Glossary of IIT Terms

The 0th axiom of IIT is that experience exists. This insight is immediate and irrefutable, and the starting point of IIT, establishing the primacy of experience in both an epistemological and ontological sense. Building from this ground truth, IIT then characterizes the essential properties of every experience (the five axioms of IIT) and the accidental properties of specific experiences.  

For more, see

• Foundations: 0th Axiom

• FAQ: What is meant by the term axiom in IIT?

• Paper: Consciousness and the Fallacy of Misplaced Objectivity (an academic paper on the “phenomenology-first” approach of IIT) 

The 0th postulate of IIT—also called the existence postulate—is that the substrate of consciousness can be characterized operationally by cause–effect power: its units must take and make a difference. 

The 0th postulate is the physical counterpart of IIT’s 0th axiom. In other words, it is IIT’s starting point in its attempt to account for experience in physical terms: it guides us to analyze the substrate of consciousness purely in terms of cause–effect power. The 0th postulate is supported by IIT’s three methodological assumptions of realism, physicalism, and atomism. 

For more, see

• Foundations: 0th Postulate

• FAQ: What is the meaning of cause–effect power in light of other notions of “causal power”?

The accidental properties of an experience are those that are true of some experiences, but—contrary to the essential properties of experience—they are not irrefutably true of every conceivable experience (hence they’re accidental and not essential). For instance, even if most of the time we hear sounds, auditory contents are accidental because there can clearly be experiences with no sound (e.g., when we are in a perfectly silent room). The research program of IIT aims to account for accidental properties of experience including space, time, objects, and local qualities such as color, pain, etc., whose underlying neural mechanisms may correspond to content-specific NCCs. 

Though accidental, these properties must nonetheless be accounted for by a theory of consciousness. 

The actual causation framework was developed following the postulates and principles of IIT. However, rather than describing the intrinsic causal powers of a substrate in a state (potential causes and effects), it was designed to uncover extrinsic, actual causes and effects of a substrate in a state. That is, given that a physical substrate (constituted of a set of operationally defined units at a particular grain) has dynamically transitioned through an actual series of states, we can use the substrate’s TPM and the actual causation framework to uncover the actual causes and effects of any occurrence (a set of units in a state) within the series of observed states.

For example, the figure below depicts a substrate observed to have transitioned through the series of states (t0: Abc, t1: ABc, t2: abC, t3: aBc). Using actual causation, we can get quantitative answers to questions such as “What was the actual cause in t1 of the occurrence bC at t2?” Was it ABc as a whole, or perhaps only A alone, or Ac together? Likewise, the framework could help us answer “What was the actual effect of unit A (ON) at t0?” or “What was the chain of occurrences that actually caused c (OFF) at t3?” or “What is the structure of actual causes and effects that bind t2 to t1 and t3, respectively?”

For details on the framework, see Albantakis, L., Marshall, W., Hoel, E., & Tononi, G. (2019). What caused what? A quantitative account of actual causation using dynamical causal networks. Entropy, 21(5), 459.

For applications of actual causation, see, for example,

• Mayner, W., Juel, BE., Tononi, G. (Forthcoming). Meaning, perception, and matching: Quantifying how the structure of experience matches the environment.

• Juel, B. E., Comolatti, R., Tononi, G., & Albantakis, L. (2019, July). When is an action caused from within? Quantifying the causal chain leading to actions in simulated agents. In Artificial Life Conference Proceedings (pp. 477-484).

• Tononi, G., Albantakis, L., Boly, M., Cirelli, C., & Koch, C. (2022). Only what exists can cause: An intrinsic view of free will. arXiv preprint arXiv:2206.02069.

Atomism states that to provide an explanation that is as complete as possible, we ought to assess cause–effect power down to the smallest constituents (“atoms”) that we can observe and manipulate. Together with realism and physicalism, atomism is one of three methodological assumptions of IIT. Note that atomism in IIT is an operational assumption, not a reductionist ontological position.

For more, see 

• Foundations of IIT

• FAQ: If IIT assumes "atomism," does this make it a reductionist theory of consciousness?

The axioms of IIT capture the essential properties of experience (i.e., of consciousness). They are not logical “propositions” but rather phenomenological properties that are immediate and irrefutably true of every conceivable experience.

Starting from the fact that experience exists (0th axiom), the axioms express five properties essential to every experience: intrinsicality, information, integration, exclusion, and composition. IIT aims to account for each of these properties by formulating them as a corresponding postulate—that is, an essential property of the substrate of consciousness. 

For more, see 

• Five Axioms: Essential Properties of Every Experience

• FAQ: What is meant by the term axiom in IIT? 

The term background conditions refers to the current state of the units that are outside the candidate substrate of consciousness. Operationally, these units are made causally inert through causal marginalization.

Background conditions can be thought of as the “context” in which the substrate is situated. In the case of the substrate of consciousness in the brain, likely background conditions include the particular state of subcortical activation, the provision of oxygenated blood, and the state of sensory relay cells in the thalamus.

For more, see 

• Intrinsicality Postulate

• FAQ: Does intrinsicality mean that the substrate of consciousness is not influenced by its environment?

• FAQ: What would be considered background conditions in the human brain?

• Koch et al. 2016 (where the term was first used in NCC research) 

The term candidate is used to describe a unit, substrate, distinction, etc. until it is confirmed that it satisfies the relevant postulates. 

For instance, in the operations to identify the main complex, an early step is to define our substrate as a set of units and apply the intrinsicality postulate (by treating the remaining units as background conditions). The substrate we define in this step is a mere candidate because it might turn out that it is reducible (if it doesn’t satisfy the integration postulate). 

This was an operation (and term) used in earlier IIT publications. This operation is now covered by conditional causal marginalization.

Marginalization is a standard mathematical operation from probability theory: averaging over the possible values of one set of variables to determine the marginal contribution of another. In IIT, we apply causal marginalization, which just means that it is applied to probabilities obtained from causal interventions (see perturbational approach & TPM). That is, when assessing the causal powers of a substrate, a certain set of unit states is averaged over (marginalized out) to leave only the contribution of the remaining unit states. This method is used to “screen off” some units (those that are averaged out) in order to isolate the causal powers among the remaining units.

