  We propose an interpretation for the meets and joins in the lattice of
experimental propositions of a physical theory, answering a question of
Birkhoff and von Neumann in [1]. When the lattice is atomistic, it is
isomorphic to the lattice of definably closed sets of a finitary relational
structure in First Order Logic. In terms of mapping experimental propositions
to subsets of the atomic phase space, the meet corresponds to set intersection,
while the join is the definable closure of set union. The relational structure
is defined by the action of the lattice automorphism group on the atomic layer.
Examining this correspondence between physical theories and infinite group
actions, we show that the automorphism group must belong to a family of
permutation groups known as geometric Jordan groups. We then use the
classification theorem for Jordan groups to argue that the combined
requirements of probability and atomicism leave uncountably infinite Steiner
2-systems (of which projective spaces are standard examples) as the sole class
of options for generating the lattice of particle Quantum Mechanics.
