Suppose you always want to talk about two things simultaneously for some reason. When you say "a set $A$" you actually mean a pair of sets $(A_1, A_2)$, when you say "a function $f : A \to B$" you mean a pair of functions $f_1 : A_1 \to B_1$, $f_2 : A_2 \to B_2$. When you say a statement $p$ is true, you actually mean $p_1$ is true and $p_2$ is true. Then from an external perspective there obviously are truth values which "are neither true nor false", namely $(true, false)$ and $(false, true)$. (Note that the partial order of truth values is not a total order here, so it is not very similar to fractional numbers.)
This is the internal language of the topos $\mathrm{Sh}(\{*\} \sqcup \{*\})$, the sheaf topos on the discrete two-point space. (A sheaf on this space is just a pair of sets.) If you take $\mathrm{Sh}(X)$ instead, for $X$ any topological space, there are as many internal truth values as $X$ has open subsets.
Be careful, however, to distinguish between internal and external statements. The truth values $p = (\mathrm{true}, \mathrm{false})$ and $(\mathrm{true}, \mathrm{true})$ are different externally, but the internal statement "$p$ is different from $\mathrm{true}$" has truth value $(\mathrm{false}, \mathrm{true})$, so it is not valid (everywhere). In fact, we can prove in intuitionistic logic (which is valid in every topos) that there is no truth value which is neither true nor false.
Also note that $\mathrm{Sh}(\{*\} \sqcup \{*\})$ is a boolean topos, that is, the internal language is in fact classical -- the law of omniscience (excluded middle) is valid internally. For example, $p \lor (\lnot p) = (\mathrm{true}, \mathrm{false}) \lor (\mathrm{false}, \mathrm{true}) = (\mathrm{true} \lor \mathrm{false}, \mathrm{false} \lor \mathrm{true}) = (\mathrm{true}, \mathrm{true})$.