In IIT, marginalization is applied with either conditional or uniform distributions, depending on what we aim to quantify:

unconditional causal marginalization 

To evaluate candidate distinctions (see Computing Φ: Step 6.1—Distinctions), we use the standard (unconditional) marginalization. Here, we marginalize out all units in the complex that are not part of the candidate mechanism or purview, assuming a uniform distribution over their states. This ensures that any remaining causal power should indeed be by the mechanism and over the purview. Furthermore, using the uniform distribution ensures that we make no assumptions about the distribution of any relevant unit, as is required when taking the intrinsic perspective.

conditional causal marginalization ("pinning")

We also use causal marginalization when applying the intrinsicality postulate (see Computing Φ: Step 2—intrinsicality), where we obtain the (cause and effect) TPMs of a candidate complex and remove any causal influence of units outside the candidate. In this case, however, the marginalization is done using a conditional distribution of substrate states obtained by conditioning on the current state of the units outside of the complex (see background conditions). Since the current state of the background units is informative of their previous state, when marginalizing out their contribution, it is important to use a distribution over their previous states conditioned on their current state. Note that conditional causal marginalization is sometimes called “pinning” for short.

For more, see:

• Computing Φ: Step 2—intrinsicality

• Computing Φ: Step 6.1—Distinctions

• IIT 4.0 (for precise mathematical definitions) 

The term causal power is generally used as a synonym for cause–effect power in IIT. However, it may occasionally appear in a more general, less technical sense—as simply the capacity to affect something or be affected by something. 

A cause–effect is another word for a distinction: a cause and an effect (i.e., a cause–effect) that are linked through a mechanism. 

Cause–effect power is the mark of the physical: the ability to “take and make a difference.” This notion is the starting point in characterizing the substrate of consciousness (0th postulate).

Ultimately, for a candidate substrate to exist as a substrate of consciousness (i.e., intrinsic entity), its cause–effect power must satisfy all five postulates—that is, it must be intrinsic, specific, unitary, definite, and structured.

For more, see:

• Foundations: Phenomenal & Physical Existence

• FAQ: What is the meaning of cause–effect power in light of other notions of “causal power”?

The cause–effect state comprises the cause state and effect state selected by the system in its current state. Following the information postulate, the cause and effect states selected are those that maximize intrinsic information. 

The cause–effect state of the system constrains which distinctions can exist in the cause–effect structure it specifies. That is, only distinctions whose purviews are congruent with the cause–effect state of the system can exist.

For more, see

• Information postulate

• Unfolding – Step 3: Apply information

See Φ-structure.

A complex (or maximal substrate) is a set of units that satisfies the postulates. In other words, it is a substrate that lays a stronger claim to existence than any candidate substrate it overlaps with. In IIT’s ontology, a complex is a substrate of consciousness, which—fully unfolded as a Φ-structure—exists as an intrinsic entity. 

A given substrate of units may “condense” into various complexes. To identify their borders, we apply the postulates systematically (see Computing Φ): we first identify the major complex (or “main complex,” “first complex”)—the candidate with the highest φs—then among the remaining units, we identify the minor complexes (“second,” “third,” etc.). In the human brain, the major complex is presumed to be the neural substrate of consciousness. 

Composition is the 5th axiom and postulate of IIT. As an axiom, it states that “experience is structured: it is composed of distinctions and the relations that bind them together, yielding a phenomenal structure that feels the way it feels.”

Formulated as a postulate, it states that “the substrate of consciousness must have structured cause–effect power: subsets of its units must specify cause–effect states over subsets of units (distinctions) that can overlap with one another (relations), yielding a cause–effect structure (or Φ-structure) that is the way it is.”

For more, see 

• Composition page

• FAQ: Aren’t there some experiences that have no structure?

• FAQ: What counts as a "component" of experience?

• FAQ: Why do we assess the causal power of all orders of mechanisms? Why not simply assess the causal power of the individual units alone?

A compound distinction is an introspectable component of a phenomenal structure, which is itself composed of elementary distinctions, which are not always introspectable. In physical terms, a compound (causal) distinction is the set of all distinctions picked out by a set of units.

A compound relation is an introspectable component of a phenomenal structure, which is itself composed of elementary relations, which are not always introspectable. In physical terms, a compound (causal) relation is the set of all relations between distinctions specified by a set of units.

This term was used in earlier IIT publications for what is now called a distinction. 

This term was used in earlier IIT publications for what is now called a Φ-structure (or cause–effect structure).

The term congruent is used to indicate whether two states are in agreement. It is especially important in two steps of unfolding:

• When assessing the existence of a candidate distinction, we make sure that its cause and effect purview states are congruent with the cause–effect state of the substrate as a whole. For example, if the substrate specifies cause state aBc (OFF, ON, OFF), and a candidate distinction has cause purview B (ON), these are congruent.

• When assessing relations, we determine two (or more) distinctions to be related only if their purviews overlap congruently. For example, if two distinctions share purview units (e.g., unit A), the state of that shared purview unit must be the same (e.g., both specify A, “ON”). 

In IIT, we often use the terms consciousness and experience interchangeably, in the sense that being conscious is synonymous with having an experience. Yet there are slight nuances. Consciousness refers to having experiences regardless of their content, while experience is always of something specific. Hence we describe a patient as “losing consciousness” but not as “losing experience,” and we might say “my experience changed” (from one content to another) but not “my consciousness changed.”

For more, see

• FAQ: What do you mean by consciousness?

• Overview: What does it mean to explain consciousness scientifically?

When we assess the irreducibility of a system or mechanism, we apply a partition, which comprises a set of specific cuts among parts. The way we apply cuts varies for systems or mechanisms. But in general, to cut means to eliminate the causal constraint of one set of units upon another. For example, for system ABCD, cutting units A and B from C and D can be achieved by eliminating the causal constraints that part AB has over CD, or those that CD has over AB, or both. 

For details, see 

• FAQ: What is a valid partition?

Degree is used to define the number of elements that are bound (i.e., overlapping) in a relation or relation face. The degree of a relation reflects the number of overlapping distinctions. For example, a relation that binds three distinctions is a 3rd-degree relation (or a 3-relation). The degree of a relation face reflects the number of overlapping purviews. A relation face that binds together all causes and effects of the distinctions making up that 3-relation—six purviews in total—would be a 6th-degree relation face. 

