Higher Order Modal Logic

In Patrick Blackburn, Johan Van Benthem & Frank Wolter (eds.), Handbook of Modal Logic. Elsevier. pp. 621-653 (2006)
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A logic is called higher order if it allows for quantification over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc. Higher order logic began with Frege, was formalized in Russell [46] and Whitehead and Russell [52] early in the previous century, and received its canonical formulation in Church [14].1 While classical type theory has since long been overshadowed by set theory as a foundation of mathematics, recent decades have shown remarkable comebacks in the fields of mechanized reasoning (see, e.g., Benzm¨
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Essays in Logical Semantics.van Benthem, J. F. A. K.

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Outlook-Based Semantics.Coppock, Elizabeth
Quantified Multimodal Logics in Simple Type Theory.Benzmüller, Christoph & Paulson, Lawrence C.

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