|Abstract||A logic is called higher order if it allows for quantiﬁcation (and possibly abstraction) over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc. Higher order logic (often also called type theory or the Theory of Types) began with Frege, was formalized in Russell  and Whitehead and Russell  early in the previous century, and received its canonical formulation in Church .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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