For more, see: Composition: Relations.

Differentiation is a consequence of the information axiom and postulate:

• On the axiom side: by virtue of being specific, each experience also differs from a large repertoire of other possible experiences. 

• On the postulate side: by virtue of being in its specific current state—and thus specifying a specific cause–effect state—the substrate of consciousness may also be in different states, which also specify different cause-effect states among a repertoire of possible ones.

A related notion is neurophysiological differentiation, which refers to the number of different states available to a substrate. This notion has been used to devise practically applicable measures of consciousness—such as the perturbational complexity index. 

Note that in IIT literature, the term differentiation has sometimes been used to help understand the axiom/postulate of information—and sometimes treated as a synonym. Most precisely, however, differentiation is not itself an essential property of experience (and thus of its substrate); it is rather a consequence of the essential property.

For more, see:

• Information postulate

• Marshall, W., Gomez-Ramirez, J., & Tononi, G. (2016). Integrated information and state differentiation. Frontiers in Psychology, 7, 926. 

A distinction is a “difference that makes a difference.” Phenomenally, a distinction is any aspect of our experience that we can “pick out” through introspection—for example, my hand, a musical tone, or the left half of visual space [1]. Physically, a (causal) distinction is the cause–effect power exerted by one subset of units over other subsets within the complex. It thus comprises a mechanism, a cause purview, and an effect purview. Like the system as a whole, distinctions must adhere to the postulates (save composition)—that is, they must be intrinsic, specific, unitary, and definite.

Distinctions, and the relations that bind them, make up the fundamental components of structures in IIT—both phenomenal structures and Φ-structures. 

For more, see Composition: Distinctions. 

______

See principle of being. 

In phenomenal terms, elementary distinctions are assumed to be the most basic components of a phenomenal structure, and as such are not introspectable. For example, if we use introspection to pick out the smallest patch of visual space we can discern—a “dot”—this patch is still not an elementary distinction because it is necessarily extended in space, which means it must be structured by lower-level components. Despite the limits of introspection, IIT assumes that any distinction we can introspect must comprise lower-level distinctions, down to an elementary level (see the assumption of methodological “atomism”).

In physical terms, an elementary (causal) distinction is simply a distinction. 

Elementary relations are assumed to be the building blocks of the introspectable components of a phenomenal structure, and as such are not always introspectable. In physical terms, an elementary (causal) relation is just a relation.

In the ontology of IIT, intrinsic entities are systems that exist for themselves—those for whom it “feels like something to be.” These systems are characterized by cause–effect power that is intrinsic, specific, unitary, definite, and structured (i.e., they are entities that fulfill the five postulates). An example of an intrinsic entity is the main complex in the brain (unfolded as a Φ-structure).

In everyday language, we refer to “entities” in a looser sense—for example, a rock or a cloud. In IIT, these are extrinsic entities because they do not exist for themselves (i.e., they do not fulfill the five postulates), yet they still may exist phenomenally—as concepts in the mind (i.e., as Φ-folds).

Some extrinsic entities—such as a rock—not only often exist phenomenally but also physically; this means that by manipulating and observing such systems, we may find that they indeed “hang together” as relative maxima of cause–effect power but not absolute maxima (i.e., they are not complexes; see Exclusion Postulate). In contrast, other extrinsic entities—such as a cloud—may be more difficult to define operationally: if we manipulate and observe them, they might turn out to not even be relative maxima of cause–effect power, though they may still feel to “hang together” as concepts in the mind.

Exclusion is the 4th axiom and postulate of IIT. As an axiom, it states that “experience is definite: it is this whole.” Formulated as a postulate, it states that “The substrate of consciousness must have definite cause–effect power: it must specify its cause–effect state as this whole set of units.”

For more, see Exclusion. 

Existence refers to the “0th” axiom and postulate of IIT. As an axiom, it states that “experience exists.” As a postulate, it states that “the substrate of consciousness can be characterized operationally by cause–effect power: its units must take and make a difference.” These form the starting point of the IIT method (see Foundations).

The term existence also appears in the ontology of IIT. Intrinsic existence characterizes entities that exist for themselves (i.e. intrinsic entities). In contrast, extrinsic existence characterizes entities that don’t exist for themselves, but may exist for us as experiencing subjects (i.e., extrinsic entities). For more, see entity.

Experience refers to “what it is like to be” [1]. This includes what it feels like to be you—or any human—in any conscious moment, whether in regular waking states, dream states, or altered states of consciousness. But it also refers to the subjective state of any intrinsic entity—whether non-human animals or potentially artificial systems. In IIT, experience is often used synonymously with consciousness; but while consciousness refers to having experiences regardless of their content, experience is always of something specific.

In IIT, when we do a phenomenological analysis of essential and accidental properties of experience, we aim to dissect a single “moment” of experience—”right here, right now.” In other words, we do not use the term experience to describe processes or events that are more than a “timeslice” of experience.

For more, see

• Overview: What does it mean to explain consciousness scientifically?

• FAQ: What do you mean by consciousness?

IIT’s explanatory identity (or fundamental identity) states that every property of an experience is accounted for in full by the physical properties of the Φ-structure unfolded from a complex, with no additional or “ad hoc” ingredients. This means there must be a one-to-one correspondence between the way the experience feels and the way distinctions and relations of the corresponding Φ-structure are structured. 

Importantly, the identity is not meant as a correspondence between the properties of two separate “things,” nor is it meant in a reductive or “eliminitivist” sense (e.g., experience is “nothing but” a Φ-structure). The identity should rather be understood in an explanatory sense: Experience remains primary in IIT, both epistemologically and ontologically; nevertheless, the properties of experience can be accounted for in operational terms of cause–effect power.

For more, see

• The Fundamental Identity 

• FAQ: How is the fundamental identity of IIT different from classical identity theories?

• FAQ: How does the fundamental identity of IIT compare to other standard approaches to explaining consciousness?

• FAQ: If IIT assumes atomism, does this make it a reductionist theory of consciousness? 

Extrinsic is a general term used in IIT in opposition to the notion of intrinsicality and the intrinsic perspective. 

When referring to cause–effect power, extrinsic means cause–effect power that is not from the intrinsic perspective of the system (or mechanism)—its “raw power” (measured by informativeness). Cause–effect power can also be extrinsic in the sense that one entity has causal power over another (rather than over itself). 

