A variant on this approach is that of Bub (1997), on which some observable R is chosen to to be always-definite; the set of definite observables is then expanded to the maximal set that avoids a KS obstruction.
Given any system such as integer arithmetic or real arithmetic, we can write T for the set of all definite terms, constituting the definite language, and T[V] for the larger indefinite language permitting variables drawn from a set V in place of some of the constant symbols.
Another approach is to let the set of definite observables vary with the state; this is the approach taken by various modal interpretations.
However, previous studies failed to identify the set of definite biomarker genes, whose expression is consistently deregulated in AD brains across different studies [ 8].
The first challenge is posed by the fact that the set of definite-valued properties let us call it 'S' may change over time.
This illustrates one of the advantages of the modal approach: it makes use of a precise set of rules that determine the set of definite-valued observables, and this makes it possible to derive rigorous results.
Each one of them proposed a group of conditions that the set of definite-valued properties should obey, and characterized this set in terms of the dynamical state |φ⟩ of the system.
Some progress has been made since Healey's book was published (see for example Reeder and Clifton 1995) but, in general, it remains difficult to see what the set of definite-valued properties is according to his approach.
The set of definite-valued observables of a system should be left invariant by the Galilean transformations: it would be unacceptable that this set changed as a mere result of a change in the perspective from which the system is described.
It should be noted that the integral in (13) is over the set of positive definite matrices (Y>0).
The second inequality above is due to the concavity of the function over the set of positive definite symmetric matrices [30, 7.6.7], and the Jensen's inequality.
