Orthogonal projections correspond to disjoint events, if one of the events happen then the other event cannot happen. Addition of orthogonal events corresponds to union of disjoint events. If two density operators are orthogonal it means that there exist an event such that the event will happen for sure if the system is represented by the first state and the event will for sure not happen if the system is represented by the second state. Orthogonal vectors correspond to orthogonal pure states. I hope this answers your first question. I do not understand your second question.

Dear Peter, I think you helped clarify things.

I presume there is then a distinction between:

(A)orthogonality of' events'/ 'projections on the same basis or of the same pure state/vector'

as denoted by orthogonality of their density operators,; ie 'spin up in x direction of spin system prepared m' versus spin down x direction of spin system prepared in m'   on the same prepared  spin system measure on the same angle, 

(B)And events on, distinct/ orthogonal bases/unit vectors ,of the same prepared spin system, distinct possible super-positions given preparation m, which are often complementary events/projections,as they are projections of distinct and orthogonal measurements/bases,/superpositions

Does (A) denote events generally measurable together,  spin up x versus spin down y , which cannot occur (ie mutually exclusive, events of the same measurement on the same prepared spin state m, generally on the same basis, x,  or commutting bases. That is events that are projections of the same measurement, x, y of the same prepared state m, that are as it were, mutually exclusive, of kolmogorov. They are projections of the same state and superpositions,

IF 'spin up occurs,  given a measurement of spin  in the  x  direction, for a spin system prepared in state m, then' spin down does not obtain, on that same x-direction measurement ,for that same spin system prepared in state m, ' .'

Whilst (B) orthogonality of vectors, in the context of holding the fixed the same prepared state m ,denotes that the measurements themselves, that is the superposition or basis vectors, pertaining to x,y, direction measurements of spin, that is  the basis vectors are  orthogonal, often relating to geometric orthogonality and where the corresponding projections are said to be complementary

; tthat is where the measurement angle x,y,are geometrically orthogonal (and the corresponding hilbert inner products denote that the vectors superpostion state are orthogonal, in that the inner product is zero) but in the logical sense, the events of those distinct bases/measurements/superopositions, such as ,

spin up , measured at x, versus spin up measured at y ,both on the same system, prepared, in state m, 

are complementary or non commutting,

that is NOT,  ';orthogonal ' projections or disjoint events of the same state and measurement, in the sense of (A), . Rather, they are ' vector or basis 'orthogonal'; in that  they are projections of distinct orthogonal vector-pure states/ bases, , but are not orthogonal projections or disjoint, even if the prepared state is the same, the measure or bases vectors are distinct, and orthogonal and thus the events are on each orthogonal vector/state, meaurements,  are not 'disjoint or projection-orthogonal',  but rather complementary, or non commuting

as generally,  as the measurements themselves are geometrically orthogonally (or non commuting) and the corresponding bases at those angles are Hilbert space orthogonal the events themselves are complementary , or non commuting,

In that the system cannot collapse into both the x, direction, superposition basis vector  and the y direction superposition  basis vector simultaneously, despite the  prepared state m,  being the same,

that is, the events/projections spin up x, spin up y,  that relate to  'geometric orthogonal spin measurements, in directions x,y,respectively or orthogonal basis, vectors/ superposition, ie the , x direction-basis-vector of spin, and the  y direction basis -vector of spin ,of the same spin 1/2 system where one considers the spin preparation state m to be the same in each case,

 :is generally diametrically opposed to the 'orthogonality of events or projections '(which are generally events on  the same basis vector, the measurements must also be the same, not just the prepared state,)

l 

Does' orthogonal additivity'  or gleasons orthgonal triple property mean orthogonality of projections, and orthogonality of vectors/pure states?

Where in the 9  events which, contained each of the  three mutually orthogonal triad / basis vectors in the sense of (B) (state or vector orthogonal, not necessarily events that ,  projection orthogonal or disjoint, )  are such that if their amplitudes mod squared sum to one, so do their function sum

. ie v= ,v1,v,2,v3,

x=, x1,x2,x3, y, ,

y=y1, y2, y3 where the y,x,v vectors are vector orthogonal in the sense of (B),

and v1,v2,v3,denote the three (A) projection orthogonal events ion vector v,

y1,y2,y3 are the three (A) projection orthogonal events on y

and x1,x2 , x3 are the three ()A projection events of state/vector x

where the vectors/states/bases x,y,v, are vectorially orthogonal in the sense of (B),

. I presume that Gleason did not merely mean:

F(v1)+F(v2)+F(v3)=1, F(x1)+F(x2)+F(x3)=1, F(y1)+F(y2)+F(y3)=1 , which (which is just probabilism normalization of mutually excluisve and exhaustive projection, orthogonal in the sense of (A) ) but also derived that if any three of of those nine events (x1,x2,x3,v1,v2,v3,y1,y2,y3) most of which which sit on mutually orthogonal vectors in the sense of (B), such that if their amplitudes mod square sum to one, the frame function does, where I am using the symbol x1 , for example to simultaneous denote that event or projection and its amplitude mod squared.

ie if  x1+v2+ y3=1 iff F(x1)+ F(v2)+F(y3)=1 where the events x1,v2, y3 are on distinct vectors , x,y,v, that would appear to be mutually orthogonal state/vectors x,y,z being orthogonal, 

And thus the events themselves, x1,v2,y3 'appear to be',  non commuting . That is,

 not 'orthogonal' in the sense of (A),orthogonal projections of the same vector/state/pure state

 that is, x1, y3, v2, are not orthogonal projections of the same vector/disjoint in the traditional mutually exclusive sense;, as  the vectors/ state, of which they, that is (x1,v2,y3) are projections,  are the distinct orthogonal states x,y,z , where ie x,y,z are 'orthogonal' vectors/state.pure state (not projections), and are orthogonal  in the sense of (B); and thus not the same state

if their amplitudes mod squared sum to one, so do their frame function values

Something that is confusing is that the cone of positive operators is self-dual. That means that operator can both be used to represent states (what we know about the physical system) and the dual objects of states, i.e. observables (measurements that lead to a real number). The consequence is that in principle we spin in such a way that if the result is +1/2 then a subsequent measurement of the spin will again result in +1/2 and we may say that the system is in a state that we may call +1/2. Note that there are many measurements that do not directly correspond to states.

Ah OK, so the hilbert space operator relates to the events or projections or observable , given both measurement and preparation being specificied, such as spin up in the x occurred direction versus spin down in x direction, occurred for a particle that was measured for for spin in direction and prepated in an eigenste of spin up in the m direction,

as well as preparation the state, PR(1)=1 spin up in m direction, (or if i measure spin up in m direction )as well, as the  conditional /preferred basis measurement states/superposition (all of the distinct angle x,y,z' angles of spin measurement,that spin system, that is prepared in an eigenstate of the m direction, could instead be measured in, ie the distinct possible preferred basis states

, the spin particle is an ' eigenstate of spin up in m direction of spin '

such that if i measure the, the spin  particle in spin in the x direction ,

 particle will instead have a preferred basis/super-position state:

  '1/2