Inquisitive possibility semantics is a version of inquisitive semantics where meanings are not assumed to be downward closed.

Dropping the assumption of downward closure we can make more fine-grained semantic distinctions, but we also lose the principled account of the connectives of the basic system.

Since it is suited to deal with non-maximal possibilities, this system has been used as a framework to investigate the semantic of free-choice imperatives and attentive might.

A close relative of this system, and for some purposes an evolution of it, is attentive semantics. Also, this system is tightly connected to Fine’s intuitionistic truth-maker semantics. A manuscript exploring in detail the connection is available on request.

References:

Ivano Ciardelli, Inquisitive Semantics and Intermediate Logics, section 6, 2009.
MSc thesis, University of Amsterdam.

Ivano Ciardelli, Jeroen Groenendijk and Floris Roelofsen, Information, issues, and attention, 2010. Extended version of this paper published in the proceedings of SALT2009.