  The finite set of subsystems of a finite quantum system with variables in
${\mathbb Z}(n)$, is studied as a Heyting algebra. The physical meaning of the
logical connectives is discussed. It is shown that disjunction of subsystems is
more general concept than superposition. Consequently the quantum probabilities
related to projectors in the subsystems, are incompatible with associativity of
the join in the Heyting algebra, unless if the variables belong to the same
chain. This leads to contextuality, which in the present formalism has as
contexts, the chains in the Heyting algebra. Logical Bell inequalities, which
contain `Heyting factors', are discussed. The formalism is also applied to the
infinite set of all finite quantum systems, which is appropriately enlarged in
order to become a complete Heyting algebra.
