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 fracitonal / non-binary / elastic notion of truth). Thanks in advance for your patience.

I've often heard whispers of vague statements like: "topos theory can used give continuous, subtile, 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).