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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 26th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded in a link to [[tripos]]. <a href="https://ncatlab.org/nlab/revision/diff/higher-order+logic/56">diff</a>, <a href="https://ncatlab.org/nlab/revision/higher-order+logic/56">v56</a>, <a href="https://ncatlab.org/nlab/show/higher-order+logic">current</a>

   Added in a link to tripos.

   diff, v56, current

2. • CommentRowNumber2.
   • CommentAuthorNikolajK
   • CommentTimeSep 28th 2020
   • PermaLink
   Author: NikolajK
   Format: MarkdownItex>A higher-order logic is any logic which features higher-order predicates One can and should probably pin it down to those logic also having quantifiers ranging over predicates. In particular, I'm thinking of FOL systems with individual predicates expressing e.g. syntactic aspects such as rules like `Bounded(P) |- Bounded(not P)` Such a logic "features" higher order predicates, but it wouldn't count as higher order logic just because of that.

   A higher-order logic is any logic which features higher-order predicates

   One can and should probably pin it down to those logic also having quantifiers ranging over predicates.

   In particular, I’m thinking of FOL systems with individual predicates expressing e.g. syntactic aspects such as rules like

   Bounded(P) |- Bounded(not P)

   Such a logic “features” higher order predicates, but it wouldn’t count as higher order logic just because of that.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 6th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexJust to record that someone anonymously added > 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 dependent type $\mathcal{P}(t)$ whose terms $P(t):\mathcal{P}(t)$ are higher-order predicates depending 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]]. They made similar additions at [[predicate logic]] and [[second-order logic]].

   Just to record that someone anonymously added

   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 dependent type $\mathcal{P}(t)$ whose terms $P(t):\mathcal{P}(t)$ are higher-order predicates depending 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.

   They made similar additions at predicate logic and second-order logic.

4. • CommentRowNumber4.
   • CommentAuthorHugo Herbelin
   • CommentTimeMay 28th 2023
   • PermaLink
   Author: Hugo Herbelin
   Format: MarkdownItexAdd link to system Fω, presented on the Pure Type Systems page. <a href="https://ncatlab.org/nlab/revision/diff/higher-order+logic/61">diff</a>, <a href="https://ncatlab.org/nlab/revision/higher-order+logic/61">v61</a>, <a href="https://ncatlab.org/nlab/show/higher-order+logic">current</a>

   Add link to system Fω, presented on the Pure Type Systems page.

   diff, v61, current

5. • CommentRowNumber5.
   • CommentAuthorHugo Herbelin
   • CommentTimeMay 28th 2023
   • PermaLink
   Author: Hugo Herbelin
   Format: MarkdownItexAdd a link to the Calculus of Constructions <a href="https://ncatlab.org/nlab/revision/diff/higher-order+logic/61">diff</a>, <a href="https://ncatlab.org/nlab/revision/higher-order+logic/61">v61</a>, <a href="https://ncatlab.org/nlab/show/higher-order+logic">current</a>

   Add a link to the Calculus of Constructions

   diff, v61, current

6. • CommentRowNumber6.
   • CommentAuthornLab edit announcer
   • CommentTimeNov 13th 2023
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexAdded reference: > * [[Colin Rothgang]], [[Florian Rabe]], [[Christoph Benzmüller]], *Theorem Proving in Dependently-Typed Higher-Order Logic -- Extended Preprint* ([arXiv:2305.15382](https://arxiv.org/abs/2305.15382)) Anonymouse <a href="https://ncatlab.org/nlab/revision/diff/higher-order+logic/62">diff</a>, <a href="https://ncatlab.org/nlab/revision/higher-order+logic/62">v62</a>, <a href="https://ncatlab.org/nlab/show/higher-order+logic">current</a>

   Added reference:

   • Colin Rothgang, Florian Rabe, Christoph Benzmüller, Theorem Proving in Dependently-Typed Higher-Order Logic – Extended Preprint (arXiv:2305.15382)

   Anonymouse

   diff, v62, current