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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 26th 2020

    Added in a link to tripos.

    diff, v56, current

    • CommentRowNumber2.
    • CommentAuthorNikolajK
    • CommentTimeSep 28th 2020

    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.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 6th 2021

    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 VV called the domain of discourse, and for each type TT and each term t:Tt:T, a dependent type 𝒫(t)\mathcal{P}(t) whose terms P(t):𝒫(t)P(t):\mathcal{P}(t) are higher-order predicates depending on tt. The system (𝒰,V:𝒰,𝒫:𝒰𝒰)(\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.

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

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