The Logical Form of Negation offers a unified analysis of the syntax of negation which reinterprets the results of much recent research by tying in a strictly syntactic account with its necessary semantic underpinnings. Couched in the Principles and Parameters framework, the analysis begins by highlighting some problematic aspects for contemporary approaches based on the Negative Criterion, a condition requiring negative elements to appear in specifier-head relation by the level of Logical Form. These shortcomings are overcome by the simple and semantically well-motivated view that negative 'quantifiers' are in fact indefinites bound by a negated existential operator. More precisely, negation provides a Boolean negative connective which is attached to the existential closure operator; the resulting negated existential binds the event variable associated with the verb, and possibly one or more indefinites. Negative 'quantifiers' are like indefinites (including polarity items) in being bound by an operator through unselective Binding; however, their featural characterization (often morphologically expressed) imposes the tighter locality constraint of Government by the operator. As a result, the proposed analysis can account for some apparently problematic issues for alternative approaches, such as the licensing of preverbal negative subjects, asymmetries of negation in reason adverbials, and the behavior of negatives in donkey-sentences. The interpretive format argued for also warrants a unified analysis for languages with and without generalized negative concord. What is more, the same analysis can be extended to cover the syntax of so-called 'monotone decreasing quantifiers', elements whose similarity with negatives has long been recognized but never accounted for by an explicit syntactic analysis. Finally, the distinction between non-referential indexing and featural characterization is shown to account for some poorly understood locality constraints on the scope of negative quantifiers, including argument / adjunct asymmetries, specificity effects and inverse linking.