State dependent additivity and state independent additivity? ;
akin to more to cauchy additivity versus local kolmorgov additivity/normalization of subjective credence/utility, in a simplex representation of subjective probability or utility ranked by objective probability distinction? Ie in the 2 or more unit simplex (at least three atomic outcomes on each unit probability vector, finitely additive space) where every events is ranked globally within vectors and between distinct vectors by < > and especially '='
i [resume that one is mere representability and the other unique-ness
the distinction between the trival
(1)x+y+z x,y,z mutually exclusive and exhaustive F(x)+F(y)+F(z)=1
(2)or F(x u y) = F(x)+F(y) xu y ie F(A V B)=F(A)+F(B)=F(A)+ F(B) A, B disjoint
disjoint on samevector)
(3)F(A)+F(AC)=1 disjoint and mutually excluisve on the same unit vector
and more like this or the properties below something more these
to these (3a, 3b, 3C) which are uniqueness properties
forall x,y events in the simplex
(3.A) F(x+y)=F(x)+F(y) cauchy addivity(same vector or probability state, or not)
This needs no explaining
aritrarily in the simplex of interest (ie whether they are on the same vector or not)
or(B) x+y =z+m=F(z)+F(m) (any arbitary two or more events with teh same objective sum must have the same credence sum, same vector or not) disjoint or not (almost jensens equality)
or (C)F(1-x-y)+F(x)+F(y)=1 *(any arbitrary three events in the simplex, same vector or not, must to one in credence if they sum to one in objective chance)
(D) F(1-x)+F(x)=1 any arbitary two events whose sum is one,in chance must sum to 1 in credence same probability space,/state/vector or not
global symmetry (distinct from complement additivity) it applies to non disjoint events on disitnct vectors to the equalities in the rank. 'rank equalities, plus complement addivitity' gives rise to this in a two outcome system, a
. It seems to be entailed by global modal cross world rank, so long as there at least three outcome, without use of mixtures, unions or tradeoffs. Iff ones domain is the entire simplex
that is adding up function values of sums of evenst on distinct vectors to the value of some other event on some non commutting (arguably) probability vector
F(x+y)=F(x)+F(y)
In the context of certain probabilistic and/or utility unique-ness theorems,where one takes one objective probability function and tries to show that any other probability function, given ones' constraints, must be the same function.
As it appears that when objective propensities are ranked globally and totally within world and between world with equality transitive within vector and outside vectors, that so long as the dimension of the simplex is at least 2 (that is at least three outcome on each and every triple) as every events lies with the same chance is ranked with the same subjective probability everywhere and some point that value will appear alongside a distinct value where they are disjoint, which can only happen if there at least three outcome ie 1/3 only lies along 2/3 on a two outcome system, any so in any three outfcome system every distinct element with the same chance thus ranked equal numerially in subjective probability in the representation, e, will lie along distinct pairs of values with the same sum, uncountably many times, constraining sets of two events with the same to have the function sum, and due local additivity where both sets of three events sum to one locally 1 in chance and propensity, the common function value in both of the two distinct vectors entails that the sum of the other two function values whill be the same, despite having distinct values individually, and despite lying on distinct vectors
w1 w2 within wolrd rank 1>F(3/8)w1=F(3/8)2w2>F(1/4)>0
F(3/8)+F(3/8)+F(1/4)=1 local additivity
0pjiff F(pt)>F(pj) ; pi=pj, iff F(pi)=F(pj) pj< pi iff F(pi)z
where F(1/3,1/3, 1/3 )=(1,3 1,3, 1,3)
F(1,0,0)= (1,0,0)
F(0,1,0)=(0,1,0)
F(0,0,1)=(0,0,1)
F:dom(sigma 2) codomain (sigma 2)
where
that for example model, subjective probability on objective probability on the entirety of the unit objective probability simplex (the set of all triples of atomic events that are non negative and sum to one);
F(x u y)=F(x)+F(y) x,y disjoint
or A,B, C ,zsit on the same vector disjoint and mutually exclusive (2) F(A)+F(B) F(C)=1
and x+y+z, although when ranked by cross world equalities and objective chances
F(A)+F(B) F(C)=1 this collapes into x+y+z iff F(x)+F(y) F(z)=1global normalization unrestricted (the gleason property) or cauchy normalization
globally (cross world in the unit 2 (triple simplex) when ranked by objective chance with equalities in the rank, cross world for
-some very strong form of cross world addivitity stronger than probabilism bordering on cauchy additivity (presuming uniqueness) by adding the 'credence/function sum of events with the same chance sum accross distinct vector to the function value of some other atomic event whose chance value=chance sum
where the only distinct between this and cauchy additivity (if at all) is that its restricted to hold only if those events are in that simplex, as many times as ones uses it (
where for cauchy additivity ) so long 1/6 1/3 1/2 in dom (F) and even if the domain is merely {1/3, 1/6, 1/2, and P(sigma=1),
is merely the domain one can flat out say that
F(1/3)=F(1/6+1/6)=F(1/6)+F(1/6) =2F(1/6) and F(sigma)=F(1)=1=6F(1/6(
so long as 1/6 is in the domain once. I presume that if ones domain is the entier simplex though the distinction would disappear; I also presume that
I also not that even without local additivity of chance or closure for objective chance or credence under unions, its will hold even if presumed only state additive if the events are ranked and flanked by the objective chance globably and cross world cross wold by objective chance and credence presumed only locally normalized state additive on each vector in subjective probability with n=3, (atomic outcomes on each vector) and no horizontal vector goes missing, as one gets global normalization
F(1-x-y)+F(x)+F(y)
the gleason property due to the rank equalities cross world, so that when the chances add to one vertically there is always a horizontal vectors wit those same chances values, and so the subjective probabilities of the column (cross world do not sum to one), yet each is assigned the same credence term by as every event in the horizontal vector, then they are going to have the same sum, the sum of the latter being forced by state dependent additivity to sum to one. so one is going to state independent normalization in any case
(using the vertices once) one get F(2x)=2F(x) and F(3x)=3(x).,...... and the any given two property F(1-x)+F(x)=1
even if objective chance not closed under local union, if ones domain is the entire simplex
on a distinct vector in the simplex in the simplex?
add subjective credences of events whose objective chances sum to the value of some other event on 'a non-commuting vector/distinct 'in the same simplex/distinct yet finite probability space in the same system.
ie =3 so long as ones domain is the entire simplex I presume that the only difference between cauchy additivity and the simplex is that its restricted to only adding up as many events as there are in the simplex on distinct vectors whose sum is that equal to that value (but if one domain is the entire simplex I presume one could form infinite strings, and it would be no different from presuming cauchy addivity. Its trivial to derive given global rank and local closure under finite unions so long as the domain is present, it does not need to be presume. I note that it does not need to be presumed, either if its not closed under unions. as this hold globally
F(1-x-y)+F(x)+F(y)=1 globally
x+y=2/3 iff F(x)+F(y)=2/3
F(0)=0, F(1)=1 F(1/3)=1/3
x+y=z+m in [0,1] cross world iff F(x)+F(y)=F(z)+F(M)
x+y>z+m in [0,1] cross world iff F(x)+F(y)>F(z)+F(M)
x+y
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