Reasoning in Quantum Theory

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.

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.

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.

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.

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.

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.

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.

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

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:

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.

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.

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:

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”.

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.

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.

Some general questions that have been often discussed in connection with (or against) quantum logic are the following: