Phi-monads are conceptually problematic and mathematically undefined in Integrated Information Theory 4.0

The Integrated Information Theory (IIT) 4.0 is an influential framework aiming to explain consciousness mathematically in terms of intrinsic existence, namely what exists for itself in a specific, maximally irreducible, structured form. IIT thus posits that the minimal substrates of consciousness are also the ultimate building blocks of what exists. These are called ‘monads’, and are defined as self-connected, single-unit systems purportedly specifying maximal values of IIT’s measure of conscious irreducible existence, namely system integrated information phi_s* (i.e., phi-monads). In this paper, we argue that IIT 4.0’s formalism cannot be applied to monads, thus threatening the theory’s own ontological ground. We show that IIT’s formalism requires that any candidate substrate of consciousness must be constituted by at least two non-overlapping, non-empty, and jointly exhaustive parts, which we refer to as the Plurality requirement. We then show that monads violate this requirement, which leads to serious mathematical and conceptual problems, since the equations needed to calculate system integrated information become undefined. Thus, our paper calls for urgent clarification and/or revision by IIT proponents concerning the status of phi-monads as ontologically fundamental building blocks and minimal substrates of consciousness.