thus, a standard orthomodular Kripkean realization is a standard realization, where Pr is orthomodular. In the case of OL, every nonstandard Kripkean realization can be naturally extended to a standard one (see the proof of Theorem 8.1.13). In particular, Pr can always be embedded into the complete ortholattice of all propositions of the orthoframe at issue. Moreover, as we have learned from the completeness proof, the canonical model of OL is standard. In the case of OQL, however, there are various reasons that make significant the distinction between standard and nonstandard realizations:
(i)
Orthomodularity is not elementary (Goldblatt, 1984). In other words, there is no way to express the orthomodular property of the ortholattice Pr in an orthoframe 〈I, R〉 as an elementary (first-order) property.
 
(ii)
It is not known whether every orthomodular lattice is embeddable into a complete orthomodular lattice.
 
(iii)
It is an open question whether OQL is characterized by the class of all standard orthomodular Kripkean realizations.
 
(iv)
It is not known whether OQL admits a standard canonical model. If we try to construct a canonical realization for OQL by taking Pr as the set of all possible propositions as in the OL-case (call such a realization a pseudo canonical realization), do we obtain an OQL-realization, satisfying the orthomodular property? In other words, is the pseudo canonical realization a model of OQL?
 
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 12. First-order quantum logics and quantum set theories

Abstract
The most significant logical and metalogical peculiarities of QL arise at the sentential level. At the same time the extension of sentential QL to a first-order logic seems to be quite natural. As in the case of sentential QL, we will characterize first-order QL both by means of an algebraic and a Kripkean semantics.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 13. Partial classical logic, the Lindenbaum property and the hidden variable problem

Abstract
Orthologic and orthomodular quantum logic are examples of total logics: their language is syntactically closed under the logical connectives and any molecular sentence always has a meaning both in the algebraic and in the Kripkean semantics. We will now investigate a form of quantum logic (first proposed by Kochen and Specker (1965b)) based on a different idea. From the semantic point of view, the crucial relation will be represented by a compatibility relation, that may hold between the meanings of two sentences. As expected, the intended physical interpretation of the compatibility relation will be the following: two sentences α and β will have compatible meanings iff α and β can be simultaneously tested.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 14. Unsharp quantum logics

Abstract
The quantum logics we have studied so far are all examples of sharp logics. Both the logical and the semantic version of the noncontradiction principle hold:
  • any contradiction α¬α is always false;1
  • a sentence α and its negation ¬α cannot both be true.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 15. The Brouwer Zadeh logics

Abstract
We now study some stronger examples of unsharp quantum logic, that have been called Brouwer Zadeh logics (also fuzzy intuitionistic logics). These logics represent natural abstractions from the class of all BZ-lattices (defined in Chapter 4). As expected, a characteristic property of Brouwer Zadeh logics is a splitting of the connective “not” into two forms of negation: a fuzzy-like negation, that gives rise to a paraconsistent behavior and an intuitionistic-like negation. The fuzzy “not” represents a weak negation, that inverts the two extreme truth-values (truth and falsity), satisfies the double negation principle but generally violates the noncontradiction principle. The second “not” is a stronger negation, a kind of necessitation of the fuzzy “not”.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 16. Partial quantum logics and Łukasiewicz’ quantum logic

Abstract
In Chapter 5, we have considered examples of partial algebraic structures, where the basic operations are not always defined. How does one give a semantic characterization for the different forms of quantum logic corresponding, respectively, to the class of all effect algebras, of all orthoalgebras and of all orthomodular posets? We will call these logics: unsharp partial quantum logic (UPaQL), weak partial quantum logic (WPaQL) and strong partial quantum logic (SPaQL), respectively. By PaQL we will denote any instance of our three logics. Since PaQL as well as partial classical logic PaCL (studied in Chapter 13) are examples of partial logics, one could expect some natural semantic connections between the two different cases. However, we will see that the basic idea of the semantic characterization of PaQL is somewhat different with respect to the PaCL-semantics.
M. Dalla Chiara, R. Giuntini, R. Greechie

Chapter 17. Quantum computational logic

Abstract
Finally we study a new form of unsharp quantum logic that has been naturally suggested by the theory of quantum computation. One of the most interesting logical proposals that arise from quantum computation is the idea to use the quantum theoretical formalism in order to represent parallel reasoning.1 As is well known, the unit of measurement in classical information theory is the bit: one bit measures the information quantity that can be either transmitted or received whenever one chooses one element from a set consisting of two elements, say, from the set {0, 1}. From the intuitive point of view, both the objects 0 and 1 can be imagined as a well determined state of a classical physical system, for instance, the state of a tape cell in a given machine.
M. Dalla Chiara, R. Giuntini, R. Greechie

Conclusions

Conclusions

Abstract
Some general questions that have been often discussed in connection with (or against) quantum logic are the following:
(a)
Why quantum logics?
 
(b)
Are quantum logics helpful to solve the difficulties of QT?
 
(c)
Are quantum logics “real logics?” And how is their use compatible with the mathematical formalism of QT, based on classical logic?
 
(d)
Does quantum logic confirm the thesis that “logic is empirical?”
 
M. Dalla Chiara, R. Giuntini, R. Greechie

Backmatter

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