From Babel to Boole: the logical organization of information decompositions

The field of partial information decomposition (PID) has become increasingly fragmented, with various proposed solutions approaching the problem from distinct perspectives. At the heart of each proposal lies what we call a PID-inducing concept —a type of information that, once formally specified, determines the PID components. Traditionally, redundancy has been the PID-inducing concept of choice, but other concepts, such as synergy, unique information and union information, have also been used. However, no cohesive logical framework exists to unify these approaches and clarify their relationships, particularly in the general case of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> information sources. In this work, we introduce a unifying framework that connects PID components to Boolean functions and identifies PID-inducing concepts with logical conditions on these functions. We demonstrate that all existing PID-inducing concepts are special cases of this general construction. We also introduce novel concepts, including vulnerable information and partner concepts , and provide a comprehensive analysis of their properties, lattice structures and interrelationships.