In this talk I will explore the logic of broad necessity. Definitions of what it means for one modality to be broader than another are formulated, and I will show, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. I will argue, moreover, that it is possible to give a reductive analysis of this necessity in extensional language: using truth functional connectives and quantifiers. This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This idea can be formulated precisely in higher-order logic, and I will examine some concrete cases in which it fails. Finally, we’ll take a look at the modal logic of broad necessity and examine some arguments for and against the thesis that its logic is S5.