  Motivated by quantum states with zero transition probability, we introduce
the notion of ortho-set which is a set equipped with a relation
$\neq_\mathrm{q}$ satisfying: $x\neq_\mathrm{q} y$ implies both $x\neq y$ and
$y \neq_\mathrm{q} x$. For an ortho-set, a canonical complete ortholattice is
constructed. Conversely, every complete ortholattice comes from an ortho-set in
this way. Hence, the theory of ortho-sets captures almost everything about
quantum logics.
  For a quantum system modeled by the self-adjoint part $B_\mathrm{sa}$ of a
$C^*$-algebra $B$, we also introduce a "semi-classical object" called the
Gelfand spectrum. It is the ortho-set, $P(B)$, of pure states of $B$ equipped
with an "ortho-topology", which is a collection of subsets of $P(B)$, defined
via a hull-kernel construction with respects to closed left ideals of $B$. We
establish a generalization of the Gelfand theorem by showing that a bijection
between the Gelfand spectra of two quantum systems that preserves the
respective ortho-topologies is induced by a Jordan isomorphism between the
self-adjoint parts of the underlying $C^*$-algebras (i.e. an isomorphism of the
quantum systems), when the underlying $C^*$-algebras satisfy a mild condition.
