We investigate a novel intuitionistic modal logic, called Propositional Lax Logic, with promising applications to the formal verification of computer hardware. The logic has emerged from an attempt to express correctness `up to' behavioural constraints --- a central notion in hardware verification --- as a logical modality. The resulting logic is unorthodox in several respects. As a modal logic it is special since it features a single modal operator fl that has a flavour both of possibility and of necessity. As for hardware verification it is special since it is an intuitionistic rather than classical logic which so far has been the basis of the great majority of approaches. Finally, its models are unusual since they feature worlds with inconsistent information and furthermore the only frame condition is that the fl -frame be a subrelation of the oe-frame. In the paper we will provide the motivation for Propositional Lax Logic and present several technical results. We will investigate...

1 An introduction to inquisitive semantics 1 1.1 Information states and the classical update perspective...... 1 1.2 Limitations of the classical picture................. 2 1.3 The inquisitive programme: propositions as proposals...... 3

Sono allora un soldatino di piombo appeso a un filo sul mare aperto I miei pensieri passano come velieri Leonardo Martellini, Velieri To my schoolteacher Mauro, who taught me how to doubt i Contents Acknowledgements............................. Overview of the thesis........................... iv v