Ignacio Cea ^1,2* Camilo Miguel Signorelli ^3,4,5

• ¹ Center for Research, Innovation and Creation, and Faculty of Religious Sciences and Philosophy, Temuco Catholic University, Temuco, Chile
• ² Philosophy Department, Faculty of Philosophy and Humanities, Universidad Alberto Hurtado, Santiago, Chile
• ³ Department of Computer Science, Mathematical, Physical and Life Sciences Division, University of Oxford, Oxford, England, United Kingdom
• ⁴ Center for Philosophy of Artificial Intelligence, University of Copenhagen, Copenhagen, Denmark
• ⁵ Laboratoire de Neurophysiologie et Biomécanique du Mouvement, Université Libre de Bruxelles, Brussels, Belgium

Integrated Information Theory 4.0 (IIT) is one of the leading frameworks in the neuroscience of consciousness (Consortium et al. 2023;A. K. Seth and Bayne 2022;Signorelli, Szczotka, and Prentner 2021). It aims to explain consciousness by mathematically formalizing its relation to causeeffect power and existence, while employing computational tools to investigate this experimentally (Zaeemzadeh and Tononi 2024;Albantakis et al. 2023;Ellia et al. 2021). In principle, IIT can be used to assess both the level and content of consciousness in any physical system, such as the brain of a comatose patient or under anesthesia (Albantakis et al. 2023;Tononi et al. 2016).More specifically, IIT Concerning IIT's explanatory strategy, it conceives consciousness as an intrinsic structure of cause-effect powers, proposing that any conscious system exists for itself as a maximally unitary whole, irreducible to its parts (Albantakis et al. 2023;Ellia et al. 2021). This is mathematically formalized and computationally analyzed in terms of several measures of "integrated information". In this article, we focus specifically on maximal system integrated information (□□ ! * ).This is used by IIT to identify, among a set of candidate systems, the one(s) that supports consciousness, and hence "exists for itself" subjectively and irreducibly. In contrast, according to IIT's assumptions, systems that do not specify □□ ! * , at best, only "exist" from the perspective of another conscious entity, and hence do not "truly exist" (Albantakis et al. 2023;Koch 2024;Tononi et al. 2022). In this way, IIT's measure of □□ ! * is conceptually tied to both consciousness and an absolute, intrinsic form of existence, thus providing a computational neuroscience framework to quantitatively address questions related to both consciousness and ontology (i.e., about existence)that have been relegated to centuries of endless philosophical debates. At the same time, we acknowledge that, unlike familiar physical frameworks (like classical mechanics or thermodynamics)which can be introduced at progressive levels of mathematical detail and complexity (e.g., from F = ma to more advanced vectorial formulations), IIT's formalism remains comparatively opaque and harder to grasp.Nevertheless, IIT still holds the potential to advance our understanding of questions related to ontology and consciousness through mathematical and computational means. But this potential is hindered by some key theoretical-ontological assumptions (i.e., relative to existence) of IIT, which lead to a problematic conceptual interpretation of its mathematical formalism, computational simulation results, and hypothetical scenarios allowed by the theory make it very problematic (Cea, Negro, and Signorelli 2024b;2023;2024a). This underscores the crucial role that conceptual interpretation plays in scientific theories employing mathematical formalisms. History shows that reinterpreting the samepre-existing formalisms can be crucial for scientific advancement. For example, non-Euclidean geometry, developed by Gauss and later generalized by Riemann into higher-dimensional spaces, was conceptually reinterpreted by Einstein in his general relativity to describe the curvature of spacetime, addressing the limitations of Newtonian gravity, including its assumption of instantaneous information transfer in gravitational fieldsA notable example is non-Euclidean geometry: initially an abstract mathematical theory, it became the foundation of Einstein's general relativity when he reinterpreted it to describe the curvature of spacetime (Renn 2007;Torretti 1996). While our aims are far more modest and we are not comparing our work to Einstein's monumental achievements, reconsidering IIT's mathematical formalism from a new conceptual perspective might help address current limitations and enhance its explanatory power concerning the relationship between brain activity, consciousness, and ultimately, the concept of existence.In the following, we first introduce the main principles of IIT (section 2), focusing on the mathematical formalization of the theory's proposed marker of intrinsic, conscious existence (i.e., maximal system integrated information □□ ! * ), its and IIT's associated conceptual interpretation based on the ontological principles of i) being, ii) true existence, iii) maximal existence and ontological iv)"Great Divide of Being". Next (section 3), we briefly explain why these ontological assumptions are troublesome and motivate revision. commitments, and why they are troublesome. Then (section 4),we propose specific amendments to these ontological assumptions to improve the theory's overall theoretical robustness and thus, its capacity to address issues about consciousness and existence from a computational neuroscience perspective. We end with concluding remarks and propose directions for future research (section 5). Grounded in the purportedly essential properties of experience (i.e., the "phenomenal axioms"), IIT proposes six "postulates of physical existence", which, according