# Encoding fuzzy logic with the topos of set-valued sheaves

One of the canonical examples used by Barr & Wells in order to motivate the use of topoi is that we can construct a theory for fuzzy logic and fuzzy set theory as set-valued sheaves on a poset (Heyting algebra) of confidence values for the fuzziness. Doing this constructs a fuzzy theory where both the membership and equality relations have more truth values than just true and false.

How would one construct a ternary approach using this mindset? In other words, is there an easy way to see a sheaf on a poset with three values as a fuzzy set theory for the truth values (true, maybe, false)? Or is this formulation even the wrong approach? Do I want a different Heyting algebra, so that the subobject classifier ends up having three elements?

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The topos with this logic is the arrow category of set: Set$^\to$. Objects consist of Set functions $A \to B$, and morphism consist of pairs of set functions forming a commutative square.
The Heyting algebra for the example given consists of elements $0 < * < 1$, with obvious meet and join, and implication is mostly 1, except $1 \Rightarrow * = *$, $1 \Rightarrow 0 = 0$ and $* \Rightarrow 0 = 0$. This example can also be considered as sheaves over the topological space {0,1}, where {1} is open, but {0} is not - the Sierpinski space. Note that the partial order of open subsets agrees with the Heyting algebra above. – J Williams Dec 9 '09 at 5:56