Higher-Order Syllogistics

We propose a distinction between bottom-up and top-down systems of natural logic, with the classical syllogism epitomizing the first and the Monotonicity Calculus the second. We furthermore suggest it useful to view top-down systems as higher-order generalizations of broadly syllogistic systems. We illustrate this view by proving a result of independent interest: we axiomatize the first-order/single-type fragment of a higher-order calculus for reasoning about inclusion and exclusion (MacCartney and Manning, 2009; Icard, 2012). We show this logic is equivalent to a syllogistic logic with All and nominal complementation, in fact a fragment of a system recently studied (Moss, 2010b).