We prove that three cornerstone properties of classical information theory, namely non-negativity of information, the chain rule, and invariance under invertible transformations, become mutually incompatible when extended to Partial Information Decomposition (PID), the framework for decomposing the information that multiple sources carry about a target into redundant, unique, and synergistic contributions. This is, to our knowledge, the first PID incompatibility result involving only direct extensions of standard information-theoretic principles, without relying on any PID-specific assumptions. It reveals that PID necessarily operates in a different conceptual realm from classical information theory, where not all familiar properties can be preserved. Additionally, we re-examine the incompatibility result of Rauh et al. involving the so-called identity property and show that it holds under weaker assumptions than originally stated. Our proofs leverage the recently developed mereological framework for PID that makes the structural origins of these incompatibilities transparent and allows the results to be formulated for any PID-inducing concept, not just redundant information. We discuss the implications of these findings and examine which of the incompatible properties might be relaxed.

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