#
Homotopy Type Theory
excluded middle > history (Rev #1)

## 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

Revision on May 12, 2022 at 05:44:11 by
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