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.

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