to the theory, define the necessary and sufficient conditions for any physical substrate to instantiate consciousness. In line with IIT's Principle of Being (PB), which states that "to be is to have cause-effect power" (Albantakis et al.2023, p. 11), these postulates are framed in terms of cause-effect power that must satisfy: i) existence, ii) intrinsicality, iii) information, iv) integration, v) exclusion, and vi) composition.The theory then mathematically formalizes these causal-physical postulates and applies them to simulated neural networks (i.e., candidate substrates/systems) 1 . As anticipated, Wwe will focus on maximal system integrated information □□ ! * , which is based on system integrated information □□ ! , a quantity that computes which measures the cause-effect power of a system as an irreducible whole (Albantakis et al., 2023;Marshall et al., 2023) foot_1 .Mathematically, □□ ! is computed as the minimum between a substrate's integrated cause information (□□ # ), and integrated effect information (□□ $ ), according to the formalism: et al., 2023, p. 17, Equations 19, 20 and 21) both describe state transitions probabilistically based on current states, but also differ because IIT's TPMs marginalize external influences to focus on the internal causal relationships of a system, and encode interventional, rather than purely observational probabilities. Thus, transitions between conscious states in a substrate are operationally tracked by its TPM and the corresponding unfolded cause-effect structure (=conscious state) at each time step. However, according to IIT's ontological narrative, what truly happens is that subjective conscious states successively cause one another irreducibly, as only consciousness "truly exists" (Tononi et al., 2022). This creates a problematic tension between IIT's operational framework and its ontological claims concerning the causality of consciousness, a point critiqued in other work (Signorelli et al., 2023). Many thanks to reviewer 1 for pressing these and other relevant points addressed below.□□ □□ (□□ $ ,Conceptually, bBoth □□ # and □□ $ quantify the difference that a partition □□ of the system makes, with respect to the probability with which the entire unpartitioned system specifies its past/cause state □□ # ( (for □□ # ), and its future/effect state □□ $ ( (for □□ $ ), given its current state s. The greater the cause/effect probabilities □□ □□ (□□ $ ( |□□) and □□ # ← (□□ # ( |□□) specified by the unpartitioned system, and the greater the difference when partitioned (evaluated by the logarithmic terms), the greater the integrated cause/effect information, and hence, its overall □□ !foot_2 .Given that IIT endorses the principle of maximal existence (PME), which posits that "what exists is what exists the most" (Albantakis et al. 2023, p. 11), the complex (i.e., consciousness substrate) is identified as the subset of interconnected units within a universe □□ , the network with maximal system integrated information (□□ ! * ), as formalized below: et al., 2023, p. 19, Equation 24) where □□ , represents the set of units available at iteration k, and S denotes a candidate subset. The subset achieving the maximum value, □□ ! * , is selected as the complex for that iteration, consistent with the PME. Iteratively, once a complex is identified, its units are removed from the universe □□ , , and the search continues within the remaining units to identify the next, non-overlapping, maximal subset. This process ensures that at the end of the iterative search, and "overlapping substrates with ) and therefore do not qualify as additional substrates of consciousness besides the main complex in our brains, IIT's ontological interpretation implies that our own bodies do not truly exist. At best, they merely exist as objects for some consciousness observing them:j ! * (□□ □□ , □□ □□ , □□ , ) = max -Í. ! □□ ! (□□ □□ , □□ □□ , □□) (Albantakis"since my body is a superset of my true PSC [i.e., neural complex], it is excluded from it-relegated to the realm of entities that only exist relatively, for an observer" (Tononi et al. 2022, 8). 4 In contrast to common computational approaches that prioritize input-output functions as critical for understanding conscious processes, IIT targets the internal causal structure of a system as explanatorily central i.e., how its constitutive mechanisms affect each other and the dynamical evolution of the whole system. Thus, while a complex does have clear boundaries defined by the subset of interacting units that maximize the value of □□ " compared to overlapping subsets, and it can receive/send input/output signals from/to units outside its boundaries, these external interactions are not constitutive of the system's intrinsic cause-effect power and consciousness. Nonetheless, a system's internal causal structure should somehow match the causal structure of the environment in perceptual experience, otherwise both individual and shared perception of the external world among different people would be impossible. This is currently an important limitation of IIT, but efforts to address it are ongoing (Mayner, Juel, and Tononi, 2024). Additionally, IIT provides, in principle, a framework to measure the internal causal structure of systems computationally through interventions and state-transition analyses. However, practical limitations, particularly in applying these methods to biological systems like the human brain, remain significant. We thank Reviewer 1 for pressing these important points.5 From IIT's perspective, only □□ " -maximal systems are conscious entities that truly exist as subjects experiencing their own existence and, potentially, an external world (Tononi et al. 2022;Cea, Negro, and