  This paper deals with topos-theoretic truth-value valuations of quantum
propositions. Concretely, a mathematical framework of a specific type of modal
approach is extended to the topos theory, and further, structures of the
obtained truth-value valuations are investigated. What is taken up is the modal
approach based on a determinate lattice $\Dcal(e,R)$, which is a sublattice of
the lattice $\Lcal$ of all quantum propositions and is determined by a quantum
state $e$ and a preferred determinate observable $R$. Topos-theoretic extension
is made in the functor category $\Sets^{\CcalR}$ of which base category
$\CcalR$ is determined by $R$. Each true atom, which determines truth values,
true or false, of all propositions in $\Dcal(e,R)$, generates also a
multi-valued valuation function of which domain and range are $\Lcal$ and a
Heyting algebra given by the subobject classifier in $\Sets^{\CcalR}$,
respectively. All true propositions in $\Dcal(e,R)$ are assigned the top
element of the Heyting algebra by the valuation function. False propositions
including the null proposition are, however, assigned values larger than the
bottom element. This defect can be removed by use of a subobject
semi-classifier. Furthermore, in order to treat all possible determinate
observables in a unified framework, another valuations are constructed in the
functor category $\Sets^{\Ccal}$. Here, the base category $\Ccal$ includes all
$\CcalR$'s as subcategories. Although $\Sets^{\Ccal}$ has a structure
apparently different from $\Sets^{\CcalR}$, a subobject semi-classifier of
$\Sets^{\Ccal}$ gives valuations completely equivalent to those in
$\Sets^{\CcalR}$'s.
