## Definition ## Excluded middle says that there is a term of the sum of a proposition $P$ and the type of functions $P \to \emptyset$. $$\frac{P\ \mathrm{type} \quad p:\prod_{a:P} \prod_{b:P} a =_P b}{q:P + P \to \emptyset}$$ ### In universes ### Excluded middle is said to hold in a [[universe]] $\mathcal{U}$ if the universe comes with a dependent function $$p:\prod_{P:\mathcal{U}} \left(\prod_{a:\mathcal{T}_\mathcal{U}(P)} \prod_{b:\mathcal{T}_\mathcal{U}(P)} a =_{\mathcal{T}_\mathcal{U}(P)} b\right) \to \left(\mathcal{T}_\mathcal{U}(P) + \mathcal{T}_\mathcal{U}(P) \to \emptyset\right)$$ ## See also ## * [[limited principle of omniscience]] * [[double negation]] * [[Whitehead's principle]]