Signorelli 2023). For instance, a □□ " -maximal brain region within a conscious neuroscientist would constitute a genuine subject. In turn, whether a patient observed by the neuroscientist is another intrinsically existing subject or merely an object within the neuroscientist's experience depends on whether the patient also possesses a □□ " -maximal brain region. In previous works, we have examined the problematic implications of these radical assumptions in detail we will t (Cea, Negro, and Signorelli 2024a;2024b;2024c;2023;Signorelli, Cea, and Prentner 2023). Here, we will briefly introduce them and direct the reader to that literature for further details.First, IIT's ontological commitments create an explanatory tension with both common neuroscientific practice and IIT's own declared goal of explaining consciousness in physical terms: why attempt to explain consciousness in neuroscientific terms if it is considered ontologically primitive, while, in contrast, any non-conscious physical entity is deemed mind-dependent? (Signorelli, Cea, and Prentner 2023). Second, IIT's ontology seems to entail that truly existing, conscious systems, can i) be eliminated from existence solely by altering external, non-existent entities; ii) be engineered out of nothing; and iii) phylogenetically originate from nothing (Cea, Negro, and Signorelli 2024b;2024a). Now, IIT could prima facie address these issues by invoking an "ontological dust" (Tononi et al. 2022) ultimately composed of minimally conscious monads (indivisible units) (Hendren et al. 2024).However, there are several problems with the latter (Cea, Negro, and Signorelli 2024a;2024c). to contradict IIT's own formalism (Cea, Negro, and Signorelli 2024c), which requires-to apply the integration postulate-that valid partitions of a system into at least two non-overlapping, non-empty parts, are possible (Albantakis et al. 2023). Otherwise, □□ ! is not computable as such, but has to be replaced in practice by the measure of intrinsic information, which is insufficient for consciousness according to IIT's postulates (Cea, Negro, and Signorelli 2024c).In sum, IIT's Great Divide of Being, along with its principles of true existence and eliminativism regarding non-conscious entities, raises several important issues 6 . To address these concerns and align better with neuroscience practice, we propose minimal theoretical conceptual adjustments for IIT to adopt causal-physical realism, which asserts that non-□□ ! * , non-conscious systems may also truly exist if they have cause-effect power. This conceptual revision allows a reinterpretation of IIT's formalism such that maximal □□ ! (i.e., □□ ! * ) is no longer the exclusionary marker of a truly existent entity, but just the marker of consciousness, while causally powerful non-conscious entities may also be acknowledged to exist. In the following, we propose that for IIT to endorse causal-physical realism and overcome the many problems we briefly sketched in the previous section, the theory should: IIT should: i) reject the principle of true existence (PTE)(and associated Great Divide of Being) (section 4.1), ii) modify the principle of maximal existence (PME) (section 4.2), and iii) endorse a realistic, not merely operational, principle of being (PB) (section 4.3).6 Another important issue is the ontological relationship, in IIT, between a physical substrate and its consciousness. , are they one entity? Different properties of the same entity? Different entities? Although a difficult question, iIn previous work we argued at length that the most plausible interpretation is that a physical substrate ontologically reduces to its intrinsic □□-structure, which in turn ontologically reduces to its inner subjective experience. In other words, what truly exists would be the subjective experience, but it could be observed from an extrinsic point of view as a physical substrate, whose experience can be described-in scientific-theoretical terms-as a □□-structure (Cea, Negro, and Signorelli 2023). Thus, although subjective experience is ontologically primary and not directly observable from the third-person perspective, IIT suggests that it can be scientifically represented by the corresponding Φ-structure, which, in principle, can be computed for any physical system modeled as a causal stochastic network, . In principle, this Φ-structure could be derived computationally from the causal properties of the physical substrate through interventions and state-transition analyses according to equations 57 and 58 in Albantakis et al. 2023 (p. 29)., althoughHowever, this is not yet feasiblepractical for actual biologicalrealistic systems,. due to combinatorial explosions in the calculations, manipulations and observations needed, making it currently practical only for idealized systems with a few units. This is a significant limitation that requires further development (e.g. Zaeemzadeh and Tononi (2024)) to enable rigorous empirical testing of IIT. Without this, IIT remains only 'testable in principle'-a shortcoming for a theory that seeks to scientifically explain consciousness. Nevertheless, practical tools like the Perturbational Complexity Index (PCI), inspired by IIT, have provided some initial empirical validation for the theory by reliably estimating consciousness levels in clinical settings (Massimini et al. 2009;Casarotto et al. 2024). We thank again reviewer 1 for highlighting this issue. According to IIT's principle of true existence (hereafter "PTE", Cea, Negro, and Signorelli 2023), only consciousness (i.e., phenomenal existence) is true existence, as it is "the only existence worthhaving-what we might call true existence" (Tononi et al., 2022, p. 8). While