62 research outputs found
A presentation of Quantum Logic based on an "and then" connective
When a physicist performs a quantic measurement, new information about the
system at hand is gathered. This paper studies the logical properties of how
this new information is combined with previous information. It presents Quantum
Logic as a propositional logic under two connectives: negation and the "and
then" operation that combines old and new information. The "and then"
connective is neither commutative nor associative. Many properties of this
logic are exhibited, and some small elegant subset is shown to imply all the
properties considered. No independence or completeness result is claimed.
Classical physical systems are exactly characterized by the commutativity, the
associativity, or the monotonicity of the "and then" connective. Entailment is
defined in this logic and can be proved to be a partial order. In orthomodular
lattices, the operation proposed by Finch (1969) satisfies all the properties
studied in this paper. All properties satisfied by Finch's operation in modular
lattices are valid in Hilbert Space Quantum Logic. It is not known whether all
properties of Hilbert Space Quantum Logic are satisfied by Finch's operation in
modular lattices. Non-commutative, non-associative algebraic structures
generalizing Boolean algebras are defined, ideals are characterized and a
homomorphism theorem is proved.Comment: 28 pages. Submitte
Residuated structures and orthomodular lattices
The variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., ℓ-groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated ℓ-groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated ℓ-groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated ℓ-groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices
On Contextuality and Unsharp Quantum Logic
In this paper we provide a preliminary investigation of subclasses of bounded
posets with antitone involution which are "pastings" of their maximal Kleene
sub-lattices. Specifically, we introduce super-paraorthomodular lattices,
namely paraothomodular lattices whose order determines, and it is fully
determined by, the order of their maximal Kleene sub-algebras. It will turn out
that the (spectral) paraorthomodular lattice of effects over a separable
Hilbert space can be considered as a prominent example of such. Therefore, it
arguably provides an algebraic/order theoretical rendering of complementarity
phenomena between unsharp observables. A number of examples, properties and
characterization theorems for structures we deal with will be outlined. For
example, we prove a forbidden configuration theorem and we investigate the
notion of commutativity for modular pseudo-Kleene lattices, examples of which
are (spectral) paraorthomodular lattices of effects over finite-dimensional
Hilbert spaces. Finally, we show that structures introduced in this paper yield
paraconsistent partial referential matrices, the latter being generalizations
of J. Czelakowski's partial referential matrices. As a consequence, a link
between some classes of posets with antitone involution and algebras of partial
"unsharp" propositions is established
A Bridge Between Q-Worlds
Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding quantum mechanics by reformulating parts of the theory inside of non-classical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, `Q-worlds'. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory
Bohrification of operator algebras and quantum logic
Following Birkhoff and von Neumann, quantum logic has traditionally been
based on the lattice of closed linear subspaces of some Hilbert space, or, more
generally, on the lattice of projections in a von Neumann algebra A.
Unfortunately, the logical interpretation of these lattices is impaired by
their nondistributivity and by various other problems. We show that a possible
resolution of these difficulties, suggested by the ideas of Bohr, emerges if
instead of single projections one considers elementary propositions to be
families of projections indexed by a partially ordered set C(A) of appropriate
commutative subalgebras of A. In fact, to achieve both maximal generality and
ease of use within topos theory, we assume that A is a so-called Rickart
C*-algebra and that C(A) consists of all unital commutative Rickart
C*-subalgebras of A. Such families of projections form a Heyting algebra in a
natural way, so that the associated propositional logic is intuitionistic:
distributivity is recovered at the expense of the law of the excluded middle.
Subsequently, generalizing an earlier computation for n-by-n matrices, we
prove that the Heyting algebra thus associated to A arises as a basis for the
internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the
"Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of
functors from C(A) to the category of sets. We explain the relationship of this
construction to partial Boolean algebras and Bruns-Lakser completions. Finally,
we establish a connection between probability measure on the lattice of
projections on a Hilbert space H and probability valuations on the internal
Gelfand spectrum of A for A = B(H).Comment: 31 page
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