A decade ago, Isham and Butterfield proposed a topos-theoretic approach to
quantum mechanics, which meanwhile has been extended by Doering and Isham so as
to provide a new mathematical foundation for all of physics. Last year, three
of the present authors redeveloped and refined these ideas by combining the
C*-algebraic approach to quantum theory with the so-called internal language of
topos theory (see arXiv:0709.4364). The goal of the present paper is to
illustrate our abstract setup through the concrete example of the C*-algebra of
complex n by n matrices. This leads to an explicit expression for the pointfree
quantum phase space and the associated logical structure and Gelfand transform
of an n-level system. We also determine the pertinent non-probabilisitic
state-proposition pairing (or valuation) and give a very natural
topos-theoretic reformulation of the Kochen--Specker Theorem. The essential
point is that the logical structure of a quantum n-level system turns out to be
intuitionistic, which means that it is distributive but fails to satisfy the
law of the excluded middle (both in opposition to the usual quantum logic).Comment: 26 page