there is no systematic philosophical defense of PTE in the IIT literature, its core intuition seems to be that true, absolute existence is self-evident and immediately known by itself, a condition only conscious, intrinsically existing entities can meet: "consciousness truly exists because it exists for itself -it exists absolutely" (Tononi et al., 2022, p. 9). Therefore, only consciousness would truly exist, as only it exists for itself.We have two worries about this intuition. First, it seems to rest on a necessity-sufficiency equivocation, conflating self-consciousness as a sufficient condition for the truth of one's existence (inspired by the Cartesian ""cogito ergo sum"") with self-consciousness as a necessary condition for existence (as implied by PTE). One could argue that self-consciousness is sufficient to prove one''s existence but reject the stronger claim that self-consciousness is required to exist. The Cartesian intuition supports the idea that ""entities that exist for themselves truly exist"" (a sufficiency claim), but this is compatible with the existence of non-conscious physical entities, not entailing that ""only entities that exist for themselves truly exist"" (a necessity claim).Second, IIT's motivation to adopt PTE may also stem from an epistemic-ontological equivocation.While consciousness may entail knowing that one exists, there is no clear reason why this knowledge is inextricably tied to existence. However, IIT conflates this epistemic fact-self-known existence (""existing for itself"")-with the ontological fact of truly existing (""existing in itself""). We see no reason why something must know that it exists in order to exist cannot exist without knowing it (e.g. why a non-conscious stone cannot exist if it doesn't know its existence?). In short, "existing in itself"(true existence) does not imply "existing for itself" (self-aware existence), and thus the latter is not a necessary condition for the former.In sum, we are happy to grant the Cartesian intuition according to which being conscious about one's existence may suffice for the truth of one's existencefoot_3 (a sufficiency claim) which IIT may safely embrace to assert the intrinsic existence of subjective experience based on its epistemic certainty and immediacy. This epistemic and phenomenological primacy of consciousness could position IIT in dialogue with the enactive approach and its neurophenomenological method, where first-person experience plays a foundational role in the scientific study of the mind (Varela 1996;Varela, Thompson, and Rosch 2016;Signorelli, Cea, and Prentner 2023). However, we see no reason to accept IIT's stronger thesis that in order to exist, an entity must "exist for itself (phenomenally)" (a necessity claim), which underlies the theory's principle of true existence and associated "Great Divide of Being". Therefore, we propose that IIT theorists set aside this necessity claim and its associated theses to open up the conceptual possibility that non-conscious systems might exist in themselves, independently from any consciousness. In other words, without the problematic necessity claim underlying the PTE, the 'Great Divide' between truly existing conscious entities and the merely relative, observer-dependent 'existence' attributed to non-conscious entities disappears. The the theory to endorse a full-blown realism about all non-conscious (non-□□ ! * specifying)-but causally powerful-physical entities, and thus, potentially resolve all the issues we briefly outlined, which arise from non-conscious systems not being regarded as truly existent which stem from non-conscious systems with causal power being excluded from true existence. In the technical terms of IIT, we propose that specifying eitherboth non-maximal □□ ! , and even or just intrinsic information (causeeffect power), may be sufficient for an entity to exist genuinely, even if not consciously. In sum, our analysis suggests that IIT needs to endorse causal-physical realism, and to achieve that, ("PRB"), according to which "to truly exist is to have causal power". This would allow IIT theorists to pursue their current neuroscientific methodology and computational framework to find the physical substrate of consciousness (i.e., complex) and unfold its cause-effect structure, without conceptually entailing the rejection of the mind-independent, genuine existence of elementary particles, molecules, and the non-conscious parts of their own brains and bodies.Future theoretical research should assess the types of entities allowed by IIT's formalism, once the "Great Divide of Being" is overcome. For instance, what is the ontological difference between entities that only specify positive values of intrinsic information but zero system integrated information, compared to entities that do specify positive values of the latter? Presumably, both exist in virtue of having cause-effect power, but only the latter present causal emergence (Hoel, Albantakis, and Tononi 2013;Mediano et al. 2022;Hoel et al. 2016).Additionally, future research could explore IIT's potential to integrate theoretical insights and empirical findings from embodied approaches, which propose that the non-neural body, far from 'existing' solely from the perspective of the conscious brain, is structurally and dynamically intertwined with it (Thompson and Cosmelli 2012; Thompson and Varela 2001) and fundamental to understanding both the origins of our mathematical capacities (Lakoff and Nunez 2000) and the very feeling of existence at the root of all consciousness. (Thompson and Cosmelli 2012; A. Seth 2021; 2024; Damasio and Damasio 2023; 2024; Thompson and Varela 2001; Cea and Martínez-Pernía 2023; Ratcliffe 2020).