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The necessity modality of intuitionist S4 is a comonad. In this paper, we study indexed necessity modalities that provide the logical foundation for a variety of applications; for example, to model possession of capabilities in policy languages for access control, and to track exceptions in type theories for exceptional computation.

Noninterference properties of the intuitionist logic of indexed necessity modalities capture the limitations on the information flow between formulas that are under the scope of necessity modalities with different indices. The impact of noninterference is seen in the unprovability of certain formulas. Noninterference is necessary for several applications. In models of capabilities, noninterference facilitates distributed reasoning. In models of exceptions, noninterference is necessary to ensure that the exceptions are tracked conservatively.

In this paper, we prove noninterference properties for indexed intuitionist necessity S4 modalities.

To our knowledge, this is the first examination of noninterference results for the intuitionist S4 necessity modality (even without indexing).