nLab
higher-order logic

Contents

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 VV called the domain of discourse, and for each type TT and each term t:Tt:T, a type of predicates on T T 𝒫(T)\mathcal{P}(T) whose terms P(t):𝒫(T)(t)P(t):\mathcal{P}(T)(t) are propositions dependent on tt. The system (𝒰,V:𝒰,𝒫:𝒰→𝒰)(\mathcal{U}, V:\mathcal{U}, \mathcal{P}:\mathcal{U}\rightarrow\mathcal{U}) consisting of the type universe 𝒰\mathcal{U}, the domain of discourse VV, and the power type functor 𝒫\mathcal{P} is a natural numbers object.

Last revised on March 3, 2021 at 12:42:53. See the history of this page for a list of all contributions to it.