When referring to an entity, extrinsic means a system does not exist for itself, i.e. it is not a complex. Extrinsic entities can still exist (extrinsically) as local maxima of integrated information (e.g., our body or a rock), or as concepts in the mind of an intrinsic entity or conscious being (e.g., a chess set or a sports team).

When referring to a perspective, extrinsic means that we’re describing or analyzing something based on what we can observe and manipulate (e.g., a scientist’s perspective on neurons). 

For more, see entity. 

The fault lines of a system refer to “weak links” among its subsets. It is a metaphorical way to refer to areas of a system that are more weakly connected, which is where we expect to find the minimum partition.

The term grain refers to the scale at which a substrate is analyzed and operations are performed on it:

• The constituent grain defines the “spatial scale” [1] of the units—whether molecules, neurons, minicolumns, etc.

• The update grain defines the “temporal scale” [1] of the units’ state—whether microseconds, milliseconds, seconds, etc.

Grains can be further described as micro, macro, or meso:

• The micro grain is that at which the units and their updates are operationally indivisible. Micro units cannot be partitioned into finer constituents, and micro updates cannot be partitioned into finer updates. Micro units are essentially “atoms” of cause–effect power.

• A macro grain is any grain coarser than the micro grain. A macro unit consists of multiple micro units, and a macro update consists of multiple micro updates. 

• Because macro grains can be built hierarchically (macro units at a given grain can be constituted of macro units at a finer grain), it is often useful to refer to these intermediate grains as meso grains.

In IIT, the grain that supports consciousness is determined following the exclusion postulate: among all candidate grains, the one that maximizes system integrated information (φs) is the one at which the complex exists. We call the units at this grain the complex’s intrinsic units. 

For the mapping of micro states into binary macro states, and the justification for why states are binary, see Marshall et al. (2024).

For more, see

• Marshall, W., Findlay, G., Albantakis, L., & Tononi, G. (2024). From micro to macro units: a mathematical framework for identifying the causal grain of a system from its intrinsic perspective. bioRxiv, 2024-04. 

• Glossary: unit
  
  

See explanatory identity.

In IIT, the term immediate is used to describe the way we as experiencing subjects relate to the properties of experience (both essential and accidental properties) and recognize the truth of these properties. The properties of experience are immediate because they are given in the experience itself [1]; they are not inferred or “mediated” by anything. Similarly, we know the truth of these properties immediately in the sense that we need not (and can not) appeal to any justification outside the experience itself.

For more, see

• 0th Axiom: Experience exists

• FAQ: If the axioms are “immediate” and “irrefutably true,” shouldn’t they also be self-evident?

In the IIT method, the principle of inference from a good explanation works hand-in-hand with inference to a good explanation. 

The IIT method starts by using introspection to identify properties of experience, and then uses inference to a good explanation to account for these properties in physical terms. This method allows us to bootstrap the fundamental identity of the theory, which offers a good overall explanation of consciousness. 

Though necessary, this “phenomenology-first” method has its limits. Some properties of experience are seemingly impossible to dissect through introspection—for instance, the experience of “redness” simply feels monolithically “red.” Yet these properties, too, must be accounted for if we are to have a complete explanation of consciousness and its contents. In such cases, we may use inference from a good explanation to probe the identity in the reverse direction—to use what we know about the neural substrate of such experiences (e.g., of color) to consider what kinds of Φ-structures this substrate would most likely unfold into, and then to check the results against our own phenomenology. Reasoning in both directions in this way is what allows us to firmly establish the identity as an overall good explanation of consciousness.

Inference from a good explanation is also what lets us reason about consciousness beyond humans. The more IIT’s fundamental identity is validated scientifically in humans, the stronger our inferences become when reasoning about the quantity and quality of consciousness in, for example, newborns, animals, and machines. 

IIT uses inference to a good explanation to refer to various inferential steps made in the theory—especially the theory’s methodological assumptions and the way we formulate the postulates based on the axioms. The term is related to the more common “inference to the best explanation” (a type of abductive reasoning), which requires multiple options of explanation in order to judge what is “best.” In IIT, we emphasize good rather than best in an attempt to be more humble—to recognize that, when it comes to consciousness, there’s no “objective” standpoint from which to determine the options and judge which is “best.” With consciousness, we must bootstrap our explanation from the inside out. IIT aims for a “good” explanation, understood in terms of the seven S’s presented here. 

Information is the 2nd axiom and postulate of IIT. As an axiom, it states that “experience is specific: it is this one.” Formulated as a postulate, it states that “the substrate of consciousness must have specific cause–effect power: it must be in this state and select this cause–effect state.”

For more, see

• Information page

• FAQ: Can’t I have a vague or generic experience, which is not “specific”?

• FAQ: How is information in IIT different from Shannon information?

• FAQ: Is my experience specific owing to the potential experiences I could be having (differentiation)?

A factor in the formula for intrinsic information. See intrinsic information. 

Integration is the 3rd axiom and postulate of IIT. As an axiom, it states that “experience is unitary: it is a whole, irreducible to separate experiences.” Formulated as a postulate, it states that “the substrate of consciousness must have unitary cause–effect power: it must specify its cause–effect state as a whole set of units, irreducible to separate subsets of units.”

For more, see

• Integration page

• FAQ: If I have two unrelated contents of experience, doesn't this mean my experience is not integrated?

Integrated information is a measure of irreducibility. Depending on what it is applied to, its exact definition and operational implementation varies. Broadly speaking, integrated information either refers to irreducible cause–effect power determined through partitioning (denoted by φ, “small phi”), or it summarizes the overall existence of an entity as the sum of the irreducibility of its components (denoted by Φ, “big phi”). Below are various ways that integrated information appears in IIT.

φs – “system phi”

Quantifies the irreducibility of the cause–effect state specified by a system: it measures to what extent a system as a whole specifies its cause and effect in a unitary way, which is not reducible to power specified by separate parts of the system. For more, see the Integration Postulate and FAQ: How do we calculate φs? 

φd – “distinction phi” 

Quantifies the irreducibility of the cause–effect power a mechanism specifies over its purviews: it measures to what extent a mechanism specifies its cause and effect within a system in a unitary way, which is not reducible to power specified by separate parts of the mechanism over separate parts of the purviews. For more, see Composition postulate: Distinctions.

