Contents

foundations

# Contents

## Idea

A higher-order logic is any logic which features higher-order predicates, which are predicates of predicates or of operations. If we think of a predicate as a function to truth values, then a higher-order predicate is a function on a power set or a function set.

Typed higher-order logic may be called higher-order type theory. Typed higher-order intuitionistic logic is often identified with the internal logic of a topos.

Higher-order logic in general could be thought of as a first order theory with dependent types. There is a type $V$ called the domain of discourse, and for each type $T$ and each term $t:T$, a type of predicates on $T$ $\mathcal{P}(T)$ whose terms $P(t):\mathcal{P}(T)(t)$ are propositions dependent on $t$. The system $(\mathcal{U}, V:\mathcal{U}, \mathcal{P}:\mathcal{U}\rightarrow\mathcal{U})$ consisting of the type universe $\mathcal{U}$, the domain of discourse $V$, and the power type functor $\mathcal{P}$ is a natural numbers object.

Last revised on May 28, 2023 at 19:53:29. See the history of this page for a list of all contributions to it.