We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of being a necessity, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about their logical behaviour. We also wield the framework to probe the conditions under which a logicist account of necessities is possible, in which the theory is completely reducible to logic.

我们在高阶逻辑版本中开发了一种必然性算子理论，该版本对细粒度现实是中立的。该理论根据必要性的原始性进行了公理化，我们展示了如何从中恢复模态哲学中的核心概念。各种问题在框架内被制定和解决，包括关于力量下的必需品的排序问题，满足各种逻辑条件的最广泛的必需品的存在，以及关于它们的逻辑行为的问题。我们还运用这个框架来探讨在何种条件下逻辑主义对必然性的解释是可能的，在这种情况下，该理论可以完全还原为逻辑。