# On incompleteness in modal logic. An account through second-order logic

## Description

The dissertation gives a second-order-logic-based explanation of modal incompleteness. The leading concept is that modal incompleteness is to be explained in terms of the incompleteness of standard second-order logic, since modal language is basically a second-order language. The development of Kripke-style semantics for modal logic has been underpinned by the conjecture that (maybe) all modal systems are characterizable by classes of frames defined by first-order conditions on a binary (accessibility) relation. However, the discovery of certain incomplete (uncharacterizable) modal systems has undermined the all-encompassing feature of Kripke-style approach. There are logics not determined by any class of Kripke frames at all In the dissertation I investigate a normal incomplete sentential modal system due to J. F. A. K. Van Benthem, and I address both the formal and the philosophical facets of modal incompleteness from the vantage point that modal logic is essentially second-order in its nature. Modal systems can be analyzed in terms of structures with a domain of second-order individuals (subsets) that are assigned under an interpretation to propositional variables within languages of sentential modal logic. Hence, the phenomenon of there being incomplete modal systems can be accounted for through the incompleteness of standard second-order logic Since second-order logic plays a crucial role in the explanation, Quine's animadversions upon second-order logic and in particular his views that second-order logic is 'Set Theory in Sheep's Clothing' are examined. Against Quine's stance I will seek to show that a vindication of second-order logic can be gotten provided a proper due is given to the sharp distinction between the logical (Fregean) notion of set which is the concern of second-order logic and the iterative notion of set which lies within the realm of set theory