Contextuality is a non-classical property of quantum physics that is believed to provide a key resource for quantum computing.  The topological approach to this property by Abramsky-Brandenburger 2011 etc. characterizes it as ``global inconsistency'' coupled with ``local consistency'', revealing it to be a phenomenon also found in many other fields and topics---including, notably, logical paradoxes arising from circularity in the reference and composition of sentences.  While this approach has yielded logical and algebraic methods of proving the ``global inconsistency'' part of contextuality, this talk targets the other, ``local consistency'' part.  We characterize it in logical terms by providing a novel system of ``local inference'' that is sensitive to the topology of contexts and whose notion of consistency is local as opposed to global.