I am wondering whether there are some weaker forms of midpoint convexity; in my system it appears that jensens equation, as far as I can see, will just fall out; if not cauchy equation or something close to it, for all rational numbers at least;
Given symmetry in an strictly monotonically increasing and continuous strict quasi convex, strictly quasi concave function over a total order on events in differences;
Just given use of of
midpoint convexity
on half the structure; and using equimatched pairs and symmetry to deliver the other half (if is there such a name for it, of jensens equation, midpoint concavity, on the entire structure)
F(1)=1;-max, for all x, x=1 F(x)0,
and if neither then F(x)>0, F(x)0.5 F(x)>0.5, else F(x)F(x +y)/2)1,x+y1 iff F-1(x)+F-1(y)1 F-1(x1)+F-1(y2)F(y)| F(x)y,F(x)=F(y)implies x=y
And non atomicity,of chances and chance difference (with perhaps a segmental additivity, or additive difference monotone continuity or archimedean condition)
where the domain is a closed bounded interval just a linear chance function.I dont think i have to even use much of my other constrinat
Central symmetry F(x)/2 +F(1-x)/2= 0.5 and standard additivity G(x)+F(x)=1, where G(x)=F(1-x) and has the same properties
It appears to be far too strong I was also wondering if there is a name for the property F-1(1-p)+F-1(p)=1 if F(1-p) +F(1-p)=1 (which if often called symmetry). JUst an injective symmetric function. And whether there is a specific definition of the properties of a mirror or reflection or square symmetric function (it appears that have some of those properties as a result of these constraints) and G being identical to F; ie all four halves are identical, slopes and diagonals and credal widths veritcal horizontally being identical withinand without any given indifference, with the diagonals likewise (although i am not sure if they have a slope of 1- exactly unless midpoint convexity is imposed; or whether it satisfies the golden ratio trianle condition; or strong quadruple condtiion I presume that would be something like the diagonal length where |F(y)-F(y+delta)| =|F(y)-F(y-delta|, and likewise for G , =sqrt2|F(y)-F(y-delta| ^2|=sqrt(2|g(y)-F(y-delta| ^2|; with the four halfs being identical, where they always cross did centre in teh same position (f(0.5). I doubt it is any unique or constant function of x however,
G(xin on both sidses, with all four being ssqrt2(F(y)-F(y+delta)= sqrt2 (y+delta)-(y-delta); with F(x)-F(x1)/|x-x1|=1 at least not without midpoint convexity (this appears to induce everything, but i thinks its too strong to impose, hence the question, given what it entails with conjoined with symmetry/equimatched pair symmetry)
This is four the two outcome case; nothing needs to be really imposed in the entangled structure, given just three outcome normalization, and the abillity to connect everything through equalities.
Perhaps at worst, are the irrational values.I presume that these are dealt with via point reflection, continuity, or imposed a multiplicative structure (or swapping strict).
Although there appears is a kind of point reflection symmetry that might actually allow you to resolve them in principle, by looking at the table (obviously one does not resolve by hand that way of course), which induces a violation of an inequality even if their irrational values which are already constrained within the finest interval of rational values fall out of line with the chances numerically. Although I am not sure. In any case, other than a mild archimedean constraint, assumed or derived (perhaps nothing but that) they can be dealt with
(and its linear as well and continuous in rationals, linear in irrationals)
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