φr – “relation phi” 

Quantifies the irreducibility of the cause–effect power specified by a set of distinctions by virtue of congruently overlapping on their cause or effect. That is, it measures to what extent a set of distinctions jointly specify their causes and effects in a unitary way, which is not reducible to power specified by separate subsets of the distinctions. 

For more, see 

• Composition Postulate: Relations

• Computing Φ: Step 6.3: Relations (with simplified formulas)

• IIT 4.0 section (with complete formulas)

φu – “unit phi”

Quantifies the irreducibility of the units constituting the substrate. φu is computed by treating units as candidate systems and computing their φs. However, as they are constituents of a system, units only have to be maximally irreducible “within” (among all their subsets)—unlike systems, which also have to be maximally irreducible “without” (among their supersets). For more, see Exclusion Postulate.

Φ – “big phi”

Quantifies the irreducibility of all distinctions and relations in the Φ-structure of an intrinsic entity; it is the sum of all φd and φr values. Φ is also referred to as structure integrated information and can be interpreted as the entity’s quantity of experience and thus of its intrinsic existence. For more, see Composition Postulate: Φ-structures and Φ. 

ΦR – “Φ-fold phi”

Quantifies the value of interrelatedness of a Φ-fold (substructure of a Φ-structure, which corresponds to a content of experience). As a local maximum of integrated information, ΦR is computed by summing the φ values of the distinctions and relations across a minimum partition.

Intrinsicality is the 1st axiom and postulate of IIT. As an axiom, it states that “experience is intrinsic: it exists for itself.” Formulated as a postulate, it states that “the substrate of consciousness must have intrinsic cause–effect power: it must take and make a difference within itself.”

For more, see 

• Intrinsicality page

• FAQ: Does the intrinsicality axiom imply a self, self-awareness, or self-report?

• FAQ: How does intrinsic in IIT relate to the notion as used in philosophy (e.g., Lewis)?

• FAQ: Does intrinsicality postulate mean that the substrate of consciousness is not influenced by its environment? 

Intrinsic difference is a measure of information that reflects the intrinsic perspective of a set of units, rather than the extrinsic perspective of, for example, a channel designer. It is the difference measure used to calculate intrinsic information in IIT. The measure was devised to satisfy IIT’s postulates of existence (causality), intrinsicality, and information (specificity). For more, see Barbosa et al. 2020.

See entity.

Intrinsic information quantifies the specific, intrinsic cause–effect power of a system as a whole or of a mechanism within a system. As such, it operationalizes the postulates of existence, intrinsicality, and information in precise mathematical terms. 

Intrinsic information is quantified using the intrinsic difference formula, which is made up of two factors, informativeness and selectivity. Conceptually, informativeness can be thought of as quantifying causal “raw power,” and selectivity as quantifying “control”: from the outside, a system may appear to have power, but from its intrinsic perspective, this power is weak if it is spread thinly across the whole system (diluted). Rather, as the product of selectivity and informativeness, intrinsic information captures the interplay between “power” and “control.” In ideal systems, these two may grow together. However, in most systems these factors introduce a tension between “expansion” and “dilution”: the more units a system has, the more “power” it may have, yet more units also introduces greater potential for degeneracy or overdetermination (less “control”), which “dilutes” this power.

Here is a simple version of the formula (see IIT 4.0 for a more technical look):

Informativeness

On the effect side, informativeness measures the “raw power” of the current state over a specific effect state, and on the cause side, it measures the power of a specific cause state over the current state. It is formalized as the log ratio of two probabilities: Pconstrained is the probability of a specific state transition (i.e., the number we read off of the relevant square of the cause or effect TPM [see this FAQ]), and Punconstrained is the probability of the same cause or effect states independent of the current state (i.e., based on chance).

Selectivity

Selectivity is quantified as the pointwise probability of an occurrence (e.g., the probability of finding the outputs of a system or mechanism in a given state) relative to the maximal possible probability of observing that state (necessarily 1). (Note that P′constrained is the same as Pconstrained on the effect side, but on the cause side, it is the “backwards” probability found by applying Bayes’ rule to the “forwards” probabilities in the TPM.)

For more, see:

• • Information Postulate

  • FAQ: How do we calculate intrinsic information?

  • FAQ: When we calculate intrinsic information, what operations do we perform on the TPM?

  • FAQ: How is information in IIT different from "Shannon information"?

  • Academic papers:

Barbosa, L. S., Marshall, W., Albantakis, L., & Tononi, G. (2021). Mechanism integrated information. Entropy, 23(3), 362.

Barbosa, L. S., Marshall, W., Streipert, S., Albantakis, L., & Tononi, G. (2020). A measure for intrinsic information. Scientific Reports, 10(1), 18803.

Marshall, W., Grasso, M., Mayner, W. G., Zaeemzadeh, A., Barbosa, L. S., ... & Tononi, G. (2023). System Integrated Information. Entropy, 25(2), 334.

Albantakis, L., ... & Tononi, G. (2023). Integrated information theory (IIT) 4.0: formulating the properties of phenomenal existence in physical terms. PLoS Computational Biology, 19(10), e1011465. 

When accounting for experience in physical terms, physical existence should be evaluated from the intrinsic perspective of an entity—from the perspective of what exists for the entity itself, not that of an external observer. This perspective is embedded in the intrinsicality postulate, and it has several consequences when accounting for experience in physical terms: 

1. We must account for the quantity and quality of a system’s cause–effect power from its own perspective, in a way that is observer-independent and non-arbitrary. 

2. We treat units outside of the system as background conditions in that they do not contribute directly to what the system is. 

3. We evaluate the cause–effect power of a system by considering as input states the uniform probability distribution (of possible states)—the one that assumes no extrinsic knowledge—rather than one that is observed or imposed extrinsically.

4. We evaluate the cause–effect power of a system by considering not only its “raw power” (informativeness) but also its degree of “control” (selectivity) from its own perspective, as captured by the intrinsic information formula.
   
   

For more, see

• Intrinsicality Axiom & Postulate

• FAQ: How does intrinsic in IIT relate to the notion as used in philosophy (e.g. Lewis)? 

