So after all, what is this thing about topos theory and non-binary truth? Disclaimer. The question below is necessarily vague. I understand neither the subject matter topos theory nor the object about which my question is (the construction of a fractional / non-binary / elastic notion of truth). Thanks in advance for your patience.

I've often heard whispers of vague statements like: "topos theory can be used to give  continuous, subtle, and perhaps a much more interesting notion of truth (not just True or False, Right or Wrong, Correct or Incorrect)". For lack of better terminology, let us refer to this mysterious phenomenon / concept as fractional truth.

Question. What is a minimal construction which already displays this phenomenon?

I have essentially no knowledge of topos theory (but a decent handle on set theory, group theory, and topology, etc., and I'm naively assuming that this should be sufficient to understand the would-be construction, which I'm also assuming can be made so simple).
 A: 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, $p \lor (\lnot p) = (\mathrm{true}, \mathrm{false}) \lor (\mathrm{false}, \mathrm{true}) = (\mathrm{true} \lor \mathrm{false}, \mathrm{false} \lor \mathrm{true}) = (\mathrm{true}, \mathrm{true})$.
