 Let us call a complete orthomodular lattice satisfying the hypotheses of Piron’s theorem a Piron lattice. Can we give any general reason for supposing that the logic/property lattice of a physical system (one for which these are isomorphic) is a Piron lattice? Or, failing this, can we at least ascribe some clear physical content to these assumptions? The atomicity of \(L\) follows if we assume that every pure state represents a “physical property”. This is a strong assumption, but its content seems clear enough. Irreducibility is usually regarded as a benign assumption, in that a reducible system can be decomposed into its irreducible parts, to each of which Piron’s Theorem applies.