To introspect means to direct one’s attention to the content of one’s own experience. Together with reasoning, introspection is our principal tool to investigate the essential and accidental properties of experience. In the IIT method, introspection is the necessary first step because the properties of experience make up our explanandum—what we aim to explain in a theory of consciousness. 

That said, introspection has its limits. For example, we may be able to introspect the experience of spatial extendedness reasonably well, but we may struggle to decompose the experience of redness—it just seems monolithically “red.” For this reason, though introspection is necessary for initial bootstrapping, a mature theory can then take us beyond the limitations of introspection, letting us apply inference from a good explanation to fill in gaps in accounting for properties of experience that we struggle to introspect. 

For more, see

• FAQ: If IIT starts from experience, isn't this method "subjective" and thus unscientific?

• FAQ: If introspection requires reflection, how can we be sure that non-reflective experiences exist at all?

• Ellia, F., Hendren, J., Grasso, M., Kozma, C., Mindt, G., P. Lang, J., ... & Tononi, G. (2021). Consciousness and the fallacy of misplaced objectivity. Neuroscience of Consciousness, 2021(2), niab032. 

In IIT, a candidate system, distinction, or relation is irreducible if the cause–effect power it specifies cannot be reduced to that specified by its parts separately. For example, if we partition system AB into two causally isolated systems (A and B), and find that its cause–effect power is unaffected, then we say that AB is reducible. To the contrary, if even the least-disruptive partition (MIP) makes a difference to AB’s cause–effect power, then AB is irreducible—its cause–effect power cannot be reduced to that of A and B.

Irreducibility is measured by integrated information (φ – “small phi”).

For more, see 

• Integration Postulate

• Integrated information (glossary)

• FAQ: How do we calculate φs? 

In IIT, irrefutable is used to describe the essential properties of experience expressed by the axioms. These properties are irrefutable because if we imagine an experience lacking one of them, we end up confirming that the experience has that property. For instance, every experience is unitary—a whole, irreducible to separate experiences (integration axiom). If we try to refute this by imagining an experience that were not unitary, we end up imagining two or more experiences, each of which is indeed a whole, irreducible to separate experiences.

IIT uses the term irrefutable much the way Descartes used “indubitable,” yet note that irrefutable is more precise since any property can indeed be doubted but perhaps not refuted.

For more, see

• • Five Axioms: Essential properties of every experience

  • FAQ: What is meant by the term axiom in IIT?

  • FAQ: If the axioms are “immediate” and “irrefutably true,” shouldn’t they also be self-evident?

Coming soon. 

Coming soon.

For now, see: Mayner, W., Juel, B.E., and Tononi, G. (2023) Meaning, perceiving, and matching: quantifying how the structure of experience matches the structure of the world. (2024). arXiv, 2024-12 

In IIT, meaning corresponds to the content of experience, which can be expressed in the maxim, ”the meaning is the feeling.” As such, a meaning is always fully intrinsic: if I see an apple on the table, hallucinate the apple, or dream of the apple, the meaning “apple” is always for me as the experiencing subject. That said, a meaning often matches the environment to some extent—that is, it often refers to external causal processes that are relevant to an organism. In physical terms, meanings correspond to Φ-folds, typically composed of sets of highly interrelated distinctions.

IIT’s view of meaning as intrinsic contrasts with views that place meaning elsewhere—for example, in the world (externalism), in the relation between the subject and the world (intentionalism or representationalism), in the functional role played by neural mechanisms (functionalism), in the information passed from the world to the subject (information processing views), or in the capacity to act on the world (enactivist views).

A mechanism is a subset of units in a complex that has maximally irreducible cause–effect power within the complex. Any set of units from the system’s powerset can be a candidate mechanism (e.g., 1st-order mechanism “A” or 3rd-order mechanism “ABC”). But among candidates, mechanisms are only those whose cause–effect power fulfills the postulates (save composition). Moreover, a mechanism must specify a cause and an effect that are congruent with the system’s cause–effect state.

Operationally, a candidate mechanism is any subset of units in a state. Ontologically, however, what exists is not the mechanism per se but rather the causal distinction the mechanism specifies. A mechanism can be understood as an “link” between a potential cause and effect—the three of which together form a distinction.

For more, see

• Composition Postulate: Distinctions

• FAQ: Why do we assess the causal power of all orders of mechanisms? Why not simply assess the causal power of the individual units alone?

• FAQ: 5. Do mechanisms emerge from their units?

The minimum partition (MIP) is used to assess the irreducibility of a system (φs), a distinction (φd), or a relation (φr). It is the partition that makes the least difference to the intrinsic information specified by a system about itself (system MIP), by a mechanism about its purviews (distinction MIP), or by overlapping distinction purviews (relation MIP). 

Note that in earlier publications, the minimum partition was also referred to as the minimum information partition.

For more, see

• Integration Postulate

• Principle of minimal existence (for why the minimum partition is used) 

In IIT, a monad is the smallest system that can exist intrinsically. A monad is constituted of a single unit with a self-loop; it specifies a single distinction, with cause and effect over itself, and a self-relation. According to IIT, if every connection among the atomic units of the universal substrate were severed, we would be left with a collection of monads—each a universe in itself, having the simplest possible experience and isolated from the rest.

The term network refers to the full set of units that are under analysis. Any subset of the network can be considered a candidate substrate (or system).

An early definition of the neural correlates of consciousness (NCC) was “the minimal set of neuronal mechanisms jointly sufficient for a specific conscious percept” (Koch 2004, p. 16). More recently, some have subdivided the term (Koch et al 2016):

• The full NCC is “the neural substrate supporting experience in general, irrespective of its specific content.”

• A content-specific NCC, on the other hand, is “the neural substrate supporting a particular content of experience (for example, faces) whether seen, dreamt or imagined.” 

These concepts have aimed to offer a theory-neutral framework for empirical studies.

The term NCC is rarely used in IIT since the theory explicitly aims to characterize the substrate of consciousness following the postulates; hence IIT rather uses the term neural substrate of consciousness (NSC) when validating the theory empirically. That said, NCC research is helpful to inform IIT’s account—especially to determine where the borders of complexes are likely to be found in the brain (full NCC) and which parts of the complexes are likely to contribute particular contents (content-specific NCC). 

