  Contextuality in quantum physics provides a key resource for quantum
information and computation. The topological approach in [Abramsky and
Brandenburger, New J. Phys., 2011, Abramsky et al., CSL 2015, 2015]
characterizes contextuality as "global inconsistency" coupled with "local
consistency", revealing it to be a phenomenon also found in many other fields.
This has yielded a logical method of detecting and proving the "global
inconsistency" part of contextuality. Our goal is to capture the other, "local
consistency" part, which requires a novel approach to logic that is sensitive
to the topology of contexts. To achieve this, we formulate a logic of local
inference by using context-sensitive theories and models in regular categories.
This provides a uniform framework for local consistency, and lays a foundation
for high-level methods of detecting, proving, and moreover using contextuality
as computational resource.
