The finite set of subsystems of a finite quantum system with variables in
Z(n)Z(n), is studied as a Heyting algebra. The physical meaning of the logical connectives is discussed. It is shown that disjunction of subsystems is more general concept than superposition. Consequently, the quantum probabilities related to commuting projectors in the subsystems, are incompatible with associativity of the join in the Heyting algebra, unless if the variables belong to the same chain. This leads to contextuality, which in the present formalism has as contexts, the chains in the Heyting algebra. Logical Bell inequalities, which contain “Heyting factors,” are discussed. The formalism is also applied to the infinite set of all finite quantum systems, which is appropriately enlarged in order to become a complete Heyting algebra.
The set of subsystems Σ( m ) of a finite quantum system Σ( n ) with variables in ( n ), together with logical connectives, is a Heyting algebra. The probabilities τ( m |ρ n )=Tr[ ( m )ρ n ] (where ( m ) is the projector to Σ( m )) are compatible with associativity of the join in the Heyting algebra, only if the variables belong to the same chain. Consequently, contextuality in the present formalism, has the chains as contexts. Various Bell-like inequalities are discussed. They are violated, and this proves that quantum mechanics is a contextual theory.
A review is given of recent work aimed at constructing a quantum theory of cosmology in which all observables refer to information measurable by observers inside the universe. At the classical level the algebra of observables should be modified to take into account the fact that observers can only give truth values to observables that have to do with their backwards light cone. The resulting algebra is a Heyting rather than a Boolean algebra. The complement is non-trivial and contains information about horizons and topology change. Representation of such observables quantum mechanically requires a many-Hilbert space formalism, in which different observers make measurements in different Hilbert spaces. I describe such a formalism, called "quantum causal histories"; examples include causally evolving spin networks and quantum computers.