The neural substrate of consciousness (NSC) corresponds to the set of neural units that contribute to an overall maximum of φs in the brain. By IIT, this set of units (with their particular grains) form the main complex in the brain and unfold into a Φ-structure whose properties account in full for all properties of the corresponding experience. 

More generally, the term NSC has some similarities with full NCC (neural correlates of consciousness), yet two differences are worth noting. First, NCC was devised as a theory-neutral term (hence “correlates”), whereas NSC is based explicitly on the explanatory identity of IIT. Second, “full NCC”  more loosely refers to all brain areas that correlate with consciousness, whereas the NSC is always the substrate of a specific experience (a single “moment”). 

This term was used in earlier IIT publications for what is now referred to as unconditional causal marginalization. It indicates that some input is made causally inert because every one of its states are equally probable.

IIT uses the term operationalize in the general sense: to translate an abstract concept into something measurable. Hence operational refers to any interventional procedure that involves manipulation and observation. 

In IIT, however, the term operational also offers a useful contrast with ontological. More precisely, the IIT method recognizes the ontological primacy of phenomenal existence and first approaches physical existence in strictly operational terms.

For example, IIT works with candidate substrates of consciousness, explicitly stipulating the spatial, state, and update grain of substrate units. These parameters are not presumptions that the substrate truly exists in this way; they are rather operational assumptions that let us bootstrap our account of experience, and ones that should be updated as our understanding of the system under investigation grows (albeit under the constraints of the postulates). More generally, the IIT method explicitly assumes physicalism and atomism; these too are meant in an operational, not ontological, sense. As such, physicalism does not mean materialism, and atomism does not mean ontological reductionism.

By first approaching the physical in operational terms, IIT aims to start from an agnostic stance about what “truly exists” physically; this way, the theory allows the operational procedures to spur our inferences about a comprehensive ontology.

Order is used to define the number of units constituting a mechanism, distinction, or purview. For example, the mechanism AB is 2nd-order because it is constituted of two units—A and B—and (assuming it is irreducible) the distinction it specifies is also 2nd-order (the order of a distinction is the order of the mechanism that specifies it). Similarly, cause purview Abc is 3rd-order because it is constituted of three units. 

For more, see Composition Postulate.

A part is a subset of units within a substrate. While the meaning is close to the commonsense notion, in IIT, the term refers specifically to the parts obtained by partitioning a system to assess its irreducibility. In this context, a part is a subset of units that has been causally separated (through partitioning) from at least one other subset (part) of the system.

For more, see:

• Integration Postulate

• FAQ: How do we calculate φs (system integrated information)?

• FAQ: Is "partitioning" the same as severing (eliminating) the connections between units?

• Academic papers:

  • Barbosa, L. S., Marshall, W., Albantakis, L., & Tononi, G. (2021). Mechanism integrated information. Entropy, 23(3), 362.

  • Marshall, W., Grasso, M., Mayner, W. G., Zaeemzadeh, ... & Tononi, G. (2023). System Integrated Information. Entropy, 25(2), 334. 

A partition is a mathematical operation used to assess the irreducibility of a candidate system (its φs), of a candidate distinction (its φd), or of a relation. A system partition comprises a set of “cuts” that sever all causal connections to or from one or more parts (for an illustration, see the Integration Postulate). For distinctions, valid partitions comprise a set of cuts through the mechanism, separating the mechanism–purview pair into non-interacting and non-overlapping subsets. Finally, we partition a relation by “unbinding” its overlapping distinctions one by one.

In all cases, among all possible partitions, we seek the minimum partition (MIP) in order to quantify the irreducibility of the system, distinction, or relation in question.

For more, see 

• FAQ: What is a “legal” or "valid" partition?

• Computing PHI…

The perturbational approach refers to IIT’s method of studying physical systems in an interventional rather than purely observational way. Using the perturbational approach, we aim to assess causal interactions (as opposed to mere correlations) by actively manipulating the system and observing the results. This approach is essential in IIT since it characterizes the physical solely in operational terms of cause–effect power (see 0th Postulate).

An exhaustive perturbational approach requires us to repeatedly set the system into all of its available input states and then observe which states the system subsequently transitions into (and with what probabilities). This allows us to build a transition probability matrix (TPM) for the system, which can be used to evaluate whether the system’s cause–effect power fulfills the postulates of the theory and is thus a substrate of consciousness.

Needless to say, it is not possible to apply the full perturbational approach in a system as complex as the brain. Yet the perturbational complexity index (PCI) aims to do so in a non-exhaustive way, thus providing a proxy of integrated information—the measure proposed by IIT to quantify consciousness. PCI is a causal measure in two senses: First, it relies on direct perturbation of the cortex (through TMS) followed by observation of the response—as opposed to mere observation. Second, it employs repeated stimulation and observation to extract the deterministic component of the cortical response to the perturbation—thus isolating the neural interactions caused by the perturbation itself from the noise of other brain activity. 

For more, see

• FAQ; Why is the perturbational approach so important in IIT?

• 0th Postulate

The perturbational complexity index (PCI) is a proposed measure of consciousness, inspired by IIT and designed as a tool to test and apply the theory. It is a number that quantifies the complexity of the global brain response (typically measured using EEG) to a local direct perturbation (typically applied using transcranial magnetic stimulation). The measure was published in 2013 by Casali and colleagues, under the supervision of Marcello Massimini, and was the culmination of experimental work in Giulio Tononi’s lab between 2000 and 2010. Since then, PCI has been refined, tested, and further developed into a measure that can be applied in clinical settings as well as experimental research. 

PCI involves two operational steps, easily remembered by the phrase “zap and zip.” First, a region of cortex is stimulated (“zapped”) with a TMS pulse and the response is recorded using EEG. This process is repeated many times to  isolate the deterministic component of the cortical response (i.e., the specific pattern caused by the stimulation) from the ongoing activity of the brain and other sources of noise. Second, the cortical response inferred from the EEG pattern is compressed using the same algorithms used to “zip” computer files. If the pattern is highly compressible, we can conclude that there is little causal interaction across cortical areas (low integration) or that the interactions are highly uniform (low differentiation). In contrast, if the pattern cannot be easily compressed, it is indicative of high integration and high differentiation—a reasonable proxy for integrated information.

