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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).

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    $\begingroup$ It looks like you are ready to bargain, so perhaps the third stage is where you will find some useful ideas. $\endgroup$
    – Andrej Bauer
    Commented Mar 12, 2021 at 14:52
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    $\begingroup$ @AndrejBauer This is really relevant to my question. Thanks! $\endgroup$
    – dohmatob
    Commented Mar 12, 2021 at 15:17
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    $\begingroup$ @SimonHenry: Did you mean A guided tour in the topos of graphs? $\endgroup$
    – Andrej Bauer
    Commented Mar 12, 2021 at 21:21
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    $\begingroup$ In some parts of the world it is legal to smoke marijuana, in others it is not. This is a non-classical truth value in $\operatorname{Sh}(S^2)$. $\endgroup$
    – lambda
    Commented Mar 12, 2021 at 22:31
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    $\begingroup$ If you want to see a couple of examples for the wondrous non-classical facts holding in specific toposes, you can have a look at these notes of mine, particularly Section 2 and Section 3. I am happy to any questions you might have concerning these notes! $\endgroup$
    – Ingo Blechschmidt
    Commented Mar 13, 2021 at 12:32

1 Answer 1

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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, $p \lor (\lnot p) = (\mathrm{true}, \mathrm{false}) \lor (\mathrm{false}, \mathrm{true}) = (\mathrm{true} \lor \mathrm{false}, \mathrm{false} \lor \mathrm{true}) = (\mathrm{true}, \mathrm{true})$.

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  • $\begingroup$ Thanks for the input (upvoted). i'm not sure I follow (first paragraph). What do you mean by "are neither true nor false" when you've only defined what it means for a statement to be "true" ? In this system, what does it mean to say something is "false" ? (This is classically defined to mean "not true", via excluded middle principle / axiom, I guess. But how to go about that here ?). $\endgroup$
    – dohmatob
    Commented Mar 12, 2021 at 18:08
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    $\begingroup$ @dohmatob: it is quite impossible for anyone to define precisely all the details of the logic in a topos in an answer on MO, so you should give a great deal of leeway to the answer. It is just trying to illustrate a point, it's not a precise definition. $\endgroup$
    – Andrej Bauer
    Commented Mar 12, 2021 at 21:25
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    $\begingroup$ But to answer your question: in this topos we always use pair of truth values. As far as the topos is concerned, "truth" is $(\mathrm{true}, \mathrm{true})$ and "falsehood" is $(\mathrm{false}, \mathrm{false})$. If you ask "Is $(\mathrm{true}, \mathrm{false})$ equal to $\mathrm{true}, \mathrm{true})$?" the answer is "$(\mathrm{true}, \mathrm{false})$". Given any truth value $(p_1, p_2)$, the value of $\neg (p_1, p_2) \land \neg\neg (p_1, p_2)$ (whose meaning is "$(p_1, p_2)$ is neither true nor false") is $(\mathrm{false}, \mathrm{false})$. $\endgroup$
    – Andrej Bauer
    Commented Mar 12, 2021 at 21:26
  • $\begingroup$ @AndrejBauer Thanks for the clarification. $\endgroup$
    – dohmatob
    Commented Mar 12, 2021 at 23:55

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