ABSTRACT
The most general ground remaining for a solution is that a relation between symmetry and the equal epistemic status of the chords can rule out paradoxical probabilities. I show that for this hope to succeed we must be able to represent the symmetry of events mathematically and there must be a mathematical constraint that I call a Justified Guarantee.
The set of symmetries of an object satisfies the mathematical axioms for being a group. There is one attempt to use groups in the literature from Gyenis and Rédei. I show that Gyenis and Rédei offer a Justified Guarantee, although not named as such, and that it fails.
I then turn to considering the hope from symmetry in full generality. I prove a number of representation theorems and then define the Principle of Symmetric Indifference. I show that solving the paradox in these terms is equivalent to defining a mathematical condition on renamings to be the Justified Guarantee. I give a mathematical proof that the natural condition to try fails to guarantee consistency. I then prove that a condition on renaming can either be justified or guarantee consistency, but not both, and that therefore the hope from symmetry is forlorn.