As reported in Tononi et al. 2016, “So far, studies using PCI have confirmed the prediction of IIT that the loss and recovery of consciousness is associated with the breakdown and recovery of the capacity for information integration. This relationship holds true across different states of sleep and anesthesia (using different agents with various mechanisms of action) and in patients with brain damage, at the level of single subjects. Importantly, once PCI is validated in participants that can report on whether they were conscious or not, the index can be used to assess the capacity for information integration in patients who are unresponsive (such as those in a vegetative state) or cannot report (such as newborn infants and non-human species).”

PCI is a rough proxy of integrated information in that it only explicitly operationalizes three of the postulates (existence, information, and integration)—and these only crudely. However, the measure has proven uniquely powerful in objectively characterizing the “level” of consciousness in humans, without the need for explicit reports or collaboration from the subject under investigation. In the coming decades, a main aim is to provide updated measures, applicable in experimental settings, that incorporate the latest improvements of IIT. 

For more, see 

Phenomenology refers to the study of experience. In IIT, the term specifically refers to the analysis of experience through introspection and reasoning to characterize its (essential or accidental) properties. Phrases such as “starting from phenomenology” and “phenomenology first” refer to the method of starting from characterizing experience itself—rather than the brain—to develop a scientific theory of consciousness.

For more, see

• What does it mean to explain consciousness scientifically?

• FAQ: What do you mean by consciousness?

• FAQ: Why not build a theory of consciousness starting from empirical neuroscientific research?

Phenomenal existence refers to the existence of experience, which is immediate and irrefutable and the starting point of IIT. See 0th axiom. 

In IIT, the term phenomenal structure refers to an experience as a structure (or sub-structure) of phenomenal distinctions and relations. For instance, when I see a blue book, my experience is composed of various phenomenal distinctions (the color blue, the shape of the book, its location in visual space, etc.), which are all related in a particular way to form a unitary content (“that blue book, right there”). 

A phenomenal structure captures both the essential properties of every experience and accidental properties of the specific experience. As such, a phenomenal structure is the explanandum of IIT—it is what we aim to account for scientifically.

For more, see 

• Ellia, F., Hendren, J., Grasso, M., ... & Tononi, G. (2021). Consciousness and the fallacy of misplaced objectivity. Neuroscience of Consciousness, 2021(2), niab032.

• FAQ: What counts as a "component" of experience? 

See integrated information. 

See integrated information. 

A Φ-fold (“phi fold”) is a substructure within a Φ-structure—a subset of its distinctions and relations. Types of Φ-folds can be categorized based on different levels of analysis.

• A distinction Φ-fold is a substructure composed of a single distinction and the relations bound to it, which form its context. The level of analysis is helpful because it lets us decompose the Φ-structure into non-redundant components—like individual “pieces” of a Lego structure.

• A compound Φ-fold is a substructure composed of the distinction Φ-folds specified by a subset of units. For example, for the subset AB within substrate ABC, the compound Φ-fold includes every distinction Φ-fold specified by any mechanism that includes unit A or B. This level of analysis is useful because a compound Φ-fold is the part of the Φ-structure that we “access” by manipulating a particular subset of the substrate. For example, if the gain of units A and B is modulated by attention, all distinctions specified by those units (and probably many of their relations) would be affected by the change—not just the AB distinction Φ-fold. 

• A content Φ-fold is composed of a subset of distinctions that are highly interrelated (regardless of the mechanisms and units that specify them). In IIT, a content Φ-fold corresponds to a content of experience (e.g., a “face”). As such, these are a salient aspect of a Φ-structure in that they “hang together well,” as measured by ΦR. 

A Φ-structure (or cause–effect structure) is the fully unfolded cause–effect power specified by a substrate of consciousness. In line with IIT’s explanatory identity, the properties of a Φ-structure correspond one-to-one to the properties of a phenomenal structure (i.e., of an experience), which allows IIT to account for both the quality and quantity of experience.

A Φ-structure is composed of causal distinctions and relations among them, and is quantified by Φ (‘big phi,’ the sum of φ [‘small phi’] of all distinctions and relations). 

For more, see

• Composition Postulate: Φ-structures and Φ

• Unfolding a Φ-structure 

IIT assumes physicalism in an operational sense: from within our experience, we can observe and manipulate the world outside of us in a way that is reliable and persisting. Thus we assume that something exists in a physical sense if it can reliably “make a difference” to something or “take a difference” from something—that is, if it has cause–effect power.

In IIT, operational physicalism is one of three methodological assumptions (together with realism and atomism), which allow us to translate the 0th axiom into the 0th postulate. 

For more, see

• Three Methodological Assumptions

• FAQ: If IIT assumes “physicalism,” does this make it a materialist theory of consciousness?

• FAQ: What is the meaning of cause–effect power in light of other notions of “causal power”?

Physical existence refers to the 0th postulate of IIT. 

This term PSC was used in earlier IIT publications for what is now referred to as a substrate of consciousness or complex. 

This is a casual term for conditional causal marginalization.

The posterior hot zone (or the posterior cortical hot zone) is an area of the human cortex that—according to ongoing empirical testing—appears to be necessary for normal waking perceptual experiences. It comprises sensory cortical areas in the parietal, temporal, and occipital lobes. The term was coined by Christof Koch and colleagues in a 2016 review article on the neural correlates of consciousness (NCC), where it was presented as the best current anatomical candidate for where the NCC should be located in the human brain. 

For more, see: Koch, C., Massimini, M., Boly, M. and Tononi, G. (2016). Neural correlates of consciousness: progress and problems, Nature Reviews Neuroscience, 17(5), 307–321.

The postulates of IIT express essential properties of the substrate of consciousness. In the IIT method, the postulates are obtained by formulating the essential properties of experience (the axioms) as physical properties of the substrate of consciousness—understood solely in terms of cause–effect power. Mirroring the axioms, the postulates are intrinsicality, information, integration, exclusion, and composition.

The postulates guide the computational steps required to identify and unfold the substrate of consciousness into a cause–effect structure.

For more, see:

• Five Postulates: Essential properties of the substrate of consciousness

• FAQ: Why are the postulates so important in IIT?

Powerset is a standard term from set theory. The powerset of S is defined as the set of all subsets of S, including the empty set and S itself. For example, the powerset of S = {x, y, z} is Ps={{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}. 

The notion of powerset is used at various points in IIT’s formalism, especially when applying the postulates of exclusion and composition.